Mathematical Methods for Surface and Subsurface Hydrosystems (Series in Contemporary Applied Mathematics ? Vol. 7) (Series in Contemporary Applied Mathematics)

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Mathematical Methods for Surface

and Subsurface Hydrosystems

Series in Contemporary Applied Mathematics CAM Honorary Editor: Chao-Hao Gu (Fudun University) Editors: P. G. Ciarlet (City University ofHong Kong), Tatsien Li (Fudun University)

1. Mathematical Finance -Theory and Practice

2.

3. 4.

5.

6.

7.

(Eds. Yong Jiongmin, Rama Cont) New Advances in Computational Fluid Dynamics -Theory, Methods and Applications (Eds. F. Dubois, Wu Huamo) Actuarial Science -Theory and Practice (Eds. Hanji Shang, Main Tosseti) Mathematical Problems in Environmental Science and Engineering (Eds. Alexandre Ern, Liu Weiping) Ginzburg-Landau Vortices (Eds. HdimBrezis, Tatsien Li) Frontiers and Prospects of Contemporary Applied Mathmetics (Eds. Tatsien Li, Pingwen Zhang) Mathematical Methods for Surface and Subsurface Hydrosystems (Eds. Deguan Wang, Christian Duquennoi, Alexandre Ern)

Series in Contemporary Applied Mathematics CAM 7

M ~hemat t ica Mit hods for Surface and Subsurface Hydrosystems editors

Deguan Wang Hohai University in Nanjing, China

Christian Duquennoi Alexandre Ern CERMlCS ENPC, France

Higher Education Press

\: N E W JERSEY

*

LONDON

*

World Scientific

SINGAPORE

*

BElJlNG

*

SHANGHAI

*

HONG KONG

*

TAIPEI

CHENNAI

Deguan Wang College of Environmental Science and Engineering Hohai University 1 Xikang Road Nanjing, 210098, China

Christian Duquennoi Le centre d’Antony Parc de Tourvoie BP 44 92163 Antony Cedex, France

Alexandre Ern Ecole nationale des ponts et chaussCes 6 et 8 avenue Blake Pascal 77455 Marne-la-VallCe, France

H*aEG El (CIP) BtE &&a~-F7kE%@@J=&*%+it

+

Mathematical Methods for Surface and Subsurface Hydrosystems / ?%@’$%, 0)

(1.3)

where 0 is a bounded, polygonal open set of R2, 80’ and dR2 are partitions of the boundary dR of R corresponding to Dirichlet and Neumann boundary conditions and nan the unit outward normal to the boundary

dR. This system is often solved by finite volumes (FV) or finite elements (FE) methods of lower order (see LeVeque [l]and Ern & Guermond [2] among others). FV ensures an exact mass balance over each element and continuous fluxes across common element boundaries. FE ensures an exact mass balance on a dual mesh but leads to discontinuous fluxes at common elements edges. However, finite element is considered as

2

P. Ackerer, A. Younes

more flexible because of its high capacity of discretizing domains with complex geometry. The mixed finite element method (MFE) keeps the advantages of both methods: accurate mass balance at the element level, continuity of the flux from one element to its neighbors one and mesh flexibility. Moreover, the method solves simultaneously the state variable and its gradient with the same order of accuracy (Babuska et al. [3], Brezzi & Fortin [4], Girault & Raviart [5] and Raviart & Thomas [S]). Therefore, MFE has received a growing attention and some numerical experiments showed the superiority of the mixed finite element method with regard to the other classic methods (Darlow et al. [7], Durlofsky [8], Kaaschieter [9] and Mod et al. [lo]). However, its implementation leads to a system matrix with significantly more unknowns than FV and FE methods. When the lowest-order Raviart-Thomas space [6, 111 is used, which is very often the case, the resolution of (1.1) and (1.2) leads to a system with one scalar unknown per edge for the hybrid formulation of the MFE [6, 111. Attempts to reduce the number of unknowns have been investigated by various authors. For rectangular meshes, mixed finite elements of lowest order reduce to the standard cell-centered finite volume method [ll] provided that numerical integration is used. Baranger et al. [12] provides similar results for triangles and Cordes & Kinzelbach [13] showed the equivalence between mixed finite element and finite volumes without any numerical integration. However, such equivalence is restricted to steady state and without sink/source terms inside the domain. Moreover, the mixed finite element method does not require a Delaunay triangulation [9] unlike a finite volume scheme. We present here an alternative formulation of the MFE which leads to a system matrix with only one unknown per element without any approximation. In the first part, the steady state formulation is derived in details for the MFE, FV and the alternative formulation. In the second part, the main results concerning the general formulation for the elliptic PDE (steady state) will be described. The detailed developments can be found in Younes et al. [14, 151 and Chavent et al. [16]. The last part is dedicated to the parabolic PDE.

2

Resolution of the elliptic case with a scalar flux related parameter

Assuming that the system is in equilibrium, the storage term in (1.1) vanishes. The system of equations (1.1) and (1.2) leads to an elliptic

A Finite Volume Formulation of . . .

3

partial differential equation. The finite volume formuIation of yields, 3

2.1

The mixed finite element formulation

In the lowest-order MFE formulation for triangular elements, the flux is approximated with vector basis functions that are piecewise linear along both coordinate directions. In any point inside element A, qA is approximated by (e.g. [9]): 3 i=l

where Qi are the fluxes across the element edges Ai taken positive outwards and wi(L-') are the three vectorial basis functions for the element A (Figure 2.1) defined by:

1

4

lifi=j

wi'nj = A

{

Oififj

(2.3)

For a triangular element, the vectorial basis function is given by:

,~

where (xi,yi) are the coordinates of the vertices of A and IAl is its area. In addition, they satisfy v . w . - -1 -

I4

(2-5)

and, on the edge Aj,

where lAil is the length of the edge Ai. Using properties (2.5) and (2.6) of the vectorial basis functions, the flux's law (1.2) is written in a variational form

/ ( K - ' q A ) . wi = -/(VU).wi A

A


4

0.6'

f

0.4 - '

f

f

f

f

f

f f

0.3'

4

fl

, f

r'

0.2''

1

#

x

I -T

--

+

\

0.5.

*

+

0.4-

k

I

t

I I

1.

o.z-,

t

4

I

4

f

Vector

0.7 I,

I I

t

0.5..

0.6-,

N

\

+

\

t

I

4

Vector w;'

W;

0.5;-

\

J

--

J

o,41-

7

J

J

z

J

0,J.T

J

J

J

J

J

J . 3

J

F l u (Qi= 2.0, Qj= -1.0, Q ) =-1)

Vector w i

Figure 2.1 Vectorial basis function and related flux.

which can be written as

with BC =

-s1

W?

. w f , U A is the average value of the state variable

KAE

over A and uf is the average value of the state variable on element edge Ai. K A is a scalar and represents the average value of K on element A . We define rij as the edge vector from node i toward node j and Lij as its length (Lij = IlrijII). As shown by Cordes & Kinzelbach [13], applying the scalar product

1

rijrik

= -(L:j 2

+ L:k - L;,)

leads to the

A Finite Volume Formulation o f . . .

5

following relation: 3

C B$ = + + r;i)/48KAIAI (T$

~ ; k

=L

(2.9)

j=1

The previous properties of the discretized flux law is used to build a system of equations with average edge value as unknown. The system of equations (2.8) gives 3

3

i=l

i= 1

Equation (2.10) is inserted in (2.1), which leads to: (2.11) System (2.8) is inverted and u is replaced by the previous formulation (2.111, K A Q4 = -~ (2.12) T ~ ~ C [ T A~ r ~k iUu ~A j ~iju;] 2 QA a IAl 3 Using the continuity of the fluxes between two adjacent elements A and say B (Figure 2.2) Q~+Q;=o (2.13)

+

+

leads then to the equation:

For continuity reasons, we have B uiA = ui

(2.15)

This equation is written for each edge of the mesh which is not a Dirichlet type boundary. If a Neuman boundary conditions is applied on edge i, (2.14) becomes:

Equation (2.14) is the discretized form of equations (1.1)and (1.2) using the mixed finite element .method in its hybrid form.


6

Figure 2.2 Triangular element A and its three neighbors.

2.2

The finite volume formulation

The main idea of the finite volume formulation consists in defining the flux by Qf = -KA(u$ - u

i ) h L&

(2.17)

8

where ug is the value of the state variable at the circumcenter of A, LA, is the length of the edge i, and L& is the distance from edge i to the circumcenter of the element A (Figure 2.3). Writing flux and variable continuity at the common edge of element A and one of its neighbors noted B yields (2.18) and therefore,

( ~-2u C ) = KAB(U$- u:) B

(2.19)

where KAB is the harmonic mean of the flux related parameter multiplied by the length of the edge and is called the stiffness coefficient.

A Finite Volume Formulation of

..

7

Figure 2.3 A triangular finite volume and its neighbor. The finite volume discretization of equations (2.1) and (2.2) is obtained by plugging (2.19) into (2.1): K A B ( UAc - ug)

+ K A C ( UAc -

UC C)

+ K A D ( U 2 - ug)=

&S

(2.20)

This equation is written for each element. The finite volume requires significantly less unknowns than the mixed formulation (one per element against one per edge) and leads to a more sparse system matrix (4 non zero values per line against 5 ) .

2.3

The re-formulation of the mixed finite element

The main idea of the reformulation is first to define a linear interpolate of the state variable by, 3

U A=

C7r$f

(2.21)

i=l

where U is the value of the state variable somewhere (in or outside the element A , not necessarily the average value over the element or the value at the circumcenter of the element), and second, to use a very generic finite volume formulation of the flux, Qf = c t ( U A- uf)

+ rt

(2.22)

Building an equation with unknowns U is then straightforward. The continuity of the fluxes between two adjacent elements and of the state variable on the common edge yields: (2.23)


8

and therefore

ui = JAUA

+ [ B U B + 'Yf + 'Y?

JA + s$

EA + t$

(2.24)

Equation (2.24) is plugged in equation (2.22) which leads to

Equation (2.25) is then plugged in the discretized mass balance equation, which leads to an equation with one unknown per element, if, of course, the values of 7rA, Jtand r t can be determined. Replacing (2.21) in (2.22) and comparing with (2.12) allows the identification of these coefficients,

(2.26)

and the discretized form of (1.1) and (1.2) is then

Note that

r"?

Ef + E?

= KABand therefore, the system equation (2.27)

of the mixed reformulation differs from the system equation of the finite volume formulation (equation (2.20)) only by the sink/source term. Without sink/source terms, both formulations are identical. Moreover, the variable U is then the state variable at the circumcenter since, from equations (2.26) and (2.21):

u=-

1

4142

+

[(TikTjk)(TijTkj)ui (rijTik)(TikTjk)uj + ( ~ i j r i k ) ( ~ i j ~ k j )= ~ kU$ ]

(2.28)

With sink/source terms, both formulations are different, except for equilateral triangles and homogeneous domain. Moreover, in that case, the velocity derived from the MFE approach varies linearly and therefore, the linear interpolation of the state variable is no more valid.


3

9

General 2D formulation for the elliptic case

We treat here the case where K is a full tensor. With the FV method, the computation of accurate fluxes with a full parameter tensor is a difficult issue, especially for discontinuous coefficient. For these methods, recent developments have been done to improve the flux computation by using locally additional constraints on the continuity of the state variable and fluxes [17-201. With MFE method, the case of full parameter tensor is treated in an elegant way leading to a system with as many unknowns as the total number of edges. Reduction of the number of unknowns can be obtained for rectangular meshes when using appropriate quadrature rules with a variant of the MFE method, the “expanded mixed method” [21,22]. For general geometry, enhanced cell-centered finite difference method was obtained from a quadrature approximation of the expanded mixed method [23]. This method is improved by adding Lagrange multiplier for non smooth meshes or abrupt changes in K [23]. The parameter tensor K is generally symmetric [24]. It commonly arises from a rotation of the locally diagonal tensor from its principal axes with respect to the computational grid and is therefore always symmetric and positive definite. K A is defined by

The principal components of K A are constant and positive over each element A , therefore det(KA) = k $ k t - (k$y)2 > 0

3.1

(34

The mixed finite element formulation

We define now lij by l i j = r;(KA)-lrij where (KA)-’is the parameter tensor defined over element A . The variational form of equation (1.2) leads now to:


10

Written in a matrix form yields 3

BAQA 23 3 = uA - ut with

B$ = / z u ~ ~ ~ ( I C ~ ) - ' (3.4) W~ A

j=1

The matrix B is given by

with

where L A can be seen as the inverse of the parameter tensor scaled by the shape of the element. Therefore, we obtain the same equations than and (2.11), i.e. 3

3

L A ~ Q i = 3 u A - - ~ uand t i= 1

i= 1

The system (3.4) is inverted and (3.7) is used to obtain:

The final system of equations is obtained using continuity properties of the flux and state variable, which yields

,

A Finite Volume Formulation of . ..

11

and therefore, the discretized equation of system (1.1) and (1.2) is

= -0% +3

Q? 3 (3.10)

This equation is slightly modified when the edge belongs to the domain boundary.

3.2

The corresponding re-formulation

The same development as for the case of a scalar parameter K is used and 7 t are (see equations (2.21) and (2.22)). The coefficients

.rrt,

Replacing this last relation in the balance equation for an element

A surrounded by three elements B , C and D (Figure 2.2) leads to the equation

(3.12)


12

Equation (3.12) differs from equation (2.27) by the value of the stiffness coefficient. Notice that for both formulations, the discrete equation is obtained without any assumption on the way of approximating gradients or equivalent flux related parameters.

The parabolic case

4

The finite volume formulation of the balance equation (1.1) is now

We present here the development for K scalar parameter K .

4.1

The standard mixte formulation

The discretization of the flux equation is described in Section 2.1. Equation (2.10) is inserted in the balance equation (4.1) which leads to

where S = 1AIL& and F = LQ,. Inverting equation (2.8) leads to:

where L l l = -IAIL-'/KA. Equation (4.2) is plugged in (4.3) which gives 3

KAb Q~ = C a i j u j - j=1

I4

(4.4)


13

where

KA a33 = - -(T12T12 -

I4

r) (4.5)

with

c=---L11 3

LTl lcl and b = -( F Sun-l) (4.6) s+3 s+3 Again, the continuity of the fluxes and the average edge value of the state variable is used to build the system of equations.

4.2

+

The re-formulation of the MFE

The same approach is used as in Section 2 but the flux formulation is modified by Q i = O

KD (5-2)

For homogenous domains, this condition is equivalent to the Delaunaycriterion [12]. Note that for the case of a uniform, anisotropic parameter tensor K, a transformation of coordinates can be performed leading to a scalar parameter K in the transformed space. In this case, the criterion for positive-definiteness of the resulting system of equations would be equivalent to the Delaunay-criterion of the grid in the transformed space [25]. Because the system matrix is not the same for both formulations, the accuracy of the solution might not be the same. The conditioning of the system matrix is not the same for both formulations, especially when one angle of an element tends to 7r/2 (see equation (5.2)). For parabolic PDE, the coefficient matrix of the re-formulation is unsymmetrical and the use of an accurate solver is then an important issue. Numerical experiments [14-161 show that the re-formulation reduces significantly the CPU time without significant loss in accuracy. This reduction of the CPU is less important for parabolic PDE but the new formulation is still more efficient.

16


Finally, the extension to a general 3-D tetrahedral discretization is not possible [15].

References [l]LeVeque R J. Finite volume methods for hyperbolic problems. Cam-

bridge Univ. Press, 2002. [2] Ern A, Guermond J L. Theory and practice of finite elements. Vol. 159 of Applied Mathematical Series, Springer-Verlag, New York, 2004.

[3] Babuska I, Oden J T, Lee J L. Mixed-hybrid finite element method approximations of second-order elliptic boundary value problems. Part I, Comput. Methods Appl. Mech. Eng. 11, 175, 1977. [4] Brezzi F, Fortin M. Mixed and hybrid finite element methods. Springer, Berlin, 1991. [5] Girault V, Raviart P A. Finite element methods for Navier-Stokes equations: theory and algorithms, in Lecture Notes in Computational Mathematics, 1986. Vol. 5 (Springer-Verlag, Berlin). [6] Raviart P A, Thomas J M. A mixed finite element method for second order elliptic problems, mathematical aspects of the finite element method, Lectures Notes in Mathematics, Springer Verlag, New York, 1977; 606: 292-315. [7] Darlow B L, Ewing R E, Wheeler M F. Mixed finite elements methods for miscible displacement problems in porous media, SPE 10501, SOC.Pet. Eng. J. 24 (1984), 397-398. [8] Durlofsky L J. Accuracy of mixed and control volume finite element approximations to Darcy velocity and related quantities, Water Resour. Res. 30, (1994) 965-973.

[9] Kaasschieter E F, Huijben A J M. Mixed-hybrid finite elements and streamline computation for the potential flow problem, Numerical methods for Partial Differential Equations, 1992; 8, 221-266. [lo] Mos6 R, Siege1 P, Ackerer Ph, Chavent G. Application of the mixed hybrid finite element approximation in a groundwater flow model: luxury or necessity? Water Resour. Res., 30, (1994) 3001-3012. [ll]Chavent G, Roberts J E. A unified physical presentation of mixed,

mixed hybrid finite elements and standard finite difference approximations for the determination of velocities in waterflow problems, Adv. in Water Resources, 14 (1991), 329-348.


17

[12] Baranger 3, Maitre J F, Oudin F. Application de la thkorie des 616ments finis mixtes & l'ktude d'une classe de schkmas aux volumesdiffkrences finis pour les probl6mes elliptiques, C. R. Acad. Sci. Paris, 1994; 319, Skrie I : 401-404. [13] Cordes C, Kinzelbach W. Comment on by Mose et al., Water Resources Research, 1996; 32(6): 1905-1909.

[14] Younes A, Ackerer Ph, Mose R, Chavent G. A new formulation of the mixed finite element method for solving elliptic and parabolic PDE with triangular elements. Journal of Computational Physics, 1999; 149: 148-167. [15] Younes A, Ackerer Ph, Chavent G. From mixed finite elements to finite volumes for elliptic PDEs in 2 and 3 dimensions. International Journal of Numerical Methods in Engineering, 2004, 59 (3), 365388. [16] Chavent G, Younes A, Ackerer Ph. On the finite volume reform& lation of the mixed finite element method for elliptic and parabolic PDE on triangles. Computer Methods in Applied Mechanics and Engineering, 2002, 192(5-6), 655-682. [17] Lee S H, Jenny P, Tchelepi H A. A finite-volume method with hexahedral multiblock grids for modeling flow in porous media, Computational Geosciences, 2002; 6: 353-379. [18] Putti M, Cordes C. Finite element approximation of the diffusion operator on tetrahedra, SIAM J . Sci. Comput., 1998; 19 (4): 11541168. [19] Edwards M. M-Matrix flux splitting for general full tensor discretization operators on structured and unstructured grids, Journal of Computational Physics, 2000; 160: 1-28. [20] Edwards M. Unstructured, Control-Volume Distributed, FullTensor Finite-Volume Schemes with Flow Based Grids, Computational Geosciences, 2002; 6: 433-452. [21] Arbogast T, Wheeler M F, Yotov I. Mixed finite elements for elliptic problems with tensor coefficients as cell centered finite differences, SIAM J. Numer. Anal., 1997; 34(2): 828-852. [22] Wheeler M F, Roberson K R, Chilakapati A. Three dimensional bioremediation modeling in heterogeneous porous media, in Computational Methods in Water Resources IX, V01.2, Mathematical Modeling in Water Resources, Russell T F, Ewing R E, Brebbia C A, Gray W G, and Pindar G F, eds., Computational Mechanics Publications, Southampton, U.K., 1992; 299-3 15.

18


[23] Arbogast T, Dawson C N, Keenan P T, Wheeler M F, Yotov I. Enhanced cell-centered finite differences for elliptic equations on general geometry, SIAM J. Sci. Comput., 1998; 19 (2): 404-425. [24] Durlofsky L J. Numerical calculation of equivalent grid block permeability tensors for heterogeneous media, Water Resources Research 1991; 17(5): 699-708. [25] Russel T F, Wheeler M F. Finite element and finite difference methods for continuous flows in porous media, in the Mathematics of Reservoir Simulation, edited by Ewing R E, SIAM, Philadelphia, 1984, 35.

19

Finite Element Modeling of Hydrosystems with Fully Saturated, Variably Saturated, and Overland Flows* Alexandre Ern Cermics, Ecole nationale des ponts et chausse'es 6 et 8 avenue Blaise Pascal, 77455 Marne-la-Valle'e, France E-mail: [email protected]

1

Introduction

The prediction of overland flow routing in hillslopes is of paramount importance during flood episodes. Owing to the complexity of the problem, in situ experiments are often difficult to perform and general conclusions are difficult to draw. This has prompted the development of numerical tools with the capability to analyze water transfer in virtual hillslopes subjected to various operating conditions. This approach has experienced a vigorous development over the past decades. Many advances have been reported in the specialized literature, concerning both the development of models describing the hydrological behavior of hillslopes through partial differential equations (PDEs) and appropriate boundary conditions (BCs) and the design of suitable numerical methods able to tackle such PDEs and BCs. Recent work in this direction includes, among others, Ogden & Watts [l],Cloke et al. [2], and Weiler & McDonne11 [3]. An important aspect of hillslope hydrology during heavy rainfall episodes is the fact that subsurface flows can saturate the soil in various regions near the surface and, therefore, contribute to the production of overland flow. There is therefore a strong motivation to improve our understanding of subsurface flow dynamics in variably saturated porous media and, in particular, of the role played by the hydraulic state of the soil in storm hydrographs. As part of an ongoing research effort in that direction, the Cooperative Research Action DYNAS (Dynamics *Part of this work has been carried in collaboration with H. Beaugendre (Cermics/Enpc), B. Augeard, I. Ginzburg, C. Kao (Cemagref), and T. Esclaffer, E. Gaume (Cereve/Enpc) in the framework of the Cooperative Research Action DYNAS sponsored by INRIA.

Alexandre Ern

20

of Shallow water-tables) has been sponsored by INRIA since 2003; see http: //www-rocq. inria.fr/estime/DYNAS for a detailed description of the project. This paper discusses numerical tools, essentially based on finite element techniques, to discretize various PDEs relevant to hillslope hydrology models, and it will also present some selected results illustrating the interaction of subsurface flow dynamics with the ground surface. Section 2 presents various finite element discretizations of Darcy’s equations. These equations can be used to model steady subsurface flows in fully saturated porous media. The emphasis in this section is set upon the fact that several finite element formulations can be designed, each leading to different physical properties at the discrete level. It is assumed that the reader is familiar with the basics of finite element methods; see, e.g., Ciarlet [4],Brenner & Scott [5], Quarteroni & Valli [6], or Ern & Guermond [7] for an introduction to this topic. Section 3 presents a model often considered to describe unsteady flows in variably saturated porous media, namely Richards’ equation. Richards’ equation requires two algebraic relations to model the hydrodynamic behavior of the soil. These relations, which relate the unsaturated hydraulic conductivity, the volumetric water content, and the hydraulic head, are discussed in this section. Section 3 also addresses in some detail the case where the water-table (i.e., that part of the soil where saturation takes place) reaches the top boundary of the hillslope, i.e., the ground surface. Since the water-table position is an unknown of the problem, its intersection with the ground surface is a priori unknown. This leads to an unsteady obstacle-type problem. Section 4 discusses the modeling of overland flow on the ground surface. Two simplified forms of the Saint-Venant equations are considered, namely the diffusive wave approximation and the kinematic wave approximation. Numerical algorithms coupling the overland flow with subsurface flows governed by Richards’ equation are also presented. Finally, Section 5 presents some selected results on hillslope behavior during heavy rainfall episodes. It discusses the impact of phenomenological parameters on model predictions and compares two approaches to simulate the subsurface flow, one neglecting the height of the overland flow in the specification of BCs for Richards’ equation and the other taking it into account.

