Subsidence above a planar reservoir

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 17, NO. B9, 222, doi:1.129/21jb66, 22 Sbsidence above a planar reservoir J. B. Walsh Adamsville, Rhode Island, USA Received 9 May 21; revised 25 Janary 22; accepted 3 Janary 22; pblished 27 September 22. [1] A tablar reservoir, circlar in plan, is located at depth parallel to the srface. Compaction of the reservoir as flid is withdrawn cases a change in the stress and displacement fields in the srronding medim. A soltion to this problem is obtained by determining the bondary conditions at the interface between the reservoir and the halfspace sing Eshelby s [1957] ct-and-weld techniqe. The displacement field in the reservoir and the srronding medim is then calclated sing the reciprocal theorem. The displacement field is fond to be the same as that calclated by Geertsma [1973a, 1973b] and by Segall [1992] Segall et al. [1994] sing another techniqe. INDEX TERMS: 321 Mathematical Geophysics: Modeling; 3299 Mathematical Geophysics: General or miscellaneos Citation: Walsh, J. B., Sbsidence above a planar reservoir, J. Geophys. Res., 17(B9), 222, doi:1.129/21jb66, Introdction [2] The extraction of liqids or gas at depth cases compaction of the reservoir and changes the stress and deformation fields in the srronding contry rock. Becase these stress changes and displacements are sometimes large enogh to case damage (see Segall [1989], Segall et al. [1994], and McGarr [1991] for a review of indced seismicity and Yerkes and Castle [1976] for srface sbsidence at oil and gas fields), the topic has been the sbject of theoretical analysis, laboratory experimentation, and direct observation in the field. Reservoirs of interest, which might contain oil, water, or gas, are natral featres and are fond to have a variety of shapes and sizes. Here, we restrict or attention to the case of a horizontal, tablar reservoir, circlar in plan. This configration is an acceptable idealization of many actal reservoirs and has been analyzed theoretically by Geertsma [1966, 1973a, 1973b], Segall [1992], and Segall et al. [1994]. [3] The approach sed by Segall [1992] and Segall et al. [1994] is in a fndamental sense, the same as that sed by Geertsma [1966, 1973a, 1973b], and both derive the stress field, as well as the srface sbsidence, reslting from the extraction of flid or gas from the reservoir. Segall and Geertsma both considered the case of a niform decrease in pressre in a horizontal tablar reservoir, and their soltions for the displacement field are identical [Segall, 1992]. Geertsma [1966, 1973a, 1973b] finds stresses and srface displacements cased by the niform compaction of a reservoir by applying the inflence fnctions derived by Mindlin and Cheng [195] which relate varios nclei of strain sorces to displacements in an elastic half-space. To find srface displacements, for example, Geertsma ses Mindlin and Cheng s expressions for the srface displacements arising from a single dilatational sorce and then calclates the field by integrating over the area of the reservoir. Segall [1992] derives the inflence fnction for a dilatational sorce from first principles; the rest of the calclation is similar to Copyright 22 by the American Geophysical Union /2/21JB66$9. Geertsma s and, as mentioned above, the reslts are identical. Both analyses show that the reservoir compacts niformly with no radial displacement, i.e., in niaxial strain. [4] I propose a different approach to this problem based on Eshelby s [1957] ct-and-weld techniqe and an application of the reciprocal theorem [e.g., Fng, 1965, p. 5]. The techniqe provides a soltion for displacements, strains, and stresses anywhere in the half-space, althogh only the vertical srface displacement field is considered here. Compaction of the reservoir and the state of stress in the reservoir are calclated as part of the soltion. 2. Analysis 2.1. Defining the Problem [5] Eshelby [1957] stdied the change in the elastic strain field which occrs when an inclsion in an infinite medim ndergoes a change in shape and size. The techniqe he sed consisted of first removing the inclsion from the srronding medim and allowing it to ndergo a stressfree transformational strain. Appropriate srface tractions were applied to retrn the inclsion to its original size and shape. The inclsion was then retrned to the cavity in the matrix material, welded, and the interfacial bondary tractions were relaxed. Eshelby showed how to calclate the constrained strain field which develops after the interfacial tractions are relaxed. [6] Eshelby s techniqe is sefl in the present problem. First, the reservoir, having volme V, is separated from the contry rock, as in Figre 1a, by ctting the solid material at the interface; flid in the reservoir and the contry rock is retained by a membrane which closes passageways crossing the interface. Flid m having density r is extracted throgh the membrane, which is then resealed. Extraction of the flid cases a decrease p f in flid pressre and a niform transformational hydrostatic strain V T /V of the body; simple elastic analysis [see Segall, 1989] shows that V T =V ¼ b ð d b s Þp f ð1aþ ETG 6-1

