NOTICE: this is the author s version of a work that was accepted for publication in Journal of Petroleum Science and Engineering Changes resulting

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1 NOTICE: this is the author s version of a work that was aepted for publiation in Journal of Petroleum Siene and Engineering Changes resulting from the publishing proess, suh as peer review, editing, orretions, strutural formatting, and other quality ontrol mehanisms may not be refleted in this doument. Changes may have been made to this work sine it was submitted for publiation. A definitive version was subsequently published in Journal of Petroleum Siene and Engineering 80,4-5, 0 DOI:

2 Poroelasti Model for Indued Stresses and Deformations in Hydroarbon and Geothermal Reservoirs Chen, Z.R. * CSIRO Earth Siene and Resoure Engineering, Bayview Avenue, Clayton, VIC 368, Australia Abstrat Fluid migration and heat transport result in hanges in pore pressure and temperature within a reservoir, whih an indue stresses and deformations in the reservoir and the bounding rok system through poroelasti and thermoelasti ouplings. Analytial and semi-analytial solutions for investigating the indued stresses and deformations are therefore extremely useful when applied to both prediting and monitoring the reservoir volume hanges and assoiated subsurfae and surfae deformation. The poroelasti and thermoelasti eigenstrains are used here to haraterise the pore pressure hange and temperature variation, respetively, in the reservoir. The indued stresses and deformations are then obtained by using the single- and double-inlusion models, and are expressed in terms of the Galerkin vetor stress funtion, whih is related to the orresponding eigentrains in a straightforward way. The differene in mehanial properties between the reservoir and the bounding roks is aounted for using the theory of inhomogeneous inlusions. The effet of the reservoir shape, elasti properties, and the distribution of pore pressure hange within the reservoir on the surfae deformation has been investigated. The magnitude and pattern of the indued surfae tilt have been ompared with those produed by a hydrauli frature. The analytial expressions obtained here for the displaement and tilt fields an serve as a useful forward model for monitoring and mapping hydrauli fratures, subsurfae fluid migration and heat transport assoiated with injetion or prodution of fluid into or from a reservoir by surfae deformation-based monitoring tehniques, suh as tiltmeter monitoring. Keywords: Reservoir; Poroelastiity; Thermoelastiity; Indued deformation; Hydrauli fraturing; Monitoring; Tiltmeter * Corresponding author. Tel.: ; fax: mail address: zuorong.hen@siro.au (Z.R. Chen.

3 Introdution Fluid prodution or injetion in a reservoir results in hanges in pore pressure and fluid mass ontent within the reservoir, whih generally indues in situ stress hanges within the reservoir and the bounding roks through poroelasti oupling. The pressure hange in the reservoir results in reservoir swelling or ompation, produing ground surfae uplift or subsidene. Heat transport and temperature variation in the rok system ours during the injetion of a ool fluid and the extration of heat from a geothermal reservoir, whih indues deformation and stress hanges in the rok system through thermoelasti oupling. Various phenomena suh as reservoir ompation, land subsidene, landslides and earthquakes have been observed in a variety of field loations (Geertsma, 973; Massonnet et al., 998; Roeloffs, 988; Segall, 985 and an be attributed to poroelasti and/or themoelasti stress hange assoiated with fluid injetion or prodution. Another important phenomenon involving fluid injetion and assoiated poroelasti swelling is produed by the pressure transient effet arising from hydrauli fraturing (Palmer, 990. Hydrauli fraturing is a powerful tehnology mainly used in the petroleum industry to stimulate reservoirs to enhane oil and/or gas prodution. During a standard treatment, the appropriate amounts of fraturing fluid and proppant are blended and pumped into a wellbore at high enough injetion rates and pressures to initiate and propagate a frature hydraulially in the diretion perpendiular to the diretion of the in situ minimum prinipal ompressive stress. During and after a hydrauli fraturing treatment, fraturing fluid leaks out of the frature through the frature surfaes into the reservoir, whih results in an inrease in pore pressure within the reservoir. The pore pressure inrease affets the in situ stress through poroelasti oupling, ausing the reservoir to expand. Pore pressures inrease muh more in the permeable reservoir than they do in the surrounding low permeability roks. This leads to a strain mismath, with the reservoir formation swelling more than the surrounding roks, and this strain mismath indues stress. For a high diffusivity reservoir, a pore pressure transient will propagate relatively fast and far from the frature. In some ases the reservoir swelling may produe a signifiant tilt signal, suh that the interpretation of the frature geometry from surfae tiltmeter 3

4 monitoring ould be in error if the reservoir swelling assoiated with the pore pressure transient is not aounted for (Palmer, 990. Therefore, for many industrial appliations, estimating or monitoring fluid flow, heat transport, and pressure and temperature hanges within the reservoir is ritial. Surfae deformation-based monitoring tehnologies suh as tiltmeter monitoring, global positioning system (GPS surveys, and interferometri syntheti aperture radar (InSAR have been suessfully applied to monitoring hydrauli fraturing, fluid flow or pressure hanges in the reservoir, and temperature hange in geothermal reservoirs (Carne and Fabriol, 999; Chen and Jeffrey, 009; Evans and Holzhausen, 983; Massonnet et al., 998; Palmer, 990; Vaso et al., 00. Muh effort has been devoted to prediting and understanding the fluid prodution- or injetion-indued stresses and deformations within the reservoir and the bounding rok system (Du and Olson, 00; Fokker and Orli, 006; Geertsma, 973; Morita et al., 989; Rudniki, 00; Safai and Pinder, 980; Segall, 985; Segall and Fitzgerald, 998; Soltanzadeh and Hawkes, 008; Soltanzadeh et al., 007. Some of the geomehanial models developed have been used as forward models in monitoring and mapping of fluid flow or pressure hange in reservoir (Du and Olson, 00; Geertsma, 973; Muntendam- Bos et al., 008; Vaso et al., 00; Vaso et al., 998. Beause they are relatively easy to implement, and are suited to and highly effiient for inverse analysis of parameter estimation, losed-form or semi-analytial forward models for prediting the deformations (uplifts and tilts indued by the pressure and temperature hanges are extremely useful. Remote monitoring of hydrauli fratures or subsurfae fluid migration and heat transport assoiated with oil extration, geothermal heat extration, ground-water use, and injetion of CO for sequestration requires effiient forward models for use in solving the inverse problem. The nuleus of strain onept has been used in establishing a forward model to relate the measured stress and deformation to reservoir deformation. With the help of the soalled nuleus-of-strain in the elasti half-spae, as introdued by Mindlin and Cheng (950b, Geertsma (973 developed a model to estimate the reservoir ompation and the aompanying subsidene assoiated with a reservoir pressure redution, assuming that both the reservoir and its surroundings are homogeneous, with idential mehanial properties and assuming the reservoir undergoes uniaxial strain. Adopting 4