2

Finite elements for fully saturated steady flows

The goal of this section is to present various finite element approximations of the equations governing steady flows in fully saturated porous

Finite Element Modeling of Hydrosystems with . . .

21

media, the so-called Darcy’s equations. We stress the fact that several mathematical formulations of the problem at the continuous level can be envisaged, each one leading to a discrete setting with different features. We study the so-called (pressure) primal formulation as well as several (pressure-velocity) mixed formulations.

2.1

Darcy’s equations

Let R denote a fully saturated porous medium with boundary dR. Let u be the flow velocity in field R and let p be the hydraulic head (or the pressure up to an appropriate rescaling). Darcy’s phenomenological law states that u=-KVp inn. (1) Here, K denotes the hydraulic conductivity (or permeability) tensor. Furthermore, mass conservation implies that

V.u=f

inR, (2) where f represents the volumetric water sources or sinks in R. To enforce boundary conditions, we assume that the boundary 80 can be partitioned into 801U 8R2 in such a way that the hydraulic head is imposed on dR1 and that a normal velocity is prescribed on 8R2. Using Darcy’s phenomenological law, the boundary conditions can be recast as follows: p=g1 -n . K V p = g2

onaa1, on 8 0 2 ,

(3) (4)

where n is the outward normal to R and where g1 and 9 2 are prescribed functions on 8R1 and 2 2 2 , respectively. In other words, mixed DirichletNeumann conditions can be imposed on the hydraulic head. * To pinpoint the key mathematical points, we shall make, in this section, the following simplifying assumptions: 0

0

The porous medium is homogeneous and isotropic; in other words, the hydraulic conductivity tensor takes the form K = 5I where 5 is a positive constant in R and I is the identity tensor. Homogeneous Dirichlet boundary conditions are imposed throughout dR.

As a result, we shall seek the solution ( u , p ) to

V.u=f

inR,

k - l u + ~ p = ~ inR, p=O on8R. Henceforth, we assume that the data f is in L2(R).

Alexandre Ern

22

2.2

The pressure primal formulation

Taking the divergence of (6) and using (5) to eliminate V . u yields ino, ondfl

-kAp=f p=O

This is the well-known Poisson problem whose weak formulation is Seek p E Ht(R2) such that

(i,

kVp*Vq=

s,

fq,

V q E Hi(R).

(10)

As an immediate consequence of the Lax-Milgram Lemma, we infer that the problem (10) is well-posed, i.e., it admits one and only one solution. Henceforth, this problem is referred to as the pressure primal formulation or simply the pressure formulation. 2.2.1

Conforming approximation

Let l h be a simplicia1 mesh of the domain R (the elements of '& are triangles in dimension two and tetrahedra in dimension three). To simplify, assume that the mesh is affine and that it covers f2 exactly, i.e., R is a polygon in two dimensions or a polyhedron in three dimensions. In the sequel, dh denotes the set of faces of the mesh (edges in two dimensions). Consider the approximation space

P2.o = { P h E Co@) n Hi(R);VK E %, PhlK E pi } ,

(11)

consisting of those discrete pressure fields that are piecewise linear, conand that vanish on do. The index c emphasizes the fact tinuous on that P&, is conforming in &(a),i.e., that Pio c Hh(C2);the index 0 indicates that the functions in Pio vanish at the boundary. The conforming approximation of the primal formulation consists of the following problem: Seek p h E P2,0 such that

n,

Clearly, this problem is well-posed. Moreover, if the exact solution is smooth enough, namely p E H2(R)n @(a),the following error bound holds: b-phllH1(R) 5 ch ~ ~ P ~ ~ H 2 ( 0 ) ~

(13)

. ..

Finite Element Modeling of Hydrosystems with 2.2.2

23

Non-conforming approximation

In this section the pressure is approximated by piecewise affine functions that are continuous only at the midpoints of edges separating adjoining elements. Consider the so-called Crouzeix-Raviart finite element space

PAC,o= (2% E L 2 ( a ) ;V K E %, PhlK E

va E .Ah, / [ P h ] a = o}, a

(14) where [Ph], denotes the jump of ph across the face a ( [ P h ] , = p h ( , is the face a is located at the boundary). The index nc emphasizes the fact that @ Hi(s2). The restriction of a function p h in to a face a E Ah being affine, the condition J,[Ph]a = 0 is equivalent to the continuity of p h at the midpoint of a. Consider the discrete problem

Since ph $! Ht(R), the integral over in the left-hand side of (12) has been replaced by a sum of integrals over mesh elements. A classical result is that the problem (15) is well-posed. Moreover, if the exact solution is smooth enough, namely p E H2(s2)n H i ( R ) , the following error bound holds:

2.3

Mixed formulation in [L2(R)Idx Ht(R)

Once the pressure field has been obtained, the velocity field can be reconstructed in [L2(a2)ldby setting u = - k V p . This equation can be equivalently written in weak form by taking the scalar product with arbitrary test functions in [L2(R)ld. As a result, the problem (5)-(7) can be reformulated as follows: Seek ( u , p ) E [L2(f2)ld x H t ( R ) such that

Is,

(k-lu

+ Vp) .

2,

=0,

vv E

[L2(O)]d,

24

Alexandre Ern

or, in a more abstract form,

i

Seek ( u , p ) E [L2(fl)ldx H i ( 0 ) such that

a(u,w)

+ b(v,p) = 0 ,

vw E [L2((s2)ld,

(18)

b(u,4 ) = Z(q) , vq E H m 7

where a ( u ,w) =

s,

L-lu.

b(v,p) =

21,

s,vp

*

21,

Z(q) = -

s,

fq.

(19) Problem (18) is one possible mized formulation of the problem (5)-(7). In (18), the mass conservation equation has been integrated by parts ' owing to the fact that u is a priori sought in a space with little regularity (uE [L2(Q)ldso that V u is not a priori meaningful in L2(sZ)).This is at variance with Darcy's law where no integration by parts has been used. We shall see that in a discrete setting, these choices have important consequences: Darcy's law holds locally on each mesh element whereas mass conservation holds only in a weak sense. *

2.3.1

Conforming approximation

The approximate problem (12) can be used to construct an approximation to the mixed formulation (18). The discrete velocity is sought in the space

[pold= {uh E [ L 2 ( 0 ) l d ; QE K'&,

u h lE ~

[pold),

(20)

consisting of those discrete velocity fields that are piecewise constant on a.Consider the discrete problem

( Seek ( u h , p h ) E [Po]' x P2,0 such that

One readily verifies that this problem is well-posed. Moreover, if p h E PCl,o solves (12), the pair (-lcVph,ph) solves (21). Conversely, if ( u h , p h ) E [Poldx P:,o solves (21), u h = -kVph and ph solves (12). Finally, if the exact solution is smooth enough, namely u E [H1(Q)ldand p E H 2(0)n H t (a), the following error bound holds: IIu - uhll[L2(n)]d + IIP - PhllHl(n) I c h (ll4l[H'(n)]d+ IlPllH.(n)>. (22)


25

To simplify, we consider a two-dimensional setting. Let N e denote the number of elements in the mesh 7 h , N u the number of edges, and N s the number of vertices. Assume that the domain 0 is simply connected (meaning that there are no holes inside 0). Owing to the Euler relations, Ne - Na+ N s = 1 , 2Na - 3Ne = N a f , where N a f is the number of boundary edges, the following holds: N e = 2Ns - N a f Nu = 3Ns- Naf

+2 M 2Ns, + 1 M 3Ns,

(24)

in the practical limit where N s >> N a f . Using this notation, the approximate problem (21) is a linear system of size 2Ne N s E 5Ns. Of course, this system is not inverted in practice; instead, the discrete pressure field p h is first recovered from (12) and then the discrete velocity field is reconstructed according to u h = -JcVph. Hence, the linear system to be solved is only of size N s . Note that the above reconstruction of the velocity field implies that Darcy's law holds locally on each mesh element. To investigate the discrete mass conservation equation resulting from ( 2 l ) , let us introduce some additional notations. Partition Ah into Ai U A: where A; and A: denote, respectively, the set of interior and boundary edges (interior means that the edge is not included in an). Similarly, let Sh denote the set of vertices in the mesh. Partition Sh into Sk U Sf where S i and Sf denote, respectively, the set of interior and boundary vertices. For K E 55, let 1K1 be its measure, SK the set of its vertices, and d K the set of its edges. For s E S K , let nK,, be the outward normal to K on the edge opposite to s and let hK,, be the distance of s to the edge in question. Let 1~ denote the characteristic function of K and let w, E P&, be the so-called hat function taking the value 1 at s and 0 at the other mesh vertices. Let 0, be the support of w,. For an interior vertex s E S i , let A, be the set of edges containing s. Finally, for an interior edge a E A;, let la1 be its length and [uh. n], be the jump of the normal component of u h across a defined as follows: if a = K1 n K2 where K1 and K2 are two distinct mesh elements with respective outward normals nl and 722, [uh.n], = ( ~ h l ~ ~ ) . n l + ( u h l K , ) . n 2 . Using the above notations, the discrete pressure and velocity fields can be written as

+

26

Alexandre Ern

Figure 1 A mesh element K E 0,.

7j,

a vertex s E S K , and the domain

Proposition 1. The solution ( u h , p h ) to (21) satisfies the discrete mass conservation equation

and the local velocity reconstruction formula

Proof. Let a E Ah and let g be an integrable function on a. Recall that the distribution 6, @ g da is defined as follows:

where D ( n ) denotes the set of infinitely differentiable functions that are compactly supported in 0. Using this definition and classical results of distribution theory, it is readily inferred that for all u h E [Pold,

Equation (26) is then a direct consequence of the above equation by applying it to the hat functions w, for all vertices s E S i . Furthermore, (27) results from the fact that for K 6 Th, UK = - k V p h I ~ and that for s E S K , VW,IK= - ( l / h ~ , s ) n ~ , s .

Finite Element Modeling of Hydrosystems with . .

2.3.2

27

Non-conforming approximation

To construct a non-conforming approximation to the mixed formulation (18), one can proceed as in the previous section. The discrete velocity field is again sought in [Pold.Since the discrete pressure is not in H ; ( a ) , the bilinear form b has to be modified into

This leads to the approximate problem

solves (15), then, Problem (31) is well-posed. Furthermore, if ph E letting uh E [Poldbe such that for all K E U ~ J K = - k V p h J ~ the , pair ( u h , p h ) solves (31). Conversely, if ( u h , p h ) E [Poldx P&, solves (31), then, for all K E '&, U ~ I K = - k V p h I ~ and ph solves (15). Finally, if the exact solution is smooth enough, namely u E [H1(R)ld and p E H2(R) n H;(s1), the following error bound holds:

z,

(32) Let wa be the function in P2c,otaking the value 1 at the midpoint of the edge a and the value 0 at the midpoint of all the other edges. Let R, be the support of w, (0, consists of the two triangles of 7 h sharing the edge a). Using the above notation, the discrete pressure and velocity fields can be written as uh =

UK KETh

1~

and

pawa.

ph =

(33)

aEAh

Proceeding as in the proof of Proposition 1, one easily verifies the following: Proposition 2. The solution ( u h , p h ) to (31) satisfies the discrete mass conservation equation

28

Alexandre Ern

and the local velocity reconstruction formula

where nKqais the normal to the edge a pointing outwards K .

2.4

Mixed formulation in

Hdiv

x L2(R)

We now consider a mixed formulation in which the velocity field is sought in a space with more regularity than [L2(R)ld,namely in Hdiv =

{ U E [L2(R)ld,V * U E L2(R) } ,

(36)

equipped with the norm

+ 1'

= ll~ll[~2(n)ld

IlUIlHdiv

*

~ l l ~ ' ( *n )

(37)

Consider the following problem: Seek (u,p) E Hdiv x L2(R) such that a(u,V )

+ b ' ( ~ , p =) 0 ,

b ' b , 4) = l ( 4 ) where

9

VV E H d i v ,

(38)

v4 E L2P)> r

b'(v,p)= -]nPv.v.

(39)

A comparison with the mixed formulation (18) is noteworthy. In (38), Darcy's law has been integrated by parts owing to the lack of regularity of the pressure ( p is sought a priori in L2(R); hence, V p is not meaningful in [L2(fl)ld) whereas the strong form of the mass conservation equation has been used. In the discrete setting, these choices will imply that mass conservation holds exactly on each mesh element while Darcy's law holds only in a weak sense. Using classical arguments, it is shown that (38) is well-posed and that its unique solution ( u , p ) verifies (5)-(6) a.e. in R and (7) a.e. on dR. F'urthermore, one readily verifies that if ( u , p ) solves (38) and only if (u,p) solves (18). 2.4.1

The Raviart-Thomas finite element

To simplify, consider a two-dimensional setting. The idea is to seek an approximation for the velocity that is uniquely determined on each element K E 7 h by its normal fluxes across the three edges a E d K . On K , introduce the polynomial space

RTo(K)= [Po(K)I2+ (.,Y)'PO(K).

(40)


29

This space is generated by the polynomials

where (xi,y i ) are the coordinates of the vertex i of K . The polynomials {WK,1 , W K , 2 , W K , ~ satisfy } many important properties. In particular, V . W K ,= ~ 1/JK1for all 1 5 i 5 3, and for all 1 5 i , j 5 3, W K , .~n

~= j& j

>

(42)

where S i j stands for the Kronecker symbol. Consider the approximation space RT:c

=

{ u h E [L2(fi2)ld;YK E %, U h l K E R T O ( K ) } .

The index nc emphasizes the fact that RT;, is not conforming in i.e., RT& @ &iv. A well-known result is the following:

(43) Hdiv,

Proposition 3. The approximation space RT; = RT;, n H d i v as equal to RT: = { u h E [L2(O2)ld; VK E %>UhlK E RTO(K); Ya E

k h , [Uh *

n], = o} .

(44)

Since the discrete velocity U h E RT: is affine on each element K E %, the last condition in (44) is equivalent to the continuity of the normal component of Uh at the midpoint of the edges a E di. Hence, RT: is a vector space of dimension N u whereas RT;, is a vector space of dimension 3Ne. In other words, dim(RT:) M f dim(RT2,). Consider the discrete problem

The degrees of freedom for the discrete velocity are the normal fluxes across the edges and the degrees of freedom for the discrete pressure are the mean-value at the mesh elements; see Figure 2. Using standard arguments, one can verify that the problem (45) is well-posed. Moreover, if the exact solution is smooth enough, namely E [H1(a)ld, V .uE H1(R), and p E H1(R), the following error bound holds:

Alexandre Ern

30

2

Figure 2 Degrees of freedom for the standard Raviart-Thomas approximation: normal fluxes across edges for the velocity and mean-value at elements for the pressure.

For a function f E L2(f2),let I I h f be its L2-orthogonalprojection onto Po, i.e., I I h f is the piecewise constant function such that

Using this notation, the discrete mass conservation equation resulting from (45) simply takes the form

The discrete Darcy's law takes a more complex form that will not be detailed here; on each element K E 7 h 7 the velocity can be expressed using the pressure in K and the pressure on the three triangles sharing an edge with K .

2.4.2

Hybrid finite elements

+

The discrete problem (45) is a linear system of size N u Ne M 5Ns. It is possible to express explicitly the solution of this system in terms of the solution of a smaller linear system. The idea is to relax the continuity constraint of the normal velocity component across edges by introducing a Lagrange multiplier on each edge. aE

Introduce the vector space spanned by functions defined on the edges and constant on each edge, i.e.,


31

Consider the approximate problem

(50) Then, one can show that the above problem is well-posed. Moreover, if ( u h , p h , X h )E RT;, x Po x A' solves (50), then u h E RT: and ( u h , p h ) solves (45). The advantage of the discrete problem (50) is that the first two equai tions are purely local. Indeed, on each element K E 7 h , let WK be the vector formed by the three components of the velocity in the local basis of RTo(K), i.e., WK = ( W K , wK,2, ~, W K , ~ where ) ~

sK

j W K , .~W K , ~ Let . AK Let BK be the 3x3 matrix with entries B ~ , i= be the vector whose three components are the three Lagrange multipliers associated with the three edges of K (the numbering of the edges is such that the ith edge of K is the edge opposite to the ith vertex). Using this notation, the discrete Darcy law can be written locally in the form

where U = (1,1, l)T.Furthermore, the local mass conservation equation takes the form

where (., .) denotes the scalar product in It3. Owing to the elementary properties of the matrix BK (see Lemma 4 below), it is readily inferred that p ( K = i ( A K ,u)-k ak-'&nKf

and hence,

7

(54)

Alexandre Ern

32

The quantity p~ (having dimension of a length) denotes the gyration radius of K defined as P”KKl =

/

K

llrK(x)I12 dx 7

(56)

where 7 r ~ ( x=) (z1-g K , l , . . . , xd - gK,d), ( 2 1 , . . . ,xd) are the Cartesian coordinates, and QK is the barycenter of K . The implementation of the hybrid method is as follows. On each edge, the continuity of the normal velocity component is enforced using (55) leading to a linear system of size N a where the only unknowns are the Lagrange multipliers at the edges. Once the Lagrange multipliers have been evaluated, the formulas (54) and (55) are used locally to evaluate the velocity and the pressure. To sum up, the hybrid mixed finite element method requires only the solution of a linear system of size N u M 3Ns; this means a saving of 40% with respect to the original size of problem (45).

Lemma 4. Let K E 77.Then, the matrix BK is invertible with

Ki

B-’ =

a2 + a 3

-a3

-a3

a1 + a 3

1:;

-a2

-a1

a1 +a2

)

+‘U@U, 3 1 ~

(57)

where ai = 2 c o t Q ~ , i1, 5 i 5 3, O K , ~ being the angle at the ith vertex of K , and 1~ = where p~ is the gyration radius of K defined in

$&,

(56).

2.5

Mixed formulation in

Hdiv

x Ht(R)

The motivation for introducing the following non-standard mixed formulation is to use test functions for both the velocity and the pressure that can be localized at the element level, i.e., test functions in [L2(f2)ld and L2(1;2),respectively. Such schemes bear a close relationship with finite volume schemes and are often termed finite volume box schemes. The results presented hereafter are borrowed from [8]. The starting point of the derivation is to seek the velocity and the pressure in spaces that yield sufficient regularity to avoid integration by parts when setting the weak formulation. Consider the following mixed formulation: Seek ( u , p ) E U(U,V)

Hdiv

x Hi(f2)such that

+ b(w,p) = 0 ,

b’(u,q ) = -Z(q)

,

VV E [L2(1;22)ld,

vq E L2(1;2).

(58)


33

Then, one can show that the above problem is well-posed. Moreover, its solution ( u , p ) solves (5)-(6) a.e. in R and (7) a.e. on dR. Finally, (u,p) solves (58) if and only if it solves (18). The terminology non-standard is employed for the mixed formulation (58) because the space where the solution (u,p)is sought is dzfferent from the space for the test functions (v,q).

2.5.1

The discrete setting

The lowest-order finite volume box scheme consists of seeking the discrete velocity in the conforming Raviart-Thomas finite element space RT; and the discrete pressure in the non-conforming Crouaeix-Raviart finite element space Test functions for both the Darcy law and the mass conservation equation are taken to be piecewise constant. This leads to the following problem:

One can show that the problem (59) is well-posed. Moreover, if the exact solution is smooth enough, namely u E [H1(R)ld, V . u E H1(R), and p E H2(R), the following error bound holds:

5 c h (II4[H'(n)]d + IIV . UlIHl(n)+ IlPllHyn)) .

(60)

The discrete problem (59) is a linear system where the number of unknowns is dim RT:

+ dim

= Nu

+ ( N u - N a f ),

(61)

and where the number of equations is dim[P0Id+ dim Po = ( d + 1 ) N e .

(62)

The equality between the number of unknowns and the number of equations results from the Euler relations (23). This equality is of course only a necessary condition for the well-posedness of (59); to prove wellposedness, a discrete inf-sup condition must also be established; see [8].

34

Alexandre Ern

2.5.2

Some properties of the finite volume box scheme

Proposition 5. The solution (uh,ph) to (59) is endowed with the following properties: 1. Local mass conservation VKETh,

(63)

V*'Uhl~=nKf,

or, in other words, v * uh = n h f . 2. Local velocity reconstruction:

VK E

lh

UhlK =

-kvPhIK

+i(nKf)rK(z)

7

(64)

where T K ( X ) is defined below equation (56). 3. Equivalence with the non-conforming approximation of the pressure primal formulation: ph is the unique solution of (15) with righthand side f replaced by &f . Proof. (1) Equation (63) is obtained by taking qh equation of (59). (2) Since uh E RT:, the following holds: VK E

lh, U ~ I K

+ i(V

=n ~ u h

*

=

1~ in the second

U ~ ) ~ K ~ K ( Z ) .

(65)

Taking V h = 1Kei where ei is the i-th vector of the canonical basis of Rd in the first equation of (59), it is inferred that I I K u ~= - k V p h I ~ . Equation (64) then results from (63) and (65). Since V q h E [Po]',it is inferred that (3) Let qh E

and the sum of boundary terms vanishes since qh E Hence,

The proof is complete.

and uh E RT;.

(67) 0

Statement (3) in Proposition 5 has important practical consequences on the optimal way to solve (59). Firstly, the non-conforming approximation of the pressure primal formulation is considered (with data IIh f

Finite Element Modeling of Hydrosystems with

..

35

instead o f f ) ; the discrete pressure is obtained by solving a linear system of size N u M 3 N s . Secondly, the velocity field is reconstructed locally from (64). This approach yields a saving of 50% in the size of the linear system to be inverted.

2.6

Summary

Table 1 summarizes the various approximations to Darcy's equations investigated in this section. The third and fourth lines in the table indicate whether Darcy's law and the mass conservation equation, respectively, are satisfied in a strong or in a weak sense. Table 1: Finite element approximations to Darcy's equations. primal conforming

primal non-conforming

mixed standard

mixed non-standard

strong weak

strong weak

weak strong

strong strong

[L2((n)ldx H,'Q) [POId

3

x

PnlC.0

Hdiv X

L2(Q)

RT; x

PO

Finite elements for variably saturated flows

This section focuses on Richards' equation to model variably saturated flows. In particular, some emphasis is set on the finite element formulation in the case where the water-table reaches the ground surface, thus leading to an obstacle-type problem for the hydraulic head.