2 ETG 6-2 WALSH: SUBSIDENCE ABOVE A PLANAR RESERVOIR niform hydrostatic tension p T over its oter bondary, where from eqation (1), p T ¼ V T 1 =V b ¼ b d b s b p f ; ð2þ and b is the ndrained compressibility of the reservoir rock. Note that drained conditions prevail only when flid is removed to effect the transformational strain in eqation (1). The pressre p T cases a decrease pf in flid pressre, where from the definition of Skempton s coefficient B p f ¼ Bp T : ð3þ Figre 1. The bondary vale problem to be solved is developed following Eshelby s [1957] ct-and-weld techniqe for analyzing inclsions. (a) A reservoir with impermeable bondaries is located in a half-space having the same elastic properties. (b) The reservoir is ct from the half-space, and a mass m of flid is removed, casing a decrease pf in flid pressre and a niform transformational strain V T /V in the reservoir. (c) The reservoir is retrned to its original size and shape by imposing hydrostatic tensile pressre p T at the bondary, casing a decrease p f in flid pressre. (d) The reservoir is retrned to the half-space, welded, and the transformational pressre is relaxed by imposing tractions p T at the interface, casing an additional change p f in flid pressre. Comparing Figre 1d with Figre 1a, we see that removing flid from the reservoir is eqivalent to imposing a transformational pressre p T to the otline of the reservoir in a niform half-space. The problem becomes that of calclating the displacement field in the half-space cased by the body forces in Figre 1d. Displacements h z and h r for a section of the reservoir at the centerline are indicated in the insert to Figre 1d. where V T /V is the volmetric strain and b d and b s are the effective drained compressibility of the poros reservoir rock and compressibility of the solid matrix material, respectively. Note that I se the convention that loss of flid and compressive strains and stresses are positive. Elementary analysis shows that volmetric strain V T /V in eqation (1a) can also be expressed in terms of the mass m extracted at density r, as follows: V T =V ¼ Bm=r V ; ð1bþ where B is Skempton s coefficient (the change in flid pressre per nit change in confining pressre for a satrated sample nder ndrained conditions). [7] As shown schematically in Figre 1c, the reservoir is then restored to its original size and shape by imposing a [8] The reservoir is now retrned to the cavity in the contry rock, and the interface is removed by welding those solid segments which had been ct; the membrane remains intact. The half-space is now in its original state except for the loss of flid in the reservoir and the body forces represented by the transformational stresses p T (see Figre 1c). The transformational stress p T is now relaxed by imposing p T at the interface, as in Figre 1d. We see that the problem of calclating the compaction of the reservoir and sbsidence of the srface has now become the problem of deriving the displacements cased by the body forces representing p T in Figre 1d, where p T is given by eqation (2). Note that compaction of the reservoir cases a frther change p f in flid pressre, bt p f cannot be evalated ntil the constrained state of strain in the reservoir has been calclated. [9] In eqation (2) and in the later steps, the operations are carried ot nder ndrained conditions where no flow of flid is allowed to or from the reservoir. The total flid transfer m occrs in the first operation, leading to eqations (1a) and (1b), and flid pressre changes with each step. The steps cold also be carried ot nder conditions of constant flid pressre [see Segall, 1992] instead of constant flid content. When sch a procedre is followed, flid flx m/r is adjsted at each step to maintain constant flid pressre. As shown later in the analysis, both procedres prodce the same reslts Reservoir Compaction [1] I follow