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GEOPHYSICS, VOL. 73, NO. 6 �NOVEMBER-DECEMBER 2008�; P. O37–O52, 15 FIGS.10.1190/1.2978164

stimating permeability from quasi-static deformation:emporal variations and arrival-time inversion

. W. Vasco1, Alessandro Ferretti2, and Fabrizio Novali2

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ABSTRACT

Transient pressure variations within a reservoir can betreated as a propagating front and analyzed using anasymptotic formulation. From this perspective, one can de-fine a pressure “arrival time” and formulate solutions alongtrajectories, in the manner of ray theory. We combine thismethodology and a technique for mapping overburden defor-mation into reservoir volume change as a means to estimatereservoir flow properties, such as permeability. Given the en-tire “travel time” or phase field obtained from the deforma-tion data, we can construct the trajectories directly, therebylinearizing the inverse problem. A numerical study indicatesthat, using this approach, we can infer large-scale variationsin flow properties. In an application to Interferometric Syn-thetic Aperture Radar �InSAR� observations associated witha CO2 injection at the Krechba field, Algeria, we image pres-sure propagation to the northwest. An inversion for flowproperties indicates a linear trend of high permeability. Thehigh permeability correlates with a northwest trending faulton the flank of the anticline that defines the field.

INTRODUCTION

Knowledge of the structures controlling fluid flow at depth is criti-al in many activities, such as petroleum production, geothermal en-rgy extraction, CO2 disposal, and groundwater utilization. Signifi-ant effort has been expended to increase understanding of fluid flown the subsurface using flow and transport, and time-lapse, geophysi-al observations. For example, new instrumentation and reliableensors have advanced the state of the art in flow and transport moni-oring, allowing for crosswell pressure tomography, multilevel sam-lers, and long-term downhole pressure measurements �Freyberg,986; Butler et al., 1999; Hsieh et al., 1985; Paillet, 1993; Cook,

Manuscript received by the Editor 31 October 2007; revised manuscript re1Berkeley Laboratory, Earth Sciences Division, Berkeley, California, U.S.2Tele-Rilevamento Europa, Milano, Italy. E-mail: alessandro.ferretti@treu2008 Society of Exploration Geophysicists.All rights reserved.

O37

995; Masumoto et al., 1995; Karasaki et al., 2000; Yeh and Liu,000; Vesselinov et al., 2001; Vasco and Karasaki, 2001; Gringartent al., 2003�.

There have been numerous efforts to use geophysical observa-ions, particularly time-lapse data, to monitor fluid flow at depthTura and Lumley, 1999; Landro, 2001�. Much of this work has con-entrated on seismic time-lapse, or four-dimensional �4D�, ampli-ude data from oil and gas reservoirs �Barr, 1973; Greaves and Fulp,987; Nur, 1989; Eastwood et al., 1994; Lee et al., 1995; Lazaratosnd Marion, 1997; Huang et al., 1998; Johnston et al., 1998;urkhart et al., 2000; Behrens et al., 2002; Arts et al., 2004�. Im-roved understanding of fluid flow in the shallow subsurface has re-ulted from shallow and crosswell time-lapse electromagnetic sur-eys �Brewster and Annan, 1994; Wilt et al., 1995; Barker andoore, 1998; Binley et al., 2001; Hoversten et al., 2003�.Geodetic data provide still another set of observations related to

uid flow in the subsurface �Castle et al., 1969; Evans et al., 1982;asco et al., 1988; Palmer, 1990; Dussealt et al., 1993; Bruno and Bi-

ak, 1994; Chilingarian et al., 1994; Castillo et al., 1997; Massonnett al., 1997; Vasco et al., 1998; Wright, 1998; Wright et al., 1998;ossop and Segall, 1999; Stancliffe and van der Kooij, 2001; Vasco

t al., 2001; Du and Olson, 2001; Hoffmann et al., 2001; Vasco anderretti, 2005�. Furthermore, current and planned instrumentation,uch as permanent seafloor sensors and high-precision pressureauges �Sasagawa et al., 2003� can generate useful displacementeasurements. In addition, developments such as side-scan-sonar

nterferometry �Chang et al., 2000� and acoustic positioning �Spiesst al., 1998� might provide useful data.

In spite of these efforts, fundamental understanding of fluid flowithin the earth is hampered by some factors. For example, there of-

en is limited data coverage, particularly with regard to flow andransport measurements. Similarly, the temporal sampling can beparse, particularly with regard to time-lapse geophysics surveys,hich can be years apart and missing important transient phenome-a. Frequently, the scale of investigation is small, of the order of aew meters; and the results might not scale up into the kilometerange, in which faults and fracture zones control flow. In addition, it

8 January 2008; published online 28 October 2008.ail: dwvasco@lbl.govm; fabrizio.novali@treuropa.com

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O38 Vasco et al.

ften is difficult to relate geophysical attributes, such as seismic am-litudes, to flow properties. There can be trade-offs between variousets of attributes, such as between mechanical properties and fluidressure estimates. Finally, in applications, one typically requiresnowledge of factors influencing subsurface flow in a timely fashiono optimize or mitigate the activities.

In this study, we use a cost-effective technique for imaging fluidow and estimating flow properties, such as permeability. We use

ime-varying measurements of deformation in the overburden asata. Such observations are of increasing importance for at least twoeasons.

First, several developments in the past few years advanced thetate of the art in geodetic monitoring. For example, downhole tilteters can provide dense time sampling of deformation at a wide

ange of depths, from the surface to as much as several thousandeters �Wright, 1998; Wright et al., 1998; Du et al., 2005�. Space-

ased techniques, such as Interferometric Synthetic Aperture RadarInSAR�, can sample surface deformation with a spatial resolutionpixel size� of meters through tens of meters and with an accuracy ofhe order of a centimeter to as much as a few millimeters �Massonnetnd Feigl, 1998; Burgmann et al., 2000; Ferretti et al., 2000, 2001�.nSAR observations might be used to image fluid-induced deforma-ion over regions tens to as many as hundreds of square kilometers inxtent �Fielding et al., 1998; Galloway et al., 1998; Amelung et al.,999; Hoffmann et al., 2001; Schmidt and Burgmann, 2003; Cole-anti et al., 2003; Vasco and Ferretti, 2005�.

Second, recent developments in the processing of time-lapse seis-ic data allow for estimates of deformation in the overburden andithin the reservoir �Guilbot and Smith, 2002; Landro and Stam-eijer, 2004; Tura et al., 2005; Hatchell and Bourne, 2005; Barkved

nd Kristiansen, 2005; Hall, 2006; Roste et al., 2006; Rickett et al.,007; Staples et al., 2007; Hawkins et al., 2007�. Such estimates of-er high-resolution, three-dimensional sampling of deformationver a producing reservoir. Based upon a geomechanical model theeformation can be mapped into pressure changes within the reser-oir �Vasco, Karasaki, and Doughty, 2000; Du et al., 2005; Vascond Ferretti, 2005; Hodgson et al., 2007�.