3.1

Richards' equation

In an unsaturated porous medium, the fluid phase contains water and air. The most general approach to model flows through such media is to address the two-phase problem by considering mass and momentum balance for each of the two phases; see, e.g., [9]. However, a simpler approach is feasible if the gas-phase always remains connected so that a single air pressure can be considered. This is the first important assump tion underlying Richards' equation. Note that entrapped air pockets are excluded from the present configuration. Let 19 be the volumetric water content and let cp be the (total) hydraulic head. Set cp=++z, (68)

+

where z is the vertical coordinate (oriented upwards) and is the socalled matrix potential. This quantity is related to the difference between

36

Alexandre Ern

air pressure and water pressure in such a way that $ < 0 in unsaturated regions and $ > 0 in saturated regions. The water-table position can be located by the isoline {$ = 0). The second assumption underlying Richards’ equation is that a generalized Darcy’s law still relates the water flow velocity to the hydraulic head gradient through the concept of relative hydraulic conductivity. In other words, it is assumed that

u = -Ic,lc,(e)vp,

(69)

where k, is the hydraulic conductivity at saturation (i.e., the hydraulic conductivity pertinent for Darcy’s equation) and kT(8) is the so-called relative hydraulic conductivity that depends on the volumetric water content 8. For the sake of simplicity, the medium is assumed to be isotropic so that a single scalar hydraulic conductivity can be considered instead of a full tensor. As a result, the water conservation equation ate + V . u = f takes the form

ate - v . (klJ%(e)(v$ + ez)) = f

(70)

1

where e, is the unit vector oriented upwards and where f represents the volumetric water sources or sinks in Q. To close the problem, it is necessary to assume that an algebraic relation links the volumetric water content 8 to the matrix potential $, say $ H 8($). The graph of the function $ H 8($) is often called the soil water retention curve. Equation (70) then yields a PDE where the sole dependent variable is the matrix potential $, namely,

ate($)- V . (kSkT(8($)>(W+ e,))

=f

.

(71)

This PDE can be written in mixed form by introducing the generalized Darcy velocity u($),

{

ate($) + v . 4.111)= f

7

u($)+ WG(~($))(V$+ e,) = 0 .

(72)

To enforce boundary conditions for (71) or (72), assume that the boundary dR can be partitioned into 801 U 8 0 2 in such a way that the hydraulic head is imposed on dQ1 and that a normal velocity is prescribed on dR2. This yields

$ = 91- x -k3kr(q$))(V$

+ez)

12.

= g2

on dR1,

(73)

on dR2,

(74)

where n is the outward normal to 0. In other words, Dirichlet and (nonlinear) Neumann conditions are imposed on $. A more complex


37

situation where the partition dR1 udR2 is not known a priori is discussed in Section 3.4. Finally, (71) or (72) is supplied with an initial condition specifying the initial value of the matrix potential, namely

$I = $I0

at t = 0 and in R .

(75)

3.2 Soil hydrodynamic functions To close (71) or (72), two algebraic relations between 8, IC,, and must be specified. In soils, the volumetric water content does not vary between 0 and 1,but between a minimum value 8, called the residual volumetric water content and a maximum value Bs called the saturated volumetric water content. Define the reduced volumetric water content $J

which, by definition, varies between 0 and 1. A model often considered in the engineering literature to specify the soil water retention curve is that derived by van Genuchten in 1980 [lo]

i.

with m = 1 Equation (77) involves two parameters, a and n. It should be noted that if the exponent n is less than 2, the function $I H $($I) is only of class C1at $I = 0 and its second-derivative explodes as $I -+ 0-, whereas for n 2 2, the function $I H g($I)is of class C2 at $I = 0. This fact can be important in numerical approaches where the volumetric water content is chosen as the main dependent variable for (71). A model to express the relative hydraulic conductivity has been proposed by Mualem in 1976 [ll],

where 1 is the pore connectivity parameter (generally set to 1 = $) and T H $ ( T ) denotes the reciprocal function of y!~ tt $($I). Combined with (77), the Mualem relation yields

lc,(i?) = $+ (1 - (1- 8 q m )

2

.

(79)

38

Alexandre Ern

Equations (77) and (79) are referred to as the Van Genuchten-Mualem (VGM) model. In the framework of the VGM model, a soil is described by five parameters: k,, a, n, O,, and Or. Recently, it was observed that relatively small changes in the shape of the soil water retention curve near saturation could significantly affect the numerical performance of variably saturated @owsimulations [12]. As a result, a modified form of the VGM model was proposed to account for a very small, but non-zero, minimum capillary height, $, < 0, in the soil water retention curve. This modification leads to less non-linearity in the hydraulic conductivity function near saturation and, hence, to more stable numerical solutions. The modified VGM model takes the following form:

+

where the parameter ,6 is defined as ,B = (1 ( - Q $ ~ ) % ) - ~in such a way that g($,) = 1. Furthermore, the relative hydraulic conductivity dependency on g is modified so that its dependency on $ is unchanged if (80) replaces (77). This leads to [ l - (l-,6-k)m]-2.

(81)

Equations (80) and (81) reduce to the original VGM model when $, = 0 (and, hence, ,6 = 1). Equation (81) bears some similarity with the well-known Brooks-Corey model [13]relating the relative hydraulic conductivity to the matrix potential in the form (82)

where $, is the minimum capillary height and e is a suitable parameter. Although the approach leading to Richards’ equation is attractive since it reduces the variably saturated flow problem to a single PDE, it requires two algebraic relations for its closure, and this presents two drawbacks. Firstly, phenomenological parameters must be supplied to specify the soil hydrodynamic functions. For instance, five parameters are required in the VGM model and six in the modified VGM model. These parameters are often difficult to determine experimentally, but can have a significant impact on model predictions. This issue will be further discussed in Section 5. The second drawback is that field experiments indicate that the functions $ H 8($), $ H k T ( $ ) , and 8 H k r ( e ) are not single-valued. Instead, an hysteresis behavior is often observed


39

depending on whether the water content in the soil is decreased or increased. Hence, the use of the above models implicitly assumes that the volumetric water content is monotone during the simulation.

3.3

Space and time discretization

Many strategies to approximate Richards’ equation are feasible depending on the choice for the primal unknown, of the non-linear iterative solver, and of the space and time discretization scheme; see, e.g., Celia et al. [14]. In the present work, we focus on some specific choices. Firstly, the matrix potential $ is retained as the primal unknown. Secondly, the socalled method of lines is employed for the time-space discretization: the problem is first approximated in space using (for instance) finite element methods, thereby yielding a system of coupled ordinary differential equations (ODEs) where the time is the only independent variable. Then, a time approximation is constructed by using the vast theory of solution techniques for ODEs; see, e.g. [15] for a thorough review. Finally, Newton’s method is used as the iterative solver to obtain an approximate solution to the non-linear discrete equations. In this section, we briefly describe two finite element approximations to Richards’ equation, namely a conforming approximation of Richards’ equation in primal form and a non-conforming approximation of Richards’ equation in mixed form.

3.3.1

Conforming approximation in primal form

Consider the primal form (71) of Richards’ equation. Up to an appropriate lifting of the boundary data g1 and a modification of the right-hand side f , we can assume that a homogeneous Dirichlet boundary condition is prescribed on dR1. Set

and let

UR

be the form (non-linear in $, linear in

4):

Then, a possible weak formulation of (71) is the following:

( Seek $ E C1([O,TI;Van,) such that, for all t 2 0,

40

Alexandre Ern

where C'([O,T];Van,)is the space of functions in time with values in Vanl that are of class C1 in time, and T is a fixed simulation time. To approximate (85), consider conforming, piecewise linear finite elements. Assume that the mesh 7 h is compatible with the partition 6'01 U dR2, i.e., that 6'01 n dR2 consists of mesh vertices (and edges in three dimensions). Introduce the finite element space

P,,anl= { 4 h E co(a); VK E $,

$h(K

E Pi; & = 0 On 6'a1}.

(86)

The conforming approximation of Richards' equation in primal form is the following:

( Seek $h E cl([O, T ] ;P,',,,,)

such that, for all t 2 0,

To approximate (87) in time, we restrict the discussion to the implicit Euler scheme. Let 6t be the time step and define

Let +fl E P,',an, be a suitable approximation to the initial value $0, e.g., the L2-orthogonalprojection of $0 onto Pianlor its Lagrange interpolant. Then, the time-marching scheme consists of generating the sequence of approximations $; E PkaS2,,n 2 1, where $; is an approximation of $ at time tn = n6t, by solving sequentially in time the following problems: For n 2 1,

( Seek $;

3.3.2

E P:,an, such that

Non-conforming approximation in mixed form

Consider the mixed form (72) of Richards' equation. This section briefly describes the extension of the finite volume box scheme presented in Section 2.5 to Richards' equation in mixed form. For the sake of simplicity, assume g2 = 0, i.e., no-flux conditions are imposed on 8 0 2 . Define the finite element spaces .I


41

and R T $ ~ , = { u h E [ L ' ( ~ ) ] ~ ;EvZ, K U Va E

E RT~(K);

n do',

[uh. n], = 0;Va E

d:,

~IK

.

u h n1, = 0 ) .

(91)

Let 4: E be a suitable approximation to the initial value $0. Then, the time-marching scheme to generate a sequence of non-conforming approximations (UE,$;) E RTtanz x P2,,an1, n 2 1, consists of solving sequentially in time the following problems: For n 2 1, such that

( Seek (uE,$z) E RT,qan2x

Proposition 6. For all n 2 1, the solution (u;, $:) with the following properties:

to (92) is endowed

1. Local mass conservation VK E

z , nK(t1( ~ ( + -; )e($;-l)) + v . u; - f)

2. Local velocity reconstruction: For all K E

UEIK

= -hn~(b($E))(V$Ek

+:Wf

-

&(e($z)

= 0 . (93)

7h,

+ e,)

e($;-l)))..K(x).

(94) 3. Equivalence with the non-conforming approximation of the primal formulation: $; can be evaluated by solving sequentially in time the following problems: For n 2 1,

( Seek $:

E Piclanl such that

[ 6t J,

Hhf$h

=

+

s,

Iw($;-l))$h,

Proof. Similar to that of Proposition 5 .

-

W hE

P&,anl . 0

3.4 Water-tableground interact ion and the obstacle problem To investigate water-tableground interaction, we consider a simplified problem: the domain R is a quadrangle, no-flux conditions are imposed

42

Alesandre Ern

on the le&, bottom, and right boundaries, and the top boundary, aQ,, is exposed to a constant rainfall with velocity v, = -ie, where i > 0 is

the rainfall intensity; see Figure 3. In this setting, a%E,coincides with the ground surface. Henceforth, it is wsumed that i 5 k,, i.e., that the rainfall intensity never exceeds the infiltration capacity of the soil.

$ 5 0 and

u

s

n = v, . n

/'

Figure 3 Setting for water-tableground interaction. A s s m e that initially the water-table does not intersect the ground surface. Then, the boundary condition is u . n = v, . n throughQut 8Q, Le. , a non-homogeneous Neumann boundary condition expressing the fact that She rainfall i d h a t e s inside the soil. As a result, the v o ~ u ~ e water content in the soil increases, and the water-table moves upwards the until it rexhes the ground surface. Once the water-table ~n&ersect~ ground surface, the following holds on ail,:

$I 5 0 ,

u n 2 v, ' n ,

+(u n - v, * n)= 0 ,

(96)

so that the ground surface can be divided into two regions

an;

corresponds to the non-saturated part of 8 2 , : 11, < 0 and u , n = v, . n; corresponds to the saturated part of 82,: = 0 and u . n >

+

v, " 12. An import an^ ~ s ~ ~ pintthis ~ model o n is that the height of the overland flow above the ground surface is neglected so &hatthe ~ o ~ d a~r y~ in 8Sa; is indeed $ = 0. A model accounting for the coupling between the water-table dynamics and the overland flowwill be discussed in Section 4. Figure 4 presents an example of profiles for the matrix potential and the normal flow velocity along the ground surface. %he point where the water-table intersects the ground surface is clearly visible and core ~ ym. It is also responds to a longitudinal abscissa of a p p r o ~ ~ a S 0.55 observed that the saturated part can be further divided into two subregions, one part still allows for some infiltration: 0;

+ = 0 and v, .n < u n < ~


+

..

43

Figure 4 Matrix potential (solid) and normal flow velocity (dash) along the ground surface. a

the other part corresponds to the exfiltration region: $ = 0 and u . n > 0.

To investigate the formulation of Richards’ equation (71) together with the obstacle boundary condition (96), we consider first a steady setting and then an unsteady setting. For the sake of brevity, only the conforming approximation to the primal form of Richards’ equation is presented; the extension to the non-conforming approximation in a mixed setting is straightforward. 3.4.1

Steady obstacle problem

Given dRg

c dR,,

let P:,anz be defined as in (86) and let

Note that working with the space Van; implies $ = 0 on the saturated part an:. The discrete steady obstacle problem consists of seeking a pair {an:, $ h } such that

dR9+ c dR, ,

(98)

P:,an; 7 ax?$($h’ 4 h ) = 0

(99)

$h E

50 u($h) . n 2 w, . n $h

W J hE

on

P:,a,$

Xl;,

on d~;2g+.

The well-posedness of (100) requires that table has reached the ground surface.

’

(100) (101) (102)

# 0, i.e., that the water-

44

Alexandre Ern

An approximate solution {an:, $h} of (98)-(102) is sought using Newton's method embedded into a fixed-point iteration to determine the intersection of the water-table with the ground surface. The following iterative algorithm is proposed to solve the problem: 1. choose an initial dR:; 2. solve problem (100); 3. check whether (101) and (102) are satisfied; 4. if (101) is satisfied and (102) is not, decrease dR$ by one or more mesh cells; go to step 2; 5. if (102) is satisfied and (101) is not, increase dC$ by one or more mesh cells; go to step 2; 6. if both (101) and (102) are satisfied, then the current pair {do:, $h} is the desired solution; one can refine the mesh and go back to step 2. Owing to the maximum principle, both (101) and (102) can not be violated simultaneously. However, in numerical approximations, this can happen. In this case, we still consider that the water-table has been correctly positioned. With this "loosened" convergence criterion, the final position of the water-table depends from whether the converged has been approached from below or above. The two resulting values give lower and upper bounds for the water-table position (typically differing from one or two mesh cells at the most). 3.4.2

Unsteady obstacle problem

The unsteady version of the above obstacle problem can be implemented using an implicit Euler scheme. For n 2 1, given (aCl$)n-l and $,"-', seek { (afl:)", $}; such that

(an;)" c an,, $2 E P;(an;)s , ( B ( i i t ) - Wh"-')) 4 h + a(an;,-($r,

$;I0

on

(103) 7

4h) = 0

(an;). ,

(104) W h E P1 .,(an;)-

7

(105) (106)

u($,hn). n L w, . n on (107) This problem can be solved using the same iterative algorithm as for the steady obstacle problem. In step 1, the initial choice is = Note that in the unsteady case, problem (105) is well-posed even if the water table has not reached the top boundary yet.


45

4 Overland flow This section briefly presents some models that can be used to describe overland flow over hillslopes. The coupling with Richards’ equation is also discussed.

4.1 Model formulation Transient flow of shallow water can be modeled by the Saint-Venant equations. To simplify, we consider a two-dimensional hillslope model; the overland flow then reduces to a one-dimensional problem governed by the following equations:

where y is the water depth, V the flow velocity, w the source term, g the gravity, S the river bed slope, and Sf the energy line slope. We assume that in the momentum equation (log), the first two terms, i.e., the inertia terms, can be neglected in comparison with the last two terms. This yields the so-called diffusive wave approximation

+

aty az(Yv) = w , a,y - Sf s = 0 .

+

This model is widely used to describe flood routing; see, e.g., [16]. The Manning-Strickler uniform flow formula is usually chosen to evaluate the energy line slope Sf [17,18], namely

v = K~R:S),

(112)

where KS is the Strickler coefficient of roughness and R is the hydraulic radius defined as the ratio between the cross-sectional flow area, A , and the wet perimeter, x. Assuming that the overland flow occurs as a thin layer with a wide rectangular section of width B yields the relation y - e($;-l))

#h

-k a(an$)-.($;,

# h ) = R($;, $:-';

#h)

7

if there exists avertex Ni on (aR$)n such that #h = # h , i , and R(@,$$-'; otherwise.

#h) = 0

5

Applications

This section presents two applications of the above finite element techniques to simulate hydrosystems, namely infiltration into a one-dimensional soil column and a two-dimensional hillslope exposed to heavy rainfall.

Finite Element Modeling of Hydrosystems with . .

5.1

49

One-dimensional infiltration

One-dimensional stationary and unstationary infiltration problems with analytical solutions are simulated to validate the subsurface flow code. 5.1.1

Stationary test case

Consider a one-dimensional steady infiltration under a constant rainfall rate, 21, = -ie,. Mass balance readily yields i = h k T ( $ ) (&$

+ 1)

(131)

Assuming $(zL) = $L, the solution to (131) is

The Brooks-Corey model (82) is used to express the relative hydraulic conductivity in terms of the matrix potential. Three types of soils are considered. Table 2 lists the saturation hydraulic conductivity, k,, and the two Brooks-Corey model parameters, $s and e, for each soil. The left plot in Figure 6 displays the relative hydraulic conductivity as a function of matrix potential near saturation. Table 2 Saturation hydraulic conductivity and Brooks-Corey model parameters for the three soils considered in the stationary infiltration test case.

3.3

5.64

11.88

Using (82) in (132) yields (see, e.g., [20])

+ 3;

where G is the hyper-geometric function G(x)= z F l ( 1 , l ;1 x). Simulations are performed with the the same infiftration rate, i = 1.224 x lo-* m/h, for the three soils. The boundary condition is $L = 0 at Z L = 0. The right plot in Figure 6 displays simulation results. The agreement with the analytical solution is excellent.

Alexandre Ern

50 1 ._ 0.8 3

-$

8 .-

0.6

TJ

0.4

2

3 ._ J c

_m

d

0.2

0

-4 -3.5 -3 -2.5 -2 -1.5 -1

-0.5

0

Pressure head h (cm)

3.5 3 2.5 2 1.5 1

Silt:!!lsniy

0.5

=

0

0

0.5

1 1.5 2 2.5 Dimensionless capillary tension

3

3.5

Figure 6 Stationary infiltration test case. Top: relative hydraulic conductivity kT versus matrix potential for the three soils. Bottom: simulated and analytical profiles of capillary tension, &+(-$-), for the three soils.

5.1.2

Unstationary test case

The following test case is discussed in Barry et al. [21]; see also [20]. Assume that the relative hydraulic conductivity is defined by the BrooksCorey model, equation (82), with parameters qSand @. Let k: denote the derivative of the function H kr($). Assume that the soil water retention curve takes the following form

+


51

with parameter

Set G

=

and

+

A(t) = 1 W [- exp(tL/y - l)],

(137)

where W is the lower portion of the Lambert function. Then, one can show that an analytical solution to Richards’ equation, $la(t, z ) , such that $la@,0) = 0 for all t , is, for z < 0,

z &(t, z ) = -- . A@)

-N

0

-45

0.2

-4

04

35

0.6 08 1 Effectivewater content

-3

-2 5 -2 -1.5 Pressurehead (m)

1.2

-1

1.4

-0.5

0

Figure 7 Unstationary infiltration test case. Simulated and analytical vertical profiles for the water content (top) and the pressure head (bottom) at the times t = 10.1, 20.1, 30.1, 40.1, 50.1, 60.1, 70.1, 90.1, 100.1, 130, 170, 200, 500, 1000 h.

Alexandre Ern

52

Simulations are performed with the clay soil considered in the previous section (13, = 0.04, OS = 0.432). The simulation domain is the interval z E [-1.6, -0.61. The boundary conditions enforce the value of the analytical solution at both ends of the interval. The initial condition is the analytical solution at t = 0.1 h. The saturation front reaches the top boundary at about t = 60 h, and the domain is completely saturated at about t = 290 h. Results are displayed in Figure 7. The agreement with the analytical solution is very good.

5.2

Two-dimensional hillslopes

Consider the two-dimensional hillslope configuration shown in Figure 8. Define &rain = iL(e, en). Let Qin(t) be the (time-dependent) infiltration flux and let Qnotin(t) be the “direct runon” flux (the water that never infiltrates); hence, Qrain

= &in@)

+ Qnotin(t).

(139)

Let Qexf(t) be the exfiltration flux and let Qrunoff(t) consist of the exfiltration and the direct runon fluxes

Qrunoff(t) = Qexf(t)

+ Qnotin(t)

= Qexf(t)

+

Qrain - Qin(t).

(140)

If the rainfall intensity is kept constant in time, the time to reach equilibrium, Te, is conventionally defined as the first time in the simulation for which I Q i n ( Z ) - Qexf(Te)l I 0 - 0 0 5 Q i n ( Z ) . (141) At equilibrium Q i n ( m ) = Qexf(m) and, hence, Qrunoff(m) = Qrain. Note that at a given time t, Qrunoff(t) is not the instantaneous water flux at the toe of the slope but the instantaneous water flux into the overland flow. L=L t L 1

2

A (035) B (1014.8)

C (39.85:ll) D (54.8530.8) A

E (54.857.8) F (39.85;8)

G (1011.8) H

0

E

Figure 8 Hillslope geometry (length unit: meter)

Finite Element Modeling of Hydrosystems with ...

5.2.1

53

Influence of the soil hydrodynamic parameters

For all the test cases presented in this section, the initial condition is the steady-state solution corresponding to a rainfall intensity of = 0.4%. The initial condition is presented in Figure 9. Furthermore, the obstacle model is used to simulate the water-table dynamics once it has reached

&

15 14 13

10 9

a

.

7

.

-0.5 0

-1

.

.

.

.

.

.

0.5

1

1.5

2

2.5

3

3.5

Psi (m)

0.1

........

I . .

............

.............

.....

-0.8 0

20

10

30

40

50

x (m)

0.002

0.0015

....... .:. .....-1. ......;............4.. ..

0,001 ........,...................... .......

J....

c S

0,0005 .......

0

........I.

~i.. ....-1. ......:..... ...... I . .

.....I.. .

-0.0005 0

10

20

30

40

50

I

Figure 9 Initial condition for the hillslope test case. Top : vertical profile of matrix potential at various points located by their horizontal coordinate 5. Meddle: matrix potential along the ground surface. Bottom: normal flow velocity along the ground surface.

Alexandre Ern

54

the ground surface. We first investigate the impact of the parameters a and n in the VGM model. The soil chosen as a reference is a so-called Yo10 Light Clay (YLC) soil; its VGM parameters are 8, = 0.23, Os = 0.55, a = 3.6 m-l, n = 1.9, and k, = 0.018 m/h. Then, the parameters a and n are varied, and a new steady-state solution is computed. In all the simulations, the soils are subjected to a constant rainfall intensity of = 2%.

t

Table 3 Time to reach equilibrium. YLC soil T, [h] 2034

I n = 1.31 I 865

n = 3.9

a = 0.3

2925

71

a = 8.0 3298

Times to reach equilibrium are reported in Table 3. The first line indicates the parameter that has been varied and the corresponding value. The impact of both a and n cn Te is evident. Figure 10 presents the hydrodynamic functions $ H O ( $ ) and $ H lc,($) for the various soils considered in the simulations. A more specific investigation of the impact of the parameter n and that of the parameter a is presented in Figure 11.

:;--."--.-----

1

0.9

.

0.7

. .

0.3

.

0.8

Ib

0.3

0.2

. .

-

0.1.

f' /'

-.

',

YLC n=1.31-n=3.9 7 '

^ _ ,

__ .- .

.

, I

/ '

-

0.2-

0 . 1 . 4.7

'

'

4.6

4.5

YLC

oL=0.3 "

4.4

+

4.3

'

-- .

a1=8.0 7-

4.2

I

4.1

0

(m)

1

0.1

0.01 C

Y 0.w1

0.wol

c ."

1 6 4 5 '

4.7

YLC

'

'

4.6 0.5

'

4.4

' 4.3

-

n=1.31n=3.94.2 0.1

.

.

o.wo1

--

YLC 01=0.3

1

a=s.o .-

6

0.7

4.6

-0.5

0.4

4.3

4.2

.-

0.1

Figure 10 Hydrodynamic functions for various values of n (left) and a (right). Top: $ H g($). Bottom: $ H k,(+).


...