5 the idea of Eshelby s inlusion theory (Eshelby, 957, Segall derived the transformational strain resulting from uniform fluid withdrawal, and alulated the stress and deformation (outside the reservoir resulting from the withdrawal of fluid from a permeable layer embedded in a fluid-infiltrated, impermeable half-spae for plane strain onditions (Segall, 985, and further for axisymmetri reservoir geometries (Segall, 99. The indued stress within the reservoir resulting from uniform pressure and temperature derease in the reservoir was investigated by Segall and Fitzgerald (998, treating the reservoir as an ellipsoidal inlusion embedded in a full spae with the same mehanial properties. Following the method of Segall (99, Du and Olson (00 onstruted a numerial model to predit surfae subsidene and reservoir ompation. Vaso et al. used Segall s method as a forward model for alulating the surfae deformation for use in analysing surfae tilt generated by fluid injetion and soil onsolidation (Vaso et al., 00; Vaso et al., 998. Generally the material properties of the reservoir and surrounding roks may be signifiantly different. The differenes in mehanial properties between them an potentially affet the stress and deformation in the reservoir and at the surfae. To aount for this effet, Rudniki (00 adopted the theory of inhomogeneities developed by Eshelby (957 to investigate the indued stress hanges within the reservoir, treating the reservoir as an elliptial inhomogeneity embedded in a full spae. Soltanzadeh et al. (007 solved the problem of a poroelasti inhomogeneity of elliptial ross-setion under plane strain onditions in a full spae. Morita et al. (989 provided a orretion fator method to determine subsidene, ompation, and indued stresses for a disk-shaped reservoir with elasti properties different from those of the surrounding rok. The orretion fators were alulated by using a finite-element model. Using the moduli perturbation method, Du et al. (009 proposed a semi-analytial method to predit the surfae deformation indued by the reservoir pressure or volume hanges in an inhomogeneous half-spae. The moduli perturbation method requires that the modulus ontrast should not be very large to guarantee a fast onvergene of the displaements to the true solution. When the depth is greater than the lateral extend and when the deformation being monitored is loated far from the ground surfae, a good approximation is to model 5

6 the reservoir as an inhomogeneity embedded in a full spae. However, the deformation field at or near the ground surfae is often needed in many important appliations suh as in surfae tiltmeter monitoring, whih is a ground surfae deformation-based tehnology for monitoring reservoir and hydrauli frature deformation. In this ase, the free boundary surfae has a signifiant effet on the near surfae stress and deformation fields, so the solution of an inhomogeneity embedded in a half spae is neessary. Following Geertsma s alulations (Geertsma, 973, Palmer (990 has investigated the reservoir expansion and aompanying uplift and tilt at the ground surfae aused by an inrease in reservoir pressure, assuming that the reservoir is under uniaxial strain and has the same mehanial properties as the surrounding rok. In this paper, the reservoir is modelled as an ellipsoidal, poroelasti inhomogeneity embedded in a semi-infinite elasti body. The poroelasti and thermoelasti eigenstrains are used to haraterise, respetively, the hanges in pore pressure (or equivalently pore fluid mass ontent and temperature within the reservoir using the single- or double-inlusion model. The differene in mehanial properties between the reservoir and the bounding roks is aounted for using the equivalent inlusion method. The indued stress and deformation fields are then obtained in a straightforward way by expressing them in terms of the Galerkin vetor stress funtion, whih is related to the orresponding eigentrains. The main purpose of this work is to investigate the indued deformations assoiated with fluid prodution or injetion in a reservoir, and to develop a forward model for monitoring and mapping of fluid flow, heat transport, pressure or temperature hange in the reservoir by using ground surfae deformation-based monitoring tehnologies.. Poroelasti and Thermoelasti Eigenstrains Figure. As shown in Figure, the reservoir is idealized as an ellipsoidal poroelasti inhomogeneity with semi-major axes a, a, and a 3 aligned with the oordinate axes 6

7 x, x, and x 3, embedded in a semi-infinite isotropi elasti body at a depth of below the free surfae. The elasti properties of the reservoir differ from those of surrounding rok. Assume that the pore pressure, fluid mass ontent, and temperature within the entire or part of the reservoir hange during fluid prodution or injetion, while outside the reservoir these values remain unhanged. The onstitutive equations for a linear isotropi thermo-poroelasti medium are (MTigue, 986; Rie and Cleary, 976, and ( ( ν ( ν α = σ σ kkδ + pδ + λ stδ, ( µ µ + ν µ + ν ( ν ( αρ0 3 m m0 = σ kk + p + 3φ 0 ( λ f λ s T, ( µ + ν B where are the strains in the solid, σ are the stresses, p is pore pressure, m is the mass of pore fluid per unit bulk volume ( m 0 in the referene state, ρ 0 is the density of pore fluid in the referene state, and T is the temperature hange. µ and ν are the K shear modulus and Poisson s ratio measured under drained onditions, α = K is the Biot pore-pressure oeffiient, B = φ K Ks K + K + K ( φ 0 f 0 s is Skempton s oeffiient, λ s is the oeffiient of linear thermal expansion for the solid, λ s is the seond thermal expansion oeffiient for the solid, λ f is the oeffiient of thermal expansion for the fluid, φ 0 is the referene porosity, and K, K s, and K f are the drained bulk modulus of porous material, the bulk modulus of the solid phase, and the bulk modulus of the fluid phase, respetively. s For quasi-stati, isothermal poroelasti problems, the deformation and pore pressure fields are oupled through the equilibrium equations for the solid (negleting body fores σ and the isothermal diffusion equation governing pore fluid flow x j = 0, (3 7

8 where the hydrauli diffusivity p is given by µ ( ν ( ν B( + ν ( ( ν Bα ν κ p = η, 3α where κ is permeability, and η is fluid visosity. 3 3 p σ kk + p = σ kk + p, (4 B t B The diffusion equation Eq. (4 an also be expressed in terms of the fluid mass ontent m as (Rie and Cleary, 976 p m m =. (5 t Generally, the equilibrium (Eq. (3 and diffusion (Eq. (4 or Eq. (5 equations have to be solved simultaneously for the pore pressure distribution p ( x,t or fluid mass ontent distribution, m ( x,t. Most omputations of the pressure or fluid mass ontent distributions have typially assumed that the rok behaves as a rigid matrix. Then the fluid diffuses through the rigid rok matrix, deoupled from the rok deformation, and the pressure or fluid mass ontent distribution is then obtained diretly by solving the diffusion equation subjet to the boundary and initial onditions posed in terms of pore pressure or fluid mass ontent (Detournay and Cheng, 99; Segall, 985. It is worth noting that the hydrauli diffusivity p generally is not a onstant but deformation dependent via the permeability whih is a funtion of the porosity. Therefore, the de-oupled treatment is only valid when the hydrauli diffusivity remains onstant or hanges only slightly throughout deformation. It is beyond the sope of the present work to solve for the pore-pressure or fluid mass ontent distribution within a reservoir. Rather, we will take them to be given, and solve for the deformations diretly as shown below, whih is enough to serve as a forward model in an inversion analysis to infer the fluid mass ontent, pressure and temperature hanges in reservoir from the surfae deformation measurements. 8