It is possible to use estimated pressure changes directly to inferow properties, such as permeability �Vasco et al., 2001; Vasco,004b; Vasco and Ferretti, 2005�. Here we take an alternative ap-roach on the basis of the idea of a diffusive traveltime associatedith a propagating pressure transient �Vasco, Karasaki, and Keers,000; Vasco, 2004a�.Although the approach requires adequate sam-ling of the pressure field in time, it is relatively insensitive to theeomechanical properties within the reservoir and their heterogene-ty. Furthermore, the approach shares the robustness of arrival-timeomography and actually leads to a linear inverse problem for flowroperties. Thus, it is possible to map deformation measurementsnto reservoir permeability in a linear fashion.

METHODOLOGY

Before we discuss the approach in detail, we outline the majorteps of our inversion methodology. The first step entails invertingeasurements of overburden deformation to estimate volume

hange within the reservoir. Next the volume change is mapped intouid-pressure change induced by production or injection. Finally,

he fluid-pressure changes p�x,t� are used to estimate flow propertiesn the reservoir using a technique similar to traveltime tomographypplied to a disturbance, which evolves according to the diffusion

quation. Because this latter step is unfamiliar to most readers, weiscuss it in more detail in this section. The first two steps are de-cribed in earlier papers �Vasco et al., 1988; Vasco et al., 1998; Mos-op and Segall, 1999; Vasco, Karasaki, and Doughty, 2000; Du andlson, 2001; Vasco et al., 2001; Vasco et al., 2002; Vasco and Fer-

etti, 2005�. Therefore we outline them only briefly inAppendix A.

diffusion equation for fluid-pressure variations

We are interested in the propagation of pressure changes within aeservoir. The pressure changes are caused by production and/or in-ection from wells located within the reservoir itself. The fluid-vol-me changes are determined by the flow rate out of, or into, the reser-oir, denoted here by the source-time function q�x,t�. The reservoirill deform because of the fluid-volume changes, and that deforma-

ion will lead to displacements in the overlying material.Our starting point is the set of partial differential equations de-

cribing the flow of an aqueous phase and a nonaqueous phasePeaceman, 1977; de Marsily, 1986�,

· ��wK�x�krw

�w� �pw�x,t� � �wgz�� �

� ���wSw�� t

� · ��nK�x�krn

�n� �pn�x,t� � �ngz�� �

� ���nSn�� t

� q ,

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here Sw and Sn denote saturations of the aqueous and nonaqueoushases, respectively; and q denotes the injection rate of the nonaque-us phase. The relative permeabilities of the aqueous and nonaque-us phases are represented by krw and krn, and the absolute perme-bility is given by K�x�. The respective densities are �w and �n, theravitational constant is g, and the porosity is ��x�. The pressure as-ociated with the aqueous phase is pw�x,t�, and the pressure for theonaqueous phase is pn�x,t�; the respective viscosities are �w andn. The above equations are coupled because the saturations are as-

umed to sum to unity

Sw � Sn � 1. �2�

Typically, in modeling CO2-brine systems, it is assumed that cap-llary effects are negligible �Pruess, 2004; Obi and Blunt, 2006� andence pw � pn � p, where p denotes the common fluid pressure.dding the two equations in 1 together and assuming that the pres-

ure in the aqueous and nonaqueous phases are equal results in

� · �K�� � p � g f�� �� ��� f�

� t� q , �3�

� ��wkrw

�w�

�nkrn

�n, �4�

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�nkrn

�n� ��ngz� , �5�

nd

� f � �wSw � �nSn � �w � ��n � �w�Sn, �6�

here we have used the fact that the saturations sum to unity, equa-ion 2. Equation 3 is a differential equation describing the evolution

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f the total fluid pressure in a heterogeneous medium resulting fromtime-varying source q�x,t� �Rice and Cleary, 1976�.Equation 3 is a single equation in several unknowns ��w, �n, Sw, Sn,

, and p�, and further assumptions are necessary to produce a usefulxpression. For example, we expand the time derivative in equation; thus

� ��� f�� t

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�� f

� t. �7�

sing the relationship in equation 6, we can write the second partialerivative on the right-hand side of equation 7 as

�� f

� t�

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, �8�

here �� � �n � �w is the density difference between water andO2 under reservoir conditions. Typically, one can neglect the com-ressibility of water and treat the water density as a constant. Theecond term can be important when the compressibility of the non-queous phase is significant. For this reason, we maintain this termn our formulation.

As will be seen, because we are interested in the “arrival time” ofhe initial pressure change, this term will vanish because Sn is insig-ificant at this time. Physically, this results from the fact that theressure change propagates much faster than the saturation change.urthermore, at the pressures within the Krechba reservoir, the den-ity of the CO2 does not vary significantly and is close to that of wa-er, between 0.85 and 0.90 gm/cm3.

When the solid constituents are incompressible, the compressibil-ty of the rock is dominated by changes in pore volume; and one typi-ally has a relationship between fluid pressure p, the bulk or overbur-en pressure po, and porosity �, of the form

p � F�po,�� . �9�

he function F usually is derived from laboratory measurements. Inhe fully general situation, porosity, permeability, and fluid density

ight be functions of fluid pressure and the overburden or confiningressure, leading to a nonlinear generalization of the diffusion equa-ion 3 �Audet and Fowler, 1992; Wu and Pruess, 2000�.

In this study, we adopt the theory of poroelasticity �Biot, 1941�, inhich the relationship between the fluid pressure, overburden pres-

ure, and change in porosity is linear; thus

� � �0 ��

Kupo � Cep , �10�

hich involves the initial porosity �0, the undrained bulk modulusu, and the constants � and Ce defined by Biot �1941�. The variablee�x� relates fluid-pressure variations to changes in porosity.Differentiating equation 10 with respect to time gives

��

� t� Ce

� p

� t�

�

Ku

� po

� t, �11�

ssuming that �0, Ce, �, and Ku are time invariant. Substituting thiselationship into equation 3, and accounting for the time derivativesand 8, results in

� fCe� p

� t� � · �K̂ � p� � Rep � q̂ � 0, �12�

here q̂ is given by

q̂ � q �� f�

Ku

� po

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�

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� t, �13�

Re � Ce� ���Sn�

� t, �14�

nd K̂ is the effective permeability

K̂ � �K . �15�

In the presence of two-phase effects, the coefficients � f and � willhange with time. However, as shown next, in this application we arenterested in the arrival time of a pressure disturbance as it propa-ates from a well. This transient pressure variation generally willropagate significantly faster than the two-phase saturation front.or this reason, we neglect the two-phase changes in � f and � and

reat them as constants. Then we can apply the Fourier transform toquation 12, resulting in the frequency-domain equivalent,

�2P � � · � P � i�DP � Q̂ , �16�

here P�x,�� is the Fourier transform of the pressure; � is the gradi-nt of the logarithm of conductivity

��x� � � ln K̂�x� , �17�

hich vanishes for constant K̂�x�;

D�x� �� fCe�x�

K̂�x��18�

s the reciprocal of the diffusivity; and Q̂ is the Fourier transform ofhe normalized source term

Q̂�x,�� �q̂�x,��

K̂�x,��. �19�

f we include two-phase effects, equation 16 will contain convolu-ions because the coefficients � f, K̂, and R in equation 12 are thenime-dependent.

n asymptotic solution to the diffusion equation

It is not possible to derive an analytic solution of equation 16 for aeneral variation in K̂�x� and Ce�x�. Now, numerical approaches arehe most common methods for approximating solutions in arbitraryarth models. Such numerical approaches are general but do not pro-ide much insight into the nature of pressure propagation within aeterogeneous porous medium. Furthermore, as shown below, aemi-analytic solution has some advantages in treating the inverseroblem, a central concern of this work. Thus, in this subsection, weighlight a semi-analytic solution to the diffusion equation, follow-ng upon the earlier work of Cohen and Lewis �1967�; Virieux et al.1994�; Vasco, Karasaki, and Keers �2000�; and Shapiro et al.2002�.