55

n

time trre 1

1 0.9

0.8

0.7 0.6

0.6 0.5 0.4 t 0.3

0.2

h

0.1

0

0.003

0.2

0.4

YLC

-I j

0.8 time trre

0.6

-

1 exf.

h 1

h 1.2

,

I

1.4 time VTe

:

e

a

0

-0.0005 0

-0.0005

1 0 m 3 0 4 0 5 0

Figure 11 Left column: influence of n on simulation results; right column: influence of o on simulation results. Top: saturated length L , normalized by length L (see Figure 8) as a function of time normalized by T,; middle: runoff and exfiltration fluxes normalized by Qrain as a function of time normalized by T,; bottom: normal velocity flux 'U . n along the ground surface.

As a second test case, we investigate the impact of the modified VGM model on simulation results. The reference soil is again taken to be the YLC soil. Two additional simulations are performed by setting the minimum capillary height to $, = -2 cm and to $, = -10 cm in the modified VGM model, Figure 12 presents the corresponding hydrodynamic functions 4 H e($) and $ H kT($). A more specific investigation of the impact of the modified VGM model on simulation results is presented in

Alexandre Ern

56

Figure 13. 1

0.1

c

&?

0.01

0.w1' -0.7

0.6

0.5

-0.4

-0.3

-0.2

0.1

0

' 0.7

' 0.6

' 0.5

vj (m)

' 0.4

' 0.3

' 0.2

' 0.1

0

vj (m)

Figure 12 Modified VG-M model: hydrodynamic functions for various values of $s. Left: $ ++ t9($). Right: $ A kT($).

100

1

0.9

80

8

0.8 c

6o

0.6

-

0.5

a

3-

5 4 0

0.7

@

20

0.4

0.3 0.2

0

0.1

w

0

time (h)

time (h)

0

10

20

30

40

50

x (m)

Figure 13 Impact of the modified VGM model on simulation results. Top left: saturated length L , normalized by length L (see Figure 8) as a function of time normalized by T,; top right: runoff and exfiltration fluxes normalized by Qrain as a function of time normalized by T,; bottom: normal velocity flux 2 1 . n along the ground surface.

Finite Element Modeling of Hydrosystems with . . . 5.2.2

57

Influence of the overland flow

Consider the two-dimensional hillslope configuration shown in Figure 14 and investigated previously in [l].A no-flow boundary condition is imposed at the bottom and left surfaces, simulating an impermeable layer. The right boundary corresponds to a stream in which the hydraulic head is prescribed. The initial condition is a horizontal water table located at the toe of the slope; see Figure 14. The problem geometry is characterized by the slope So = lo%, the depth to the impermeable layer D = 1m, and the length L = 50 m. Simulations are run with a constant rainfall intensity such that $ = 0.6%. The soil hydrodynamic properties are Or = 0.069, Os = 0.435, a = 0.326 m-l, n = 3.9, and k, = 5.0 m/h; they correspond to the sand considered in [l].

water table

Figure 14 Hillslope geometry to compare the obstacle model with the overland flow model.

Figure 15 Comparison of the obstacle model with the model accounting for overland flow. Left: relative saturated area. Right: exfiltration flux. Figure 15 compares the results obtained with the obstacle model to those obtained with the model accounting for overland flow. The Strickler coefficient of roughness is set to K s = 10 Both models

58

Alexandre Ern

yield very similar results, especially for the time evolution of the relative saturated area. Slight differences in the exfiltration flux are observed for sufficiently long times; these differences arise mainly near the toe of the slope.

References [l]Ogden F L, Watts B A. Saturated area formation on non convergent

hillslope topography with shallow soils: a numerical investigation, Water Resour. Res., 36(7), 1795-1804, 2000. [2] Cloke H L, Renaud J P, Claxton A J, McDonnell J J, Anderson M G, Blake J R, Bates P D. The effect of model configuration on modeled hillslope-riparian interactions, J. of Hydrology, 279,167181, 2003. [3] Weiler M, McDonnell J. Virtual experiments: a new approach for improving process conceptualization in hillslope hydrology, J. of Hydrology, 285, 3-18, 2004.

[4] Ciarlet P. The Finite Element Method for Elliptic Problems. NorthHolland, Amsterdam, 1978. [5] Brenner S, Scott L R. The Mathematical Theory of Finite Element Methods. Vol. 15 of Texts in Applied Mathematics, Springer-Verlag, New York, 1994. [6] Quarteroni A, Valli A. Numerical Approximation of Partial Differential Equations. 2nd edition, Vol. 23 of Series in Computational Mathematics, Springer-Verlag, New York, 1997.

[7] Ern A, Guermond J L. Theory and Practice of Finite Elements. Vol. 159 of Applied Mathematical Series, Springer-Verlag, New York, 2004. [8] Croisille J P. Finite volume box schemes and mixed methods, Math. Mod. Numer. Anal., 31(5), 1087-1106,2000. [9] Baer J, Bachmat Y. Introduction to Transport Phenomena in Porous Media. Kluwer Academic Publishers, 1990. [lo] van Genuchten M Th. A closed-form equation for predicting the hydraulic conductivity of unsaturated soils, Soil Sci. SOC.Am., 44, 892-898,1980. [ll]Mualem Y. A new model for predicting the hydraulic conductivity of unsaturated porous media. Water Resour. Res., 12, 513-522, 1976. [12] Vogel T, van Genuchten M Th, Cislerova M. Effect of the shape of the soil hydraulic functions near saturation on variably-saturated


59

flow predictions, Advances in Water Resources, 24(6), 133-144, 2001. [13] Brooks R H, Corey A T. Hydraulic properties of porous media, Hydrol. Pap., 3,Colorado State University, Fort Collins, p. 27, 1964. [14] Celia M A, Boulotas E T, Zarba R L. A general mass-conservative numerical solution for the unsaturated flow equation, Water Resour. Res., 26(7), 1483-1496, 1990. [15] Lambert J. Numerical Methods for Ordinary Differential Systems. Wiley, New York, 1991. [16] Moussa R, Bocquillon C. Criteria for the choice of flood routing methods in natural channels, J. of Hydrology, 186, 1-30, 1996. [17] VanderKwaak J E, Loague K. Hydrologic-response simulations for the R-5catchment with comprehensive physics-based model, Water Resour. Res., 37(4), 999-1013,2001. [18] Liu Y B, Gebremeskel S, De Smedt F, H o h a n n L, Pfister L. A diffusive transport approach for flow routing in GIS-based flood modeling, J. of Hydrology, 283, 91-106, 2003. [19] Borah D K, Prasad S N, Alonso C V. Kinematic wave routing incorporating shock fitting, Water Resour. Res., 16(3), 529-541, 1980. [20] Ginzburg I, Carlier J P, Kao C. Lattice Boltzmann approach to Richards’ equation, Proceedings of the CMWR Conference, North Carolina, 2004. [21] Barry D A, Parlange J P, Sander G C, Sivaplan M. A class of exact solutions for Richards’ equation, J. of Hydrology, 142, 29-46, 1993.

60

Sharp Front Modeling Patrick Goblet Centre d’Informatique Gkologique Ecole Nationale Supkrieure des Mines de Paris 35, rue Saint-Honor6 77305 Fontainebleau, France E-mail: [email protected]

Introduction

1

This paper discusses some recent orientations in the modeling of the concentration fronts resulting from the solution of the solute transport equation (dispersion equation). The objective is to show the main apporaches and to discuss their advantages and shortcomings. It is beyond the scope of this paper to describe in detail the techniques, and the reader is referred to the bibliography for a detailed description of each particular approach.

Sharp front problem statement

2

The dispersion equation describes the movement of a solute (tracer) in an aquifer. This movement is the result of three main mechanisms ( [lo]): 0

0

0

Advection is the global movement of a volume of water containing tracer at the average pore water velocity. Molecular diffusion is the spreading of tracer in immobile water under a concentration gradient (Brownian motion) Dispersion is an additional spreading mechanism due to water velocity fluctuations around the mean value. It is ususally described by a diffusive (fickian) model.

The dispersion equation writes:

dC V(DVC - UoC) = W Z where

C is the solute concentration (tracer mass/unit volume)

Sharp Front Modeling

61

D is the dispersion tensor (m2/s) UD is the Darcy velocity (m/s) w

is the kinematic porosity

t is the time (s) The D tensor includes the molecular diffusion coefficient and the dispersion coefficient. It writes, in axes aligned with the local velocity: aL

D= 0 0

Q T ~and Q T ~ are

0 0

0 QT~

0

+

IUD~ doI

wr2

transverse dispersivities (m)

do is the tracer diffusion coefficient in the porous medium

I is the unit tensor Equation (1) is made of an advective and a diffusive term. The relative importance of each term depends on the value of the dispersivity tensor. As dispersivity is in fact a global representation of local velocity heterogeneity, it can be considered as reflecting our ignorance of the actual velocity field. The role played by dispersion is therefore reduced as the knowledge of the velocity field is refined. For this reason, in a well characterized Aow field, advection may become the main transport mechanism. On the other hand, solute transport is not sufficient to describe the fate of chemical substances in an aquifer: most solutes are subject to chemical interactions with the immobile medium as well as with other species. The dispersion equation is now merged in more general models which describe the movement of a chemical system in groundwater. This leads to source and sink terms in the transport equation of each species, which often create sharp transition zones (redox front are a typical example). The numerical simulation of sharp concentration fronts is a problem faced in many field studies. Most classical numerical techniques (Finite Volumes, Finite Elements) perform well when the dispersion front is spread over several grid blocks. This constraint can be expressed in terms of a Peclet Number:

Pe

UDAX

= -5

D

2

(3)

Patrick Goblet This constraint may not be met in practical applications involving large scale models, because it would require either inacceptable values of the dispersivity tensor, or an unpractically fine mesh. As a consequence, better approximation techniques are continuously looked for.

Classification of approaches

3

Most numerical techniques used for the solution of the advectiondispersion problem fall into one of the following categories: 0

0

0

Eulerian techniques: the transport equation is solved in a fixed coordinate system. This means that the advective and dispersive terms must both be discretized. This can be done by Finite Elements, Finite Differences, Finite Volumes, . . . Lagrangian techniques: the advective term is treated in a moving coordinate system. The dispersive term can be treated by various techniques. Eulerian-Lagrangiantechniques: both terms are generally discretized in a fixed coordinate system. However, some techniques are borrowed from the lagrangian approach to facilitate the treatment of the advective term.

Particle techniques are a particular class of approaches which do not necessary rely on a discretization of the equation, although they may use one. These techniques are interesting by themselves, but also because they bring some ideas which can be used in the Lagrangian or Eulerianlagrangian approaches. In this paper, we shall first present a few ideas pertaining to the particle approach technique. Then we shall present approaches which aim at improving the performance of Eulerian techniques. Lagrangian approaches will then be briefly presented. Finally the more recent developments in the Eulerian-Eagrangianfield will be discussed.

4

Particle techniques

The general idea behind so-called “particle techniques” is to follow the movement of a small volume of water. This volume carries a certain mass. It is tracked along the flow lines at pore water velocity, which accounts for the advection mechanism. As the velocity field in the general case is not uniform, but rather varies continuously along the flow path, time must generally be discretized and the velocity field must be


63

approximated. The position of a particle at time t + At is obtained from its position at time t by the following relations:

~ (+ tAt) = ~ ( + t )V,At y(t

+ At) = y ( t ) + VgAt

(4)

The precision of this scheme depends on the time step, but also on the precision of the velocity field. When, as is generally the case, no analytical expression is available for the velocity field, it must be derived from an approximate solution of flow, obtained e.g. by Finite Elements or Finite Volumes. The precision on the velocity field can then be increased in two ways: 0

0

Increasing the order of representation of the velocity. Mixed Finite Elements are a classical approach to achieve this goal without increasing the degree of approximation of the hydraulic head. Formulating the flow problem in terms of stream function rather than in terms of hydraulic head. This approach is only possible in steady state, and when no hydraulic source terms are present.

Both approaches yield a velocity field which has a continuous normal component accross element interfaces. This property is very valuable to allow the tracking of particles in strongly varying velocity fields. The description of dispersion in the framework of particle techniques can be done in two ways, which can be related to the main “historical” techniques: 0

Method Of Characteristics (MOC, see e.g. [B])

0

Discrete Parcels Random Walk Technique (DPRW, see [l]).

The MOC technique is based on a fixed grid. Each time step is split into an advective and a diffusive step. At the end of an advective step, particles are counted in each grid block. The obtained mass is then averaged on the grid block, yielding a concentration value. The dispersion operator is then solved on the fixed grid. The new concentrations are spread again in particles, and a new advective step is performed. In the DPRW technique, no mesh is required for the description of dispersion (a mesh may however be required to describe a non uniform flow field). The redistribution of particles in space at the end of each advective step is done according to a Brownian motion, which describes the diffusive nature of dispersion. Particles techniques can be found in the literature under various flavours (see e.g. [14] for an object-oriented implementation in fracture networks). They are implemented in several classical water management softwares. MOC methods often suffer from numerical diffusion, due to

64

Patrick Goblet

the repetitive mechanism of scattering-gathering of particles. This drawback does not exist in DPRW methods, because each particle retains its identity throughout the whole simulation. A large number of particles is necessary to achieve a sufficient precision, especially in 3 D simulations. On the other hand, particles are only requested in the regions of the studied domain where tracer is effectively present. This contrasts with classical discretized equations, for which the equations must be solved even in tracer free regions. This makes particle techniques well suited for the simulation of short and localized contaminations. Two important ideas of particle techniques have been retained in the Eulerian-lagrangian techniques which will be discussed later: the splitting of operators (solving advection and dispersion in two separate steps), and the development of accurate techniques to track water volumes along characteristic lines of flow.

5

Improving eulerian approaches

As stated earlier, eulerian techniques perform well when the ratio between advective and dispersive flux permits the development of a relatively smooth front, i.e. typically a front spread over several grid blocks. When this condition is not met, a problem arises, due to the difficulty to describe appropriately the sharp concentration variation around the front: spurious oscillations (overshoot-undershoot) develop and are p r o p agated (see Figure 1). The solutions proposed in the literature to this problem try to tackle it according to various strategies: by an optimization of the definition of the upstream concentration, either in Finite Element Schemes (Streamline Upstream schemes, e.g. SUPG ([13]), space-time approaches (SUFG, [12])), or in Finite Volume schemes (Monotone Upstream Centered Schemes for Conservation Laws, MUSCL, [IS]) by an increase of the degree of approximation, either in a Finite Element Framework, or in a Spectral Element framework ([S]) by discontinuous approximations of the concentration field. We shall first discuss in this section the basic principle of upstream weighted approaches and show examples of this approach. Spectral elements will then be presented and their use illustrated on an example.

5.1 Improving a 1D Finite Difference Scheme A standard approximation of the advection-dispersion equation on a 1D Finite Difference grid for a fixe grid block size Ax and a time step At

65


140 120

100

80

60 40

20 0

0

60

40

20

100

80

120

Distance (m)

Figure 1 Instable propagation of a dispersive front. (see Figure 2) writes:

where Ci is the value of concentration at node i for some time between tn and tn+l,and Ci-1/2 (resp. Ci+1/2) are concentration values at some point between points i - 1 and i (resp. i and i 1).

+

i- 1 I

i

Ax

i+ 1 I

I

i-112

Figure 2 tion.

i+1/2

1D Finite Difference Approximation of the Dispersion equai

One simple choice, which gives the best accuracy for this scheme, is a centered approximation in space as well as in time:

Ci-1/2= ci+1/2=

Ci-1+ Ci 2

Ci

+ Ci+l 2

66

Patrick Goblet

and

This scheme gives a second order accurate approximation,which is stable under the condition: U oAX P e = -5 2

D

Pe is the Peclet Number. When this condition is not met, numerical solutions exhibit oscillations. 5.1.1

First order stabilizing scheme

The classical cure to this problem is to replace the centered spatial approximation of velocity in the advection term (eq. 6) by a so called “upwind” approximation: the concentration is taken upstream of the cell boundary.

This scheme is stable and monotonous, but only first order accurate. It can be shown simply that it is equivalent to introducing into the centered scheme an extra diffusion term (numerical diffusion term) equal to

As a general rule, it can be shown that a linear, monotonous stabilizing scheme cannot be 2nd order accurate. Note that this upstream scheme is classically used in classical Finite Difference softwares. These can therefore be expected to introduce a numerical diffusion with a longitudinal component equal to half the grid block size. 5.1.2

Higher order stabilizing schemes

The basic idea of higher order schemes is to adapt the scheme to the smoothness of the concentration profile: far from the front, a centered scheme can be adopted. Close to the front, a variable formulation, intermediate between a fully upstream and a fully centered formulations, is used. This type of scheme, which depends on the concentration profile, is by nature non linear. A classical example of such approach is the family of MUSCL schemes (see [16]). Discussing this type of scheme is beyond the scope of this presentation. Let us just show a simple example of a “quasi-second order” scheme.


67

-

Front propagation in an homogeneous medium Pe = 100 Variable upstreaming scheme

Figure 3 Solution of the 1D dispersion equation using an adaptative upstream scheme. We start by defining an indicator which is related to the concentration gradient variation: for a node i, this indicator is the ratio between the gradient upstream of the node and the average gradient at the node:

ai is close to 0.5 in smooth concentration zones, close to 1 on the top edge of a front, and close to 0 at the foot of a front. We use ai as a variable weighting factor to interpolate the upstream concentration: Ci-l/2 = Qi-lCi-1 Ci+1/2=z aiCi

+ (1 -

+ (1 -

ai-l)Ci

ai)Ci+l

(12)

This yields a centered scheme in smooth concentration regions, an upstream scheme on top of the front, and a “downstream” scheme at the foot of the front. Figure 3 shows the propagation of a dispersive front calculated on a 120 grid blocks mesh according to this scheme. The solution is compared to a reference solution obtained on a 12000 grid blocks mesh. The apparent (real+numerical) dispersivity for this scheme is 3 cm, showing a numerical dispersivity of 2 cm, a very low value compared to the grid block size of 1 m.

68

5.2

Patrick Goblet

Improving a 1D Finite Element Scheme

The abstract nature of Finite Elements does not allow to introduce the same kind of “intuitive” upstreaming as in Finite Differences or Finite Volumes. The same type of result can be obtained using asymmetric test functions for the advection term. This idea was first introduced, at least for solute transport problems, by Huyakorn ([7]). Figure 4 shows an example of an asymmetric test function. The degree of asymmetry can be related to the Peclet Number.

Figure 4 Asymmeric test function. Variants of this type of scheme are widely applied in simulators. The asymmetric test functions can be generalized in 2 or 3 space dimensions. Upstreaming is then performed along the velocity vector (Streamline Upstream Petrov Galerkin, SUPG, see e.g. [13]). Basically, the stabilization introduced by these schemes may be interpreted in terms of numerical diffusion. A more general approach has been proposed e.g. in [12], in which time is treated as an extra dimension (Space-Time Finite Elements). The advantage of this approach is that upstreaming is performed not along flow lines, but along characteristics, which drastically reduces the spatial numerical diffusion. This technique is termed SUFG (Streamline Upstream Full Galerkin). While upstream appproaches provide often a very efficient way to simulate sharp fronts, they are generally subject to a Courant-FriedrichLevy (CFL) condition on the time step:

c = -Ax

VAt

b, then - < 0, if u < b, then - > 0. That is to say, dt dt u + b when t + 00. It can be seen in Figure 2.1. Now, the question is how to determine the value of ‘b’ when red tide breaks out.

Ecological Simulation of Red Tides in Shallow Sea Area

103

t

Figure 2.1

Integral line of u for equation (2.1).

From equation (2.1) it can be found that the larger ( b - u)is, the du is small, larger - is. Imagining that ‘b’ is very large and the initial ‘u’ dt what would happen to the population of phytoplankton? Apparently, is very large at the beginning, and it will drop when the value of ( b - u)decreases. The factors influencing ‘b’ value are exactly those mentioned above. The nutrients (pollutants) in the water are the main reasons that lead to algae over-grow. The equilibrium point of the phytoplankton concentration is the value when phytoplankton concentration is unchanged with time. Under certain environment conditions phytoplankton concentration meets with the equilibrium point at the end and the differences between the equilibrium point and the concentration of phytoplankton determine the changing rate of phytoplankton concentration. The equilibrium point is essentially dependent on the environmental factors. This is used to explain the phenomenon of HAB. dt

3 3.1

Red tide ecological dynamic model Physical/Biological Coupling

Despite the diverse array of HAB species and the many hydrographic regimes in which they occur, one common characteristic of such phenomena is that physical oceanographic forces play a significant role in bloom dynamics. Furthermore, the interplay or coupling between the physics and biological “behavior”, such as respiration, photosynthesis, physiological adaptation, or prey-predator relation holds the key for un-

104

Deguan Wang

derstanding red tide outbreaks. This physical/biological coupling can occur at both large and small scales. And the biological and physical mechanisms that account for enhanced phytoplankton biomass are diverse, as they must be accounted for the different physical and chemical gradients associated with the regions. Many suggested mechanisms relate to differences in nutrient availability and light. High nutrient concentrations are often found at or below the pycnocline. Physical processes which can influence nutrient exchange across a front include interfacial friction and Ekman pumping[lO][ll], baroclinic eddies[l2I,and tidal ~tabilization/destabilization[~~]. On the other side, the growth and accumulation of individual harmful algal species in a mixed planktonic assemblage are exceedingly complex processes involving an array of chemical, physical, and biological interactions. So the practical implications of the basic research on these aspects are large with respect to our ability to understand and predict the patterns of blooms. There is thus a clear need to develop realistic physical models for regions subject to HAB events, and to incorporate biological behavior and population dynamics into those simulations. Models of red tide algae blooms have been developed from several different perspectives. K a m y k ~ w s k i [ ~examined ~ ] [ ~ ~ ] the response of a swimming dinoflagellate to internal waves and showed that accumulation of motile and non-motile cells may occur due to an internal wave field, with the accumulation of vertically migrating cells being most significant. These models consider only the physics of the wave field and the swimming behavior of the phytoplankton, without regard to the phytoplankton response to nutrients or light. Others have examined the response of phytoplankton to the flow field of Langmuir cell^[^^][^^] or to 2-dimensional, cross-frontal circu1ation[l81,to name just two of many physical systems. The primary method for exploring the details of the interactions of HAB populations with coastal circulation should be through the incorporation of biological and physical field data into circulation models. The physical dynamics of the model can be constrained by the equations used to describe the flow and by physical data gathered in field programs. Numerical experiments can then examine the distribution and fate of cells under a variety of forcing mechanisms. HAB cells can initially be treated as passive tracers, but it will most likely be necessary to include factors such as grazing or physiological adaptation in the models to accurately simulate observed cell distributions. It would then be possible to show, for instance, how cell division or predator behavior will cause cell accumulation and blooms, or to evaluate the relative importance of physical losses (advection,sinking) and biological losses (grazing, cyst formation) in bloom termination. The flexibility of such models allows certain aspects of the field data to be explored in a way that would be impossible through direct sampling. Another advantage is that the behavior and


105

physiology of a single HAB species can be better understood and simulated than that of an entire community. This is an area long neglected in HAB research, and one that should be strengthened in the coming years. The insights to be gained from modeling studies will do much to advance our general understanding of the dynamics and consequencesof HABs. In this section, we established a set of red tide ecological model, which include hydrodynamic simulation, water quality simulation and phytoplankton-zooplankton ecological simulation.