9 Aording to the definition of eigenstrain or stress-free transformation strain, the * eigenstrain due to the hanges in pore pressure and temperature an be obtained diretly by letting σ = 0 in Eq. ( as ( ν ( + * α = pδ + λ stδ. (6 µ ν Or, equivalently, the eigenstrain an be expressed in terms of a uniform hange in pore fluid mass ontent m as * B Bφ 0 = mδ + λ s ( λ f λ s Tδ. (7 3ρ0 ρ0. Inhomogeneous inlusions. The single-inhomogeneity problem Figure. Consider an inlusion embedded in an infinite homogeneous isotropi elasti medium D, as shown in Figure. The inlusion undergoes an eigenstrain. * The indued elasti fields aused by eigenstrain in the inlusion an be written as (Mura, 987 * where ( x = 0 u * ( x C ( x G ( x x = jlmn mn l dx, (8a i, { } x * ( x = C ( x G ( x x + G ( x x d mn mn ik, lj jk, li, (8b { pqmn mn kp, ql } * * ( x = C C ( x G ( x x dx ( x σ +, (8 for x D and C are the elasti moduli. The Green s funtion G ( x x is the displaement omponent in the x i diretion at point x when a unit body fore in the x j diretion is applied at point x in the infinite medium. Unless speifially indiated, the onventional summation onvention for the repeated indies is used, whereby repeated indies indiate summation over the values,, 3, 9

10 and indies preeded by a omma denote differentiation with respet to the Cartesian oordinates orresponding to the index following the omma. It has been shown by Eshelby (Eshelby, 957 that for an ellipsoidal inlusion with uniform eigenstrain the indued strain in the inlusion is uniform and an be expressed as ( * x = (9 S where S is alled Eshelby s tensor whih is a funtion of the inlusion shape and of elasti properties of the matrix (in the ase of an isotropi matrix, it is a funtion of Poisson s ratio only. S is symmetri with respet to i j and k l but not with respet to ( (, i.e., S = S ji = Slk S. Detailed losed-form expressions of Eshelby s tensor for various ellipsoidal inlusions suh as spherial, penny-shaped, and rod-shaped have been doumented by Mura (987, and by Kahanov et al. (003. The sub-domain is alled an inhomogeneity if it has elasti properties differing from those of the rest of the domain D (matrix. The inhomogeneity is alled an inhomogeneous inlusion when it undergoes its own eigenstrain. It was pointed out by Eshelby (957 that the problem of an ellipsoidal inhomogeneous inlusion an be transformed into an equivalent inlusion problem when the orret equivalent eigenstrain is hosen. Consider that an ellipsoidal domain with elasti moduli embedded in an infinite homogeneous medium with the elasti moduli * an eigenstrain. The indued stress an be found as σ * ( = in, * C σ = in D. C * C C undergoes In the equivalent inlusion method, the inhomogeneous inlusion is simulated by an inlusion in the homogeneous medium with equivalent eigenstrain indued stress an be obtained as ** ( σ = in, C ** (0. Then the 0

11 σ = in D. C Here, the eigenstrain is introdued as a fititious one for the simulation. The indued strain inside the inlusion, aording to Eq. (8b, is { } dx ( ( x = C ( x G ( x x + G ( x x mn mn ik, lj jk, li ( The equivaleny between Eqs. (0 and ( leads to * * ** ( = C ( σ = C in. (3 ** The equivalent eigenstrain an be obtained by solving the integral equation (Eq. * (3. For the ase of an uniform eigenstrain, the onsisteny onditions are {( } * ** * * mn mn mn mn mn C C S + C = C. (4 Components of Eshelby s tensor oupling an extension and a shear or one shear to another are all zero, so the equivalent shear strains an be found diretly from Eq. (4: µ = ( µ µ S + µ (,3,3,no sum over i, j = (5 The equivalent normal eigenstrain an be found through a simple matrix operation as where C = E ( + ν ( ν ν ν ν 33 = ν ν ν C C, (6 33 [( C S + C] ν ν, and ν S S S33 S = S S S33. S 33 S33 S3333 The onsisteny onditions (Eq. (4 is valid for an inlusion in an infinite body. Beause of the effet of the free boundary surfae, the indued strain from a onstant eigenstrain in the inlusion in a semi-infinite medium is not uniform any longer, whih an be expressed as ~ ( = ( x S x, (7

12 where ( x S ~ an be defined as a generalized Eshelby s tensor for the inlusion embedded in a semi-infinite medium. The generalized Eshelby s tensor depends not only on the shape of the inlusion and the elasti properties of the matrix but also on the position and the distane of the inlusion from the free surfae. Thus Eq. (4 needs to be modified so as to determine a orret equivalent eigenstrain, when dealing with the ellipsoidal inhomogeneous inlusion in a semi-infinite medium, by using the equivalent inlusion method {( ( x } ( x C C S % + C = C. (8 * ** * * mn mn mn mn mn ** In this ase, the equivalent eigenstrain mn is not uniform. Consider an inhomogeneous spherial inlusion whih undergoes a uniform dilatational eigenstrain = δ. The indued strain distributions along the x3 -axis are shown in Figure 3. The solid lines are the finite element modelling results, while the dashed lines are analytial results obtained with using the uniform equivalent eigenstrains determined by Eq. (4. First, it an be seen that the indued strains are not uniform inside the inlusion. While the effet of the free boundary surfae on the deformation field inside the inlusion beomes weak as the depth inreases, and nearly disappears when the depth of the inlusion is two times its radius. This is in agreement with Seo and Mura s results who ompared stress field obtained numerially for the ase of a homogeneous inlusion and by use of Green s funtion and elliptial integrals (Seo and Mura, 979. Seond, the nonuniform equivalent eigenstrains have less effet on the deformation field outside the inlusion than they do on deformation inside the inlusion, and the effet dereases as the depth inreases. So the effet of the free boundary surfae on the indued strain and stress in the inlusion is negligible when the depth of the inlusion to the free surfae is two times the harateristi size of the inlusion. In this S ~ x is well approximated by Eshelby s ase the generalized Eshelby s tensor ( tensor S auray., and thus Eq (4 is able to predit an equivalent eigenstrain with great