Geertsma and Segall and assme that differences in the elastic properties of the reservoir and the contry rock are sfficiently small that the region comprising the reservoir and the srronding contry rock can be considered to be a homogeneos, elastic half-space. (I show in section 3 that this restriction can be relaxed.) The deformation field reslting from the stress system in Figre 1d can be fond conveniently for a homogeneos half-space by applying the reciprocal theorem [Fng, 1965]. [11] As an example, the reciprocal theorem is applied to calclate the displacement field at the center of the reservoir in the constrained state. First, the dilatational component of the displacement field is fond by positioning at (, c), as in Figre 2a, a ncles of eqal doble forces F aligned along the three orthogonal axes. We are interested in the gross deformation of the reservoir, and so these doble forces are

3 WALSH: SUBSIDENCE ABOVE A PLANAR RESERVOIR ETG 6-3 Figre 2. The displacement field in the reservoir is fond by applying the reciprocal theorem first to the stress systems in Figres 1c and 2a and then repeated with Figres 1d and 2b. (a) Eqal doble forces, separated by h, are aligned along the coordinate axes at the horizon of the reservoir, casing vertical displacements w D and w + D at pper and lower srfaces of the reservoir and radial displacement v D at the periphery. (b) A tensile ncles of doble forces having the components shown in the figre prodces vertical and horizontal displacements (w T, w + T, v T ) at the periphery of the reservoir (not shown). separated by distance h, where h has been assmed to be mch smaller than other characteristic dimensions. The ncles of doble forces prodces vertical displacements w D (r, c h/2) normal to the pper srface of the reservoir, w D + (r, c + h/2) at the lower srface, and radial displacement v D (a, c) at the periphery. As shown in Figre 1d, interfacial pressre p T prodces displacements w P (, c h/2) and w P + (, c + h/2) at the pper and lower srfaces of the reservoir at the origin and radial displacement v P (h, c) near the axis of symmetry. We are interested in the axial strain z averaged over the thickness of the reservoir, and so compaction w P + w P is represented as z h. For simplicity, we consider the radial strain r also as averaged over a radial distance h, althogh it will be fond that the gage length chosen for r is not important. [12] The reciprocal theorem reqires that the work done by forces in Figre 2a acting throgh displacements in Figre 1d mst eqal the work done by forces in Figre 1c acting throgh displacements in Figre 2a; that is, Fhð z þ 2 r Z a Þ ¼ 2pp T w þ D w D rdr 2pahp T v D ða; cþ: ð4þ Expressions for the axial and radial displacements prodced by a dilatational sorce are given by Okada [1992, Table 2] (note that Okada s convention for the sign of z is opposite to that here). Theseqfnctions ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi have in the denominator qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi terms of the form R 2 ¼ r 2 þðzþcþ 2 or R 1 ¼ r 2 þðz cþ 2, or powers of these radicals. I find that terms containing R 2 have a negligible effect on eqation (4) for tablar regions where h/c, h/a 1; retaining only R 1 terms in Okada s expression, I find that eqation (4) becomes simply Fhð z þ 2 r giving Þ ¼ FpT ð1 2n Þh 2 4mð1 n Þ Z a rdr h i 3=2 ; ð5þ r 2 þ ðh=2þ 2 z þ 2 r ¼ pt ð1 2n Þ : ð6þ 2mð1 n Þ Note that Poisson s ratio n in eqations (5) and (6) is the ndrained vale, in agreement with the ndrained operations carried ot on the reservoir in Figre 1. [13] The calclation is repeated sing the force system in Figre 2b and Okada s expression for displacements for the tensile (in his terminology) ncles of strain. Repeating the operations as described above reslts in

4 ETG 6-4 WALSH: SUBSIDENCE ABOVE A PLANAR RESERVOIR z 2n r ¼ pt ð1 2n Þ : ð7þ 1 n 2mð1 n Þ Combining eqations (6) and (7) gives z ¼ pt ð1 2n Þ 2mð1 n Þ r ¼ : ð8þ As indicated by eqation (8), compaction at the center of the reservoir occrs withot radial strain. The integrations needed to evalate the displacement components at points away from the center are more difficlt than that in eqation (5), and they have been relegated to Appendix A. The calclations show that, in fact, deformation throghot the reservoir is homogeneos, i.e., a niform compaction occrs withot radial deformation, as given by eqation (8). [14] Having established the mode of deformation experienced by the reservoir, we are now able to calclate