3.2 3.2.1

Governing equations Hydrodynamic simulation

A two-dimensional, depth averaged, hydrodynamic mathematical model was applied for simulation of hydrodynamics of water body. The conservative (divergence) form of the coupled two-dimensional shallow water equations and advection-dispersionequation is given by

where q is conserved physical vector; f ( q ) and g ( g ) are flux vectors in the 5 and y direction, respectively and S(g) is source/sink term vector. They can be expressed respectively as:

where,& = 0,272 = f v h - gh(soz

+ gh(sOy

-

+

sfv)

ar,, + sfz)+ (%+ -) dY

,S3= f u h

(%+ h) . aY

h is water depth; u and v are depth-

averaged velocity components in the x and y directions, respectively; g is gravitational acceleration; sox and sfzare bed slope and friction slope in the x direction respectively; soyand sfvare bed slope and friction slope in the y direction respectively. 3.2.2

Three-component ecological simulation

This section presents an overview of the three-component ecological model. Although we look phytoplankton and zooplankton as one of water quality constituents at first, in the loading terms we take their behavior into account. The equations for various water quality constituents are based on the key principle of the conservation of mass. This principle requires that the mass of each water quality constituent being investigated must be accounted for in one way or another. We trace each water

106

Deguan Wang

quality constituent from the point of spatial and temporal input to its final point of export, conserving mass in space and time.

A mass balance equation for dissolved constituents in a water body must account for all the material entering and leaving through direct and diffuse loading; advective and dispersive transport; and physical, chemical, biological transformation and prey. These aspects are concerned in loading terms. Mathematical Representation of Physical System Input: Estuary or Lake Geomorphological Data Output: 2-D Finite Volume Network

2-D Hydrodynamic Simulation Input: Hydraulic/HydrologicData and Boundary Conditions Output: Velocity. Depth. and Water Surface Elevation

Water Quality Simulation Input: water ~ ~ l i~t y~andt a Boundary Conditions Output: ConstituentConcentrations

Ecological Simulation: Input: Initial Spatial and Temporal Distributions. Output: Spatial and Temporal Distributions of PhytoplanMon Population Estimates

Figure 3.1 Red tide ecological model system. We use a three-component to represent concentrations of nutrient ( N ) , phytoplankton ( P ) and zooplankton (2) in a physically homogeneous oceanic mixed layer. Phytoplankton, aquatic plants that are mostly unicellular, take up nutrients from the water in order to photosynthesise. The phytoplankton are grazed upon by the animal zooplankton, which in turn provide sustenance for the higher trophic levels of the food web. Figure 3.1 illustrates the structure of the model, whereby arrows indicate flows of matter through the system, and arrows not starting or not ending at a compartment represent the input to and the losses from the modelled system. The general mass balance equation is given by d -dC _ - -u-(C) at dX

d

- .-(C)

OY

+

where C is concentration of the water quality constituents. are diffusion coefficients in the x and y directions.

Ex and Eg


107

When the explicit functional forms are included, the model becomes:

d

dN

d dY

- = -u-(N) at dX

- v-(N)

+ (3.3)

dP

-=

at

d --(UP)

-

dX

d -(wP)

dY

(3.4) dZ

d

- = --(UJ) at dX

d

+EZ

a

-(U,Z) dY dZ d

-

I)&(

[ z( z )+ &

dZ

+sz

(3.5)

S N ,Sp and Sz is loading term. It can be expressed as:

sp

sz

N a P-rP-e+Nb+cP dP2 =Z - dZ" p2 P2 1

XP2 z- (s P2

p2

+

+k)P

(3.7)

+

where E N ,Ep and EZ are diffusion coefficients on the assumption of isotropic. The parameter definitions of above equations are given in Table 3.1, together with the value of each parameter originally used by Steel and Henderson[lgI.

3.3

General description of numerical methods

The sets of equations listed above are highly non-linear, one can hardly obtain the analytical solution for those equations unless simplified under several reasonable assumptions. The numerical solution technique is the only method to obtain the approximate solution to those sets of equations. There are many numerical methods that can be applied for this purpose. The general frame of the numerical solution procedure and the types of numerical methods are shown in Figure 3.2. In order to give some impression about numerical model, two examples are given in following sections.

108

Deguan Wang Table 3.1 Parameter definitions

3.3.1

Finite element model[20][21][22]

In the Auid dynamics, the finite element method based on the Galerkin weighted residual method is mostly used method in descritizing the set of equations. 1. Governing equations Equation (3.1) can be expanded into following form: (3.9)

(3.10)

(3.11) where, U = ha, V

= hv;


109

Fundamental Conservation Laws

Differential Formulation

Integral Formulation

.1

.1

Analytic Solution Method

Finite Difference Method

Uniform Grid Method

Finite Volume Method

Grid Transformation

Non-uniform Grid

Residual Method

Finite Element Method

Variations 1

Boundary Element

Method

Method

Selection of Solution Method

Program Code

Figure 3.2 General frame of the numerical solution procedure and the types of numerical methods. After some modifications, above expressions can be reduced into following forms,

P P Tzxb =

pgU @ T v

(3.16)

C2h2J i E F

(3.17)

c2h2

TZYb

= pgV

T~~~

= c,-wJwI Pa P

T,,,

= C,--WlWl

Pa

P

sino COSO

(3.18) (3.19)

110

Deguan Wang

where c is the Chezy coefficient; c, is wind drag coefficient; pa is the density of the air; w is the wind speed; Q is the wind direction. 2. Finite element representation of the governing equations (1) Trial function (3.20) (3.21) where U , V , h are the unit width discharges and water depth at any point in the element; U I , VI,h I are the unit width discharges and water depth at the nodes of the element; (PT and QT are the shape functions for the unit width discharge and water depth respectively, for triangular element with 6 nodes, they are as

Q,=

(3.22)

Q=

(3.23)

where = ARe{

l

[f(ki, z) - f(h,-z>

- m a , z) + f(k2, -z)ldto},

(2.68a)

where

(2.68b) 1 e-l/(t-to) = 0, the integrand is non-singular and to+t (t - to)3/2 the integral can be easily evaluated via any standard quadrature scheme. Physically, the solution is expected to become periodic as t increases. In other words, the effects of an initial condition on the solution are diminished after a certain time (memory time, tc) elapses. Li et a2. (2002b) and Li and Jiao (2002b) reported further studies on two-dimensional tidal water table fluctuations. Because lim

2.10

Recent developments and other effects

There have been many other studies on the tidal water table fluctuations, some of which are listed Table 2.1.

Subsurface Pathways of Contaminants to Coastal Waters:

157

Table 2.1 Main contributions Jeng, Teo, Seymour,

aquifers bounded bv rhythmic shorelines. Tide-induced groundwater fluctuation in a coastar leakv confined aquifer system extending under the sea. Analytical studies of groundwater-head fluctuations in Li and J i m , 2001b a confined aquifer overlain by a semi-permeable layer with storage. Li, Jiao, Luk and Che- ‘ride-induced groundwater level fluctuation in coastal 2003 Li and Jiao, 2001a

2000 Townley, 1995 Trefry, 1999

3

I Plex Porosity. The effects of landward boundary conditions. Periodic forcing in composite aquifers.

Implications for contaminant transport and transformation in tidally influenced coastal aquifers

As demonstrated above, the tides affect greatly the coastal ground-water. The water table fluctuation is the manifestation of such effects in the shallow unconfined aquifer and has been studied extensively. The fluctuation leads oscillating groundwater flow in the near-shore area of the aquifer, enhancing the water exchange and mixing between the aquifer and coastal sea/estuary. In the following, we will illustrate the importance of these local flow, exchange and mixing processes for chemical transport and transformation in the near-shore aquifer and the associated chemical fluxes to coastal water.

3.1

Tide-induced flushing and dilution effects on chemical transport processes

Li et al. (1999) developed a model of SGD that incorporates the net groundwater discharge, and the outflows of the tide-induced oscillating groundwater flow and wave-induced groundwater circulation (Figure

I58

Ring Li

L l ) , i.e., D S G D = B, -i-L>w

+ L)t.

(3.1) Since a large portion of the high rate SGD (a,and D,) is of marine origin, questions remain as to how much the discharge contr~bute~ to the transfer of ~an~-derived pollutants to the ocean. Using the "box" model described below, Ei et a,?. (1999) examined the i ~ ~ p o r t aof SGD, especially Dw and Dt, on the process of cliemical transfers horn the aquifer to the ocean.

Figure 3.1 A box model of chemical transfer from the aquifer to coastal sea. The model includes three water bodies: coastal sea (CS), brackish aquifer (BA), and freshwater aquifer (FA). Chemical transfers occui between the water bodies as shown by arrows in Figure 3.1. The chemicals are assumed to be strongly absorbed by sand particles in fresh groundwater and LO desorb into brackish round water. Phosphate, a ~ m and cadmium are important constitue~~ts with this behavior. T h e mass balance for FA can be described by FFA-zrb = FFA-out

if 8

=I

Seq,

dS F F A -= ~ 0~and ~ VFA-- = FFA-~,if S < Seq, dt FFA-zn = DnCm and Seq

= kilolczn,

(3.2a) (3.2b) (3%) (3.24

where F F A - and ~ ~ F F A - ~are ~ ~the : input and output mass flux for FA, respectively (the alon~-shoredistance of the water body is taken to be a unit meter); S is the amount of absorbed chemical and the ~ u ~ s c ~egi p t denotes the ~ q u ~ l i ~state; r ~ u V'A ~ n is the effective volume of the FA; is the di~tribL~tion coefficient (the dimensions of 6' have been cha~ged to ME-" and so K d is ~ o ~ - ~ i ~ i e n s i o and n a l )Czn ; is the input chemical r i ~where the concentration. Equation (3.251) expresses an e ~ u ~ l i bstate

Subsurface Pathways of Contaminants to Coastal Waters: . . .

159

maximum adsorption has been reached and hence the output flux equals the input flux. At a non-equilibrium state, the output flux is reduced to zero due to adsorption and S increases at a rate equivalent to the input mass flux. The chemical input to BA includes FFA-ozlt,Fcs (mass flux due to the incoming seawater) and FSI resulting from seawater intrusion. The chemical adsorbed on sand particles tends to desorb in seawater. Thus, seawater intrusion produces an input flux to BA, and the magnitude of this flux is related to the speed of seawater intrusion and the amount of adsorption S. The output mass flux is due to SWGD. The governing equations are listed below: (3.3a)

FSGD= ( D , -I-D ,

Fcs = ( D ,

+ Dt)CBA,

+ D , + D t ) C c s and

(3.3b) (3.3c) (3.3d)

where VBAis the volume of BA and CBAis the chemical concentration in BA. VSIis the volume of intruded seawater. Ccs is the chemical concentration in the ocean and, for the contaminants considered, is usually small compared with CBAand can be neglected.

Transfer fluctuations of land-derived pollutants to the ocean due to D, and Dt Chemicals such as phosphate and ammonia are land-derived pollutants, for example, as a result of nutrient leaching from the agricultural fertiliser. Sediments in the fresh water aquifer, as a temporary storage for these chemicals due to high adsorption, become the immediate source of chemical to the brackish aquifer when seawater intrusion occurs and the chemical desorbs into the brackish groundwater from the sediment. In this section, we present a simulation to illustrate how the local groundwater circulation and oscillations affect the transfer of landderived pollutants. In the simulation, the distribution coefficient, K d , was assumed to be 400. The FA is assumed to be in an equilibrium state and seawater intrusion occurs between t = 0 and 10 d. The saltwater front retreats shoreward between t = 10 d and 20 d. The salt wedge moved at a constant rate, equal to 5% of the net groundwater discharge rate. The net groundwater discharge rate ( D n ) was set to be 3.75 m3/d/m, and the sum of D , and Dt is 90 m3/d/m (24xD,). The inland chemical concentration (Ci,)was 1 kg/m3. The volume of the BA is 500 m3/m. During seawater intrusion, the output mass flux from

160

Ling Li

the FA is described by (3.3a) and during the retreat of the salt wedge, is given by (3.3b). The time that it takes for the FA to reach the equilibrium state after the retreat of the salt wedge can be estimated by P S I t S I / F F A - o u t '

FFA-out

Figure 3.2 Simulated rates of the transfers of land-derived chemicals to the ocean. The simulated rate of chemical transfer to the ocean is shown in Figure 3.2. Also plotted in the figure are the results from a comparison simulation with D, and Dt neglected. It is clearly shown that a large increase of the transfer rate resulted from the seawater intrusion and the local groundwater circulation/oscillating flows. The first factor (i.e., seawater intrusion) contributes to an extra and excessive source of the chemical. The second factor (i.e., the local groundwater circulation and oscillating flows) provides the mechanism for rapid flushing of the BA, resulting in increased chemical transfer to the ocean. Without the second factor, the large impulse of chemical input to the ocean would not occur as demonstrated by the comparison simulation (dashed curve in Figure 3.2). The increase of F ~ G D is very substantial, more than 20 times as high as the averaged rate. As the salt wedge retreats, the transfer rate decreases to zero since the inland chemical is all adsorbed in the FA. The local processes do not change the total amount of the chemical input to the ocean, which is determined by the inland source. The tide-induced flushing effect is further illustrated by the following simulation based on a simple one-dimensional mass transport model,

dc

d2C

dc

- = Dc- V-, at ax2 ax

(3.4a)

~ u b s ~ ~ r Pathways ~ace of Contaminants to Coastal Waters:

--

161

with

(

'I/ = Ki,, -i d2KkAoexp( -kx) cos w t - kx

+ -"> 4 ,

(3.4b)

where the first term of the RHS is the net groundwater flow rate and the second term represents the oscillating flow induced by tides (based on the analytical solution, ( 2 . 2 ) ) . The initial concentration is specified according to an existing plume as shown in Figure 3.3. The b o u n ~ condit~onsfor the chemical transport are: c = 0 at the inland boundar~, and c = 0 for V > 0 and d c / d x = 0 for r/ < 0 at the seaward boundary. The following parameter values are used in the simulation: i, (regional hydraulic gradient) = 0.01, A0 = 0, 1 and 2 m, T (tidal period) = 0.52 d, K == 20 m/d, = 10 m, rze = 0.2, a = 3 m ( L ) , = aV)?L (distance of the landward boundary from the shore) = 150 xn.

Figure 3.3 Tidal effects on transport of a contam~antplume. The results displayed in Figure 3.4 show that residence time of the chemical in the aquifer decreases due to tidal oscillations (LHS panel of Figure 3.4). The tidal effects also lead to dilution of the exit chemical concentration significantly (RHS panel of Figure 3.4). Such dilution may reduce the impact of chemicals on the beach habitats.

3.2 Tide-induced mixing of fresh groundwater and seawater In this section, we will address how the fresh groundwater discharges to the ocean. Previous studies, neglecting the tidal effects, predict that the freshwater overlies the intruded seawater and discharges to the ocean with little mixing with the saltwater. The limited mixing, driven by the density effects, occurs along the saltwater wedge. A s~mulationwas conducted using SeaWat (htt p://.rat er .usgs.gov/o~/seawat/; by Weixing Guo and ~hristianD. Langevin) to examine the tidal effects on the freshwater discharge. ~ensity-dependentgroundwater flow in a coastal aquifer subject to tidal oscillations was simulated with a set of parameter values re~resentingthe shallow aquifer conditions. The simulation grid is shown in Figure 3.5.

Ling Li

16% I00 90 80

70 c4

t./ \= 60

3 P

B

R? *

rfo

3 30 20 10

0 0

50

100 time (d)

150

0

70 40 60 80 Dist from the shore (m)

100

Figure 3.4 Tidal effects on chemical transport in a coastal aqu~fer.LHS panel: mass ~ e n ~ a ~in n the i ~ gaquifer (in percentage of the initial m ~ versus time under different tidal conditions. 8 panel: chemical concent~ationprofiles in the aquifer at different times (dark for the results shortly afier the sirn~lationstarted, light a bit mediaq medium much later, cyan near the end of the simulation). ashed lines are for results with tidal effects ( A = 2 rn) and solid lines for results w ~ t ~ i tidal o~~t effects.

Figure 3.5 The grid used in the sirnulation.

The s i ~ ~ ~ ~was at~ run o nfirst without the tidal oscillationas until t?, steady state was reached. The result of the salinity d ~ s t r i b u t ~in~ the n aquifer shows the traditional view of the groundwater discharge as discussed above (the top panel of Figure 3.6). The tide was then ~ ~ t r o d into the simula~~on~ which continued to run for 100 tidal cycles and state. The result shows a very ~ ~ ~salinity ~ r reached a quasi-§~ead~ ~ ~ s t r ~ b u thorn i o n that without the tidal eEects (middle panel of Figure 3.6). First, a saline plume wias formed in the upper part of the beach. The freshwater dischar~edto the sea through a ~ube/channelbetween

Subsurface Pathways of Contaminants t o Coastal Waters: . . .

163

this upper saline plume and the intruding saltwater wedge. Secondly, the freshwater discharge tube contracted and expanded as the tide rose and fell (shown in the attached animation)) suggesting considerable mixing activities. Such mixing is also indicated by the salinity gradient shown in the bottom panel of Figure 3.6. These simulated salinity profiles are consistent with recent results from laboratory experiments.

Fresh water

70

12

74

16

78

82

80

84

86

88

90

Nm)

Figure 3.6 Salinity distribution in the near-shore area of the aquifer. Top panel: without tidal effects; fresh groundwater discharging t o the sea without much mixing with underlying seawater. Middle panel: tidal effects leading to the formation of the upper saline plume and the freshwater discharging, and considerable mixing between the freshwater and seawater. In analysing the simulated flow and mass transport process, we are particularly interested in (a) how the mean (advective) transport of salinity is affected by the oceanic oscillations, and (b) whether the oceanic oscillations (water exchange) cause diffusive/dispersive transport of salinity. This diffusive transport represents the local, small scale mixing. For this purpose, the following decomposition is taken,

+ c(z, z , t ) = C(z, z ) + c y z ,

u ( z ,z , t ) = U ( z ,2)

U I ( Z )2,

t),

2,t ) ,

(3.5a) (3.5b)

where u and c are the raw data of flow velocity and salinity; U and C are averaged flow velocity and salinity over the tidal cycle (24 hrs); and u' and c' are the tidally fluctuating flow velocity and salinity. The total

164

Ling Li

mass transport of salinity can then be determined,

The first term represents the transport due to the mean flow (advection). The second transport component is the dif€usive/dispersive flux. In Figures 3.7 and 3.8, we show calculated mean transport flux and diffusive flux. It is interesting to note that the two fluxes exhibit different patterns; and the magnitude of the mean transport flux is one order of magnitude larger than the diffusive flux. Based on the calculated difhsive/dispersive flux,one can estimate the apparent diffusion/dispersion coefficient (the local mixing intensity parameter), (3.7) where VC is the mean salinity gradient that can be estimated from the measured salinity data. These results show that the tidal effects affect the near-shore groundwater flow and transport processes significantly, leading to increased exchange and mixing between the aquifer and the ocean. Such effects can also alter the geochemical conditions (redox state) in the aquifer and affect the chemical reactions. As shown numerically below, the exchange enhances the mixing of oxygen-rich seawater and groundwater, and create an active zone for aerobic bacterial populations in the near-shore aquifer. This zone leads to a considerablereduction in breakthrough concentrations of aerobic biodegradable contaminants at the aquifer-ocean interface.

3.5t]; 70

t ;,; 72

1;;

f I

I

74

76

: :, :. :. :. :_ : . ., 78

80

. , ,

.

,

. ..

,

82

,

.

.

.

.

.

. .

. .

84

. ,

86

x (m)

Figure 3.7 The mean mass transport flux.

.

.

, .

I

88

I

90

~ubsu~face Pathways of Conta~n~nants to Coastal ~ ~ t e r s. .: I

165

0.5

I 1.5

2.5 3

Figure 3.8 The fluctuating mass transport flux.

3.3 Tidal effects on chemical reactions ~~~~~~~~~

As d~s~ussed above. ocea71 tides induce local oscillating ~ r o ~ ~ d flows in the near -shore aquifer. These flows enhance exc~aiigesand mixing between oxygen-r~chseawater and g r o ~ d w a t e rand , create an active zone for aerobic bacterial ~opulation in the aquifer, leading to s ~ g n i f i c reduction of biodegradable contam~nants. OW and PHT3D were used to model contaminar~ttransport radation in coastal aquifers affected by tidal oscillations. Two mobile chemicals were included in the s i ~ u ~ a t ~oxygen o n : as the electron tor and toluene as a representativeb~odegradablecontamina~t.An aerobic bacterium was included as an immobile phase. The biodegradation process was oxygen-~imited(i.e., sufficient substrate) The inland ~ontaniinantsource was specified at the cells near the water table. Other c~7id~t~ons and the parameter values used in the simu~ationare shown in Figure 3.9 and listed and in Table 3.1. a

>--,

Figure 3.9 Schematic diagram of the model set-up and boundary conditions ~ ~ ~ ~ e r n einn the t e dsimulation. The s ~ ~ u l a t i owas n run first without tides until a steady state of chemical conce~itrationswas reached. Tides started after that. An ani-

Ling Li

166

Fable 3.1 Parameter values used in the...simtdatioBParameter Longitudinal dispersivity(m) Transversal dispersivity (in) Horizontal hydraulic conductivity (m/d) Vertical hydraulic condiictivity (m/d) Effective porosity Tidal range (in), diurnal tide

Value 0.01 0.001 16.8 8.4 0.25 2 m [-1 1]

Toluene

0.05

Oxygen

Aerobes

Figure 3.10 Image plots of the steady state concentrations for toluene (top panel), oxygen (middle panel) and bacteria (bottom panel).

mation of the simulation results is attached on the CD. The image plots of steady state concentrations for toluene, oxygen and bacteria are shown in Figure 3.10. Due to the lack of oxygen, little degradation of toluene occurred except in the smearing diffusive layers. Correspondingly, little growth of bacteria can. be observed. The chemical concentrations at a high tide after 5 tidal cycles were shown in Figure 3.11. The tidal effect is clearly evident: first it created an oxygen-rich zone near the shoreline, which led to biodegradation of toluene. Secondly, it enhanced the mixing process. The smearing layer was thickened. The results at the low

Subsurface Pathways of Contaminants to Coastal Waters:

167

tide show similar patterns and changes in the chemical concentrations, In short, the simulation demonstrates that tidal oscillations lead to the formation of an oxygen-rich zone in the near-shore aquifer area. Aerobic bacterial activity sustained by the high O2 concentration in this active zone degrades the contaminants. These effects, largely ignored in previous studies, may have significant implications for the beach environment. Toluene

Oxygen

Aerobes

10

SO

60

x(m)

Figure 3.11 linage plots of the concentrations for toluene (top panel), oxygen, (middle panel) and bacteria (bottom panel) at the high tide after five tidal cycles.

