Encyclopaedia of petroleum science and engineering, Volume 12

ENCYCLOPAEDIA OF PETROLEUM SCIENCE AND ENGINEERING (VOL. 12) S.L. Sah

KALPAZ PUBLICATIONS

ENCYCLOPAEDIA OF PETROLEUM SCIENCE AND ENGINEERING

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ENCYCLOPAEDIA OF PETROLEUM SCIENCE AND ENGINEERING (VOL. 12)

S.L. Sah

lH

KALPAZ PUBLICATIONS DELHI·110052

Encyclopaedia of Petroleum Science and Engineering

@S.L. Sah

ISBN :978-81-7835-618-1

All rights reserved. No Part of this book may be reproduced in any manner without written permission. Published in 2007 in India by Kalpaz Publications C-30, Satyawati Nagar, Delhi-110052 E-mail: [email protected] Phone: 9212729499 Lasser Type Setting by: Quick Media, Delhi Printed at : Salasar Imaging System, Delhi

Dedicated to the Geophysicists, Geologists, Engineers, Scientists, Universities, Organisations, Teachers, Students, and other working in different disciplines ofpetroleum science and engineering

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"Help us to harness the wind, the water, the sun, and all the ready and renewable sources of power. Teach us to conserve, preserve, use wisely the blessed treasures of our wealth-stored earth. Help us to share your bounty, not waste it, or pervert it into peril for our children or our neighbours in other nations. You, who are life and energy and blessing, teach us to revere and respect your tender world" Prayers of Thomas J. Carlisle

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(CONTENTS) Preface

11

1. Seismic Earth Modeling

15

2. Seismic Earth Imaging

78

3. Three Dimensional (3-D) Seismic Exploration, Processing and Interpretation

81

Appencfuc.-A : Utilization of Natural Gas for Power Generation

210

Appendix-B : Environment and Pollution Control

209

Appendix-C : Ozone Depletion and Greenhouse Effect

217

Appendix-D : Role of Electronics in the Industrial Growth

221

Appendix-E : World's Famous Quotations

231

Appendix-F : News in Focus

239

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Preface "Energy and persistence conquer all things. " -Benjamin Franklin India, usA, China, Japan and South Korea have decided to make common cause as the world's major buyers of oil. They make up half the oil conswnption of80 million barrels per day (BPD), while the USA alone accounts for a fourth of world oil consumption. Their coming together, augurs well for the world economy, which could hit a speed-breaker if prices continue to be unstable. This cartel of buyers can coordinate placement of large orders as well as futures contracts to check against price fluctuation. It could also reduce the impact of USA politics in the Persian Gulf on oil prices. At present, non-OPEC producers account for 60 per cent of total output, as OPEC produces below potential for a variety of reasons. When prices are driven up by rising demand as well as political instability, OPEC tries to work this situation to its benefit by reining in output, taking advantage of the inelastic nature of demand for oil. Such cutbacks are not motivated by economics alone; there is also resource nationalism on the part of oil-producing states. OPEC world have uppermost on its mind that the share of imports in USA oil conswnption, now 50 per cent, is on the rise. Oil transactions may lose some of their political edge, once the USA enters the market as part of a team of buyers who deal with OPEC countries on a different political footing. USA 'geopolitics in the Gulf region is, in part, motivated by a desire to keep oil prices low and stable. It has not been very successful in its efforts. India, China, Japan and South Korea should collectively persuade the USA to be less of a bull in a China shop in the Gulf region, in order to leverage maxinwm results as a buyers' cartel. World oil consumption is expected to rise by 38 million BPD between 2006 and 2030 to 118 million BPD, with non-OECD Asia accounting for 43 per cent of this rise. World oil conswnption rose by 1.2 million BPD in 2005, of which the rise in OECD consumption was just 0.1 million BPD. The transport sector is expected to account for half of the future increase in oil consumption. India would have to look at high oil prices and political instability in

12

Encyclopaedia of Petroleum Science and Engineering

West Asia as a long-tenn variable in economic and financial management. Oil imports have been growing at 45 to 50 per cent over the last two years, rising from $29 billion in 2004-05 to $43 billion in 2005-06. At present, our remittances, capital flows and reserves can comfortably take case of a current account deficit of about 3 per cent of our gross domestic product, or aBout $20 billion. Foreign exchange reserves are now at $160 billion, about $40 billion higher than annual imports. While this looks like a comfortable sum today, reserves are growing only at 10 percent per annum, while our imports are growing at 30 per cent in a year. Exports could grow at 15 to 30 per cent, depending on conditions and shocks in the world economy. The global economy grew by 5.1 per cent in 2004 and 4.5 per cent in 2005, despite ·the rise in oil prices. Our reserves act not only as a cushion against uncertainty but also as agents of longtenn growth to the extent that they can absorb a large current account deficit. Part-one of this encyclopaedia gives about the seismic earth modeling. Seismic modeling involves generating travel times and amplitudes of seismic wave propagating through a specified subsurface reflectivity model that is associated with a specified velocity-depth model. It can be done either in a computer or in a laboratory. In theory, any migration algorithm can be driven in the opposite direction to perfonn modeling. In particular, we can think of migration and modeling as extrapolations in depth and time, respectively. Just as there are several approaches to solving the wave equation for migration, there also are several types of modeling techniques. There are modeling techniques based on the Kirchhoff integral, finite-difference, and f-k domain solutions to the wave equation. The algorithms based on the scalar (acoustic) wave equation, which describes P-wave propagation, are suitable for structural mode)ing in which amplitudes are not as important as travel-times. The algorithms based on the elastic wave equation, which describes both P-wave and S-wave propagation, are suitable for detailed stratigraphic modeling in which amplitudes are as important as traveltimes. Modeling based on one-way wave equations does not include multiples, while modeling based on full wave equations include multiples in the solution. Part-two of this encyclopaedia gives about the seismic earth imaging. Strong lateral ·velocity variations associated with complex overburden structures require earth imaging in depth. Earth imaging in depth is