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The approach, which is implemented in the frequency domain, isased on the notion of a high-frequency asymptotic solution, similaro methods used in the study of electromagnetic or elastic waveropagation �Friedlander and Keller, 1955; Luneburg, 1966; Klinend Kay, 1979; Aki and Richards, 1980; Kravtsov and Orlov, 1990;eller and Lewis, 1995�. However, the definition of “high-frequen-

y” is relative to spatial and temporal variations in the backgroundeservoir pressure in its undisturbed state, before injection or pro-uction. In other words, though the transient pressure disturbanceight take days or months to propagate away from a producing well,e can consider it as a high-frequency variation when comparedith the background reservoir pressure, which can vary over years,ecades, or even longer. Naturally, if there are competing variationsver the timescale in question, such as interference with nearbyells, we must account for that as a secondary source.As noted by Virieux et al. �1994�, an asymptotic solution to equa-

ion 16 is a power series in 1/�,

P�x,�� � e��i�� �x� n�0

An�x�

�� i��n. �20�

his form of the series might be deduced on physical grounds byonsidering a large argument expansion of the modified Bessel func-ion of zeroth order, the solution to the diffusion equation for a ho-

ogeneous medium �Virieux et al., 1994�. The motivation for a se-ies in powers of 1/� is similar to that for asymptotic methods usedn modeling hyperbolic wave propagation.

In particular, when � is large, the first term in the series �expres-ion 20�

P�x,�� � A0�x�e��i�� �x�, �21�

r its time-domain equivalent obtained by inverse Fourier trans-orming equation 21,

p�x,t� � A0�x�� �x�

2 t3e�� 2�x�/4t, �22�

Virieux et al., 1994; Vasco, Karasaki, and Keers, 2000�, will accu-ately represent the solution to the diffusion equation. The expres-ion is similar in form to the solution of the diffusion equation in aomogeneous medium �Carslaw and Jaeger, 1959�, but with spatial-y varying coefficients A0�x� and � �x�.

To obtain explicit expressions for � �x� and An�x�, which are re-uired to fully specify the solution, the sum 20 is substituted intoquation 16. The operators in equation 16 might be applied term byerm to the series. Substitution of the expansion 20 into equation 16roduces an expression containing an infinite number of terms. Eacherm will contain 1/� to some power, and we could consider sets oferms for any given order. As shown in Virieux et al. �1994� andasco, Karasaki, and Keers �2000�, the functions � �x� and A0�x� areetermined by the terms of order �1/���2 and �1/���1, respec-ively. Because we are interested only in the phase, � �x�, we consid-r only terms of order �1/���2 � � in the subsection that follows.

We note that the expressions 21 and 22 are point-source solutionsor a source that acts at a single location in space and at a single time.n applications, such as the synthetic and field examples discussedelow, we use a step-function source, which is a more realistic repre-entation of fluid flow rates. Such sources can be obtained by convo-ution with the point-source solution. In fact, any physical source cane modeled by convolution with a known source-time function. Al-

ernatively, one could deconvolve the known source-time functionrom the observations and construct the equivalent point-source re-ponse. In applications below, in which the source is approximatedy a step function, we simply work with the time derivative of theressure. By linearity, this is equivalent to the pressure values result-ng from a delta-function source.

There is an additional issue related to presence of the overburdenressure term in the source function q̂�x,t� �equation 13�. This terms not localized in space and could act over an interval of time. In thepplications below, we assume that the overburden stress does nothange significantly with time; and thus the derivative in equation 13anishes. However, as shown in Vasco �2004a�, it is possible to cor-ect for the overburden pressure changes, allowing for the use of theoint-source expressions 21 and 22. The corrections provide a quan-itative measure of the importance of variations in overburden stressn the changes of reservoir fluid pressure.

he eikonal equation, trajectories, and arrival time ofhe maximum pressure change

If we consider terms of order �� i��2 � �, we arrive at a scalar,onlinear, partial differential equation for the phase � �x�,

�� �x� · � � �x� � D�x� � 0 �23�

Virieux et al., 1994; Vasco, Karasaki, and Keers, 2000�. Equation3, known as the eikonal equation, governs many types of propaga-ion processes �Kline and Kay, 1979; Kravtsov and Orlov, 1990�;nd there are efficient numerical methods for its solutions �Sethian,996�. The eikonal equation relates the phase function � �x� to theow properties Ce�x� and K�x�, as contained in D�x� �equation 18�.We also might solve equation 23 using the method of characteris-

ics �Courant and Hilbert, 1962�. In the method of characteristics, so-utions are developed along particular trajectories, which are denot-d by X�l�, where l signifies position along the curve. Equations forhe characteristic curves are a set of ordinary differential equations,hich depend on D;

dX

dl� s �24�

d�

dl� D , �25�

here s � �� �Courant and Hilbert, 1962�. From equation 25, wean write the phase function as an integral,

� �x� � � �X�l�

Ddl , �26�

here X��l�� is the trajectory from the injection well to the observa-ion point x.

The trajectory is found by solving equation 24 using a numericalechnique. The method for determining the trajectory X�l� from aource point to a given observation point is known as the shootingethod for a two-point boundary-value problem �Press et al., 1992�.lternatively, we can determine the phase, and the trajectory, by

imply postprocessing the output of a numerical reservoir simulator.his approach, which eliminates the need for ray tracing, is dis-ussed in more detail in Vasco and Finsterle �2004� and Vasco2004a�. It makes use of the fact that

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Estimating permeability from quasi-static deformation O41

� � 6Tmax, �27�

here Tmax is the time at which the pressure is a maximum. The rela-ionship 27 is obtained by differentiating equation 22 with respect toime and setting the resulting expression to zero.

One can postprocess the results of a reservoir simulation to extracthe arrival time of the peak of the pressure curve and hence � �x�, us-ng equation 27. The trajectories are found by integrating the differ-ntial equation 24 using a second-order Runge-Kutta techniquePress et al., 1992�. This is a much simpler procedure than ray tracingnd essentially involves marching up the gradient of the phase fieldbtained from the simulator output �Vasco and Finsterle, 2004�.

linear inversion algorithm for flow properties

The ultimate goal is to estimate flow properties on the basis of a setf measurements of deformation of the overburden. As discussed inppendix A, we use deformation data to estimate pressure variationithin the reservoir induced by fluid injection or withdrawal. Thisrocedure is illustrated by the synthetic and field examples below.or now, assume that we have estimates of pressure changes within

he reservoir for a series of time intervals. In other words, we havenapshots of the pressure changes for a sequence of times. If the timeampling is dense enough, we can estimate the time at which theressure change reaches a peak for points throughout the reservoir.hus, using equation 27, we can estimate � �x�, for x within a regionf the reservoir.

As an example, let us denote the phase estimate at point xi by theuantity � i and the integral by

� i � � �Xi

Ddl , �28�

here Xi denotes the trajectory associated with the � i observationoint. Given estimates of � i at points throughout the reservoir, wean use equation 28 to infer D�x� and consequently the flow proper-ies that define this quantity �see equation 18�.