the change in flid pressre which accompanies compaction. Simple elastic analysis shows that the stresses (s z, s r ) acting on the reservoir dring compaction reslting in the deformation given by eqation (8) are z ¼ p T r ¼ p T n : ð9þ 1 n The mean pressre acting on the reservoir is, from (9), p T [1 + n /3(1 n )], and so the increase p f in flid pressre dring compaction is p f ¼ Bp T 1 þ n 31 ð n Þ : ð1þ [15] Smming the decreases in pressre in eqations (2) and (3) and the increase in eqation (1), I find the net decrease p f in flid pressre to be pf ¼ p T b þ B 21 2n ð Þ : ð11þ b d b s 31 ð n Þ The stresses (s z c, s r c ) imposed by the srronding contry rock on the reservoir dring compaction are given by eqation (9) less the transformation stress p T : or, eqivalently s c z ¼ s c 1 2n r ¼ pt ð12aþ 1 n s c z ¼ s c r ¼ p 1 2n b f þ B 21 2n ð Þ : ð12bþ 1 n b d b s 31 ð n Þ We see in eqations (12a) and (12b) that no net change in vertical stress occrs dring compaction, whereas the radial Figre 3. A vertical force F is applied to the srface of a half-space casing a vertical force F is applied to the srface of a half-space casing vertical displacements w F and radial displacements v F. stress decreases as flid pressre is redced. These stresses are sperimposed on the in sit tectonic stress state, which is assmed to be naffected by reservoir compaction Srface Sbsidence [16] Vertical displacement at a point on the srface reslting from reservoir compaction is analyzed by positioning a vertical force F at that location. To se a simple case as an example, the force is placed at the origin as in Figre 3, prodcing displacements (v F, w F ) in the half-space; expressions for the displacements [see, e.g., Timoshenko and Goodier, 1951, p. 365] are v F ¼ F 1 2n 1 z 4pm r R 1 þ 1 r 2 z 1 2n R 3 1 w F ¼ F 2 1 n þ z2 4pm R R 3 ; ð13þ where R 2 = r 2 + z 2. Vertical compaction w F + w F of the reservoir can be fond by noting that, to sfficient accracy, w þ F w F ¼ h F=@zÞ z¼c h ð 1 2n Þc ¼ F 4pm R 3 c þ c3 3R 5 c ; ð14þ where R c 2 = r 2 + c 2. [17] Applying the reciprocal theorem to the force and displacements in Figres 1d and 3, I find that the srface sbsidence w(,) at the axis of symmetry prodced by reservoir compaction is given by Fwð; Þ ¼ p T p Z a w þ F w F rdr 2pah vf ða; cþ : ð15þ Introdcing eqations (13) and (14) into (15) and integrating reslts in an expression for maximm srface sbsidence w at the axis of symmetry, as follows: w ¼ wð; Þ ¼ p T h=m ð 1 2n Þð1 c=r a Þ; ð16þ where p T is given by eqation (2) and R a 2 = a 2 + c 2.

5 WALSH: SUBSIDENCE ABOVE A PLANAR RESERVOIR ETG 6-5 Figre 4. The changes in the volme of the sbsidence bowl is fond by applying the reciprocal theorem to these two stress systems. (a) The reservoir, shown by the dashed otline, is in a very large elastic body simlating the Earth loaded by a niform pressre p T. (b) The reservoir is positioned in the same elastic body in (a) is loaded as in Figre 1(d), changing the volme of the body by V sb. [18] The integrations analogos to eqation (15) for positions away from the axis of symmetry are more complicated, and so this calclation has been pt in Appendix A. I find that the srface displacement field calclated following my procedre is exactly the same as that in Geertsma s and Segall s analyses Volme of the Sbsidence Bowl [19] Geertsma [1973a, 1973b] derived an expression for the total volme of the depression created by sbsidence by integrating vertical displacement over the infinite area of the srface. The reciprocal theorem offers a convenient and more general method to carry ot this calclation. Consider the reservoir to be an inclsion in an elastic body which is very large, as in Figre 4, bt not infinite like the half-space nder consideration heretofore. The stress system in Figre 4a is the same as that described by Figre 1c, except that the halfspace has been made finite by a ct made a very great distance from the reservoir. The same elastic body in Figre 4a is loaded in Figre 4b by a niform external pressre p T, casing a change V in the volme of the reservoir. Applying the reciprocal theorem gives p T V sb ¼ p T V : ð17þ The volme change V reslting from the niform pressre change p T is simply V ¼ b p T V;