Preface

13

achieved by depth migration. Only depth migration algorit:lum implement the additional thin-lens term that explicitly accountS for lateral velocity variations. A velocity-depth' model is the seismic representation of an earth model in depth. An earth model and the earth image created from it are an inseparable pair of products of seismic inversion. To obtain an earth image in depth, one has to first estimate an accurate earth model in depth. Aside from earth modeling and imaging in depth, depth migration also used to verify and update velocity depth models. Depth migration is used in an iterative way. 3-D prestack depth migration often is done using the Kirchhoff integral or the eikonal equation. The output of prestack depth migration consists of image gathers, which may be linked to moveout-corrected CMP gathers with the vertical axis in depth. Image gathers consist of traces iri, their migrated positions. A stack of image gathers represents the earth image in depth obtained from prestack depth migration. If the velocity-depth model used in prestack depth migration is correct, then, events on an image gather would exhibit a flat character with no moveout. Part-three of this encyclopaedia gives about the three dimensional (3-D) seismic exploration, processing, and interpretation. Subsurface geological features of interest in hydrocarbon surveys are three dimensional in nature. A 2-D seismic section is a cross-section 9f a 3-D seismic response. 3-D migration of 3-D data provides an adequate and detailed 3-D image of the subsurfllce, leading to a more reliable' interpretation. A typical marine 3-D survey is carried out by shooting .closely spaced parallel lines (line shooting). A typical land or shallow water 3-D survey is done by laying out a number of receiver lines parallel to each other and placing the shotpoints in the perpendicular direction (swath shooting). The direction that is perpendicular to the in-line direction in a 3-D survey is called the cross-line direction. The line spacing in 3-D surveys can be 50 m or less. This dense coverage requires an accUrate knowledge of shot and receiver locations. The area extent of a 3-D survey almost always is larger than the areal extent of the object. In 3-D data processing, traces are collected as common-cell gathers (bins). After stacking, the 3-D data volume is often, but not always, migrated in two stages. The 3-D data volume then is available to the interpreter as vertical sections in both the in-line and cross-line directions and as horizontal sections (time slices). The time slices allow the interpreter to generate contour maps for marker horizons. The interactive environment

14

Encyclopaedia of Petroleum Science and Engineering

provides an effective and efficient means for interpretation of the sheer volUme of 3-D migrated seismic data. The aim of this encyclopaedia is to make the interconnections among the different disciplines ofpetrolewn science and engineering like: seismic earth modeling, seismic earth imaging, and three dimensional (3-D) seismic exploration, processing and interpretation. At the end of this encyclopaedia six appendixes have been included. These appendixes will give more information to readers about the petrolewn science and engineering. This encyclopaedia will help to promote understanding and conmnmication among users. It is suitable for geophysicists, geologists, scientists, universities, organizations, teachers, students and other working in different disciplines of petrolewn science and engineering. The another will be grateful for COmnlents and criticism which might help to improve the later edition of this encyclopaedia. Some of the material of this encyclopaedia has been taken from the books and the papers published in different journals. I am thankful to all of them who have contributed to the development of this encyclopaedia.

October, 2006

SL.Sab

1 Seismic Earth Modeling Introduction Seismic representation of an earth model in depth usually is described by two sets of parameters : (I) layer velocities, and (2) reflector geometries. Depth migration is the ultimate tool for delineation of reflector geometries. If layer velocities are determined accurately, reflector geometries can be recovered by iterative depth migration. Difficulties in estimating layer velocities with a required level of accuracy make the earth model estimation a challenging task for the geophysicist. Nearly all of the practical methods of layer velocity estimation are based on ray theory, and more specifically, on inversion of seismic travel-times. Velocity estimation methods include Dix conversion of rms velocities, inversion of stacking velocities, coherency inversion, and analysis of image gathers from prestack depth migration. Velocity variations within the earth may be characterized in two ways: (I) structure-dependent, and (2) structureindependent. A structure-dependent earth model comprises geological formations with interfaces that coincide with distinct velocity contrasts. We encounter structure-dependent earth models in areas with extensional and compressional tectonics, and especially in areas with salt and overthrust tectonics. A structure-dependent earth model usually req~ires a layer-by-Iayer escimation oflayer velocities and delineation of reflector geometries that coincide with the layer boundaries themselves. A structure-independent earth model comprises geological formations with interfaces that do not necessarily coincide with distinct velocity contrast.

DetaDed Encyclopaedia This encyclopaedia is arranged in alphabetical order. The detailed encyclopaedia is given below :-

16

Encyclopaedia of Petroleum Science and Engineering

Calibration to WeD Tops The depth structure maps derived from time-to-depth conversion or layer-by-Iayer inversion invariably will not match the well tops. The sources of discrepancy between the estimated reflector depths and the well tops include limitations in the methods for interval velocity estimation, mispicking of time horizons inputs to depth conversion, and limitations in the actual depth conversion itself within the context of ray tracing through an earth model that includes complex layer boundaries. For the depth structure maps to be usable in subsequent reservoir modeling and sinmlation, it is imperative to calibrate them to well tops. Consider a seismically derived depth structure map z, (x, y) based on time-to-depth conversion, say, for the layer boundary associated with the top-reservoir. Also consider Nw well tops Zw (xi' Y) for this horizon at locations (xp Yj)' i = 1,2, .... , N", Since the velocity-depth model derived from time-to-depth conversion is supposed to be consistent with the input data, i.e.. the time structure map T (x, y) created from the interpretation or'the time migrated volume of data, we have T(X,y) = 2

(x,y) Vs(x,Y)

Zs

...(1)