For a dense sampling of points, we can calculate the trajectoriesirectly from the estimates of phase, using equation 24,

dX

dl� � � . �29�

his, in effect, linearizes the inverse problem for D as given byquation 28; because in estimating Xi directly from the phase values,e have removed the dependence of the trajectory on the unknownistribution of D. Another way to see this is to return to the eikonalquation 23 and substitute in the estimate ��x� for the phase field.he result of

���x� · � ��x� � D�x� �30�

s a linear equation that can be solved directly for D�x�. The linearityollows from the fact that we can estimate the phase field directly ando not consider it to be an unknown quantity.

By subdividing the earth model into a set of grid blocks, we caniscretize the integral 28. Given a set of phase estimates derivedrom field data, a set of data constraint equations will result. The re-

ulting equations for the flow properties might be written in the formf

� � Gy , �31�

here � is a vector of phase estimates obtained from the deforma-ion data; G is a coefficient matrix, which contains the trajectoryengths in each grid block of the reservoir model; and y is the vectorf flow property model parameters. They will consist of estimates ofD in each grid block. If it is possible to estimate Ce for the region ofnterest, one then can solve for the effective permeability in each cellf the reservoir model.

Because the phase estimates � and the matrix coefficients of Gontain errors, we do not solve equation 31 directly. Rather, we usehe method of least squares to estimate the model parameters, theomponents of the vector y �Lawson and Hanson, 1974�, by mini-izing the sum of squares of the residuals. Because the equations

ould be inconsistent and underdetermined, we also incorporate reg-larization or penalty terms into the inversion �Parker, 1994�. Theseenalty terms consist of a norm penalty, which biases the solution to-ard a given initial model y0, and a roughness penalty, which favorsodels that are smoothly varying. The penalty terms are quadratic

orms defined over the space of model parameters.Thus, we minimize the sum

R � � � Gy 2 � Wn y � y0 2 � Wr Dy 2, �32�

here z 2 signifies the L2 norm of the vector z, Wn and Wr are scalaroefficients controlling the importance of the penalty terms in rela-ion to the importance of fitting the data, and D is a matrix that ap-roximates a differencing operator. Because the quantity R is a qua-ratic function of the model parameters, the condition for it to be ainimum is a linear system of equations. We solve this system of

quations using the iterative solver first proposed by Paige and Saun-ers �1982�.

APPLICATIONS

umerical illustration

As noted above, there are several steps in the estimation proce-ure, each involving some degree of approximation. For example,he phase field, ��x�, is inferred from the pressure estimates, whichre derived from the deformation data. To illustrate the steps in thispproach, and to examine how well the technique works under fa-orable conditions, we consider a set of synthetic deformation val-es. The test case was designed to correspond roughly to the actualeld case that we discuss below.A reference permeability model, shown in Figure 1a, was used to

imulate the pressure response resulting from the initiation of fluidnjection. �The effective permeability estimates produced by an in-ersion algorithm will be shown in Figure 1b.� As in the field case,he injection started suddenly and continued for more than twoears. For this test, we modeled the flow rate using a step function.he reservoir model consisted of a 20-meter-thick layer, lying at aepth of 2000 meters. The numerical reservoir simulator TOUGH2Pruess et al., 1999� was used to compute the evolution of pressureithin the reservoir resulting from the fluid injection. The code

olves the pressure equation 12 numerically, using an integral finite-ifferences technique �Pruess et al., 1999�.

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The pressure variations within the reservoir are used to computeeservoir volume change assuming a linear poroelastic constitutiveodel �Rice and Cleary, 1976; Segall, 1985�. Thus, the volume

hanges v f in the reservoir are given by

v f � ��1� p , �33�

here � p are the pressure changes; and � �1 is the inverse of the co-fficient matrix � , which is given in Appendix A �equation A-3�. Inalculating the volume change within the reservoir, we can accountor stress transmission within the overburden. This is done for a ho-ogeneous poroelastic overburden using the method described inegall �1985�. More complicated models of the overburden can be

reated using more sophisticated numerical codes �Vasco, Karasaki,nd Doughty, 2000�. However, factors such as layering do not influ-nce significantly the modeling of deformation resulting from vol-me change at depth �Battaglia and Segall, 2004�.

The volume change within the reservoir was used to compute theeformation within the homogeneous overburden using a generali-ation of Maruyama’s �1964� method �Vasco et al., 1988; Vasco,

Distance east (km)

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Log permeability multiplier

Log permeability multiplier

Reference model

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3000

12

3

3000

–0.5 1.5

–0.5 1.5

3 14

5200 7400 9600 11,800 14,000

Inversion estimates

a)

b)

igure 1. �a� Spatial permeability variation used to generate a set ofurface displacement values. The grayscale denotes the logarithm ofermeability variations within the reservoir model. �b� Logarithm ofermeability multipliers resulting from the inversion of the phasealues. The permeability estimates are obtained by solving the linearystem of equations defined by the necessary equations for a mini-um of R in equation 32.

arasaki, and Doughty, 2000�. In particular, we computed the rangehange, the change in distance to a hypothetical point in space, asiscussed in Vasco et al. �2002� and Vasco and Ferretti �2005�.ange change is measured by analyzing radar reflection data fromrbiting satellites and provides a very accurate measure of surfaceisplacement at high spatial densities and over large regions of thearth �Massonnet and Feigl, 1998; Burgmann et al., 2000; Ferretti etl., 2000; Ferretti et al., 2001; Colesanti et al., 2003�. For the numeri-al illustration, we calculated range change on a 40-by-40 grid ofalues at the surface for 14 irregularly spaced time intervals �24, 59,4, 129, 164, 199, 269, 304, 409, 549, 584, 689, 724, and 864 days�.he time intervals correspond to those in the actual set of range-hange observations described below. The range-change values forour of the intervals are shown in Figure 2. Note the smoothly vary-ng changes, with a peak range change of approximately 1.0 cm.

The first step in the inversion procedure involves estimating theolume change within the reservoir from the range-change observa-ions. This step is discussed extensively in other papers �Vasco et al.,988; Vasco et al., 1998; Mossop and Segall, 1999; Vasco, Karasaki,nd Doughty, 2000; Du and Olson, 2001; Vasco et al., 2001; Vasco etl., 2002; Vasco and Ferretti, 2005� and is outlined in Appendix A. Itequires the solution of a linear inverse problem �equation A-1�, inhich the unknown parameters are grid-block fractional volume

hanges, and the observations are range-change data.Arow of the coefficient matrix � � relating the observations to theodel parameters consists of the response of the overburden to a

ractional change in the volume of a grid block. The grid blocks aredentical to those used in the forward calculation, with a thickness of0 m. The entries of depend on the spatial distribution of elasticroperties of the overburden. For this illustration, we assume a ho-ogeneous elastic half-space, which is characterized by its Pois-

on’s ratio �assumed to be 0.25�. Details of the computation are dis-ussed in Vasco et al. �1988, 2002� and Vasco and Ferretti �2005�.