6 ETG 6-6 WALSH: SUBSIDENCE ABOVE A PLANAR RESERVOIR and so eqation (17) can be written V sb ¼ Vb p T ; ð18þ where V is the volme of the reservoir. The volme change V of the reservoir as it compacts is, from eqation (8), V ¼ V pt ð1 2n Þ : ð19þ 2mð1 n Þ Combining eqations (18) and (19) reslts gives the relationship between the volme change of the sbsidence bowl and the volme change of the reservoir, as follows: V sb ¼ 3V 1 n : ð2aþ 1 þ n Geertsma s [1973] expression for the volme V G sb of the sbsidence bowl is V G sb ¼ 2V ð 1 n Þ ð2bþ where n can be interpreted as the drained or ndrained vale. Note that V G sb is less than V sb for all practical vales of Poisson s ratio. 3. Discssion [2] Geertsma [1966, 1973a, 1973b] and Segall [1992] and Segall et al. [1994] analyzed the srface sbsidence reslting from withdrawing flid or gas from a tablar reservoir. The reservoir was considered to be a planar, circlar sorce of dilatational strain ncleii in a homogeneos, linearly elastic half-space. Displacements in the halfspace were calclated by either sing the expressions for dilatational sorces in Mindlin and Cheng s [195] cataloge (Geertsma) or by deriving these expressions (Segall). Here, I have developed a different way of analyzing the problem sing a techniqe involving Betti s reciprocal theorem. The analytical reslts agree with those derived by Geertsma and by Segall. Althogh the techniqe does not provide any greater efficiency than the established procedre for analyzing the present problem, it may offer some advantages in other calclations. One elementary example, presented in the Analysis, is the calclation of the volme of the sbsidence bowl Compaction of the Reservoir [21] As shown here, and previosly by Geertsma and by Segall, the reservoir compacts niformly nder niaxial strain. Uniform, niaxial strain conditions are a conseqence of the tablar configration of the reservoir, i.e., for bodies where h/c 1. Eshelby [1957] shows that the constrained strain in any ellipsoidal body in an infinite, elastic space (or, in a half-space, in ellipsoidal bodies whose dimensions are mch less than the distance to the srface) is niform when the transformational strain is niform; the strain is not niaxial, however, nless h/a 1. If the body is ellipsoidal, bt not tablar, and it is not distant from the srface, the constrained strain field is neither niform nor niaxial [Seo and Mra, 1979]. [22] Geertsma defines the compaction coefficient c m as the rate of change of axial strain with flid pressre for deformation nder niaxial strain conditions. Geertsma [1973a, p. 6] did not differentiate between drained and ndrained conditions in his derivation of the compaction coefficient, bt apparently drained elastic parameters were intended; his expression is c G m ¼ h=hp f ¼ þ n d 1 n d! ðb d b s Þ: ð21þ [23] I find from eqations (8) and (11) that the compaction coefficient (introdcing the identity B =(b d b )/(b d b s )) can be written c m ¼ 1 1 þ n ðb 3 1 n d b s Þ 1 þ 2 1 2n bd b 1 ð22þ 3 1 n and frther maniplations (J. Rdnicki, personal commncation, 1998) of the identities relating drained and ndrained parameters yields c m ¼ 1 1 þ n d ðb 3 1 n d b s Þ: ð23þ d Comparing eqations (21) and (23), we see that the expressions for the compaction coefficient are the same, indicating that both drained and ndrained procedres are valid. [24] The compaction of the reservoir can also be expressed in terms of the volme m/r of flid (at reservoir density r ) extracted; one finds sing eqations (1b), (2), (11), and (23) the relationship between total prodction volme m/r and the decrease p f in reservoir pressre: B 1 nd 1 þ n p f ¼ m=r ð VÞ : ð24þ b d b s 1 þ n d 1 n Combining eqations (21), (23), and (24) leads to a simple expression relating compaction and prodction volme: h=h ¼ B 1 þ n ðm=r 3 1 n V Þ: ð25þ [25] In the procedre I sed, the steps were carried ot nder ndrained conditions, where the flid content of the reservoir remained fixed after the initial withdrawal and flid pressre changed with each step. Alternatively, drained conditions cold have been followed, reqiring that flid pressre be held fixed with the flid content of the reservoir changing accordingly. Carrying throgh the analysis sing drained conditions, one finds that the final change m in the flid content is given by eqation (25) derived following the ndrained procedre, as expected. [26] Note that introdcing eqation (1b) into (25) reslts in a relationship between the constrained volme change V of the reservoir and the stress-free transformational change V T : h=h ¼ V=V ¼ 1 1 þ n V T =V : ð26þ 3 1 n b