Where Vs (x, y) can be either the average or rms velocity map associated with the horizon Zs (x, y). We consider vertical rays rather than image rays as in equation (1). There exists a calibration velocity Vc (x, y) such that, at a well location (x,., Yj) it satisfies the relation given by the following equation :T

_ 2 Zw (Xj,Yj) (x.,y.) V.( ) , , c Xj,Yj

...(2)

Combine equations (1) and (2) we get the following equation:-

Vc (Xj,Yj) Vs(x;,yj)

(Xj,Yj) z,(Xj,Yj)

Zw

From the knowledge of the well top derived

Zw

...(3)

(x,., Yj) and the seismically

reflector depths Zs (x,., Yj) at the well location, equation (3) gives a calibration factor c(x" Yj) = Vc (x,. y)IVs (xi' Yj) computed at each of the well locations. Next, apply kriging or some other interpolation technique to the sparsely defined calibration factors c(xi' Yj)' i = 1,2,...... , N w to derive a

17

Seismic Earth Modeling

cahbration factor map c(x, y) specified at all grid locations (x, y). Kriging is a statistical method of determining the best estimate for an unknown quality such as c(x, y) at some location (x, y) using a sparse set of values such as c(x y;) specified at locations (xi' y;) (Sheriff, 1991). The fmal " step in calibration is to scale the depth structure map Zs (x, y) the calibration factor map c(x,y) given by the following equation:-

ze(x, y)

=

c(x, y) ZS' (x, y)

...(4)

Where zc(x, y) is the calibrated depth structure map. From equations (3) and (4), the calibrated depth Zc coincides with the well top Zw at well location (x,,, y;). Calibration to well tops is done only after the completion of model building, and just before well planning and reservoir modeling. When estimating an earth model by following a layer-by-Iayer inversion procedure, depth horizon associated with the (n - l)st layer should not be calibrated before estimating the model for the next layer n. This is because seismically derived layer velocities almost never match with well velocities. The discrepancy between the two is attributable to several factors, including the limited resolution in velocities estimated from seismic data and seismic anisotropy. Additionally, the high-frequency variations in the well velocities are absent from the seismically derived velocities. The calibrated depth maps can be used to create a solid model of the earth. See fig. 1. Each layer is represented by a solid with its interior populated by specific layer parameters. See fig. 2. These may include compressional-wave and shear-wave velocities, densities, and rock physics parameters such as porosity, permeability, pore pressure, and fluid saturation. When populated by the petrophysical parameters, the solid associated with the reservoir layer represents a reservoir model. For the purpose of reservoir modeling, the solid for the reservoir layer usually is downscaled in the vertical direction by dividing it into thin slices with a thickness as small as 1m, i.e. much ,less then the threshold for vertical seismic resolution. Additionally, the solid for the reservoir layer is upscaled in the lateral direction by dividing each thin slice into fmite elements with a varying size upto 250 m on one side. The reservoir model is eventually fed into a reservoir simulation scheme to predict the geometry of the fluid flow from the given reservoir parameters. Depth Structure Maps We have a set of time horizon maps and a corresponding set of internal velocity maps. In principal, a time migrated image can be

18

Encyclopaedia of Petroleum Science and Engineen·ng

(a)

(b) Fig. 1. (a) Depth horizonts derived from image-ray depth conversion ofthe time horizons. (b) Solid model created from the depth horizons in (a) (AfterYelmaz, 2001.

Seismic Earth Modeling

lseOi" '

lmt'-< -

Fig. 2. Explosion of the solid model shown in fig. l(b).

19

20

Encyclopaedia of Petroleum Science and Engineering

converted to a depth section by mapping the amplitudes along image rays. This notation also can be employed to convert time horizons into depth horizons. The process is done layer by layer starting with the shallowest horizon. A comprehensive mathematical discussion on imageray tracing is given by Hubral and Krey (1980). See fig. 3. Consider an image ray .that departs the nth surface at point SII with coordinates (x,., YII, ZII) and emerges at the right angle at the earth's surface at point So with coordinates (xu Yo' Z = 0). Our goal is to determine the coordinates of the output point (XII' YII' ZII) on the depth map from image-ray depth conversion of the input point (xu Yo' til) on the time map. To achieve this goal, we want to trace the image ray from the point of emergence (xu Yo' 0) back to its point of departure (x,., Y,., ZII).

Fij!;. 3. Principles of image-ray depth conversion

Suppose that the first n-l horizons have already been converted to depth, and that next we want to convert the nth horizon to depth. Since the earth model is known for the first n-l layers, then we know the coordinates of the inter-section point Si of the image ray with the frrst layer, (xl'YI' ZI)' where by definition of the image ray, XI =xo andYI = Yo. By using Snell's law, we can determine the direction of the ray as it departs point Si reaching point S2 on the next surface. As the image ray moves from one surface to the next, we add up the time it takes to travel. When it reaches the (n - 1)st layer, the elapsed two-way time tit-I is given by the following equation ; -

t

= 11-1

II-1 M 2~_k

7

Vk

...(1)

21

Seismic Earth Modeling

Where vk is the interval velocity of the Jcth layer and Ils k is the distance between the inter-section points of the image ray, Sk_1 and Sk the (k - 1)st and kth surface given by the following equation : !l~k = [(Xk - Xk-i)Z + (Yk - Yk_I)2 + (Zk - zk_I)21"2 ...(2) Now, we examine the situation when the image ray departs the (n - l)st layer at point Sn_1 on the way to the nth surface. Again, by Snell's law we know the direction of the ray. We also know the elapsed time tn - tn-I from the (n - l)st surface to the nth surface since we know the total elapsed time tn-I from equation (1) and the total elapsed time tn from the input time horizon read at point (xo'Yo' zO>o Finally, we know the internal velocity VII of the nth layer from the interval velocity map. Therefore, we can calculate the elapsed distance IlsII along the raypath as it departs the pint Sk-I on the (k - 1)st surface in the direction dictated by Snell's law. The quantity Ilsn is given by the following equation : -