13 Figure 3.. The double-inhomogeneity model The single-inhomogeneity onfiguration an be used to model the elasti fields where the pore pressure and temperature hanges our within the entire reservoir. While, if only part of the reservoir (rather than the entire reservoir experienes hanges in pore pressure and temperature, the double-inhomogeneity model (Hori and Nematnasser, 993 is more useful. Figure 4. Consider a double-inhomogeneity model V = Λ +, onsisting of an ellipsoidal inlusion Λ whih ontains an ellipsoidal inhomogeneity, embedded in an infinite homogeneous elasti solid D. V (or denotes both the ellipsoidal region and its volume, depending on the ontext. The volume fration of the inhomogeneity relative to V is f = V. The elasti moduli of the inner inhomogeneity, annulus Λ = V, and the infinite solid D V are denoted by The inner inhomogeneity undergoes an eigenstrain C, C, and C, respetively.. This double-inhomogeneity V = Λ + is used to model the reservoir where pore pressure and temperature hanges our within the part of the reservoir only. As shown in Figure 4, using the equivalent homogeneous inlusion method, this double-inhogeneity problem, Figure 4(a, an be replaed by the equivalent double-inlusion problem, Figure 4(b, with proper hoie of eigenstrains of uniform elastiity C, respetively. ( and ( defined over the regions and Λ Applying the equivalent homogeneous inlusion method, the following onsisteny onditions are obtained: σ σ [ ] = C [ ( x ( x ] ( [ x x ] = C [ ( x ( x ] ( ( x C ( x ( x = ( x C ( ( =, for x in (9a, for x in Λ (9b 3

14 where the indued strains due to the eigenstrains are given by ( ( ( ( x Γ ( x x dx + Γ ( x x dx Γ ( x x dx = V where Γ ( x x = G ( x x + G ( x x [ ] C mn jm, ni im, nj., for x in V (0 Substitution of Eq. (0 into Eq. (9 leads to integral equations, whih an be solved for the loal elasti fields in V. One the equivalent eigenstrains ( and ( are determined from Eqs. (9a and (9b, the original double-inhomogeneity problem, Figure 4(a, an be replaed by the equivalent homogeneous inlusion problem, Figure 4(b, and the elasti fields outside the inlusion V an be obtained diretly by using the single-inlusion solution. Unlike the single-inlusion and single-inhomogeneity problems, the indued strain and stress fields inside the double inlusion generally are not uniform due to the presene of the annulus Λ = V, even if the eigenstrains is uniform. So the onsisteny onditions (Eqs. (9a and (9b need to be solved numerially for the loal elasti fields of the double inlusion. However, average field quantities taken over the double inlusion an be estimated analytially and expliitly. With this solution, the exat losed-form expressions for the far-field solution an be obtained. The average onsisteny onditions are (Appendix A ( V ( [( C C Smn + Cmn ] mn + ( C ( = C Smn Smn mn C Λ f f V ( C C ( S S ( mn V f V ( ( C C Smn ( Smn Smn + Cmn mn = C Λ Λ mn f mn +, (a (b 4

15 from whih the average equivalent eigenstrains and ( mn an be ( mn determined. Then the average equivalent eigenstrains taken over V are obtained by Λ ( ( f ( = f +. ( V Λ 3. Elasti fields expressed in terms of Galerkin vetors In addition to the Green s funtion method (Eq. (8, the indued elasti fields due to * the eigenstrains in the inlusion an be expressed in alternative form, in terms of Galerkin vetor stress funtions F as (Mindlin, 936; Yu and Sanday, 99 ( = ( ν Fi, jj Fk ki µ x (3a u i, ( = ( ν ( F i, kkj + Fj, kki Fk, k µ x (3b σ ( = νfk kmmδ Fk k ( v( Fi kkj Fj kki µ,, +, +, λ kkδ x (3 for x D, and the Galerkin vetor funtions are written as (Yu * where ( x = 0 and Sanday, 99 F where g ( x, x ( x ( ν [ ν ( x g ( x, x ( x g ( x x ] µ = x 4π d kk, (4 is a basi Galerkin vetor at point x due to a enter of dilatation at point x, and g ( x, x are basi Galerkin vetors at point x due to a double fore in the j diretion ( i = j or a double fore in the j diretion with moment in the diretion normal to the ox plane ( i j at point x. x i j The tilt (rotational deformation field, ω, an be expressed as ( = ( ( F, F, µω x ν (5 i kkj j kki The detailed expressions for various types of nulei of strain an be found in Mindlin (936 for the ase of full-spae solutions, Mindlin and Cheng (950a for half-spae solutions, and Yu and Sandy (99 for joined half-spaes solutions. 5

16 Aording to Eqs. (3 and (5, only high-order derivatives of the Galerkin vetors are involved in determining the elasti fields. Further, the sequene of integration and differentiation in Eq. (4 an be interhanged, thus for pratial appliations, the analytial expressions for the derivatives of the basi Galerkin vetors g ( x, x an be obtained in advane so as to improve the omputational effiieny in alulating the indued elasti fields. Then, the integration operation an be handled by subdividing the arbitrarily shaped inlusion into a number of small uboidal elements eah of whih has a uniform distribution of eigenstrains and superposition of solutions for eah element. While, in order to improve the numerial effiieny and auray, the disrete orrelation/onvolution and fast Fourier transform algorithm proposed by Liu and Wang (005 an be used to evaluate the elasti field inside and around the inlusion. For points x remote from the inlusion, the basi Galerkin vetors in Eq. (4 an be taken outside the sign of integration to obtain where F ( x µ [ ν g ( x x ( ( ( ] ( x dx g x x x d, kk, ν = x 4π (6 x is the referene point of the inlusion (the enter of an ellipsoidal inlusion, for example. Then, the far-field elasti fields an be obtained diretly by finding the high-order derivatives of the basi Galerkin vetors and substituting them into Eq. (6, and then into Eqs. (3 and (5. For a reservoir with a poroelasti eigenstrain, the Galerkin vetor an be obtained by substituting Eq. (6 into Eq. (6 as where ( ( ν F ( x α P = 6 ν (, (, δ 8π g x x g x x, (7 ( ν P = p x d is the total work assoiated with pore pressure hange. Equivalently, if the poroelasti eigenstrain expressed in terms of the fluid mass ontent hange (Eq. (7 is used, the Galerkin vetor an be expressed as F ( x µb M = 6 ν (, (, δ π ν ρ g x x g x x, (8 ( 0 6