Distance east (km)

12.0

3.03.0

1 cm

1 cm

1 cm

14.0 Distance east (km)

12.0

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3.03.0 14.0Distance east (km)3.0 14.0Distance east (km)

Distancenorth

(km)

Distancenorth

(km)

24 days 128 days

408 days 842 days

igure 2. Four snapshots of range change �24, 128, 408, and42 days� induced by pressure change within the reservoir. Filledquares indicate a decrease in range, the distance between the pointn the surface and a hypothetical point in space. The referencequare at the top of each part signifies a change of 1.00 cm.

pocshalcvor

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ce

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a

Ffij

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Estimating permeability from quasi-static deformation O43

Next the estimated volume changes are mapped into reservoirressure changes. This requires estimates of mechanical propertiesf the reservoir rock. In particular, we require the bulk modulus, asontained in � �1 in equation 33, throughout the reservoir. Wehould note that, for the arrival-time approach taken here, we do notave to take this step. Rather, we can work directly with the fraction-l volume change in estimating the arrival time. As long as there is ainear relationship between the volume change and the pressurehange, and the relationship does not contain time derivatives, as in aiscous medium, the arrival-time estimates will be identical. This isne advantage of working with a kinematic or traveltime approach,ather than working with amplitudes themselves.

The estimated pressure changes for four time intervals �24, 128,08, and 842 days� are shown in Figure 3. The peak pressure changes centered on the injection well and decays rapidly with distance.he pressure change is somewhat asymmetric, reflecting the hetero-eneity within the reservoir. To emphasize the propagation of theransient pressure change, we estimated the time derivative of theressure from the sequence of 14 time intervals. The resulting timeerivatives, normalized by the peak values in each grid block, arehown in Figure 4. One can observe clearly the propagation of theeak pressure derivative away from the injection well as a functionf time.

From the pressure derivatives, we estimate the arrival time of theeak over the simulation grid. This means that, using the pressureime series for each grid block, we estimate the time derivative andhe time at which the derivative reaches a peak value. The resultingistribution of sampled arrival times is shown in Figure 5. The distri-ution of traveltimes is not symmetric about the injection well, an in-ication of the heterogeneous propagation of the pressure distur-ance. The pressure arrival times, shown in Figure 5, form the basicata for an inversion for flow properties. They are used to estimatehe phase field and thus compute the elements of the vector � inquation 31. Furthermore, as noted above, the phase-field estimatesight be used to define the trajectories Xi in the integral 28 or its dis-

Distancenorth

(m)

Pressure change (MPa)

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30003000 5200 7400 9600 11,800 14,000

0 50 Pressure change (MPa)

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0 50

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(m)

Pressure change (MPa)

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0 50 Pressure change (MPa)

12,000

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30003000 5200 7400 9600 11,800 14,000

0 50

24 days 128 days

408 days 842 days

igure 3. Estimates of pressure change within the reservoir resultingrom an inversion of range-change values in Figure 2. The grayscalendicates changes in pressure from the background values. The in-ection well is indicated by the unfilled star in each part.

rete equivalent. The trajectories are used to calculate the matrix co-fficients of G in equation 31, resulting in a linear inverse problem.

The trajectories, or raypaths, computed from the phase field alsore shown in Figure 5. The regularized linear inverse problem entailsinimizing the quadratic form R, as defined in the expression 32.he linear system of equations, comprising the necessary equations

or a minimum of R, is solved using the iterative linear solver LSQRPaige and Saunders, 1982�. Note that we can recover only the flowroperties in regions that contain sampled phase data, the convexull of the points in Figure 5. We have no way of knowing how fluid-ressure changes propagate outside the region containing arrival-ime observations.

The effective permeability estimates produced by the inversionlgorithm are shown in Figure 1b. The model contains the large-

Distancenorth

(m)

Normalized derivative

12,000

10,200

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6600

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30003000 5200 7400 9600 11,800 14,000

0 1.0

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(m)

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(m)

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30003000 5200 7400 9600 11,800 14,000

Normalized derivative0 1.0

Normalized derivative0 1.0

Normalized derivative0 1.0

igure 4. Time derivatives of estimates of pressure change. The de-ivatives are normalized by the peak value attained in each gridlock.

Distance east (m)3000 5200 7400 9600 11,800 14,000

12,000

10,200

8400

6600

4800

3000

Distancenorth

(m)

igure 5. Solid squares denote phase or arrival-time estimates usedn the inversion for permeability. Solid curves represent trajectoriessed in the inversion. The trajectories are calculated from the phaseeld, as indicated in equation 29.

slwsietwwi

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wWptstcrofBmm

Af

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iNdgtits

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dmccpe2

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Fadrw

O44 Vasco et al.

cale features, which are present in the reference model. In particu-ar, there are high permeabilities to the north and west of the injectionell and low permeabilities to the southeast. The overall pattern is a

moothed version of the reference model �Figure 1a�, reflecting thenfluence of the roughness and norm penalties. Because of the limit-d extent of the phase data, it is not possible to estimate flow proper-ies outside a region surrounding the injection well. Stated anotheray, it is not possible to infer flow properties beyond the region overhich the pressure changes have propagated during the observation

nterval �0–864 days�.Another limiting factor is the decay of pressure with distance: Af-

er the disturbance has propagated a certain distance from the well,he changes will be small, producing no observable deformation athe surface. The sampling shown in Figure 5 identifies the regionver which estimates are reliable. This region corresponds roughlyo the area of significant permeability deviations from the back-round model �Figure 1b�.

The regularization penalty terms in R �equation 32� will trade offith the fit to arrival times. By increasing the regularization weights

n and Wr, we can degrade the arrival-time match. Thus, after com-leting the inversion, it is important to examine the fit to arrival timeso ensure that the times were not overfit or underfit. The match ishown in Figure 6 in terms of deviations from a reference arrivalime, calculated using a homogeneous starting model. In general, thealculated deviations from reference predictions appear to match theeference deviations rather well. The inverse problem is linear, andne can compute the model parameter resolution matrix as part of aormal model assessment �Aki and Richards, 1980; Parker, 1994�.ecause the focus of this study is to introduce an approach for esti-ating flow properties from deformation data, we do not computeodel parameter resolution estimates.

Observed deviation (square-root days)

0.00

–0.50

–1.00

–1.50

–2.00

–2.50–2.50 –2.00 –1.50 –1.00 –0.50 0.00

Calculateddeviation(square-rootdays)

igure 6. Observed deviation of traveltimes from those predicted byuniform permeability model, the background model. The observedeviations are plotted against deviations predicted by the inversionesult. For perfect agreement, the observed and predicted deviationsould plot along the 45° line shown in the figure.

pplication to permanent scatterer observationsrom the Krechba field, Algeria

In this section, we describe an application of the methodology toange-change data associated with CO2 injection in the Krechbaeld, Algeria. The Krechba, Teg, and Reg are gas fields distributedlong a north-northwest-trending anticline. The producing layer10.2 is a roughly 20-meter-thick layer of quartzose, fine-grained

andstone of Lower Carboniferous age. The production zone is over-ain by at least one kilometer of dark-gray to black, subfissile mud-tones. Above the mudstone is another kilometer of interbeddedudstones and sandstones.The gas produced from these fields contains a high mole fraction

f CO2, which must be removed and disposed of for economic andnvironmental reasons. The CO2 gas is separated from the hydrocar-ons and reinjected into the reservoir at three nearby wells, KB-501,B-502, and KB-503. The CO2 injection wells are in the same for-ation as the producers. The horizontal wells are located on theanks of the anticline, at depths of roughly two kilometers. The wellB-501, which we consider in this study, lies roughly 8 km to the

ast-northeast of the production wells, about 100 m lower in eleva-ion.