Ils n =

Wn

"2 + (tn -

tn-I)

••.(3)

Finally, the coordinates of the point sn that we need to know to perform the time-to-depth conversion are given by the following equations :

=

xn-I + Ils n cos n

....(4)

Y n = Yn-I + Ilsn cos J3

...(5)

+ Ilsn cos Y

...(6)

Xn zn

= zn-I

Where n, J3 and yare are directional cosines of the ray at point Sn-I' The directional cosines are known by the application of Snell's law at point Sn_1 with known coordinates (xn-I' Yn-I' zn-I)' Given the depth and interval velocity maps for the frrst n-l horizons, and the time and interval velocity maps for the nth horizon, we can trace an image ray associated with the time tn (xo' yO> on the time map and derive the depth value ZII (x n, YII) on the depth map. The depth maps are compatible with the time maps. There can be subtle differences because of velocity variations that would give rise to the departure of image rays from the vertical. To quantify the differences between the time mps and depth maps, we can calculate the modulus Mil of the lateral displacement vector between the points So and Sn as follows : -

...(7) The most significant displacement between the vertical rays and image rays is at the fault zones. The displacement vector also has a directional azimuth +n which is given by the following equation:

22

Encyclopaedia of Petroleum Science and Engineering

ell

=

tan-I Yn - Yo

n

...(8)

Xn - Xo

eIln is measured from the inline x direction. The most significant azimuthal

variations are along the fault zones.

Earth Modeling in Depth Structure-independent earth models often are associated with low-relief structures or stratigraphic plays involving a depositional sequence with facies changes. A structure-independent earth model may be estimated initially by Oix conversion ofrms velocities without requiring a layer-bylayer analysis. The simplest method for estimating layer velocities is Oix conversion of rms velocities (Oix, 1955). The method requires the rms velocities associated with the layer boundaries that are included in the earth model to be constructed. The rms velocities ideally are estimated by pre-stack time migration. Alternatively, a smoothly varying from of stacking velocities estimated from dip-movement corrected data may be a reasonable substitute for rrns velocities. Less desirably, stacking velocities themselves with a fair degree of smoothing applied may be used in lieu of rms velocities. The Oix conversion formula is valid for horizontally layered earth models with constant layer velocities and small offsets. For an earth model with dipping layer boundaries and layer velocities with vertical and lateral variations, more accurate methods are required such as stacking velocity inversion, coherency inversion and image-gather analysis. Stacking velocity inversion (Thorson et al., 1985) requires time horizons picked from unmigrated CMP-stacked data and stacking velocities at analysis locations. Assume that a velocity-depth model already has been estimated for the first n - 1 layers, and that we want to estimate the layer velocity for the nth layer below a CMP location. For a trial constant velocity assigned to the nth layer, the method involves normal-incidence time-to-depth conversion of the time horizon associated with the base of the nth layer, then modeling of the nonzero-offset traveltimes associated with the CMP reflection event that corresponds to the base of the nth layer and determining the moveout velocity by fitting a hyperbola to the modeled travel-time trajectory. This procedure is repeated for a range of constant trial velocities, and the velocity that yields the minimum discrepancy between the actual stacking velocity and the modeled moveout velocity is assigned to the nth layer below the CMP location where the stacking velocity inversion is being performed Coherency inversion (Landa et aI., 1988) also requires time horizons picked from unmigrated CMP-stacked data. However, in lieu of stacking

Seismic Earth Modeling

23

velocities as for stacking velocity inversion, coherency inversion requires analyzing CMP gathers themselves. Again, assume that a velocity-depth model already has been estimated for the fIrst n - I layers, and that we want to estimate the layer velocity for the nth layer below a CMP location. For a trial constant velocity assigned to the nth layer, coherency inversion involves normal-incidence time-to-depth conversion of the time horizon associated with the base of the nth layer, then modeling of the nonzerooffset travel-times associated with the CMP reflection event that corresponds to the base of the nth layer, and computing the semblance within a CMP data window that follows the modeled travel-time trajectory. This procedure is repeated for a range of constant trial velocities and the velocity that yields the highest semblance value is assigned to the nth layer below the CMP location where the coherency inversion is being performed. Time horizons used in normal-incidence time-to-depth conversion as part of the stacking velocity inversion and coherency inversion procedures are picked from unmigrated CMP-stacked data. Alternatively, time horizons interpreted from the time-migrated volume of data can be unmigrated to obtain the time horizons equivalent to the time horizons picked from the unmigrated data. We circumvent the picking of prestack reflection traveltimeyin coherency inversion by measuring the discrepancy between the modeled and actual traveltimes by way of semblance. Similarly, we avoid the picking of prestack reflection traveltimes in stacking velocity inversion by measuring the discrepancy between the modeled and actual stacking velocities. Stacking velocity inversion and coherency inversion both take into account vertical velocity gradients which may be available from sonic logs. The methods also honour ray bending at layer boundaries. While Dix conversion assumes a hyperbolic move-out for the reflection event that corresponds to the base of the layer under consideration, both interval velocity estimate from coherency inversion and stacking velocity inversion are based on non-hyperbolic CMP travel-time modeling. Both stacking velocity inversion and coherency inversion can be considered accurate for velocity-depth models with smoothly varying reflector geometries and lateral velocity variations greater than the effective cable length associated with the layer boundary under consideration. As for conventional stacking velocity estimation, the accuracy in interval velocity estimation from Dix conversion, stacking velocity inversion, and coherency inversion are all influenced by the reflector depth, magnitude of velocity, and the cable length. SpecifIcally, the deeper the reflector,