17 M m x d is the total fluid mass hange. where = ( It an be shown that different distributions of pressure hanges p ( x (or fluid mass ontent hange m ( x whih produe the same total work P (or the same total fluid mass hange M must results in the same far-field deformation (Appendix B. Sine a enter of dilatation is a ombination of three equal, mutually perpendiular double fores without moment, only three basi Galerkin vetors, i.e., the three double fores without moment, are neessary to determine the elasti fields due to the inlusion whih undergoes normal eigenstrains only. For the ase of the inlusion whih undergoes general eigenstrains, the other six basi Galerkin vetors for double fores with moments are required. Analytial expressions for all the basi and other types of Galerkin vetors an be found in Mindlin (936 for the ase of full-spae solutions, Mindlin and Cheng (950a for half-spae solutions, and Yu and Sandy (99 for joined half-spaes solutions, whih enables a more general desription of ases with ompliated stresses and deformations. 4. Numerial examples and disussion 4. Effet of reservoir mehanial properties and shape on surfae deformation Consider a sphere-shaped reservoir subjet to a uniform pore pressure hange p =.0 MPa. The reservoir has a radius of 00 m and its enter is 600 m below the ground surfae. The analytial solution and finite element modelling results for the surfae deformation are ompared. The effets of the Young s modulus and Poisson s ratio of the reservoir on the surfae uplifting are shown in Figures 5a and 5b, respetively. The mehanial properties of the reservoir and bounding roks are listed in the orresponding figures. It an be seen that both the differenes in Young s modulus and Poisson s ratio between the reservoir and bounding roks an have a signifiant effet on the surfae uplifting. In addition, the Biot pore-pressure oeffiient α and Skempton s oeffiient B an also have a signifiant effet on the 7

18 deformation fields beause there is a diret linear relationship between them aording to Eqs. (3, (7 and (8. Figure 5. The effet of reservoir shape on the surfae deformation is shown in Figure 6. The volume of the reservoir remains onstant but its shape hanges with the ratio of a3 a. As an be seen, the surfae displaements and tilt inrease as the ratio of a3 a dereases. A straightforward explanation for the effet of the reservoir shape an be made by simply ignoring the effet of mehanial properties mismath. Substituting Eq. (6 into Eq. (9, the indued strains due to a uniform hange in pore pressure p in a sphere-shaped reservoir ( a = a = a3 and in a thin oblate spheroid-shaped ( a = a a reservoir an be obtained, respetively, as 3 ( ν µ ( ν α = = 33 = 6 ( ν µ ( ν α p a = = π 8 a p 3, a a a3, ( = = (9a 33 ( µ ( ν α ν p π a = 3. a a a3 a ( = (9b Eq. (9 indiates that a uniform hange in pore pressure results in a state of dilatational strain in a sphere-shaped reservoir, but a long thikness deformation dominated-state in a thin oblate spheroid-shaped reservoir ( = 33. The strain 33 in Eq. (9b is greater than that in Eq. (9a, and inreases as a3 a dereases, whih results in larger deformations. As a3 a 0, i.e., the thikness of the reservoir is far less than its lateral extent, α ν p = and 33 approahes a maximum of, whih atually is the µ ν 0 ( ( result obtained for uniaxial strain in the pressurized part of the reservoir, a ommonly used simplifying assumption (Geertsma, 973; Palmer, 990. So the simple assumption of uniaxial strain state an be used as a good approximation for a thin reservoir having thikness muh less than lateral extent. 8

19 Figure Surfae tilt produed by fratures and pressure transient effet The elasti field produed by a frature in a semi-infinite medium an be solved analytially by modelling the frature as an equivalent inlusion with an eigenstrain ( x nib = j ( x + n b ( x j i δ ( S x, (30 where S is the frature surfae, n is the normal to the frature surfae, b is the displaement of the upper surfae relative to the lower surfae of the frature, and δ ( S x is a shorthand notation for i.e., ( S x ( S x = δ ( x x ds( x δ, (3 δ is infinite when x is on S and zero when it is not. S For example, for a horizontal tensile frature with onstant opening b, the eigenstrain is = bδ ( S x 33, and all other eigenstrain omponents are equal to zero. While for a vertial tensile frature with onstant opening b striking east-west ( x diretion, the orresponding eigenstrain is = bδ ( S x and all others are zero. Substitution of Eq. (30 into Eq. (4 results in the Galerkin vetor as F ( x µ = 4π ( ν S ν nkb k ( x g ( x, x nib j ( x + n b ( x j i g ( x, x δ ( S x The Galerkin fore vetor provides a very onvenient method to model the elasti field produed by a frature in a semi-infinite medium. dx (3 To ompare the magnitude and pattern of the surfae tilt fields produed by the pore pressure transient with those produed by a hydrauli frature, the reservoir with a dilatational eigenstrain arising from a uniform hange in fluid mass ontent is onsidered. The mehanial properties of the reservoir and bounding rok listed in Figure 6 are used in the omputation. The surfae tilt vetors produed by a horizontal frature and a vertial frature with a volume of 80 m 3 are shown in Figures 7(a and 9

20 7(b, respetively. The frature geometry is listed in the orresponding figure. The surfae tilt vetors produed by the fluid injetion and assoiated poroelasti swelling are shown in Figures 7( and 7(d for a spherial ( a = a = a 3 = 00m and an oblate spheroid ( a = a = 00 m, a 3 = 5m reservoir, respetively. The reservoir has a referene porosity of 0.004, and the fluid has a referene density of injeted fluid volume is 80 m kg m. The As is shown in Figure 7, the pattern of the surfae tilt vetors resulting from fluid injetion and assoiated poroelasti swelling forms a radial pattern, pointing outwards from the enter (Figures 7( and 7(d, resembling the tilt field produed by a horizontal frature (Figure 7(a. In the ase of the spherial reservoir, the surfae tilt magnitude (Figure 7( is small ompared with the tilt produed by a frature of the same volume (Figures 7(a and 7(b. The poroelasti signal, if not orreted for in the analysis, will have a small effet on the hydrauli frature mapping in this ase. In the ase of the oblate spheroid-shaped reservoir, the magnitude of the surfae tilt (Figure 7(d, however, is quite omparable with those produed by the fratures. So in this ase, an inversion of the surfae tilt measurements may overestimate the size (volume of a horizontal frature, or even ause inorret interpretations on both the orientation (dip and strike and the volume of a vertial or inlined frature if the effet of fluid injetion and assoiated poroelasti swelling are not orreted for. Figure Effet of the distribution of pressure hange within reservoir As shown in Figure 8, we onsider a reservoir loated 600 m below the ground surfae. The disk-shaped reservoir has a radius of 300 m and thikness of 0 m. Consider a pressure hange of.0 MPa ourring in part of the reservoir. In ase (a, the pressure hange ours in a irular area of a radius of 00 m, while in ase (b, it ours in an elliptial area with semi-axes of 50 and 40 m. So the total areas (or volumes subjet to the pressure hange are the same in these two ases. Case (a ould our when there is a horizontal hydrauli frature, while ase (b ould our 0