Mud losses during the drilling of numerous wells indicate flownto pre-existing fractures, dipping at approximately 81°, striking43°W. The direction is subparallel to the regional stress directionetermined for the Reg and Teg fields and observed throughout Al-eria. The stress directions in the gas fields are remarkably consis-ent, striking 315°, a sign that the horizontal stress field is highly an-sotropic. Breaks in the topography of the structure map of the top ofhe reservoir layer, as determined by seismic imaging, indicate pos-ible faults on the flanks of the anticline, trending north-northwest.

As part of a CO2 sequestration research-and-development pro-ram, Berkeley Laboratory and Tele-Rilevamento Europa �TRE�xplored the use of Interferometric Synthetic Aperture Radar �In-AR� data for monitoring CO2 injections. One goal of this work was

o identify flow paths at depth and geologic features controlling flow.he injection of CO2 into the water column induces multiphase flowecause the reservoir is initially water filled. At reservoir pressure,he CO2 behaves as a supercritical fluid, with a viscosity and densityomewhat different from water. The well-head injection pressureas variable; on average a pressure of 15 MPa was maintained after

he start of pumping, with peaks approaching 18 MPa early in thenjection and about seven months into the injection. At these pres-ures, supercritical CO2 should have a density greater than 0.85rams/cm3.In the region of interest, the reservoir does not vary significantly in

epth, and the density differences do not impact the flow. Further-ore, the advective flow almost certainly dominates the geochemi-

al changes at depth for the two to three years of injection that weonsidered. We model the CO2-water system as an equivalent singlehase for the pressure calculations, which should be acceptable giv-n the reservoir pressure conditions and time interval �Kumar et al.,005�.

This study utilized satellite radar images of the European Spacegency �ESA� Envisat archive from July 12, 2003, through March9, 2007. Two satellite tracks, number 65 and number 294, surveyedhe region during the CO2 injection, containing 26 and 18 images, re-pectively. The data were processed using a permanent scattererechnique developed by Politecnico di Milano and TRE �Ferretti etl., 2000, 2001; Colesanti et al., 2003�. In this approach, stable scat-

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Estimating permeability from quasi-static deformation O45

erers are identified using a statistically based analysis of the phasend amplitude characteristics of energy scattered from the earth’surface. A subset of scatterers is identified as permanent scattererandidates and used to estimate atmospheric and orbital errors.

The atmospheric and orbital biases in the signal phase are correct-d for, and estimates of range change are derived from the phasehifts between pairs of back-scattered radar signals. The rangehange represents the change in distance from a point on the earth’surface and the nominal satellite orbit. Thus, range change is thehange in the length of a hypothetical line connecting the observa-ion location and the sensor location. Estimates of range change arenfluenced by variations in atmospheric moisture in space and time,orrections for topography, and land-surface changes such as grow-ng vegetation.

Given a sequence of images, one can estimate the average range-hange velocity over a given time interval, or the average change inistance to the satellite orbit over the given time interval. Similar toifferential Global Positioning System �DGPS� data, all InSAR ob-ervations actually are differential measurements with respect to aeference point, which is assumed to be motionless. For example, inigure 7 we plot the average range velocity for track 65 and track94, corresponding to the region around the CO2 injector KB-501. Inhis image, one can see the range change resulting from uplift gener-ted by the CO2 injection at the well. The quality of the data for thisegion is high because of the surface characteristics of the region; therea is primarily rock outcrops and desert with little shifting sand.he peak range velocity in Figure 7 is greater than 5 mm/year,hich is several times larger than the estimated standard deviation ofmm/year.Noise in the estimates is discernible if we compare the range ve-

ocity inferred for tracks 65 and 294. Each image contains notablecatter, and there are differences between the two range-velocity es-imates. It should be noted that the time span covered by the two dataets is not identical, and that the number of data available is less thanypically is used in permanent scatterer InSAR �Ferretti et al., 2000,001; Colesanti et al., 2003�. This could lead to differing atmospher-c corrections resulting in a long period spatial variation between thewo tracks. Furthermore, the angular view of the earth’s surface islightly different for the two data sets. The well location is situatedifferently in each track. The location of KB-501 is much closer tohe edge of the image in track 65.

In an effort to improve the signal-to-noise ratio, we stacked theange-change estimates for tracks 65 and 294, interpolating themnto a common time sampling. The time sampling was identical tohat in the numerical study, consisting of 14 time intervals �24, 59,4, 129, 164, 199, 269, 304, 409, 549, 584, 689, 724, and 864 days�.urthermore, we averaged the scatterers spatially, using a movingindow with a radius of 10 points. The resulting stacked and aver-

ged estimates are shown in Figure 8 for four time intervals �24, 128,08, and 842 days�. The range change clearly increases with time, tos much as a peak value of more than 1 cm, and appears to migrate tohe northwest.

The range-change data then were inverted for volume changeithin the reservoir, using the approach described in Vasco et al.

2002� and Vasco and Ferretti �2005�, and outlined inAppendix A. Inssence, the linear system A-1 is solved for the volume changes inhe reservoir. The Green’s function gi�xk,y� used in the inversion �seequation A-2� corresponded to a homogeneous half-space. Theverburden is composed of layers of mudstone and quartzose sand-

tone. Such layering does not influence significantly the estimates ofolume change �Battaglia and Segall, 2004�, and a half-space models thought to be an acceptable approximation.

Because the reservoir itself was two kilometers deep and varied inepth only by a few tens of meters in the region of interest, we ap-roximated it by a horizontal layer. Because we did not have any di-ect pressure measurements, there were only deformation data, u�t�n the inversion. Including roughness and model norm penalty termsegularized the inversion. Furthermore, volume changes were penal-zed on the basis of their distance from the injection well KB-501.

Track 65

Track 294

Distance east (km)

Distanc

eno

rth(km)

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eno

rth(km)

Range velocity (mm/year)

Distance east (km)

Range velocity (mm/year)

10.0

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0.0 2.0 4.0 6.0 8.0 10.0

5.0

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5.0

igure 7. Range velocities estimated from scenes associated withwo tracks �65 and 294� for two satellite passes. Dark shades indicaterange decrease resulting from uplift from CO2 injection at depth.he surface projection of the injection well is indicated by solid lines

n each part.

Totw

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uoipmmq

ttptbfttafireK

fgestpmetpei

Ffsi

FcMpp

Fe3b

O46 Vasco et al.

his means that volume changes near well KB-501 were favoredver more distant changes. Thus, only the well location contributedo the inversion result; no information about flow rates within theell was available.The volume change then was mapped into reservoir pressure

hange, based on equation A-5. Laboratory tests indicated that a val-

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Distance east (km) Distance east (km)

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igure 8. Range-change estimates obtained by resampling the valuesor tracks 65 and 294, averaging values using a moving window, andtacking the values for the two tracks. The surface projection of thenjection well is indicated by solid lines in each part.