24

Encyclopaedia of Petroleum Science and Engineering

the larger the layer velocity above, and the shorter the cable length, the less accurate is the interval velocity estimate. To estimate, update and verify velocity-depth models for targets beneath complex overburden structures, such as those associated with overthrust and salt tectonics, ultimately, we have to do image-gather analysis (Reshef, 2001). An image gather is the output from prestack depth migration and is a true CDP gather at a surface location. Stacking of image gathers yields an earth image in depth. If the velocity-depth model is correct, then events on an image gather are flat. In this respect, an image gather can be considered like a moveout-corrected CMP gather, except the vertical axis on an image gather is in depth. An event on an image gather with a move-out indicates an erroneously too low or too high velocity. By examining a panel of image gathers from the same location but with different constant trial velocities for the layer under consideration, one can pick the velocity that yields a flat event and assign it as the velocity of the layer above. Image gathers also can be used to make residual corrections to velocity estimates at analysis locations. This normally is done by fIrst converting the gather to the time domain, performing residual moveout velocity analysis, and converting back to the depth domain. The resulting residual correction should favourably improve the power of the stack obtained from image gathers and yield an updated velocity-depth model. The model updating based on the image-gather analysis usually is repeated until residual move-outs on image gathers are reduced to mininrum. Generalised Linear Inversion (GLI) If given an observed data set d, then we have to estimate a set of parameters p which are used to construct a model d' of the observed data set d' such that the difference between the observed data set d and the modeled data set d' is minimum based on a specifIc norm. We need a model equation that relates the modeled data with the model parameters to be estimated as given below : -

d' = Lp

...(1)

where d' is the modeled vector, p is the model parameter vector, and L is the matrix that relates the modeled data vector to the model parameter vector. The error vector e is defIned as the difference between the modeled data vector and the observed data vector by the following equation :-

e = d-d' Substitute equation (1) into equation (2) we get

...(2)

Seismic Earth Modeling

e

=

d-Lp

25 ...(3)

The cumulative squared error S is expressed as

S

=

ere

...(4)

Where T is for transpose. By substituting the value of e in equation (4) we have : or

S = (d-Lpl(d-Lp) S = tfId-pTLTd-tfILp+pTLTLp

...(5) ...(6)

Differentiate equation (6) with respect to p and observe the requirement for least-squares minimization which is as/ap = 0, we get as: -tfTL + pTLTL = 0

...(7)

Apply matrix transpose and rearrange the terms we have: (LTL)p = LTd

...(8)

Equation (8) yields the desired least-square solution which is given as:-

...(9) Where LTL is the covariance matrix and (LTLt' LT is the leastsquares (also called generalized linear) inverse of L. Equation (9) represents the generalized linear inverse (GU) solution to the parameter vector p. This solution is widely used in many stages of seismic data analysis, e.g., deconvolution, residual statios corrections, refraction static-s corrections, and the discrete Radon transform. The constrained solution is given by the equation ; p = (LTL + ~l)-' L 7*d

...(10)

Where ~ is called the damping factor and I is the identity matrix. In some applications; the generalized linear inverse problem is formulated in the frequency domain. Then, the unconstrained solution is given by the equation:p = (L'rLt'L7*d

...(11)

and the constrained solution is given by the equation : p = (L'rL + ~l)-' L'rd

...(12)

Where the asterisk (*) denotes complex conjugate, and p, d and L are complex. In geophysical applications, techniques to solve for the parameter vector p in equations (9), (10) or (II), (12) include Levinson recursion, conjugate gradient, Gauss-Seidel and singular-value decomposition.

Encyclopaedia of Petroleum Science and Engineering

26

Generalised Linear Inversion (GLI) Formalism of Deconvolution

Deconvolution is fundamentally a data modeling technique. Specifically, we model a I-D seismogram for a minimum-phase estimate of the source wavelet, to predict multiples, and ultimately obtain an estimate of white reflectivity series. We shall consider designing a leastsquares inverse filter j(t) that converts a wavelet w(t) to a desired form d(t) such that the difference e(t) between the actual outut yet) and the desired output d(t) is minimum in the least-squares sense. The zero-delay unit spike is a special case of the desired output d(t). Other forms of d(t) can also be considered, such as a zero-phase band-limited wavelet. The model equation for deconvolution is given by the following equation : -

Y (t) = w(t) *j(t)

...(1)

Consider the discrete form of equation (1), with w( t) represented by the m-Iength time series (wo' wI' w2' •••••••••••• wm-I)' andj(t) represented by the n-Iength time series (fo,J;,J;, ............... f,,-I). Equation (1) can be expressed in matrix form as : -

Yo YI Y2

Wo

0

WJ

Wo

W2

WJ W2

0

2

fo

fi fz

Wo

WJ ...(2)

W2 Wm-I

0

Wm-I Wm-I

0

Ym+n-I

Wm-I

fn-l

Equation (2) is the complete transient convolution. Define the output vector on the left-hand side by y, the coefficient matrix on the right-hand side by L, and the filter vector by f Equation (2) can take the compact form as : -

Y

=

If

...(3)

To obtain the least-squares solution for the filter vector!, we have:-

(LTL)f = LTd

...(4)

Consider the special case of a three-point wavelet (wo' wI' w 2 ). Set up the L matrix of equation (2) for this special case:-

Seismic Earth Modeling

L

27 Wo

0

wt W2 0

Wo

0

0 0 Wo

wt W2 0

...(5)

wt W2

Its transpose LT is given by the equation:

LT =

(% ~

wt Wo 0

W2

0

WI

W2

Wo

wt

~J

...(6)

Now multiply equations (5) and (6), we have:-

LTL =

(

w5+~+~

wtWo + w2wt

Wowt + wtW2

w5+~+~ Wowt + WI W2

WOW2

Compute the first three lags of the autocorrelation (ro' rl' r2) of the wavelet (wo' wI' w2), where ro w 2 +w 2 +w 2 012

r l = WOWI + WIW2

r2 = wOw2 ...(8) The elements of the covariance matrix LTL given by equation (7) are the fIrst three autocorrelation lags of the wavelet (wo' WL W2) given by the equation (8). For the general case, we have:1j