4 Conclusions Coastal water pollution is a serious environmental problem around the world. Most contaminants are believed to be sourced from the land. To develop sound strategies for coastal water pollution control, we must be able to quantify the sources, pathways and fluxes of the contaminants to the coastal zone. Traditionally, terrestrial fluxes of chemicals to coastal water have been estimated on the basis of river flow alone. However, recent field observations indicate that contaminants entering the coastal

168

Ling Li

sea with groundwater discharge can significantly contribute to coastal marine pollution, especially in areas where serious groundwater contamination has occurred. To determine the fluxes of chemicals to coastal water, it is important to quantify both the transport processes and chemical reactions on the pathway. There has been a large amount of research work conducted on how the chemicals may be transformed during the transport along the surface pathway, i.e., the role of a surface estuary. In contrast, little is known about the chemical transformation in the near-shore area of a coastal aquifer prior to chemicals’ discharge to coastal water. In this chapter, we first reviewed a large volume of work on tideinduced groundwater oscillations in coastal aquifers, focussing on various analytical solutions of the tidal water table fluctuations. In the second part, we discussed the effects of tides and other oceanic oscillations on the chemical transport and transformation in the aquifer near the shore, drawing an analogy to the surface estuary-subsurface estuary. The discussion, based on several on-going studies, illustrated the important role that a subsurface estuary may play in determining the subsurface chemical fluxes to coastal waters. Although the tidal influence on the water table dynamics has been subjected to numerous studies, the effects of tides on the fate of chemicals in the aquifer have not been investigated adequately. Quantification of these effects is clearly needed in order to 0 provide better understanding of the pathway of land-derived nutrients and contaminants entering coastal waters, leading to improvement of strategies for sustainable coastal resources management and develop ment ; 0 provide useful information for integrating the management of u p land and lowland catchment areas; and 0 develop sound risk assessment methods and mitigation plans for coastal and estuarine pollution.

Acknowledgement A shorten version of this paper has been published by CRC Press, i.e., Li, L., D. A. Barry, D.-S. Jeng, and H. Prommer, 2003, Tidal Dynamics of Groundwater Flow and Contaminant Transport in Coastal Aquifers, in Coastal Aquifer Management - Monitoring, Modeling, and Case Studies, A. Cheng and Driss Ouazar (Edi), CRC Press, 2004.

References [l]Ataie-Ashtiani B, Volker R E, Lockington D A. Tidal effects on sea water intrusion in unconfined aquifers, J. Hydrol., 216, 17-31, 1999.


169

[2] Baird A J, Horn D P. Monitoring and modelling groundwater behaviour in sandy beaches, J. Coastal Res., 12, 630-640, 1996. [3} Baird A J, Mason T, Horn D P. Validation of a Boussinesq model of beach groundwater behaviour, Mar. Geol., 148, 55-69, 1998. [4] Barry D A, Barry S J, Parlange J Y. Capillarity correction to periodic solutions of the shallow flow approximation, In Mixing Processes in Estuaries and Coastal Seas, C. B. Pattiaratchi (ed.), Coastal and Estuarine Studies, 50, AGU, Washington, DC, 496-510, 1996. [5] Bear J. Dynamics of Fluids in Porous Media, Elsevier, New York, 1972.

[6] Bokuniewicz H. Groundwater seepage into Great South Bay, New York, Estuar. Coastal Mar. Sci., 10, 437-444, 1980. [7] Boufadel M C. A mechanistic study of nonlinear solute transport in a groundwater-surface water system under steady and transient hydraulic conditions, Water Resour. Res., 36, 2549-2565, 2000.

[8] Buddemeier R W (Ed.). Groundwater discharge in the coastal zone, LOICZ/RS/96-8, Texel, The Netherlands. 1996. [9] Burnett W C, Taniguchi M, Oberdorfer J. Measurement and significance of the direct discharge of groundwater into the coastal zone, J. Sea Res, 106-116,2001. [lo] Cable J, Bugna G, Burnett W, Chanton J. Application of 222 Rn and CH4 for assessment of groundwater discharge to the coastal ocean, Limnol. Oceanog., 41, 1347-1353,1996. [ll] Carslaw H S, Jaeger J C. Conduction of heat in solids, Clarendon

Press, Oxford. 1959. [12] Cartwright N, Nielsen P. Groundwater dynamics and salinity in

coastal barriers, Proceedings of 1st International Conference on Saltwater Intrusion and Coastal Aquifers Monitoring, Modelling and management, April 23-25, Essaouira, Morocco, 2001. [13] Cartwright N, Li L, Nielsen P. Response of the salt-freshwater interface in a coastal aquifer to a wave-induced groundwater pulse: field observations and modelling, submitted to Water Resour. Res. 2003. [14] Church T M. An underground route for the water cycle, Nature, 380, 579-580, 1996. [15] Cooper H H. A hypothesis concerning the dynamic balance of fresh water and salt water in a coastal aquifer, J. Geophys. Res., 71, 4785-4790,1959.

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[16] Dracos T. Ebene nichtstationare Grundwasserabflusse mit freier Oberflache. Swiss Federal Technical Laboratory of Hydraulic Research and Soil Mechanics, Rep. No. 57, 114, 1963. [17] Enot P, Li L, Prommer H, Barry D A. Effects of oceanic oscillations on aerobic biodegradation in coastal aquifers. Geophysical Research Abstracts, European Geophysical Society, 3: 18, 2001. [18] Ferris J G. Cyclic fluctuations of water level as a basis for determining aquifer transmissibility, IAHS Publ., 33, 148-155, 1951. [19] Glover R E. The pattern of fresh water flow in a coastal aquifer, J. Geophys. Res., 64, 439-475, 1959. [20] Godin G. The Analysis of Tides, Liverpool University Press, Liverpool. 1972. [21] Hanslow D, Nielsen P. Shoreline set-up on natural beaches, Coastal Res., 15, 1-10, 1993.

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[22] Haynes D, Michael-Wagner K. Water quality in the Great Barrier Reef world heritage area: Past perspectives, current issues and new research directions, Mar. Pollut. Bull., 41, 7-12, 2000. [23] Henry H R. Salt intrusion into freshwater aquifers, J . Geophys. Res., 64, 1911-1919, 1959. [24] Huyakorn P S, Andersen P F, Mercer J W, White H 0. Salt intrusion in aquifers: Development and testing of a three-dimensional finite element model, iWater Resour. Res., 23, 293-319, 1987. [25] Ippen A T, Harleman D R F. Tidal dynamics in estuaries, In Estuary and Coastline Hydrodynamics edited by A. T. Ippen, McGrawHill Book Company, Inc., New York. 1966. [26] Jeng D S, Li L, Barry DA. Analytical solution for tidal propagation in a coupled semi-confinedlphreatic coastal aquifer, Adv. Water Resour., 25 (5): 577-584, 2002. [27] Jeng D S, Teo H T, Barry D A, Li L. Two-dimensional approximation for tidal dynamics in coastal aquifers: Capillarity correction, submitted to J . Eng. Mech. (ASCE). 2003. [28] Jeng D S, Teo H T, Seymour B R, Barry D A, Li L. New approximation for tidal dynamics in coastal aquifers: I. Sloping beaches, submitted to Adv. Water Resour. 2003. [29] Jiao J J, Tang Z H. An analytical solution of groundwater response to tidal fluctuation in a leaky confined aquifer, Water Resour. Res., 35, 747-751, 1999. [30] Johannes R E. The ecological significance of the submarine discharge of groundwater, Mar. Ecol. Prog. Ser., 3, 365-373, 1980.


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[31] Kohout F A. Cyclic flow and salt water in the Biscayne aquifer of southern Florida, J. Geophys. Res., 65, 2133-2141, 1960. [32] Lanyon J A, Eliot I G, Clarke D J. Groundwater level variation during semi-diurnal spring tidal cycles on a sandy beaches, Aust. J. Mar. Freshwater Res., 33, 377-400, 1982. [33] Li H L, Jiao J J. Tide-induced groundwater fluctuation in a coastal leaky confined aquifer system extending under the sea, Water Resour. Res., 37, 1165-1171, 2001a. [34] Li H L, Jiao J J. Analytical studies of groundwater-head fluctuation in a coastal confined aquifer overlain by a semi-permeable layer with storage, Adv. Water Resour., 24, 565-573, 2001b. [35] Li H L, Jiao J J. Analytical solutions of tidal groundwater flow in coastal two-aquifer system, Adv. Water Resour., 25, 417-426, 2002a. [36] Li H L, Jiao J J. Tidal groundwater level fluctuations in L-shaped leaky coastal aquifer system, J. Hydrol., 268, 234-243, 2002b. [37] Li H L, Jiao J J, Luk M, Cheung K. Tide-induced groundwater level fluctuation in coastal aquifers bounded by L-shaped coastlines, Water Resour. Res., 38, 6.1-6.8, 2002.

[38] Li H L, Jiao J J. Tide-induced seawater-groundwater circulation in a multi-layered coastal leaky aquifer system, J. Hydrol., 274, 211-224,2003a. [39] Li H L, Jiao J J. Influence of the tide on the mean watertable in an unconfined, anisotropic, inhomogeneous coastal aquifer, Adv. Water Resour., 26, 9-16, 2003b. [40] Li L, Barry D A, Pattiarachi C B. Numerical modelling of tideinduced beach water table fluctuations, Coastal Eng., 30, 105-123. 1997a. [41] Li L, Barry D A, Parlange J Y, Pattiaratchi C B. Beach water table fluctuations due to wave runup: Capillarity effects, Water Resour. Res., 33, 935-945, 1997b. [42] Li L, Barry D A, Stagnitti F, Parlange J Y. Submarine groundwater discharge and associated chemical input to a coastal sea, Water Resour. Res., 35, 3253-3259, 1999. [43] Li L, Barry D A. Wave-induced beach groundwater flow, Adv. Water Resour., 23, 325-337, 2000. [44] Li L, Barry D A, Cunningham C, Stagnitti F, Parlange J Y. A twodimensional analytical solution of groundwater responses to tidal loading in an estuary and ocean, Adv. Water Resour., 23, 825-833, 2000a.

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[45] Li L, Barry D A, Stagnitti F, Parlange J Y, Jeng D S. Beach water table fluctuations due t o spring-neap tides, Adv. Water Resour., 23, 817-824, 2000b. 1461 Li L, Barry D A, Stagnitti F, Parlange J Y . Groundwater waves in a coastal aquifer: A new governing equation including vertical effects and capillarity, Water Resour. Res., 36, 411-420, 2000c. [47] Li L, Enot P, Prommer H, Stagnitti F, Barry D A. Effects of nearshore groundwater circulation on aerobic biodegradation in coastal unconfined aquifers, Proceedings of 1st International Conference on Saltwater Intrusion and Coastal Aquifers Monitoring, Modelling and management, April 23-25, Essaouira, Morocco, 2001. 1481 Li L, Jeng D S, Barry D A. Tidal fluctuations in a leaky confined aquifer: localised effects of an overlying phreatic aquifer, J. Hydrol., 265, 283-287, 2002a. [49] Li L, Dong P, Barry D A. Tide-induced water table fluctuations in coastal aquifers bounded by rhythmic shorelines, J. Hydraul. Eng. (ASCE), 128, 925-933, 2002b. [50] Li L, Cartwright N, Nielsen P, Lockington D. Response of coastal groundwater table to offshore storms, submitted to Water Resour. Res. 2003. 1511 Moore W S. Large groundwater inputs to coastal waters revealed by 226Raenrichment, Nature, 380, 612-614, 1996. 1521 Moore W S. The subterranean estuary: a reaction zone of ground water and sea water, Mar. Chem., 65, 111-125, 1999. 1531 Nielsen P. Tidal dynamics of the water table in beaches, Water Resour. Res., 26, 2127-2134, 1990. [54] Nielsen P, Aseervatham R, Fenton J D, Perrochet P. Groundwater waves in aquifers of intermediate depths. Adv. Water Resour., 20, 37-43, 1997. [55] Nielsen P. Groundwater dynamics and salinity in coastal barriers, J. Coastal Res., 15, 732-740, 1999. [56] Nielsen P, Perrochet P. Watertable dynamics under capillary fringes: experiments and modelling, Adv. Water Resour., 23,503-515,2000. [57] Parlange J Y, Stagnitti F, Starr J L, Braddock R D. Free-surface flow in porous media and periodic solution of the shallow-flow a p proximation, J. Hydrol., 70, 251-263, 1984. [58] Parlange J Y, Brutsaert W. A capillary correction for free surface flow of groundwater, Water Resour. Res., 23, 805-808, 1987.


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[59] Portnoy J W, Nowicki B L, Roman C T, Urish D W. The discharge of nutrient-contaminated groundwater from developed shoreline to marsh-fringed estuary, Water Resour. Res., 34, 3095-3104, 1998. [60] Raubenheimer B, Guea R T, Elgar S. Tidal water table fluctuations in a sandy beaches, Water Resour. Res., 35, 2313-2320, 1999. [61] Robinson M A, Gallagher D L. A model of groundwater discharge from an unconfined aquifer, Ground Water, 37, 80-87, 1999. [62] Simmons G M. Importance of submarine groundwater discharge and sea water cycling to material flux across sediment/water interfaces in marine environments, Mar. Ecol. Prog. Ser., 84, 173-184, 1992. [63] Sun H. A two-dimensional analytical solution of groundwater response to tidal loading in an estuary, Water Resour. Res., 33,14291435, 1997.

[64] Townley L R. The response of aquifers to periodic forcing, Adv. Water Resour., 18, 125-146, 1995. [65] Trefry M G. Periodic forcing in composite aquifers, Adv. Water Resour., 22, 645-656, 1999. [66] Turner I. Water table outcropping on macro-tidal beaches: A simulation model, Mar. Geol., 115, 227-238, 1993. [67] Turner I L, Coates B P, Acworth R I. Tides, waves, and superelevation of groundwater at the coast, J. Coastal Res., 13, 46-60, 1997. [68] Volker R E, Rushton K R. An assessment of the importance of some parameters for sea water intrusion in aquifers and a comparison of dispersive and sharp interface modelling approaches, J. Hydrol., 56, 239-250, 1982. [69] Wang J, Tsay T K. Tidal effects on groundwater motions, Trans. Porous Media, 43, 159-178, 2001. [70] Younger P L. Submarine groundwater discharge, Nature, 382, 121122, 1996. [71] Zekster I S, Loaiciga H A. Groundwater fluxes in the global hydrological cycle: past, present and future, J. Hydrol., 144, 405-427, 1993. [72] Zhang Q, Volker R E, Lockington D A. Influence of seaward boundary condition on contaminant transport in unconfined coastal aquifers, J. Contam. Hydrol., 49, 201-215, 2001.

174

Studies on Retrieval of the Initial Values and Diffusion Coefficient of Water Pollutant Advection and Diffusion Process I

Tingfang Wang Chengdu Meteorological Center of Airfore, Chengdu, 610041, China

Sixun Huang, Huadong Du Institute of Meteorology, PLA Universty of Science and Technology, Nanjing, 211101, China

Gui Zhang College of Science, PLA Universty of Science and Technology, Nanjing, 211101, China Abstract

In this paper, the variational adjoint method with Tikhonov regularization principle is applied to retrieving the initial concentration filed and the diffusion coefficient of a 3-D water pollutant model. The adjoint models are deduced for the complete and incomplete observation of water pollutant concentration respectively. Then a particular algorithmic scheme is designed for 1-D model as an example. Finally, a series of ideal numerical tests are performed to show the validity of the method. The numerical r+ sults show that the adoption of Tikhonov regularization principle can be used to eliminate oscillation in retrieval and the difficulty of short of information for the incomplete observations can be overcome efficiently by introduction of the background field of diffusion coefficient.

1

Introduction

Since the Industrial Revolution, the environment has been destroyed seriously by human activities. Release of all kinds of pollutants causes both air and water environments to be worsen and worsen. In water environment, not only river but also groundwater has been polluted according t o the observations in the last several decades. Therefore the advection

Studies on Retrieval of the Initial Values and . . .

175

and diffusion problem of water pollutant is attracting scientists to research and the hydrodynamic models need to describe transport process of water pollutant more accurately. The problem includes the transport process and the initial concentration distribution of the pollutant. The transport process can be divided into two sides, i.e. advection and diffusion. When change of water current velocity is small, diffusion is mainly produced by turbulence-a scientific problem far to be resolved (Hu 1995), but it is mainly due to shear dispersion for the inhomogeneity of velocity in current section when the change is obvious, and the diffusion is just about the total effects of them in this paper. On the other hand, it is worth determining initial distribution of pollutant, such as pollutant source of groundwater and so on. Consequently, the present studies are concerned with two aspects, i.e. to jointly retrieve the d i h sion coefficient and initial pollutant concentrations by a transport model with certain observations. We discuss the problem in the two cases of complete observations and incomplete ones obtained at the end time respectively. For lack of observational data, the 3-D retrieval model about the two cases are deduced theoretically, but numerical tests are only for l-D ideal ones. The variational adjoint method is applied in this paper (F. X. Le Dimet, J. Q. Yang, 2002). The retrieval of multi-parameters is usually so ill-posed that yields failures in calculation (TIKHONOV A N et al., 1977; KIRSCH A., 1996), which mainly results from short information, not from number of dimensions. Therefore, the Tikhonov regularization principle is applied to diminishing oscillations in the retrieval (Huang sixun et al., 2001, 2003, 2004; Xiao Tingyan et al., 2003). In the case of incomplete observations, a stabilized functional about the background field is adopted to remedy the short of observation information. Finally, the validity of our method is verified by a set of ideal numerical tests.

2

The 3-D model of water pollutant advection and diffusion process

The 3-D model of water pollutant advection and diffusion process with fixed boundary conditions is

[ g + ui-

dXi

=-

axi

(k)a-::

+ f(x,t ) ,

176

Tingfang Wang, Sixun Huang, Huadong Du, Gui Zhang

where C is the concentration of pollutant, time t E (O,T),x, U are 3-D vectors of position and water current velocity, k,I are diffusion coefficient and length of space domain, C = (0, ZI) x (0,Zz) x (O,Z3),ni is the 2-D position vector obtained from x by deleting the component xi,and f is a source or sink of pollutant. The Einstein sum stipulation is adopted in the above equation and the following text, and i ranges from 1 to 3 (mentioned in the following text anymore). It is worthwhile to mention that this model can describe almost all kinds of the processes of advection and diffusion including those about air, water even silt in river.

3

Retrieval in the case of complete observat ions

The present problem is to determine the initial distribution of pollutant Co(x)and diffusion coefficientk ( x ) such that the following functional is minimal,

(C - Cobs)2dx +

J[Co,k] = -

's

2

R

$J( k i g )

2

dxdt = min!

(3.1)

R

here Cobsis observations of distribution of pollutant. Retrieval of multi-parameters is usually an ill-posed problem which leads to application of the regularization principle. The second term in J is referred to as stabilization functional and y1 is the regularization parameter. As the first order norm of C , the stabilization functional can overcome ill-posedness and diminish oscillation in calculation. Next the adjoint model, adjoint boundary conditions of the tangent linear model (TLM) and gradient formula of the functional about ki,Co are derived which are divided into four steps as follows.

Step 1 Derivation of TLM Suppose that Ki, COwith the corresponding solution C of (3.1) are disturbed to &,60in the following way

with the corresponding solution 6. Define

ki = lim a+o

i& ~

- ki

a

,. , C=

lim

a-0

-.ZI-C a

Studies on Retrieval of the Initial Values and .

177

Then, the TLM is obtained as follows,

Step 2 The first order variation of J From the definition, we have

J'[k, CO;E, CO]=

1

Vc0J . &dx

c

+

/

VkiJ kidxdt.

(3.3)

$2

On the other hand, according to (3.1), the first order variation of J is

=

/ ( C - Cobs). kdxdt

1[ n

dxdt

n

R

=

g)2]

/

+ 71 12 axi Eaxi Z+ kiki (

-)]

(C - Cobs)- 71-d (ki,dC axi axi

. edxdt

here ki,C be zeroes in C as (3.2).

Step 3 Derivation of the adjoint equations Multiplying the first equation of (3.2)by P (called as adjoint variable), and integrating in 0, yields


178 and

sT

J $ . Pdxdt = Jc P ( x ,0)Codx - n C-dxdt,

n

J U i g

. Pdxdt = -

J&

-1; n

C-(UiP)dxdt,

n

n -

s Li

(kig)

Pdxdt = -

(his)

J ( E) C-

ki-

dxdt,

n

. Pdxdt = J k i zdz Pd xd dCt ,

n

A

n

here let P be zero on the boundary and at the end time. Thus (3.5) may be written as

compared with (3.4), the adjoint equation is

I

g +&pip) +& P

I t 9 = 0,

P

lxi=0

=P

@a%)

= (C - C o b s ) - Y 1axi L

(k=), 2 axi (3.7)

IXi&

= 0.

0 = c x (0,T)

Step 4 The formula for the gradients of J It follows from (3.3), (3.4) and (3.6) that 0=

J

c

Vc,J Codx

+ J VkiJ . hidxdt - J P ( x ,O)Co(x)dx n

c

Then the gradients of the functional J with respect to Ic, ~0 are

( vc,J = P(X,O),


179

The next task is to seek the optimal parameters through iteration algorithm. Let R" = (ka,Con), then the iterative scheme is

4

Retrieval in the case of incomplete observations

The present problem is to determine the initial distribution of pollutant Co(x) and diffusion coefficient k ( ~ from ) an observation Cobs(x)at end time T , such that the following functional is minimal,

+-2

s

(ki - k!)'dxdt = min!

n

For the short of observations information, the third term of (4.1) is introduced. It is a stabilized functional about the background field restriction of the diffusion coefficient k. Steps 1 and 2 are very similar to the case of complete observations, i.e. the form of TLM and definition of the first order variation of J are (3.2) and (3.3) respectively, but the expression of J includes an additional term of constraint on k from the background field. Thus


180

Multiplying the first equation of (3.2) by P (called as adjoint variable), and integrating in R, yields

and here

P ( x ,T)CT(x)dxn

c -

c

1

C catd x d t ,

n

here let P be zero on the boundary and at the end time. Thus (4.3) may be written as

1

c

P ( x ,T)CT(x)dx-

J c

+n

P ( x ,O)Co(x)dx J { - C [ %

Compared with (4.2), the adjoint equation is


181

It follows from (3.3), (4.2) and (4.4) that 0=

s

c

Vc, Jeodx +

s

Vki J . kidxdt -

R

s

c

P ( x ,O)Co(x)dx

Then, the gradients of the functional J are

Figure 4.1 is the flow chart of retrieval. In addition, since in both two cases, an original model and the associated adjoint model are mixed problems of PDE of advection and diffusion with forcing, and with t and U reverse, the adjoint model becomes same as the original model. Therefore, a difference scheme from Keller box scheme with second order accuracy and stability (Lu Jinfu et a1.,1987) is given below for l-D original model. The l-D original model in a rectangle domain of time and space can be written as

a(1,T ) = { ( x ,t ) l O I x I1,O 5 t 5 T } The discrete grids in R(1, T ) are taken as follows

Introducing the notations

~j

= j h , j = 0 ,N x , N X . h = 1

t,

= nr,n = 0 , N t , N t . r = T

-

.

(4.8)

182


caculate V , J , Vco^J from (3.8) or (4.6)

t

*

caculate k, Co from (3.9)

1 Is functional fiom (3.1) or (4.2) decreasing? I

k, Coare new guessing values

Is error of k, Co small than expected value?

Yes k, C, are inversion output, caculation is finished

Figure 4.1 Flow chart of adjoint retrieval.

then the original equation turns to (4.11)

Then the original model is transformed into one composed of (4.10) and (4.11), with the Keller box scheme as follows


183

The time distribution of C at T can be obtained through iteration using initial and boundary conditions, and the adjoint variable P is computed to get the gradient of J.

5

Numerical tests

To test the effect of the above theory and algorithm, a series of tests are performed. The model used here is as follows, which has been used in the literature(Min Tao et a1.,2003).

[ g +U(x,t)-ax ”( ax =

E)

k(x)-

+f(.,t),

C(5,O)= 5 3 , C(0,t ) = 0,

C(1,t ) = e t .