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igure 9. Pressure estimates resulting from an inversion of range-hange data shown in Figure 8. The grayscale indicates pressures inPa, with dark tones indicating a pressure increase. The surface

rojection of the injection well is indicated by solid lines in eachart.

e of Ku of 10.0 GPa was representative of the mechanical behaviorf the overburden. The inferred reservoir pressure changes for timentervals of 24, 128, 408, and 842 days are shown in Figure 9. Theeak pressure change of approximately 20 MPa is compatible witheasurements of reservoir pressure from a downhole sensor. Theigration of pressure change to the northwest is evident from the se-

uence of snapshots.From the sequence of pressure changes, we estimate the phase or

raveltime of the disturbance resulting from the onset of CO2 injec-ion. Because the flow rate was nonuniform, there were severaleaks in the time series of the pressure derivative. We focused on theime associated with the first peak of pressure derivative in each gridlock. A more sophisticated approach, which we might attempt in auture study, involves deconvolving the flow-rate function from theime series of each grid block to produce the response to a step func-ion in flow rate. In Figure 10, we plot the normalized time derivativessociated with the first peak after 305 days of pumping. From thisgure, it is evident that the pressure variation has propagated moreapidly to the northwest. In Figure 11a, the square root of the estimat-d arrival time is plotted for the region surrounding the injection wellB-501.The arrival-time values form the basic data set for the inversion

or flow properties. The estimates were constructed for a 40-by-40rid of cells defining the reservoir model. However, it is possible tostimate arrival times only for those grid blocks to which the pres-ure change had propagated during the observation interval. The ac-ive grid blocks are shown in Figure 11b, in which the block size isroportional to the square root of the arrival time. As noted in theethodology section, the phase field defines the trajectories X�l�, as

vident in equation 29. The trajectories associated with the phase es-imates also are shown in Figure 11b, as curves connecting the phaseoints to the injection well. The trajectories are used to construct co-fficients of the matrix G, which follow from a discretization of thentegral 28 to a summation. The approach is identical to that used in

Distancenorth

(km)

Normalized time derivative0

1 2 3 4 5 6 7 8 9 10

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2

1000igure 10. The normalized time derivative calculated from pressurestimates. The derivative is calculated for the pressure field after05 days of injection. The derivative values have been normalizedy the peak pressure in each grid block.

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Fprlc

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Estimating permeability from quasi-static deformation O47

raveltime tomography �Aki and Richards, 1980�. However, the in-erse problem is linear because we have used the phase-field esti-ates to calculate the trajectories directly.A regularized inversion of the linear system, equation 31, equiva-

ent to the necessary equations for a minimum of R in equation 32,as used to estimate the permeability variations in the region sur-

ounding the pumping well �Figure 12�. The coefficients Wn and Wr

ere found by trial and error, the result of some trial inversions. Theoal was to find a smooth model that fit the estimated arrival timesithin the expected error. The permeability multipliers produced by

he inversion algorithm suggest a linear, high-permeability zone cut-ing across the injection well and trending northwest. This feature isompatible with the migration of deformation to the northwest,hich is notable in Figure 8. The orientation of the high-permeabili-

y zone is parallel to the trend of the structural anticline that defineshe field.

Furthermore, the location of the high-permeability feature corre-ates with a break in topography, which suggests a fault cutting the

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0 30Square root of arrival time (days)

1000 2800 4600 6400 8200 10,000

Distance east (m)

Distanc

eno

rth(km)

a)

b)

igure 11. �a� Arrival time of the pressure transient, calculated fromressure derivative estimates. �b� Solid squares denote phase or ar-ival-time estimates used in the inversion for permeability. Solidines represent trajectories used in the inversion. The trajectories arealculated from the phase field, as indicated in equation 29.

astern side of the anticline. The dashed line in Figure 12 indicateshe inferred fault. The fault could extend southward; it is difficult toonstrain the trace from the seismic structure map. Thus, it seemshat the flow is controlled by a fault, and the permeability model is inccordance with the geologic structure of the reservoir. As statedbove, we can estimate permeability only in areas that contain arriv-l-time observations, indicated by the filled squares in Figure 11b.

Next we examine the fit to the arrival-time data provided by theermeability model �Figure 13�. Specifically, Figure 13 displays thet to the phase data, which were derived from the deformationrange-change� measurements. Thus, the “data” actually are esti-ates produced first by inverting the range change for pressure

hange and then by estimating the arrival time from the time-varying

Distance east (m)

Log permeability deviation

1000 10,000

–1.5 1.0

10,000

2000

Distancenorth(m)

igure 12. Logarithm of permeability multipliers resulting from thenversion of phase values in Figure 11b. The surface projection of thenjection well is indicated by the solid line. The surface projection offault, inferred from seismic data, is indicated by the dashed line.

Observed deviation (square-root days)–6.00 –4.00 –2.00 0.00 2.00 4.00

4.00

2.00

0.00

–2.00

–4.00

–6.00

Calculateddeviation(square-rootdays)

igure 13. Observed deviation of traveltimes from those predictedy a uniform permeability model, the background model. The ob-erved deviations are plotted against deviations predicted by the in-ersion result. For perfect agreement, the observed and predicted de-iations would plot along the 45° line shown in the figure.

ptebt

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O48 Vasco et al.

ressures. Each inversion will introduce some level of approxima-ion and bias caused by the inclusion of regularization terms and byrrors in the actual measurements. Overall, the traveltimes predictedy the permeability model appear to agree with the observed travel-imes.

Finally, we consider the influence that variations in the range-hange data have on permeability estimates. In particular, there areifferences in the range-change velocity estimates for tracks 65 and94, which are visible in Figure 7. To examine the stability of the per-eability estimates, we conducted an inversion in which data was

sed only from track 65. The resulting phase estimates are shown in

Track 65

Track 294

Distanc

eno

rth(km)

10

2

1

Square root of arrival time (days)0 30

Distance east (km)

Distance east (km)

Distanc

eno

rth(km)

Square root of arrival time (days)

10

21 10

0 30

a)

b)

igure 14. �a� Phase changes associated with an inversion of range-hange data from track 65. �b� Phase changes associated with an in-ersion of range-change data from track 294.

art a, Figure 14a. The general features of these estimates—thesymmetric distribution of the phase, the faster propagation to theorthwest, and the overall extent of the contours—agree with the av-raged range-change estimates �Figure 11a�. Furthermore, the phasestimates from track 65 are in qualitative agreement with estimatesrom track 294 �Figure 14b�.

However, there are some differences in detail among the variousstimates. An inversion of the phase data from tracks 65 and 294 re-ults in the permeability multipliers shown in Figure 15. Again, the

Track 65

Track 294

Distance east (km)

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eno

rth(km)

Log permeability multiplier

10,000

20001000 10,000

–1.5 1.0

Distance east (km)

Distanc

eno

rth(km)

Log permeability multiplier

10,000

20001000 10,000

–1.5 1.0

a)

b)

igure 15. �a� Permeability deviations from the uniform backgroundodel obtained from an inversion of phase changes associated with

rack 65 �see Figure 14�. �b� Permeability deviations from the uni-orm background model obtained from an inversion of phase chang-s associated with track 294.

gwodawrls

mhvptsptIccutA

erTtflFmta

ftslfswtas

oisStscstef

mcza

srtos�ctsIpcm

ocmdv

E

doooo

w

fg

rat

w

Estimating permeability from quasi-static deformation O49

eneral features of the permeability estimates, such as the north-est-trending linear, high-permeability anomaly and the magnitudef the deviations, are the same for the two inversions. The primaryifferences in the results are the southeast extent of the high-perme-bility trend. There also is a slight shift of the anomalies to the south-est in the inversion of data from track 294, relative to the inversion

esults from track 65. This shift might result from use of the sameook vector for the two inversions. In reality, the satellites were inomewhat different positions for the two tracks.