Ii

rn_1

ro

1j

1j

ro

rn-2 rn -2

...(9)

ro Where (ro' rl' r2, ••••••• rn_ l ) are the fIrst autocorrelation lags of the input wavelet series (wo' wI' w2 ' •••••• wn_ I ). For the special case of a desired output vector d is given by the equation :-

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Encyclopaedia of Petroleum Science and Engineering

d =

do dl d2

...(10)

d3 d4 Then the matrix product LTd is given by:-

J

wodo + l11 dl + W2 d 2 LTd = Wodl + l11 d2 + ~d3 [ Wod2 + l11d3 + ~d4 Now compute the fIrst three lags of the cross-correlation (go' gl' g2) of the desired output (do, di, d2, d3, d4 ) with the input wavelet (wo' wI w2) which are given as:go = wodo + wldl + w2d2 gl = wodl + w ld2 + w2d3

...(12)

g2 = wOd2 + w ld3 + w2d4 The elements of the matrix LTd given by equation (11) are the fIrst three lags of the cross-correlation of the desired output (do, dl' d2, d3, dJ with the input wavelet (wo' wI' w2) given by equation (12). The general case, we have:-

...(13)

gn-I where (go gl' g2' .... , gn-I) are ~ fIrst n lags of the cross-correlation of the desired output d with the inp:ut wavelet w. By substituting equations (9) and (13) into equation (4), we have:

Seismic Earth Modeling

29.

Equation (14) is the discrete form of the classic Wiener-Hopfintegral equation to estimate the least-squares shaping filter f that converts an input wavelet w into a desired form d. Practical implementations of equation (4) often require adding a small fraction E of the zero-lag of the auto-correlation to the diagonal elements of the matrix LTL given by:-

(LTL+EI)f = LTd

...(15)

where E is called the prewhitening factor and I is the identity matrix. A typical length of the deconvolution filter f in equation (14) is between 40 and 80 samples. This makes the size of the autocorrelation matrix LTL to be between 40 x 40 and 80 x 80. If the autocorrelation is computed from an input seismogram represented by a single trace, then the length of the input data vector is typically 1000 samples. The ratio of the length of the input seismogram used in computing the autocorrelation lags to the filter length should be no less than 8.

Interval Velocity Maps The Dix equation, which relates rms velocities to interval velocities, is used to derive interval velocity maps. RMS velocities are most appropriately estimated from prestack time-migrated data. The type of velocity that can be most reliably estimated from CMP data is the velocity used to apply norrnal-moveout correction. To stack the data we also substitute NMO velocities for stacking velocities. The use of NMO velocities as stacking velocities is based on the small-spread hyperbola assumption. To further substitute stacking velocities for rms velocities is only allowed if the CMP data are associated with horizontally layered earth. To justify the use of stacking velocities as rms velocities, we frrst need to correct for the dip effect on stacking velocities by way of dipmov~out (DMO) correction. In the case of a 3-D survey, we also need to correct for the source-receiver azimuthal effects on stacking velocities by way of 3-D DMO correction. This means that it is the stacking velocity field derived from 3-D DMO-corrected data that should be considered as a plausable substitute for the rms velocity field. But then the DMO velocities are associated with CMP gathers in their unmigrated positions. Strictly, we need the moveout velocities not only corrected for dip and azimuth effects but also estimated from gathers in their migrated positions. This is because the rms velocities used in Dix equation are defmed for a horizontally layered earth model. Thus the desired strategy is that velocities derived from 3-D prestack time migration should be substituted for rms velocities.

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Although prestack time migration velocities are most described to substitute for rms velocities, the interpreter may be compelled to use whatever velocity functions that may be available. These may have been derived from velocity analysis applied to DMO-corrected data or even to eMP data without DMO correction. Under those circumstances, the velocity functions picked at analysis locations need to be edited for any dip effect by either eliminating the suspect functions altogether or by smoothing. Whatever the sources of information, the interpreter starts with a set of velocity functions, each made up of a set of time-velocity pairs and associated with analysis locations over the survey area. The analysis grid typically varies from 500 x 500m to 2 x 2k:m. Hence, there may be as many as 400 velocity functions per 100 k:m.-squared of the survey area. The grided time horizons are intersected with the velocity functions and, for each horizon, velocity modes are extracted from the velocity functions coincident with the horizon times at the locations of the velocity functions themselves. These velocity nodes are then used as control points input to a gridding algorithm to create horizon-consistent rms velocity maps. There may be a need for further editing and smoothing of the rms velocity grids. Finally, the horizon-consistent rms velocity values and the horizon times at each grid point are used in Dix equation to compute the interval velocity values, which are then used to create the horizon-consistent interval velocity maps. Once again, there may be further need for editing and smoothing of the internal velocity maps to remove any geologically implausable velocity variations. Inversion Methods for Data Modeling

In practice seismic inversion has a broader scope of applications which can be grouped in two categories: (1) data modeling, and (2) earth modeling. What we do in seismic data processing is based largely on data modeling. An observed seismic wavefield can be described in two parts: (1) travel-times, and (2) amplitudes. Seismic amplitudes are more prone to the detrimental effects of noise as compared to traveltimes. Hence, in seismic inversion, we almost always treat traveltimes and amplitudes separately. When modeling the observed data, we either model the traveltimes or amplitudes. When modeling the earth, we use the traveltimes, as in structural inversion, or amplitudes, as in stratigraphic inversion. The important points of applications of seismic inversion for data modeling are given below:-

1. Deconvolution is based on modeling a one-dimensional (l-D) seismogram by optimum Wiener filtering for a minimum-phase estimate of the source wavelet, to predict multiples, and obtain an estimate of white reflectivity series.