R = ((5,t)IO 5 z 5 l , o 5 t 5 l}, 1 here U ( z ,t ) = - ( x 2 10

3 + t 2 )f(x, , t ) = -z2(x2 + t 2 ) e t . 10

If diffusion coefficient k(x) = A x 2 , the model has the analytic solution C ( x ,t ) = x 3 e t , and then the observation at t = 1 is C ( x ,1) = ex3. In tests, we disturb k(x) by its 10 percent as an initial guess, and the corresponding integration of the model is taken as an observation with error, but the observation C(x,1) at the end time is assumed to be true. Test 1: The complete observation C in the whole temporal and spacial domain is provided. In test, the regularization parameter y1 is

184


set to be y1 = 0, i.e. the regularization method is not used. From Figure 5.1 it can be seen that the profile of the retrieved diffusion coefficient approaches that of the true value with small amplitude oscillation. From Figure 5.2 it can be seen that the error between the prediction of C from the retrieved initial value and the true value is three orders of magnitude less than that between the prediction of C from the initial guess and the true value. For sufficient observation, the retrieval is satisfactory. 0.09 I

I

X

Figure 5.1 Comparison of the true, guess and retrieved profiles of diffusion coefficient in Test 1.

Test 2: The observation C in the whole temporal and spacial domain is still provided as in Testl. But the regularization parameter y1 is set to be y1 = From Figure 5.3 it can be seen that oscillations of the error of diffusion coefficient C is eliminated, and the profiles of retrieved and true diffusion coefficients fit well. From Figure 5.4 it can be seen that the error between the prediction of C from the retrieved initial value and the true value is four orders of magnitude less than that between the prediction of C from the initial guess and the true value. Test 3: Only the observation C at the end time t = 1 is provided. Again the regularization parameter is set to be y1 = 10V6, while the background parameter 7 2 = 0. From Figure 5.5 and Figure 5.6 it can be seen that the retrieval results are unsatisfactory being short of observation. Test 4: The test is the same as Test 3 except 7 2 = 1. From Figure 5.7 and Figure 5.8 it can be seen that the retrieval results are ameliorated greatly. The profile of the retrieved diffusion coefficient rather


-

185

5

0.02

12

25

Figure 5.2 Comparison of the errors between the guess and retrieved field of pollutant on cent ratio^ in Test 1.

0.07

-

0.06

-

8

10

12

14

16

18

20

X

s o true, ~ guessing and retrieved profile of Figure 5.3 ~ o ~ ~ a rofi the dispersion ~ o e ~ ~in iTest e ~2.t

a ~ ~ r o ~that c h eof~t.he true value, and the error between the prediction of C from the retrieved initial value and the true value is two orders of

4 2

0 -2

25

Copt- color

xi04

0 02 0 -0 02

-0 04 25

Figure 5.4 ~ o ~ ~ a r i of s othe n errors of guess~ngand r e ~ ~ ~ e ~ e ~ ~ ~ ~ lconcentrat~on ~ ~ t ~ in n Test t 2.

s othe n true, guessing an retrieved profile of Figure 5.5 ~ o ~ ~ a r i of d ~ s ~ e ~ coeEcient s ~ o n in Test 3.

~agnitu~ less e than that between the ~ ~ e ~ i c tof i oCn from the initial guess and the true wlue.

Studies on Retrieval of the Initial Va1u.es and .

1.137

I

C.-.C&

0.02

-

'-12

Figure 5.6 ~ Q m ~ a r i sof o nthe errors of guessing md r e t r ~ e ~ efield d of ~ o l ~ ~ tcon6entration ant in Test 3.

0.08 0.07 0.05 0.05 .c-: 0.04

0.03 0.02 0.01

'0

2

4

6

8

10

12

14

16

18

20

X

F.igure 5.7 ~ o ~ ~ aofr the ~ true, s o ~ e s s i n gand retrieved profile of ~ ~ ~ ~coefficient e ~ s in ~ Test o n4.

The above retrieval results are compared in Table 5.1. The error in the table is root-mean-square error(^^^^)^ For ~ o n ~ e n ~ ~ ait tis~ root on,

Tingfang Wang, Sixun Huang, Huadong

188

c-cobs

4 2 0

-2 25

002

-

0 -0 02

-0 04,

Figure 5.8 ~ o ~ p a r i s of o nthe errors of guessing and retrieved field of ~ o l l ~ t aco~centrat~on nt in Test 4.

mean square of errors at the grids in the whole temporal and spatial dosion coefficient, it is root mean. square of errors at the grids in the whole spatial domain; The table shows that retrievals with complete ob~ervationsare successful, and retrievals with incomplete observations in the presence of background are ako successfuI, while retrieval with i n c o ~ ~ lobservations et~ in the absence of background is unsuccessful. The retrieval results also demon st rat^ that for inverse problem, not only the r e ~ ~ ~ a r ~ z but a t i oalso n introduction of background is i ~ ~ ~ when the informat~onis sparse.

o nthe error nornis in all retrieval tests Table 5.1 ~ o ~ p a r i s of

I error norm in

Test

of retrieved field

1

error norm of retrieved field in Test 2 ._....._-.__Ierror norm of retrieved field

in TeBt 4

I

0.049

I

0.0019

I


6

189

Conclusions

In this paper, the variational adjoint method with the Tikhonov regularization principle is applied to retrieving the initial concentration filed and the diffusion coefficient of a 3-D water pollutant model. The adjoint models are derived for the complete observation of water pollutant concentration and incomplete observation at the end time respectively. Then a particular algorithmic scheme is designed for 1-D model as an example. Finally, a series of ideal numerical tests are taken to show the validity of the method. The main conclusions are as follows: (1) Tikhonov regularization principle can be used to overcome illposedness of the problem and diminish the oscillation in retrieval especially. (2) Retrieval of complete observations is very successful and accurate due to abundance of information. (3) Retrieval of incomplete observations are very difficult due to lack of information; but the difficulty of short of information can be overcome efficiently by introduction of the background field on diffusion coefficient. Numerical tests are only for 1-D problem in present paper, and multidimensional problems will be studied later.

Acknowledgement This work was supported by the National Natural Science Foundation of China (Grant Nos. 40175014 ) and the Shanghai Association Fundation For Science & Technology (Grant Nos. 02DJ14032 ). The authors wish to express their appreciation for the careful and thoughtful review by the reviewers and comments by the editor.

References [l]Du H D, Huang S X, Shi H Q. The Theoretical Analysis and Nu-

merical Experiments in the Retrieval for the 1-D Semi-geostrophic Shallow Water Model. Journal of Hydrodynamics Ser. A, 19(1): 3845, 2004.

[2] Fang H X, Huang S X, Wang T F. Retrieval Study of 1-D Dropsonde's Motion. Journal of Hydrodynamics Ser. A, 19(1): 53-60, 2004. [3] Le Dimet F X, Yang J Q. Models and Data for the Water Cycle. Mathematical Problems in Environmental Science and Engineering. Beijing, Higher Education Press, 41-65, 2002.

190


[4]Hu F. Turbulence, Intermittence and Atmospheric Boundary Layer. Beijing, Science Press, 1995. [5] Huang S X, Han W, Wu R Sh. Theoretical Analyses and Numerical Experiments of Variational Assimilation for One-Dimensional Ocean Temperature Model. Science in China, Ser. D, 9: 903-911, 2003. [6] Huang S X, Wu R Sh. Physical Mathematical Problems of Atmospheric Science. Beijing, Chinese Meteorological Press, 2001. [7] Kirsch A. An Introduction to The Mathematical Theory of Inverse Problems. Springer-Verlag, 1996. [8] Lu J F, Guan Zh. Numerical Solutions of Partial Differential Equation. Beijing, TsingHua University Press, 1987.

[9] Min Tao, Zhou X D. An iterature method of the inverse problem for the dispersion coefficient in water quality model. Journal of Hydrodynamics Ser. A, 18(5): 547-552, 2003.

[lo] Tiknonov A N, Arsenin V Y. Solutions of Ill-posed Problems. V. H. Winston and Sons, Washington D.C., 1977. [ll] Xiao T Y, Yu Sh G, Wang Y F. Numerical Method of Inverse Problems. Beijing, Science Press, 2003.

191

Application of Tabu Search Method to the Parameters of Groundwater Simulation Models Jing Chen, Zhifang Zhou College of Civil Engineering, Hohai University, Nanjzng 210098, China Abstract Tabu search method, a meta-heuristic approach, with which simulated annealing and genetic method axe extensively used in combinationaloptimization. This paper successfully applied Tabu search method to the inverse analysis of hydrogeological parame ters and validate its effectiveness in the simulation of the groundwater field affected by cutoff wall in Hanjiang dike. This method, which is based on human memory, has shown the predominance of random search.

1

Introduction

The word Tabu (or Taboo) comes from Tongan, a language of Polynesia, where it was used by the aborigines of Tonga island to indicate things that cannot be touched because they are sacred. According to Webster‘s Dictionary, the word now also means “a prohibition imposed by social custom as a protective measure” or of something “banned as constituting a risk”. These current more pragmatic senses of the word accord well with the theme of Tabu search. The risk to be avoided in this cme is that of following a counter-productive course, including one which may lead to entrapment without hope of escape. On the other hand, as in the broader social context where “protective prohibitions” are capable of being superseded when the occasion demands, the “Tabus” of Tabu search are to be overruled when evidence of a preferred alternative becomes compelling. Engineering and technology have been continuously providing examples of difficult optimization problems. The roots of Tabu search go back to the 1970’s; it was first presented in its present form by Glover (1986); the basic ideas have also been sketched by Hansen (1986). When we come to its application in groundwater, it must be emphasized that

192

Jing Chen, Zhifang Zhou

Zheng and Wang (1996) applied both Tabu Search method (TS) and Simulated Annealing (SA) to determine parameters for a hypothesis one dimensional problem for the first time. Chung and Chun(2002) focus on the application of Tabu Search to groundwater parameter zonation and they had got a good result.

2

General description of tabu search

[4]TabuSearch was applied to optimize ground water parameters in this study. TS allows moving to the worse solution among the neighborhood. The better solution of recent iterations are recorded in a Tabu list to reject moving back to these solutions again within a given number of iterations, which can prevent being trapped by local optimum. Tabu Search seeks the optimal solution by moving from the current solution to the best solution within its neighborhood. If the best solution within the neighborhood is not listed on the Tabu list, the current solution is replaced with the new selected solution. The new solution could be worse than the previous solution. TS keeps moving from one solution to its neighborhood to find the optimal solution until the stop criterion is met. The main components of TS include initial solution, movement, Tabu list, aspiration criteria, and principles to stop searching. These components are briefly introduced as follows.

2.1

Initial solution

The optimization process improves solutions by moving from a current solution to its neighbor solutions. Thus, an initial solution should be given before the searching process. Different initial solutions may lead to different local optimal solutions in a nonlinear problem. However, it is difficult to have the best guessed initial solution. An initial solution may be randomly selected. Thus, a robust algorithm needs to result in minimum deviation.

2.2

Neighborhood and movement

The process of optimization is to move from one solution(X) to another solution within its neighborhood, N ( X ) . Different definitions of neighborhood may be made by different analysts or for different problems. After defining neighborhood for a problem, the neighborhood for a s e lution can be easily determined, and then the best solution among the neighborhood is selected as the next stop. Tabu Search records the optimal solution among the previous explored solution space. If the new

Application of Tabu Search Method to the Parameters of .. .

193

selected solution is better than the recorded solution, the latter is replaced with the new one. When the algorithm stops searching , the final recorded solution is the found optimal solution.

2.3

Tabu list

Tabu list records better solutions in recent iterations and prohibits moving back to these solutions to avoid being trapped by local optimums. Thus, the length of the list cannot be too short. Otherwise, the searching process may fall into a cycle and may not have chances to find a global optimum. On the other hand, the Tabu list should not be too long. Otherwise, the searching process could become inefficient and may be limited to a small feasible space and thus reduce the chance to find a global optimum. There is no universal principle to determine the length of Tabu list. Glover(1990) suggested a magic number of 7, which can be the first guess.

2.4

Aspiration criteria

The Tabu list records better solutions of previous iterations, and it is normally designed to record only parts of decision variables. For example, a solution set contains two decision variables, X and Y . When changing X1 to X2 with Y = Y1 finds the best solution within the neighbor, I )The , Tabu list will record that X = X I is Tabued. hood of N ( X 1 Y However, when Y changing to Y2 with X = X I has the better value of objective function, the Tabu is relaxed and the solution is allowed to , move to ( X I Yz).

2.5

Principle to stop searching

Different principles to stop searching process can be applied. These may include: (1)the recorded optimal solution reaching required level; (2)given maximum iteration; and(3)the number of successive iterations on which solutions are not improved. There is no common practice for the use of the above criterion. It is dependent.

3

Where can we use tabu search method?

(1)Application of Tabu Search to Parameter Structure and Parameter

Values It has been mentioned above that Zheng and Wang (1996) applied both Tabu Search method (TS) and Simulated Annealing (SA) to determine parameters for a hypothesis one dimensional problem for the first

194


time. Chung and Chun(2002)focus on the application of Tabu Search to groundwater parameter zonation and they had got a good result. They all have proposed a combinatorial optimization model that identifies the best parameter structure. But they indicate that this method can be extended readily to include both parameter structure and values. This paper just applied it to the optimization of parameter values. (2) A n Inverse-Simulation Approach to Determine Optimal Strategies for Developing Public Water-Supply System How to find the best strategies for managing groundwater resources is often accomplished with the method of optimization. For example, the approach to find best pumping strategies in a coastal aquifer, well location and pumping rate are optimized with respect to (1)minimizing impacts to nearby surface water; (2)preventingsaltwater intrusion contamination due to overpumping, etc. If the pumping rate is assumed to be known and unchanged, the well placement becomes our goal of optimization. Typically “trial and error” approach is used for finding best strategies, but the chances to get what we need diminish because of large number of potential option. By using Tabu Search method, the better solution of recent iterations are recorded in a Tabu list to reject moving back to these solutions again within a given number of iterations, which can prevent being trapped by local optimum. Figure 3.1 shows the well placement Changjiang estuary where the problem of optimization can be solved with Tabu Search method. Figure 3.2 shows that the definition of neighborhood set associated with the grid of estuary.

Figure 3.1 The Well placement of Changjiang estuary.

Application of Tabu Search Method to the Parameters of .

195

Figure 3.2 Definition of neighborhood set.

4

Application of tabu to the inverse analysis of hydrogeological parameters

4.1 Evaluation model The mathematical model for the problem is given as follows:

the object function is,

where hi,t and h$ are calculated and observed hydraulic heads at time t ant at grid i that has a ground water observation well; n is the number of ground water observation wells; Kh is the hydraulic conductivity; s3 is the specific storage coefficient; and w is the net vertical,srecharge, respectively. In this study, ground water heads and Kh are all variables S3w are assumed to be known. By using Tabu search method, the better Kh is continually hunted within its scale. The hydraulic heads can be solved by FEM program. After several iterations, the value of object function draws near zero.

196

4.2


Discrete variables and neighbors

Because Tabu Search method comes of problems of discrete path networks(such as TSP problem), continuous parameters can be searched only after it becomes discrete. Thus far we have used the term neighborhood but not given precise definition. For a discrete variable we define its surroundings as its neighborhood. For example, given a certain searching radius, a one-dimensional variable may have two neighbors while a two-dimensional variable may have four or eight neighbors. A three-dimensional variable may have six or 26 neighbors. Of course, this is not the only way to define a set of neighbors. The neighbors of a two-dimensional variable may be circularity while a three-dimensional variable has a sphere as its neighbors.

4.3

Tabu search procedure

Step1 (Initialization) Select a starting solution X ; Evaluate the object function, Set the neighbor of X , N ( X ) ; Set the Tabu T empty; Step2 (Choice and termination) Evaluate the object function of neighbor, Determine the best solution X ’ of N ( X ) ; Step3 (Update) Find the N ( X ) , and update Tabu list T ; Let X = X’; Go back to Step 2 to continue the search until a stop criterion is satisfied.

4.4

result

Cutoff wall, as a seepage control measure, has a good effect on the improvement of safety condition of dikes. But whether the cutoff wall has a bad effect on the groundwater environment or not has been widely concerned. In the flood season, the cutoff wall effectively prevents seepage and protects dikes, while in the dry season it gets in the way of the natural hydraulic relation of both sides of dike. Now we study the procedure of Tabu search with the simulation of the groundwater field affected by cutoff wall in Hanjiang dike. The hydraulic conductivity of aquifer, the hydraulic conductivity of cutoff wall and the specific storage coefficient associated with certain region are to be optimized. The region we study is shown in Figure 4.1, where there are a cutoff

Application of Tabu Search Method to the Parameters of . . .

197

coutoff wall

cross secti

cross section No.5

Figure 4.1 The sketch map of placement of wells. wall and 25 wells, GK1-GK25. Cutoff wall, as a seepage control measure, has a good effect on the improvement of safety condition of dikes. It is made of concrete with thickness of 30cm. The value of hydraulic conductivity is assumed to be 1.0 x 10-7cm/s 9.9 x 10-7cm/s while the hydraulic conductivity of the aquifer is assumed to be 5.1 x 10W4cm/s 7.28 x The specific storage coefficient is 0.05 0.25. TABU method “searches” the best solutions in the range of these parameters. We choose the heads of GK12, GK13, GK14, GK15 as observation heads, the water level of Hanjiang and the heads of another wells are assumed to be known. After the initial parameters are given, the FEM program simulates the hydraulic heads in the study region. The value of object function can be calculated. Then we determine the neighborhood of the current solution, and also, the objective value of neighborhood can be calculated. TABU method is “searching”the best solution in the range of these parameters and the calculated heads are drawing near the observed heads. When the stop criterion is satisfied, the best solutions are got. The hydraulic conductivity of the aquifer is K = 6.96 x 10W4cm/s. The hydraulic conductivity of the cutoff wall is 1.02 x 10W7cm/s.The specific storage coefficient is 0.021. The value of object function is shown in Figure 4.2. Figures 4.2 and 4.5 show the optimal simulated and observed hydraulic heads of No.14 No.16 weIls, from which we can see the optimal simulated heads are almost the same with observed heads.

-

-

-


198

-Quadrangledmesh -Triangular mesh

01 0

100

200

300

400

500

600

700

Figure 4.2 Variation of object function in the optimal searching procedure.

365 -

36.5

36 -

36

355 -

35.5

35 -

35

345 -

34.5

Figure 4.3 Optimal simulated and observed heads of No.16 well.

~~

35.5

-

35 34.5 34 -

--Optimal

Simulated Head


Application of Tabu Search Method to the Parameters of . . 36.2

36.5

36 35.8

36

35.6 35.4

35.5

35.2 35

35

34.8 34.6 34.4

199

34.5

1

i


5

Summary

(1) This paper applied Tabu search method to the inverse analysis of hydrogeological parameters. The result shows that Tabu search method is well suited for solving the inverse problem. (2) The soul of Tabu search method is its "protective prohibitions". The better solution of recent iterations is recorded in a Tabu list to reject moving back to these solutions again within a given number of iterations, which can prevent being trapped by local optimum. (3) Tabu search method is well suited for solving the proposed combinatorial optimization model. It has shown its predominance in solving TSP problem, Job-shop problem, etc. For the area of groundwater modeling, we think it has more extensive foreground.

References [l]Zheng C, Wang P. Parameter Structure Identification Using Tabu

Search And SimulatedAnnealing. Advances in Water Resources. V01.19, No.4, 215-224, 1996. [2] Ching-Pin, Chun-An Chou. Application of Tabu search To Ground Water Parameter Zonation. Journal of The American Water Resources Association. vo1.38. No.4. 1115-1125. [3] Alain H, Eric T. Dominique de Werra. A Tutorial On Tabu Search.

Report. [4]Glover F. "Heuristics for Integer Programming Using Surrogate Constraints," Decision Sciences, Vol8, No.1, 156-166. Seminal work on tabu search and scatter search. 1977.

200


[5] Glover F. “Tabu Search - Part I,” ORSA Journal on Computing, Vol.1, No.3, 190-206. First comprehensive description of tabu search. 1989. [6] Glover F. “Tabu Search - Part 11,”ORSA Journal on Computing, V01.2, No.l,4-32. The second part of this comprehensive description of tabu search introduces additional mechanisms such as the reverse elimination method. 1990.

[7] Glover F. “Tabu Thresholding: Improved Search by Nonmonotonic Trajectories,” ORSA Journal on Computing, Vol. 7, No.4, 426-442. A description of a specialized form of tabu search known as tabu thresholding. 1995.

20 1

Several Problems in River Networks

Hydraulic Mathematics Model Xiaoming Xu Department of Applied Mathematics, Hohai University, Nanjing, 210098, China

Deguan Wang College of Environment Engineering, Hohai University, Nanjing, 21 0098, China E-mail: [email protected] Abstract The U.S. National Weather Service (NWS) FLDWAV model has capacity in simulating variety of hydraulic phenomena in river systems. However it is originally designed for simulating onedimensional unsteady flow in dendritic channel systems and is not full suitable for river networks. The modified relaxation iterative method was proposed when simulating the unstedy flow and the water quality in river networks of Shanghai with more than 400 named rivers. The advantage of this improved model is that it can satisfy for any complex river networks. The convergence of Netwon iterative method using in this model for nonlinear systems was also theoretically proved and Netwon down-hill method was applied to the case of unconvergence situation in original numerical method. A combined Gauss elimination with partial pivoting method with compress-storage (GEPP-CS) technique was developed to solve the linear systems.

1

Introduction

Shanghai, the most economically developed city in China, is located in Yangtze delta area with complicated river networks. The total river number is about 33000 with various scales and more than 170 gates, locks and pumping stations on the river systems. The rivers are strongly influenced by tides, typhoon and heavy rains. More than 400 named channels were simulated in this paper.

Xiaoming Xu, Deguan Wang

202

For the purpose of managing water resources in the Shanghai river networks, a proper water quantity numerical simulation model must be selected. The NWS FLDWAV hydraulic model (F'read and Lewis, 1998) was employed as the basic models. Although the FLDWAV hydraulic model has the capacity of simulating variety of hydraulic phenomena in the river systems which includes lots of waterworks. However this model was originally designed for simulating l-D unsteady flow in tree-type river systems, so it was not full suitable for looptype river networks such as Shanghai river networks. In order to establish Shanghai river networks hydraulic mathematical model, substantial improvements were made to strengthen the FLDWAV hydraulic model capacity. These improvements include: (a) the relaxation iteration method of branch flow in tree-type river systems was extended to the loop-type river networks, (b) the convergence of Newton iterative method was theoretically proved for nonlinear systems and two methods were proposed for Newton iterative method to deal with unconvergence situation in numerical computation, (c) new solution technique with small rounding off and high stability was set up for the linear systems.

2

2.1

Saint-Venant equations and relaxation iterative method

Saint-Venant equations and discretization

Preissman weighted four-point implicit numerical scheme is used to solve the Saint-Venant equations of l-Dunsteady flow (Read, 1985)

dA

dQ

at

dx

-+--q=o

dQ d Q2 - + -(P-) dt dx A

dh + gA( d X + S f )= O

where A is the cross-section area of flow, Q is discharge, t is time, x is the longitudinal mean flow-path distance measured along the center of the water course, q is the lateral inflow or outflow per lineal distance along the watercourse, 0 is the momentum coefficient for nonuniform velocity, g is the gravity acceleration constant, h is the water level, and S f is the boundary friction slope. Nonlinear algebraic equations with hS+', hy;: and 9::; as unknown variables for a river have the

&I+',

Several Problems in River Networks Hydraulic . . .