21 when there is a vertial hydrauli frature in the reservoir. The double inhomogeneity model introdued in Setion. needs to be used to obtain the indued elasti fields. Figure 8. Aording to Eq. (6, the eigenstrain orresponding to the pressure hange of.0 MPa is { } T = Then, the average equivalent eigenstrains an be determined from the average onsisteny onditions (Eq.(9 as follows. For ase (a ( = { } T, ( Λ = { } T. For ase (b ( = { } T, ( Λ = { } T. Apparently, due to the effet of the inner inlusion shape (the area where pressure hange ours, the distribution of the average equivalent eigenstrains within the reservoir for the two ases are different. So, the near-field deformation and stress distributions must be different for the two ases. Further, substituting the average equivalent eigenstrains ( and ( into Eq. Λ ( gives the average equivalent eigenstrains taken over the reservoir, whih are the same for the two ases, as

22 V = { } T. Aording to Eq. (7, it an be onluded that the far-field deformation and stress distributions must be same for the two ases. The surfae tilt fields for ases (a and (b are shown in Figure 9(a and 9(b, respetively. As an be seen, the surfae deformations in the near field (the area lose to the origin ( r 400m for the two ases are obviously different both in their pattern and magnitude. However, the far-field surfae deformations are very similar. Figure Conlusions The stresses and deformations indued by the hange in pore pressure or temperature assoiated with fluid migration and/or heat transport within a hydroarbon or geothermal reservoir have been investigated. The reservoir is represented by an ellipsoidal poroelasti or thermoelasti inlusion embedded in a semi-infinite elasti body. Analytial solutions for the indued stresses and deformations are obtained by using the theory of inlusion and inhomogeneity, and are expressed in terms of the Galerkin vetor funtion. The Galerkin vetor funtion representation used in this paper is able to suffiiently desribe more general deformation fields in infinite, semiinfinite, and joined semi-infinite solids with various interfae onditions. For an ellipsoidal inlusion embedded in a semi-infinite elasti body, the indued strain within the inlusion that undergoes a uniform eigenstrain is not uniform due to the presene of the free boundary surfae. However, if the depth of the inlusion is more than two times greater than its harateristi size, the effet of free boundary surfae on the strain distribution within the inlusion is very weak and an be ignored. In this ase, a uniform effetive eigenstrain determined by the equivalent inlusion

23 method an be used to aount for elasti property ontrasts without ompromising the auray. Both the shape and mehanial properties of the reservoir an have a signifiant effet on the indued deformations. The surfae tilt vetors resulting from fluid injetion and assoiated poroelasti swelling produe a radial pattern, pointing outwards from the enter, resembling the tilt field produed by a horizontal frature. The different distributions of pressure hange within the reservoir whih produe the same average equivalent eigenstrains indue different near-field deformations, but produe the same far-field deformations. The obtained analytial formulae for the indued stresses and deformations are useful and onvenient for developing analytial or semi-analytial forward models in an inversion analysis for monitoring and mapping hydrauli fratures, subsurfae fluid migration and heat transport assoiated with fluid injetion or prodution in hydroarbon or geothermal reservoirs by surfae deformation-based monitoring tehniques. Aknowledgements The author would like to thank Dr. Rob Jeffrey for valuable disussions and omments, and the support of this work. Furthermore, the author thanks CSIRO for support of this work and for granting permission to publish. 3

24 Appendix A: Consisteny onditions for the double-inhomogeneity problem As shown in Figure 4, by using the equivalent homogeneous inlusion method, this double-inhomogeneity problem an be replaed by the equivalent double-inlusion problem with proper hoie of eigenstrains and Λ of uniform elastiity C, respetively. ( and ( defined over the regions Applying the equivalent homogeneous inlusion method, the following onsisteny onditions are obtained: ( ( = C ( ( = C ( ( σ x x x x x, for x in (A ( ( = C ( ( = C ( ( σ x x x x x. for x in Λ (A The indued strains due to the eigenstrains 00 are given by (Shodja and Sarvestani, ( = Γ ( d + Γ ( ( ( x x x x x x dx V, for x in V (A3 Γ x x dx ( ( where Γ ( x x = G ( x x + G ( x x [ ] C mn jm, ni im, nj. The volume averages of the indued strain taken over is expressed, in terms the average equivalent eigenstrain, as ( = Γ ( d + Γ ( x x x x x x dx ( ( V ( Γ x x dx ( where ( denotes the volume average of ( over., (A4 The first term in Eq. (A4 is given by (Hori and Nematnasser, 993 4

25 V ( d S ( S ( V S ( Γ x x x = +, (A5 Λ ( ( ( and the seond and last terms in Eq. (A4 are given, respetively, by ( d S ( Γ x x x =, (A6 ( ( ( d S ( Γ x x x =. (A7 ( ( Substitution of Eqs. (A5-(A7 into Eq. (A4 yields ( = S ( + S ( V S ( x (A8 ( ( Λ The volume average of the indued strain over V is expressed as ( = f + ( f = S ( V f + ( f ( ( x V Λ Λ (A9 Substitution of Eq. (A8 into Eq. (A9 leads to ( f ( ( ( x = S ( V + S ( V S ( Λ Λ f Λ (A0 Volume averages of the onsisteny onditions Eqs. (A and (A taken over and Λ are ( ( = C ( ( = C ( ( σ x x x x x, (A ( ( = C ( ( = C ( ( σ x x x x Λ Λ Λ Λ x. Λ (A 5

26 Substitution of Eqs. (A8 and (A0 into Eqs. (A and (A leads to the average onsisteny onditions ( C C Smn ( C ( ( ( + + ( C C S V S = C f f ( mn mn mn Λ ( C C S ( V S ( + ( mn mn mn f ( C C S ( V S ( V S ( + C = C f ( mn mn mn mn Λ, Λ (A3 (A4 Appendix B: The far-field deformation fields The indued displaement aused by an eigenstrain in an inlusion embedded in an infinite isotropi, elasti body an be expressed as (Eshelby, 957 u ( ( ν ( x x ( x x g = ( x d, (B 8π r k i x jk ν, and ( where gk = ( ( δlk + δi j δ ji + 3lil jlk g = r Define the kernel funtion F ( x x k k ( x x ( x x l = x x r. i i i. Let point x be the enter of the inlusion, and rewrite the kernel funtion as F ( x x = ( x x x + x k F k onserve spae, only one omponent of the kernel funtion ( x x here. The kernel funtion ( x x F F is ( x Expanding ( x x ( x x x x ( x = ( v ( x x r r. To F is heked 3 g x x =. (B r F in a Taylor series gives 6