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Seismic Earth Modeling

2 We model traveltime deviations on moveout-corrected CMP gathers to estimate surface-consistent shot and receiver residual statics. 3. We model refracted arrival times to estimate surface-consistent shot and receiver intercept time anomalies, and obtain shot and receiver refraction statics. 4. One type of formulation of the discrete Radon transform is by genralised linear inversion. The discrete Radon transform is used to model a CMP gather so as to attenuate multiples and random noise, while compensating for missing data and fmite cable length in recording.

5. We model the seismic signal represented by reflection events assumed to be linear from trace to trace and at attenuate random noise uncorrelated from trace to trace by using spatial prediction filters. 6. Based on the same data modeling concept, we design spatial prediction filters to perform trace interpolation.

7. Data modeling also can be used in the design of a threedimensional (3-D) dip-moveout correction operator which accounts for irregular spatial sampling and undersampling of recorded data. Most data modeling applications are based on the theory of generalized linear inversion. Inversion Procedures for Earth Modeling Practical methods for estimating layer velocities and delineating reflector geometries can be appropriately combined to form inversion procedures to construct earth models in depth from seismic data. See table 1. Four such combinations are given in this table. These

Table 1. A set of inversion proc;edures for earth modeling in depth to estimate layer velocites and delineate reflector geometries. Layer Velocities

Reflector Geometries

Dix conversion of rms velocities stacking velocity inversion coherency inversion image-gather analysis

vertical-ray time-to-depth conversion (vertical stretch) image-ray time-to-depth conversion (map migration) poststack depth migration prestack depth

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migrationcombinations are ordered from top to bottom with an increasing level of accuracy. Also, for a given combination the methods for layer velocity estimation and reflector geometry delineation are compatible. In practice, you may wish to choose other combinations of the methods from the left-hand and right-hand columns. Also, you may be compelled to apply an inversion procedure that involves multiple combinations.For instance, in areas where salt tectonics has caused formation of diapiric structures the earth model may be estimated in three parts: (1) the overburden above the salt diapir, (2) the salt diapir itself, and (3) the substratum. You may then use coherency inversion combined with poststack depth migration to estimate the overburden model, and imagegather analysis combined with prestack depth migration to defme the base-salt geometry and estimate the substratum model. The primary consideration in the choice for an inversion procedure is the degree of lateral velocity variations and the complexity of reflector geometries. A mild-to-moderate lateral velocity variation is associated with a zero-offset diffraction response that is represented by a skewed, but almost hyperbolic traveltime trajectory. A strong lateral velocity variation is associated with a zero-offset diffraction response that is represented by a distorted, non-hyperbolic traveltime trajectory. A severe lateral velocity variation is associated with a zero-offset diffraction response that is represented by a complex, multivalued traveltime trajectory. A set of inversion procedures for earth modeling in depth are given below:1. Vertical stretch is a combination of Dix conversion of stacking velocities to estimate layer velocities and vertical-ray time-todepth conversion of time horizons picked from a time-migrated volume of data to delineate reflector geometries. This is a procedure appropriate for cases with negligible ray bending at layer boundaries, gentle dips, and lateral velocity variations judged to be within the bounds of time migration. 2. Map migration is a combination of stacking velocity inversion to estimate layer velocities and image-ray time-to-depth conversion of time horizons picked from a time-migrated volume of data to delineate reflector geometries. This is a procedure appropriate for cases with moderate ray bending at layer boundaries, moderate vertical velocity gradients. and moderate lateral velocity variations. 3. Poststack depth migration is a combination of coherency inversion to estimate layer velocities and posts tack depth

Seismic Earth Modeling

33

migration to delineate reflector geometries. This is a procedure appropriate for cases with significant ray bending at layer boundaries and significant vertical velocity gradients, and strong lateral velocity variations with sharp changes in reflector curvatures. 4. Prestack depth migration is a combination of image-gather analysis to estimate and update layer velocities, and stacking of image gathers to delineate reflector geometries. This is a procedure appropriate for cases with significant ray bending at layer boundaries, and severe lateral velocity variations associated with salt and over-thrust tectonics. These inversion methods are used to estimate an initial earth model in depth. Seismic inversion also is used to update the estimated model. A common application of inversion to estimate the errors in the initial model parameters, i.e., layer velocities and reflector depths, is reflection traveltime tomography. Tomographic inversion involves perturbing the model parameters by a small amount so as to match the modeled reflection travel-time with the observed traveltimes. Refraction traveltime tomography and reflection traveltime tomograpy both are based on the assumption that the perturbation required to update the model parameters is very small compared to the spatial variations in the model parameters themselves. In practice, tomography is best used strictly to touch-up a carefully estimated earth model based on some plausable geological constraints. It should never be used by itself to estimate the model. Map Processing

A map is defmed as a 2-D surface g(x, y). Depending on the quantity being mapped, g (x, y) may have many types of units, e.g., gravitation attraction (m Gal), magnetic intensity (gamma), elevation, or times picked along marker horizons from seismic data. The positive x-axis points eastward and the positive y-axis points northward. A discrete map function is represented by a grid of mesh points over the x - y plane. These mesh points are spaced commonly at equal intervals in the x and y directions. For many types of mapping, g(x, y) is a smooth function and such a map can be analyzed in the Fourier transform domain. However, there are situations (e.g., isochron and structure maps) in which the map function has discontinuities that represent faulting. Maps usually are created from irregularly spaced observation valves. Thus, the

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Encyclopaedia of Petroleum Science and Engineering

map function at a particular grid point must be computed by some fitting procedure. See fig. 4. Void grid points have been filled with the arithmetic

Fig. 4. A time map of a seismic horizon (After Yilmaz, 2001).