203

follow form:

here i = 1 , 2 , . . . , N - 1 ( N is the number of cross sections), superscript n 1 means time tn+l,there are 2 N - 2 equations with 2 N unknown variables. Adding to upstream and downstream boundary conditions, nonlinear systems with 2 N variables is formed:

+

F(x)= 0

(2.5)

where F(.) is a function from R2N into RZN.

2.2

Newton iterative

Newton’s method for nonlinear systems is generally expected to give quadratic convergence. Suppose upstream and downstream conditions are water level time series, then the Newton’s equations of step lc has the form: A ( k ) z= b ( k )

(2.6)

where A(k)is the Jacobian matrix of equations (2.5) (superscript n is omitted). L

+1

O

Initial approximation value take as at time tn’s. The solutions of linear systems (2.6) are cross sections water levers and discharges at time tn+l.


204

2.3

River networks relaxation iterative method

Since Saint-venant equations is only suitable for a single river. For treetype river systems ( Figure 2.1), the tributary flow at each confluence is treated as lateral flow q which is estimated by solving equations (2.1 and 2.2) for each river. Relaxation iterative method about junction discharges is proposed

gi

where is the new estimated confluence lateral flow for the next iteration, qii is the computed discharge at the confluence in the previous iteration, qE:w is the previous estimated confluence lateral flow, Q is a weighting factor(0 < Q < l), Ic is branch river number. Convergence is attained when Iqii - qEil < EQ.

/

Figure 2.1 Tree-type river systems. However when the relaxation iterative method in the FLDWAV was applied to looptype water networks (Figure 2.2), such as Shanghai water networks (Figure 2.3), the discharges at the confluence of loop-type rivers can not be balanced. For solving this problem, improved iterative method (Xu et al., 2001) was proposed. In the new relaxation iterative method the iterative must be operated for every confluence of the river networks. Therefore the problem of the discharge unbalance was well solved.

Several Problems in River Networks Hydraulic

.

205

/

Figure 2.2 Loop-type water networks.

Figure 2.3 Shanghai water networks.

3

The convergence of Netwon iterative method

Netwon’s iterative method is usually to give quadratic convergence, hence it is applied in the FLDWAV model to solve the nonlinear systems (2.5). However, there are two problems about Netwon’s iterative method. The first problem is the convergence of Netwon’s iterative method. According to convergence theorem of Netwon’s iterative method, all of the Jacobian

206


matrix A ( k )for each time step and for each river are required continuous and nonsingular at z*, where x* is the solution of equations (2.5). Xu and Wang (2001) proved the matrix A@) meeting, therefore the Netwon’s iterative method was convergent in theoretical for equations (2.5). The second problem is the local convergence of the Netwon’s iterative method. That is the each time step initial approximation dn) must in the convergence area N6 of ( n 1) time step solution z*:

+

However N6 is unknown. In the FLDWAV model, the solution of n time step is taken as the initial approximation of ( n + l ) time step. When the solution of n time step is not in the area of N6, the iterative series may unconvegence. To solve the problem in numerical computation, two methods were proposed in the modified model. One is reduction time step, At/2, At/4,. . . is taken as new time steps until convergence. Another method is Netwon’s down-hill method:

where [ F ’ ( z k ) ] - ’ is W k is relaxation factor (0 < W k < l ) , W k will be chosen as 1, 1/2, 1/4, - ” , until 11 F(Z(’+’)) II b then - < 0, if u < b, then - > 0. That is to say, dt dt u --t b when t -+ 00. Now, the question is what the value of ‘b’ is when red tide breaks out. Let us study the equation again. The fact can be found that the larger ( b - u) is, the larger is. Imagining that ‘b’ is very large and the initial ‘u’ is small, what would happen to the population of phytoplankton? Apparently, $ is very large at the beginning, and it will drop when the value of ( b - u) decreases. The next question is about the factors influencing ‘b’ value. These factors are exactly those mentioned at the beginning of the paper. The nutrients (pollutants) in the water are the main reasons that lead to algae over-grow. So in the following, the impacts of environmental factor (nutrient) on phytoplankton population will be studied using simple ecological model. In section 2, nutrient concentration in water environment is assumed constant. Based on the information in section 2, the

%

Study on the Character of Equilibrium Point and . . .

213

temporal variability of nutrient concentration is discussed in section 3.

Phytoplankton population model with constant nutrient concentration

2 2.1

Mat hematical model

Based on that of Steele and Henderson (1981), one-component model was used to represent concentrations of phytoplankton ( u ) in a physically homogeneous oceanic mixed layer. Phytoplankton and aquatic plants that are mostly unicellular, take up nutrients from the water for photosynthesis. When a constant nutrient concentration is assumed, the model becomes:

du dt

N

a

- -u - (s

e+Nb+cu

+ k)u

where u denotes the density of population for population and N denotes the concentration of nutrient. The parameter definitions and the value are given in Table 2.1, together with the value of each parameter originally used by Steele and Henderson (1981). The equations of the model are described in detail (for linear and quadratic closure) by Edwards and Brindley (Edwards and Brindley, 1996; Edwards and Brindley, 1999); here brief outline of the model is given. A physically homogeneous mixed layer is assumed, within which volumetric concentrations of N and u are uniform. Units of nutrient ( N ) and u are gCm-3, with time t measured in days, and all parameters are positive. The ratios to convert back into units of nitrogen or chlorophyll, as used by Steele and Henderson (1981), are as follows: Table 2.1 Parameter definitions and default values Parameter a / b gives maximum u growth rate light attenuation by water u self-shading coefficient half-saturation constant for N uptake cross-thermocline exchange rate 1~ sinking loss rate

Symbol

Default value 0.2 m-

0.03 gCm0.05 day0.04 day-1

1 g carbon 20 mg chlorophyll = 10 mmol nitrogen The mixed-layer depth is kept fixed at 12.5 m, and it is an autonomous dynamical system. The water below the mixed layer is assumed to have zero phytoplankton. The next step is to analysis the formulation of the model.

Jue Yang, Deguan Wang, Ying Zhang

214

At firstly, the steady state ( d u l d t = 0) is assumed. The following solution can be retrieved (when u f0): u1=

+

+

N u - b(s k ) ( e N ) c(s k)(e N )

+

+

(2.2)

Equation (2.1) becomes:

du dt

--

-(s

+

~ ) C U [ U- U I ]

b+cu

du If u1 < 0, - < 0 when initial value of u is greater than zero. That is dt to say, if there's enough time the phytoplankton will be extinct. So the

condition N

> be(s Ic) can be regarded to maintain certain phyto-

+ Ic)

+

u - b(s

plankton population. It can be explained that only when the nutrient concentration is greater than a critical value in ocean environment could phytoplankton exist. Secondly, the positive equilibrium point u1 is discussed. If u < u1 du du then - > 0; if u > u1 then - < 0. So u1 is steady (Figure 2.1). dt dt

*

I

t

Figure 2.1 Integral line of u for equation (2.1). Thirdly, note the limit value of u1, equation (2.2) can be rewritten as follows: U - _b u1= c(s k ) ( R 1) ,c

+

+

Study on the Character of Equilibrium Point and . . . From equation (2.4), it can be found that u1 when N

2.2

215

+

+ 00.

Results and discussion

In this section, the positive equilibrium point was achieved and the processes of phytoplankton population were observed along a linear nutrient gradient using four-order Runge-Kutta method. The nutrient concentrations ( N ) 0.001, 0.01, 0.1 and 1 were chosen for simulation. For du N=0.001, phytoplankton population is dropping because - < 0. Its dt du dropping line is shown in Figure 2.2. For N=0.01, 0.1 and 1, - > 0, dt their increasing lines are shown in Figure 2.3. For the four N values, the equilibrium points are -0.32079,0.88889, 3.7735 and 4.89374, respectively. For N=0.001, the phytoplankton concentration drops to 0. For other three N values, the phytoplankton concentrations reach their own equilibrium points at last. Apparently, the phytoplankton concentrations increase very fast at first, and then slow down when they approach their equilibrium points (see Figure 2.3). The dropping rate of phytoplankton concentration for N=0.001 is shown in Figure 2.4 and the increasing rates for other three N values are shown in Figure 2.5. These results showed that the larger N is, the faster phytoplankton concentration increasing rate is.

0'012 0.01

a

0.008

8

-8 f

0.006

k 0.004 c

a

0.002 0

1

2001

4001

6001

8001

10001

12001

time

Figure 2.2 Phytoplankton concentration dropping line for N=0.001.


216

N= 1

N=0.1

N=0.01

1

2001

4001

6001

8001

10001 12001 14001 16001 18001 time

Figure 2.3 Phytoplankton concentration increasing line for N=0.01,0.1 and 1. 0.0007 0.0006 0.0005

f4

0.0004

% 'E. 0.0003 & 0.0002 0.0001

0

1

2001

4001

6001

8001

10001 12001 14001 16001 18001 time

Figure 2.4 Phytoplankton concentration dropping rate for N=0.001. 0.25 I 0.2

0.15

0.1

0.05

0

1

2001

4001

6001

8001

10001 12001 14001 16001 18001 time

Figure 2.5 Phytoplankton concentration increasing rate for N=0.01, 0.1 and 1.

'

3 3.1


217

Phytoplankton population model with non-constant nutrient concentration Mathematical model

In this section, the phytoplankton model in section 2 was used, but N value varies as described in equation (3.1):

N a dN u dt e+Nb+cu du N a - - -___ u - (s dt e N b + cu - - --___

+

+ NO - N ) +k)u

where NOde'notes N concentration below mixed layer. Other parameters and conditions are the same as in section 2. The equilibrium point and its stability are consistent with the results discussed in section 2. The initial N and u value are set to be 0.01. NO is set to be 0.01, 0.1, 1 and 10.

3.2

Results and discussion

Both the phytoplankton concentration and the equilibrium point are changing with time for NO = O.Ol(see Figure 3.1 and Figure 3.2). It is indicated that when the equilibrium point becomes smaller the phytoplankton concentration begins to drop. At last they meet at 0.00389. For other three NOvalues, the change logarithm lines of nutrient, equilibrium point and phytoplankton concentration are showed in Figure 3.3, Figure 3.4 and Figure 3.5, respectively. The change lines of the increasing rate are showed in Figure 3.6. Apparently, the changing trend of equilibrium point is consistent with that of nutrient. The phytoplankton concentration and the equilibrium point meet at the end. In Figure 3.5, before the point A1, A2 and A3, the changing rates are very fast and after them the rates decrease. The characters can be observed easily in Figure 3.7, which includes the nutrient concentration, the equilibrium point and the phytoplankton concentration logarithm line for No=l. A is the critical point for u. Before it, the line of phytoplankton u is steeper than the line after it. For detailed study, we set the initial N=0.1 and use N0=0.01 to simulate. The line of phytoplankton concentration-u is shown in Figure 3.8. The nutrient line is shown in Figure 3.9.


218 0.014

I

0.002 0

1

1001 2001 3001 4001 5001 6001 7001 8001 9001 1000111001 12001

time

Figure 3.1 Time variation of phytoplankton concentration €or No=O.Ol. 1 0.8

.-

0.6

B

3

z!

0.4

." 0.2

2

0

-0.2 -0.4 time

Figure 3.2 Time variation of equilibrium point for No=O.Ol. 10

No= 10 1

0.01

No=O.l 0.001

1

2001

4001

6001

8001

10001 12001 14001 16001 18001 time

Figure 3.3 . Time variation of nutrient concentration for N0=0.1, 1, 10.

Study on the Character of Equilibrium Point and

-.

219

10 No= 10

No=O.l "

0.01

"

"

"

'

~

"

I

'

,

"

'

/

Figure 3.4 Time variation of nutrient concentration for N0=0.1, 1, 10. 10 N o = 10

No= 1

No=O.l

'

I

2001

4001

(

6001

~

~

8001

10001 12001 14001 16001 18001 time

I

I

S '

l

~I

I

I 8

8

,

Figure 3.5 Time variation of phytoplankton concentration for No=O.l, 1, 10. 0.3

0.25 0.2

'

0.15

g

0.1

3

.s" .-8

0.05

0 1

2001

4001

6001

8001

10001 12001 14001 16001 18001

1

-0.05 time

Figure 3.6 Time variation of nutrient concentration for N0=0.1, 1, 10.


220

Nutrient N

0.0011 ~ " " " " " " " " " " " " " " " " " " " " " " " " " ' " " ' " ' " " ' ~ 1 501 1001 15012001 2501 3001 3501 4001 4501 5001 5501 6001 6501 7001 7501 time

Figure 3.7 Time variation of u, N , Equilibrium point for No=l. 0.09 I

0.08 0.07

3!

-8'

0.06 0.05 0.04

B 0.03 E4

0.02 0.01

O

1

L

"

"

2001

"

" " 4001

"

" " " ' 6001 8001

10001 12001 14001 16001 18001

time

Figure 3.8 Time variation of phytoplankton concentration for initial N=0.1 and No=O.Ol. 0.12 0.1

0.8

0.04

0.02 0 1

2001

4001

6001

8001

10001 12001 14001 16001 18001 time

Figure 3.9 Time variation of nutrient concentration for initial N=0.1 and No=O.Ol.


4

221

Conclusions

The character of equilibrium point has been investigated using a simple Nutrient-Phytoplankton model. The equilibrium point is used to represent the phytoplankton concentration when its changing rate is zero. For phytoplankton, nutrient is the only factor concerned. The nutrient concentration ( N )is constant in section 2 and varies with time in section 3. From the results, some conclusions can be made: 1. The changing trend of equilibrium point is determined by the change of nutrient; 2. The phytoplankton concentration approaches the equilibrium point at the end; 3. The larger the difference between the equilibrium point and the phytoplankton concentration is, the faster the changing rate of phytoplankton u is. In terms of red tide, if the algae’s steady positive equilibrium point is made very high by some environmental sources (e.g. nutrient), the difference between the concentration of algae in water and the equilibrium point could be very large, and then the increasing rate of the algae could be very large. This is the first phase of red tide. When the sources are poorer, the steady positive equilibrium point of algae is lower, and then the increasing rate of it is lower. This is the secondphase of red tide. The equilibrium becomes smaller than the practical concentration of algae because the sources are exhausted. So the algae concentration drops. This is the third phase of red tide. At last, the algae’s concentration meets the equilibrium point, and red tide disappears. The process can be seen in Figure 3.9. In practical environment, factors affecting red tide are very complicated. When hydraulic conditions are concerned, the variability of the equilibrium point could become very complicated. This study only deals with the key environmental factors such as nutrient. From the results, the phytoplankton concentration and the changing trend are determined by the equilibrium point. Further work needs to be done on the impacts of other environmental factors on the equilibrium point.

222


References [l]Edwards A M, Brindley J. Oscillatory behaviour in a three-

component plankton population model. Dyn Stab Syst., 11: 347-70, 1996. [2] Edwards A M, Brindley J. Zooplankton mortality and the dynamical behaviour of plankton population models. Bull Math Biol., 61: 303-39, 1999.

[3] Steele J H, Henderson E W. A simple plankton model. Am Nat., 117: 67691,1981.

223

A Numerical Simulation of Thermal Discharge into Tidal Estuary with FVM Jie Zhou, Deguan Wang Environmental Engineering and Science School, Hohai University, China

Haiping Jiang Pearl River Water Resources Commission, Ministry of Water Resources, China

Xijun Lai Nanjing Institute of Geography and Limnology, Chinese Academy of Sciences, China

Abstract The two dimensional shallow water equations with heat convection and diffusion equation is proposed to simulate thermal pollution of Guangdong Nansha Thermal Power which is planed to be built. Finite volume method (FVM) and flux difference splitting(FDS) scheme is applied, and the result shows that this method can keep high precision when it is applied to simulate heat transferring in estuary.

Thermal pollution, which is environmental pollutions because of thermal waste from modern industry and life, take a great deal effect on atmosphere and water. For instance, thermal waste water from firepower or steel factory do great harm to water environment. Since water temperature becomes higher, there is less dissolved oxygen in water. Water body-will be in an oxygen-lack status, and fishes and other biology in water will be hard to live on. If temperature of water in estuary makes an ascent, it will hold back the fishes from going upstream to spawn. Then distribution of biotopes will change, and the balance of old system will break. Spatial distribution of thermal pollution beyond tolerance criterion in water, and its effects on biological environment are the key point

Jie Zhou, Deguan Wang, Haiping Jiang, Xijun Lai

224

of national environmental management and conservation. At present, there is no easy way to forecast the pollution distribution. Its general method is numerical simulation and laboratory experiment. As a numerical method, finite volume method (FVM) is an efficient, high precision, well-fitted boundary method. The method is applied to simulate distribution of the thermal pollution from to-be-built Nansha firepower of Gangdong province.

1

Two dimensional shallow water model to simulate thermal discharge

1.1 Main idea When hot wasted water from the power discharge into estuary, it immediately dispersion away from the port with the force of tide flow. Generally, heat transport depends mainly on convection effect. So hot waste transport law is governed by tidal force. Precision of tidal hydre dynamics scheme becomes the key point in the model. Two dimensional shallow water equation will be applied to simulate the tidal flow with FVM . Since in the study that focus on far or middle thermal field, tidal force take more effects on the velocity field much than thermal waste does, so thermal waste’s effect will be ignored. Thermal waste transport includes several procedures, such as convection and dispersion, exchange with atmosphere, radiation of short waves or long waves, etc. Convection and dispersion is main procedure, exchange must be taken into account because of large area of estuary. But radiation can be ignored.

1.2 1.2.1

Governing equations Hydrodynamics equations

Conserved form of two dimensional non-static shallow water equations is used to describe the flow. The vector equations can be written as:

where q = [h,hu,hvIT , h, hu, hv are conserved physical variables respectively, f(q) = [hu, hu2 gh2/2, huvIT are fluxes in x-direction; g ( q ) = [hu, huv hv2 gh2/2IT are fluxes in y-direction; h is depth; u and v is vertical depth-averaged velocity in x and y direction; g is gravitational acceleration. b(q) is source/sink term, written as:

+

+

+

b(Q)= 10, gh(S0z - Sfz),gh(S0, - S,,)lT

(1.2)

A Numerical Simulation of Thermal Discharge into . . .

225

where S,, and S f , is bed slop and friction water head slop in x direction respectively, while So, and Sf, is in y direction.

1.2.2

Thermal waste transport governing equation

d(hAT)

dt

+

ax

dAT d +-(Di,h-)

8Y

d dAT + d ( h8vYA T ) = --(oixh=) dX KAT - -+ Si PCP

d(huAT)

aY

where AT is thermal waste temperature increment reference to environmental water body; Di,, Di, is dispersion coefficient in x and y direction respectively; C, is specific heat of water; p is density of water; Si is source/sink term,

+

K = 15.7 [0.515 - 0.00425(TS - T d ) +0.00005l(T~- Td)2](70+ 0.7W:) where K is heat diffusion coefficient at water surface, whose evaluation is recommended by “technical rules on surface water environmental evaluation (HJ/T 2.3-93)”, T, is temperature at water surface; T d is dew point temperature; W, is wind velocity at 10m above water.

1.3 Numerical methods Since the Riemann invariant remains constant along their corresponding characteristic curves and flux will not change when rotate it, twodimensional problem can be solved in a one-dimensional system. Flux difference scheme-an approximate solver proposed by Roe[’] is applied to evaluate the fluxes, such as flow quantity, momentum, temperature increment across the edge of elements. Zhao et a1.[’] used this method to calculate the normal flux of sediment transport and experiential formula of sediment transport to evaluate deposition and erosion successfully. This scheme is tried here to solve the heat transport problem in shallow water system. Discretization can be found in reference [2].

1.4 Boundary conditions 1.4.1

Scope for calculation

Based on measured tides and the locations of port to fetch fresh cold water and heat outfall port, upstream boundary is decided to be at estuary of Dongjiang river, downstream boundary locate at Dahu cross section

226


t

N

Figure 1.1 Scope and mesh. of Pear river estuary. Local refined irregular quadrilateral element mesh is applied, and the control volume of FVM is the element itself. Coarse mesh elements locate at far field where there is less effects of thermal drainage, whose spatial step is about 600m. While refined elements are at near area of thermal waste outfall port and fetch water port, whose spatial step is between 25m and 80m. There are 1359 nodes and 1133 elements in mesh, see Figure 1.1. 1.4.2

Boundary conditions

There are five boundary conditions for hydrodynamics system, which are measured tidal head and discharges at Dasheng, Zhangpeng, Sishengwei, Sanshakou, Dahu hydrology station respectively. When calculating heat field, radiation open boundary conditions is applied at outflow boundaries. Since upstream boundary is far away enough so that the thermal

A Numerical Simulation of Thermal Discharge into . .

227

discharge can not effect its temperature, so temperature increment at upstream boundary is taken to be zero.

2

Validation of hydrodynamics model

Measured data at hydrology stations of low water period from year 2000 to year 2001 validates the hydrodynamics model. To validate the model, downstream boundary is extended to Shajiao temporally. Tides and discharge validation is taken at Dahu cross section. Results of validation are shown as Figure 2.1 and Figure 2.2. Calculated water head and discharges agree with the measured ones very well. So the parameters of the model are well adjusted, and the calculation method is trusty. And FVM method is fit for calculation at Pear river estuary.

-calculated 0 measured

-1.5

timeihour

Figure 2.1 Water head validation at Dahu station.

-calculated o measured

timeihour

Figure 2.2 Discharge validation at Dahu station.

228

3


Case study

Different tide types and different power capacity cases are calculated. The power station will be built in two stages. At first stage, equipped with 2x200MW power engines, thermal waste water discharge will be about 15.75 m3/s. At second stage, equipped with 4x200MW power engines, discharge will be about 31.51m3/s. Cases listed as Table 3.1. Total simulation time is 14 tidal periods. Then distribution of temperature increment at different cases during the 14 tidal periods is simulated. Most extent of contours in the same case is obtained to evaluate the effect scope of waste discharge. And averaged contours during all periods are to evaluate the effect degree, shown as Figure 3.3 and Figure 3.4. More attention is taken to the temperature increment near the fetch cold fresh water port to cool the power engines. Tide is the main factor that takes effect on the temperature. Higher tide, higher temperature. The highest temperature increment at fetch port is about 3"C, shown as Figures 3.2 and 3.3. Table 3.1 Cases to be calculated stage

tide type

waste discharge (rn3/s)

waste temperature (oC)

stage

second stage

low water period

15.75

10.0

floodwater period

15.75

10.0

low water period

31.5

10.0

floodwater period

31.5

10.0

I-

24

48

72

96

120

144

168

timehour

Figure 3.1 Temperature increment near fetch port during low water period at first stage.

A Numerical Simulation of Thermal Discharge into . . -

24

48

72

96

120

144

229

168

timehour

Figure 3.2 Temperature increment near fetch port during floodwater period at first stage.

Figure 3.3 Averaged contours of temperature increment during 14 floodwater tide periods.

Figure 3.4 Averaged contours of temperature increment during 14 low water tide periods.

230

4


Conclusions

Rather than finite element method and finite difference method, FVM can ensure matter transport keeping conserved and high precision. From the calculated results, it is noticed that effect scope of the waste discharge is governed by which is stronger between tidal flow and runoff flow. At low water period, tidal flow is stronger than runoff is, and the waste goes upstream more far away than it does at floodwater period.

References [l] ROE P L. Approximate Riemann solvers, parameter vectors and difference schemes [J]. Computational Phys. 43, 1981.

[a] Zhao D H, Shen F X, Yan Z J, Lu G H, A 2D sediment transport model based on the FVM with FDS for tidal rivers [J].Hydrodynamics, Ser. A. 19: 98-103, 2004. (in Chinese)

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