CONCLUSIONS

Fluid-pressure changes within a reservoir or aquifer can produceeasurable deformation within the overburden. Such deformation

as been observed in both geodetic data and time-lapse seismic sur-eys. The resulting deformation can be used to infer volume andressure changes within the reservoir. Although it is possible to maphe estimated pressure changes directly into flow properties, in thistudy we use an alternative approach on the basis of the idea of aressure “arrival time.” The primary advantage of this approach ishat it is not sensitive to mechanical properties within the reservoir.n other words, if the reservoir behaves elastically over each time in-rement, the arrival-time estimate can be derived from the volumehange within the reservoir and does not require mapping the vol-me change into pressure change. Thus, we do not need to estimatehe value of Ku, and its variation, within the reservoir �see equation-6�.An additional advantage of this approach is that the phase field is

stimated from the deformation data and can be used to compute theaypaths or trajectories for the tomographic traveltime inversion.his, in effect, linearizes the inverse problems; and one then has all

he tools of linear inverse theory available. Thus, the estimation ofow properties, on the basis of deformation data, is a linear problem.urthermore, one can compute useful measures for model assess-ent, such as the model parameter resolution matrix. We will illus-

rate this in a follow-up paper on the analysis of range change associ-ted with injection into well KB-502.

Application of the technique to InSAR range-change estimatesrom the Krechba field inAlgeria indicates that the approach is prac-ical. The linear, high-permeability feature produced by the inver-ion is in accordance with the local and regional geology. Specifical-y, the feature lies at the southern extension of a northwest-trendingault on the flank of the structural anticline defining the field. Thisuggests that a fault is controlling the flow of CO2 from the injectionell. The possibility exists that the fault is deforming in response to

he injected fluid. This mechanism might involve changes in perme-bility resulting from deformation, a subject not addressed in thistudy.

One limitation of this approach is the need for adequate samplingf the deformation field in time and space. In the near future, a grow-ng number of satellite platforms, mounting high-resolution radarensors, will become available �eg., Radarsat-2, Terrasar-X, Cosmo-kymed�, providing additional coverage. Dense spatial sampling in

hree dimensions also can be obtained through the use of time-lapseeismic data, although at a higher cost. Permanent seafloor sensorsould be used to improve the sampling in time. In addition, it is pos-ible to apply the arrival-time methodology to pressure estimates ob-ained directly from time-lapse seismic data, bypassing entirely thestimation of deformation. An extension of this work would allowor permeability changes to accompany deformation; that is, there

ight be coupling between deformation and flow properties. Suchoupling might occur when the CO2 is channeled into a narrow faultone. We plan on considering pressure-dependent flow propertiesnd their estimation in a future study.

ACKNOWLEDGMENTS

This work was supported by theAssistant Secretary, Office of Ba-ic Energy Sciences, and the GEOSEQ project for theAssistant Sec-etary for Fossil Energy, Office of Coal and Power Systems, throughhe National Energy Technology Laboratory of the U. S. Departmentf Energy under contract DE-AC02-05CH11231. The permanentcatterer �PS� data were processed by Tele-Rilevamento EuropaTRE�, a spin-off company of Politecnico di Milano, worldwide ex-lusive licensee of the Polimi PS TechniqueTM. The authors wish tohank ESA for all satellite data used in this study and the entire TREtaff for supporting the SAR data processing. The In Salah CO2 Jointndustry Project �BP, StatoilHydro, and Sonatrach� is thanked forrovision of production, injection, and subsurface data. Additionalomputations were carried out at the Center for Computational Seis-ology, Berkeley Laboratory.

APPENDIX A

DEFORMATION IN THE ELASTIC OVERBURDEN

When porosity changes occur, induced by fluid pressure buildupr decay in a reservoir or aquifer, there is a corresponding volumehange. The volume change induces deformation in the surroundingaterial, which is transmitted through the subsurface. In thisAppen-

ix, we note how such deformation might be used to estimate reser-oir pressure changes.

stimation of volume change

The first step in the estimation procedure is to use the deformationata to infer variations in pressure. This aspect is discussed thor-ughly by Vasco et al. �2001�. We state only the final result, a systemf linear equations relating volume changes within each grid blockf a reservoir model v f�t� to pressure changes � p�t� and deformationbservations u�t�. Thus,

� u�t�� p�t� � � �

��v f�t� �A-1�

here coefficients of the matrices and � are given by

ijk � �Vj

gi�xk,y�dV , �A-2�

or observation point xk, deformation components i and reservoirrid block j, and

� ijk � Ce�1�� jk �

B

��

Vj

Gii�xk,y�dV� , �A-3�

espectively; where B is Skempton’s pore-pressure coefficient �Ricend Cleary, 1976; Wang and Kuempel, 2003�, � is the density, andhere is no summation over the repeated index i.

The integrand in expression A-2, gi�xk,y�, is the Green’s function,hich is the point response of the elastic medium at location x .

k

Ta

cg

oksriK

Mp

aflso

w

�Cc

Nuct

A

A

A

A

B

B

B

B

B

B

B

B

B

B

B

B

C

C

C

C

C

C

C

C

C

dD

D

D

E

E

F

F

F

O50 Vasco et al.

herefore, it is medium dependent, as discussed in Vasco, Karasaki,nd Doughty �2000�. The integral is evaluated over the volume of the

j-th grid block, Vj. The kernel Gii in the integral in expression A-3 isonstructed from the Green’s function solution gi, as described in Se-all �1985�,

Gii � �2� � ���gi

�xi. �A-4�

Note that one could allow the poroelastic coefficients Ce�1 to vary

ver the aquifer or reservoir, assuming that such information isnown. The system of equations A-1 typically is unstable with re-pect to perturbations, numerical or otherwise; and some form ofegularization is required. Thus, the system is solved using a regular-zed least-squares algorithm, as discussed in Vasco, Karasaki, andeers �2000� and Vasco et al. �2001�.

apping estimates of volume change into changes inressure and overburden stress

Once estimates of the volume change within the reservoir arevailable, by solving the system of equations A-1, we map them intouid pressure and overburden stress changes. The mapping is de-cribed in more detail in Vasco et al. �2001� and is based on the workf Segall �1985�. The fluid pressure change is given by

� p�x,t� � C�v f�x,t� �B2

3��

Vi�1

3Gii�x,y�v f�y,t�dV ,

�A-5�

here

C� �B2

���

2

3

�1 � �d��1 � �u���u � �d�

� 3Ku� , �A-6�

u and � d are the undrained and drained Poisson’s ratios �Rice andleary, 1976�, Ku is the undrained bulk modulus, and the diagonalomponents of the overburden stress are given by

� ii �B

��

V

Gii�x,y�v f�y,t�dV �BKu

�v f�x,t� . �A-7�

ote that all mappings are linear, as is the inverse problem for vol-me change, equation A-1. The second term in equation A-5 ac-ounts for propagation of stress within the overburden in the calcula-ion of reservoir pressure change.

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