mean of the map function. Before map creation, some correction may be applied to observed data, followed by various typFs of editing. Some of the very short wavelengths in the seismic horizon map result from nearsurface effects that are manifested as residual statics. Some moderately long wavelengths correspond to structural undulatjons that exist in the area. Most of the features with different wavelengths are not spatially isolated, but are' superimposed This characteristic is common for all types of map functions, e.g., gravity, magnetic, elevation or time. To separate the effects of different features from each other, we must analyze them in terms of wavelength. Simple 2-D smoothing and wavelength filtering are techniques for separating anomalies. Vertical derivatives and analytic continuation also are useful for enhancing certain anomalies so that they appear more pronounced on the map. The 2-D amplitude spectrum of a map is an excellent tool for recognizing not only the wavelength content, but also the orientation of

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Seismic Earth Modeling

various components. The most useful display is the colour contour plot of the amplitude spectrum from which various bands of wavelengths are distinguished clearly. Pink represents long, beige represents moderate and yellow represents short wavelength anomalies. For a 2-D real function, such as a map, the amplitude spectrum is anti-symmetric. Thus, only a pair of quadrants (the first and second) of the amplitude spectrum needs to be displayed. A simple 2-D smoothing operation is the easiest way to obtain a map that represents the region anomaly. See fig. 5.

Fig. 5. A smoothed version of the contour map in fig. 4.

Smoothing basically is done by computing the average value of the grid points that fall onto a ring of some desired radius. The center of the ring coincides with the output point. For n concentric rings with m; points over the ith ring, the average value gj of the quality gij that is being mapped is given by:...(1)

.. Thus, the cumulative average g over n rings is given by :-

g

=

1~_

-

£..Jgj

n ;=1

...(2)

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Encyclopaedia of Petroleum Science and Engineering

Weighting factors, which depend on the distance from the center of the rings, often are used in smoothing algorithms. Equation (2) becomes as:n

g

=

L wigi

...(3)

i=1

Where wi are the weights. The residual anomaly is defmed by : -

g

=

go- g

...(4)

Where go is the grid value at the center of the rings. Fig. 6 shows the regional anomaly obtained by using IS rings. In general, the more rings, the more smoothing ofthe data. Gridding involves fitting a locally plane surface to a set of control points around each grid output point. Consider a planesurface fit in the least-squares sense which is given as:-

Fig. 6. a regional anomaly map (After Yilrnaz, 200 1).

g (x,y)

=

ao + a\x + aaY

...(5)

The least-squares error is given by :M

L = L(gj - gj)2

...(6)

i=\

Where g is the observed value at the gird point (x, y), and M is the member of observations at and around that grid point. For local plane

Seismic Earth Modeling

37

fitting, M usually is set to 9 points. We want to fmd a set of (a o' al' a2) for which L is minimum, the condition is given by : -

8L 8ao

_ 8L _ 8L -0 - 8a, - 8a2 -

...(7)

From equations (5) and (6), we have : M

L

=

~)g; -ao -a,x; -a2y;f

...(8)

;=, From equation (7), we get the following set of simultaneous equations : ~g ...(9) ~ao + ~a,x + ~aaY ~oX + ~a,x2 + ~azXY

ug

...(10)

~aoY + ~a,xy + ~azV

~yg

...(11)

r

When put into matrix fonn, we obtain : -

(:

u u2

~

~

~y2

~y

a, a2

J (~g ug J =

...(12)

~yg

Equation (12) is solved for the set of coefficients (ao, a-i, aa). Real data consist of anomalies of various shapes and orientation that are superimposed on each other. In the transfonn domain, the real anomalies can be separated in terms of their wavelength contents and orientations. This cannot be achieved in the space domain. The transfonn domain provides a way to apply various filtering operations to a map. f\ band of wavelengths can be passed regardless of orientation by the radial filter transfer function. See fig. 7. In practice the transfer functions must be tapered in the neighbourhood of cutoff wavelengths for optimal performance. +k,

(a)

(b)

(c)

Fig. 7. (a) Transfer function for band-pass filter. (b) Transfer function for directional filter. (c) Transfer function for directional band-pass wavenumber filter.

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Encyclopaedia of Petroleum Science and Engineering

Once the transfer function is designed to suit the purpose, the filter itself can be applied to the map in the transform or the space domain. In the transform domain, the transfer function is multiplied with the 2-D Fourier transform of the map. Subsequent inverse transformation yields the filtered map. To apply the filtering in the space domain, first inverse Fourier transform the filters transfer function to obtain its 2-D impulse response. Two-dimensional convolution of this impulse response with the map yields the filtered map. Before applying the 2-D filters, the possible bands of the wavelengths present on the amplitude spectrum of the map to be filtered must be determined. The bands that were determined are used as cut-off wavelengths of the radial filters transfer function. A given map can be scanned with a suite of low-pass filters and several filtered maps can be produced, each with a potentially unique interpretational value. As the bandwidth of the filter is increased, more and more of the short wavelength anomalies are included, making the output less and less regional in character. A filter scan is not limited to low-pass filtering only. High-pass, band-pass, and band-reject filters can be applied to maps to achieve a particular interpretational goal. Directional filters scan the map of interest at various angles to emphasize a particular trend that may exist in the data. In some cases, a certain band of wavelengths may have one dominant trend that is different from that of another band of wavelengths. This situation may imply a change in the tectonic setting over the geologic history in the area.

Model Building First time it is not possible to build an earth model in depth correctly. The velocity-depth ambiguity that is inherent to inversion makes it very difficult getting the right answer, i.e., the true geological model at fi~ time. Limitations in the resolving power of the methods to estimate layer velocities that arise from the band-limited nature of the recorded data and fmite cable length used in recording further compound the problem. Finally, travel-time picking that is needed for most velocity estimation techniques and time-to-depth cohversion as well as picking depth horizons from depth-migrated data to delineate reflector geometries are all adversely affected by noise present in the data. In first attempt, we can estimate an initial model and then update this model to get the acceptable final model. We shall consi