27 F ( x x = F ( x x x + x x x x x y y z z x x = ( v + f 3 x + f y + f z + O r x x y y z z x x 3 ( x x x x y y z z x x f 5 x + f y + f z + O r x x y y z z x x where f f f r + ( ( x x x x =, f ( x 3 x r 3 x = ( ( y y x x =, f ( x 3 y r ( x x x, r ( 3 y = ( ( z z x x =, f ( x 3 z r These funtion are always of order O (. ( y y x, r ( 3 z = ( z z x. r, (B3 For a point loated in the far field suh that x x R << x x = r ( R is the x harateristi size of the inlusion, the terms involving x in Eq. (B3 are x x always negligible ompared to. So in the far field ( r >> F ( x x = ( x x k F k R, we have (B4 Substitute Eq. (B4 into Eq. (B leads to the far-field displaement u i ( x ( x x ( ν Fk = ( jk x d. (B5 8π Then, for an inlusion entered at = ( 0,0 eigenstrain u = x, whih undergoes a uniform dilatational δ, the far-field displaement for the full spae problem is V ( ( + ν x x = 3 4π ( ν R, ( V ( + ν x u x = 3 4π ( ν R, u ( 3 where R x x ( x 3 ( + ν x3 3 ( ν R = + +, and V is the volume of the inlusion. V x =, (B6 4π The displaement fields for the half spae problem an be derived simply from the orresponding full spae solution (Eq. (B6 aording to the result of Davis (003. 7

28 Furthermore, the far-field tilt for the orresponding half-spae problem an be obtained as where R x + x + ( x + ( + v x( z 6V + 6V + v y z + ω =, ω =, (B7 5 π R π R =. 3 5 ( ( It should be noted that, beause it is derived from a salar potential, the result of Davis (003 is only valid for the ase of irrotational deformation fields (for example, the elasti field due to a dilatational eigenstrain. While the Galerkin vetor funtion representation used in this paper is suffiiently general to represent all possible deformation fields in an elasti body. 8

29 Referenes Carne, C. and Fabriol, H., 999. Monitoring and modeling land subsidene at the Cerro Prieto geothermal field, Baja California, Mexio, using SAR interferometry. Geophysial Researh Letters, 6(9: -4. Chen, Z.R. and Jeffrey, R.G., 009. Tilt monitoring of hydrauli frature preonditioning treatments, 43rd U.S. Rok Mehanis Symposium and 4th U.S.-Canada Rok Mehanis Symposium, June 8, July, rd U.S. Rok Mehanis Symposium and 4th U.S.-Canada Rok Mehanis Symposium. Taylor and Franis/Balkema, Asheville, NC, United states, pp. Amerian Rok Mehanis Assoiation. Davies, J.H., 003. Elasti field in a semi-infinite solid due to thermal expansion or a oherently misfitting inlusion. Journal of Applied Mehanis-Transations of the Asme, 70(5: Detournay, E. and Cheng, A.H.D., 99. PLANE-STRAIN ANALYSIS OF A STATIONARY HYDRAULIC FRACTURE IN A POROELASTIC MEDIUM. International Journal of Solids and Strutures, 7(3: Du, J. and Olson, J.E., 00. A poroelasti reservoir model for prediting subsidene and mapping subsurfae pressure fronts. Journal of Petroleum Siene and Engineering, 30(3-4: Du, J., Philip, Z., Warpinski, N.R. and Mayerhofer, M., 009. Surfae deformationbased reservoir monitoring in inhomogeneous media, 43rd U.S. Rok Mehanis Symposium and 4th U.S.-Canada Rok Mehanis Symposium, June 8, July, rd U.S. Rok Mehanis Symposium and 4th U.S.-Canada Rok Mehanis Symposium. Taylor and Franis/Balkema, Asheville, NC, United states, pp. Amerian Rok Mehanis Assoiation. Eshelby, J.D., 957. The Determination of the Elasti Field of an Ellipsoidal Inlusion, and Related Problems. Proeedings of the Royal Soiety of London Series a- Mathematial and Physial Sienes, 4(6: Evans, K. and Holzhausen, G., 983. On the Development of Shallow Hydrauli Fratures as Viewed Through the Surfae Deformation Field: Part -Case Histories. Journal of Petroleum Tehnology, 35(: Fokker, P.A. and Orli, B., 006. Semi-analyti modelling of subsidene. Mathematial Geology, 38(5: Geertsma, J., 973. Land Subsidene above Compating Oil and Gas Reservoirs. Journal of Petroleum Tehnology, 5(JUN: Hori, M. and Nematnasser, S., 993. DOUBLE-INCLUSION MODEL AND OVERALL MODULI OF MULTIPHASE COMPOSITES. Mehanis of Materials, 4(3: Kahanov, M., Shafiro, B. and Tsukrov, I., 003. Handbook of elastiity solutions. Kluwer Aademi Publishers, Dordreht ; Boston, xiii, 34 p. pp. Liu, S.B. and Wang, Q., 005. Elasti fields due to eigenstrains in a half-spae. Journal of Applied Mehanis-Transations of the Asme, 7(6: Massonnet, D., Holzer, T. and Vadon, H., 998. Land subsidene aused by the East Mesa geothermal field, California, observed using SAR interferometry (vol 4, pg 90, 997. Geophysial Researh Letters, 5(6: MTigue, D.F., 986. THERMOELASTIC RESPONSE OF FLUID-SATURATED POROUS ROCK. Journal of Geophysial Researh-Solid Earth and Planets, 9(B9:

30 Mindlin, R.D., 936. Fore at a point in the interior of a semi-infinite solid. Physis-a Journal of General and Applied Physis, 7(: Mindlin, R.D. and Cheng, D.H., 950a. NUCLEI OF STRAIN IN THE SEMI- INFINITE SOLID. Journal of Applied Physis, (9: Mindlin, R.D. and Cheng, D.H., 950b. THERMOELASTIC STRESS IN THE SEMI-INFINITE SOLID. Journal of Applied Physis, (9: Morita, N., Whitfill, D.L., Nygaard, O. and Bale, A., 989. A QUICK METHOD TO DETERMINE SUBSIDENCE, RESERVOIR COMPACTION, AND INSITU STRESS-INDUCED BY RESERVOIR DEPLETION. Journal of Petroleum Tehnology, 4(: Muntendam-Bos, A.G., Kroon, I.C. and Fokker, P.A., 008. Time-dependent inversion of surfae subsidene due to dynami reservoir ompation. Mathematial Geosienes, 40(: Mura, T., 987. Miromehanis of defets in solids. Mehanis of elasti and inelasti solids ; 3. Kluwer Aademi Publishers, xiii, 587 p. pp. Palmer, I.D., 990. Uplifts and tilts at earth's surfae indued by pressure transients from hydrauli fratures. SPE Prodution Engineering, 5(3: Rie, J.R. and Cleary, M.P., 976. SOME BASIC STRESS DIFFUSION SOLUTIONS FOR FLUID-SATURATED ELASTIC POROUS-MEDIA WITH COMPRESSIBLE CONSTITUENTS. Reviews of Geophysis, 4(: 7-4. Roeloffs, E.A., 988. HYDROLOGIC PRECURSORS TO EARTHQUAKES - A REVIEW. Pure and Applied Geophysis, 6(-4: Rudniki, J.W., 00. Alteration of regional stress by reservoirs and other inhomogeneities: Stabilizing or destabilizing? Ninth International Congress on Rok Mehanis, Vol 3, Proeedings: Safai, N.M. and Pinder, G.F., 980. VERTICAL AND HORIZONTAL LAND DEFORMATION DUE TO FLUID WITHDRAWAL. International Journal for Numerial and Analytial Methods in Geomehanis, 4(: 3-4. Segall, P., 985. STRESS AND SUBSIDENCE RESULTING FROM SUBSURFACE FLUID WITHDRAWAL IN THE EPICENTRAL REGION OF THE 983 COALINGA EARTHQUAKE. Journal of Geophysial Researh-Solid Earth and Planets, 90(NB8: Segall, P., 99. INDUCED STRESSES DUE TO FLUID EXTRACTION FROM AXISYMMETRICAL RESERVOIRS. Pure and Applied Geophysis, 39(3-4: Segall, P. and Fitzgerald, S.D., 998. A note on indued stress hanges in hydroarbon and geothermal reservoirs. Tetonophysis, 89(-3: 7-8. Seo, K. and Mura, T., 979. ELASTIC FIELD IN A HALF SPACE DUE TO ELLIPSOIDAL INCLUSIONS WITH UNIFORM DILATATIONAL EIGEN STRAINS. Journal of Applied Mehanis-Transations of the Asme, 46(3: Shodja, H.M. and Sarvestani, A.S., 00. Elasti fields in double inhomogeneity by the equivalent inlusion method. Journal of Applied Mehanis-Transations of the Asme, 68(: 3-0. Soltanzadeh, H. and Hawkes, C.D., 008. Semi-analytial models for stress hange and fault reativation indued by reservoir prodution and injetion. Journal of Petroleum Siene and Engineering, 60(: Soltanzadeh, H., Hawkes, C.D. and Sharma, J.S., 007. Poroelasti model for prodution- and injetion-indued stresses in reservoirs with elasti properties 30

31 different from the surrounding rok. International Journal of Geomehanis, 7(5: Vaso, D.W., Karasaki, K. and Kishida, K., 00. A oupled inversion of pressure and surfae displaement. Water Resoures Researh, 37(: Vaso, D.W., Karasaki, K. and Myer, L., 998. Monitoring of fluid injetion and soil onsolidation using surfae tilt measurements. Journal of Geotehnial and Geoenvironmental Engineering, 4(: Yu, H.Y. and Sanday, S.C., 99. ELASTIC FIELDS IN JOINED HALF-SPACES DUE TO NUCLEI OF STRAIN. Proeedings of the Royal Soiety of London Series a-mathematial Physial and Engineering Sienes, 434(89:

32 Captions of figures Figure. An ellipsoidal inhomogeneous inlusion (the reservoir in a half spae. Figure. An ellipsoidal inlusion with semi-major axes a, a, and a 3. Figure 3. The strain distributions along the x3 -axis. Figure 4: (a. A double inhomogeneity V = Λ + is embedded in an isotropi infinite homogeneous elasti solid D. The elasti moduli of the inner inlusion, annulus Λ = V, and the infinite solid D V are denoted by C, C, and C, respetively. The inner inlusion undergoes an eigenstrain ; (b. Using the equivalent inlusion method, the double inhomogeneity is replaed by an equivalent doubleinlusion problem with proper equivalent eigenstrains. Figure 5. The effet of (a: Young s modulus; and (b: Poisson s ratio of reservoir on the surfae uplifting. Figure 6. The effet of the shape of reservoir on the surfae deformation: (a horizontal displaement; (b vertial displaement; and ( tilt. Figure 7. Surfae tilt vetors produed by (a a horizontal frature; (b a vertial frature; fluid injetion and assoiated poroelasti swelling in ( a spherial reservoir; and (d an oblate spheroid-shaped reservoir. Figure 8. A disk-shaped reservoir experienes a pressure hange of.0 MPa within (a a irular area; and (b an elliptial area. Figure 9. Surfae tilt field produed by the disk-shaped reservoir experienes a pressure hange of.0 MPa within (a a irular area; and (b an elliptial area. 3

33 o x ( x y ( x (Reservoir z ( x 3 D (Surrounding Rok Figure. An ellipsoidal inhomogeneous inlusion (the reservoir in a half spae. D z ( x 3 a 3 a o a y ( x x ( x Figure. An ellipsoidal inlusion with semi-major axes a, a, and a 3. 33

34 (a = a (b = 3a Figure 3. The strain distributions along the x3 -axis. 34

35 Bounding roks D,C Λ,C,C V = Λ + Λ,C (,C ( V = Λ + D,C Reservoir (a (b Figure 4: (a. A double inhomogeneity V = Λ + is embedded in an isotropi infinite homogeneous elasti solid D. The elasti moduli of the inner inlusion, annulus Λ = V, and the infinite solid D V are denoted by C, C, and C, respetively. The inner inlusion undergoes an eigenstrain ; (b. Using the equivalent inlusion method, the double inhomogeneity is replaed by an equivalent doubleinlusion problem with proper equivalent eigenstrains. 35

36 (a (b Figure 5. The effet of (a: Young s modulus; and (b: Poisson s ratio of reservoir on the surfae uplifting. 36

37 (a horizontal displaement. (b vertial displaement. 37

38 ( Tilt Figure 6. The effet of the shape of reservoir on the surfae deformation: (a horizontal displaement; (b vertial displaement; and ( tilt. 38

39 (a surfae tilt vetors produed by a horizontal frature with zero leakoff. (b surfae tilt vetors produed by a vertial frature with zero leakoff. 39

40 ( surfae tilt vetors resulting from fluid injetion and assoiated poroelasti swelling in a spherial reservoir. (d surfae tilt vetors resulting from fluid injetion and assoiated poroelasti swelling in an oblate spheroid-shaped reservoir. Figure 7. Surfae tilt vetors produed by (a a horizontal frature; (b a vertial frature; fluid injetion and assoiated poroelasti swelling in ( a spherial reservoir; and (d an oblate spheroid-shaped reservoir. 40