7 WALSH: SUBSIDENCE ABOVE A PLANAR RESERVOIR ETG 6-7 This relationship is particlarly interesting becase it reprodces an expression Eshelby [1957] derived for an infinite medim. Eshelby showed sing an elegant application of elastic theory that the constrained volme change was related by eqation (26) to the stress-free transformational change for a body of any shape ndergoing a hydrostatic transformation. Elastic analysis of half-space problems involves modifying soltions (which involve only R 1 terms) for the infinite space by sperposing R 2 terms (called image terms) to accont for the effect of the free srface. As discssed in the derivation of eqation (5), I fond that R 2 terms were negligible for tablar reservoirs where h/c 1 and h/a 1, and so the half-space and infinite space soltions are identical. Eshelby s relationship (26) will not prevail for reservoirs located close to the srface Sbsidence [27] Sbsidence at the srface is proportional to reservoir compaction h/h; maximm sbsidence w is given by eqation (16): w ¼ 21 ð n Þhð1 c=r a Þ; ð27þ or by the eqivalent expression involving the compaction coefficient: w ¼ 2c m hð1 n Þð1 c=r a Þp f : ð28þ Expressions (27) and (28) were derived assming ndrained conditions; carrying ot the analysis for drained conditions shows that sbsidence is given by eqation (28) with the factor (1 n ) replaced by (1 n d ). The ndrained conditions correspond to the immediate response of the half-space if mass m of flid were drawn instantaneosly from the reservoir, whereas drained conditions correspond to the long-term response. [28] Generally, sbsidence is predicted by introdcing vales of compaction h estimated from field measrements of reservoir pressre and laboratory measrements of the compaction coefficient. Note in eqation (25) that compaction can also be estimated from measrements of flid flx (m/r V ); sing eqations (25) and (27), I find that maximm sbsidence is given by w =h ¼ ð2b=3þð1 þ n Þðm=r V Þð1 c=r a Þ ð29þ For fields where prodction volme m/r is known, (29) offers an additional method for estimating sbsidence, one which is independent of the traditional method. [29] This analysis of the deformation of the reservoir and the srronding medim involves the assmption that the elastic properties of the reservoir are the same as those of the srronding rock. A qalitative examination of the process shows that a relaxation of this assmption does not invalidate the conclsions. I show above that the halfspace soltion is the same as the soltion for the infinite space for tablar reservoirs. Eshelby [1957] considers inclsions and inhomogeneities ndergoing transformations in infinite media, and he shows that ellipsoidal bodies can be treated sing the eqivalent inclsion method. In the context of the present analysis, application of Eshelby s method shows that compaction h shold be calclated sing elastic parameters for the reservoir rock in eqations (23) and (25) and sbsidence w is calclated sing Poisson s ratio for the contry rock, as in eqation (27) or (29), for example. The constrained state of strain in the eqivalent inclsion is niaxial strain, as derived here, even when the elastic properties of the reservoir and the contry rock are different Volme of the Sbsidence Bowl [3] In eqations (2a) and (2b) we see that Geertsma s expression nderestimates the volme of the sbsidence bowl for all practical vales of Poisson s ratio. The mistake in Geertsma s analysis is not in his expression for vertical srface displacements bt in the integration of these displacements over the infinite area of the half-space. The assmption that the Earth is a half-space is appropriate when one is interested in the displacement field in the vicinity of the sorce, i.e., at distances which are small compared with the diameter of the Earth. Within this region, corrections de to the finite size of the Earth for calclations of displacements are negligible. At great distances from the sorce, displacements vary inversely with the sqare of the distance from the sorce. Calclation of volme change reqires integrating these displacements over area, which increases with the sqare of distance from the sorce. Althogh displacements at great distances are infinitesimal, they are not small enogh, and integrating them over an infinite area introdces a finite error. Wong and Walsh [1991] describe a similar error in previos calclations of edifice volme over magma chambers and present a more detailed explanation than that here. The difference between eqations (2) and (29) is only of academic interest becase the volme of the sbsidence bowl cannot be measred in practice: displacements only modest distances from the reservoir are so small that they are lost in the noise. Appendix A A1. Reservior Compaction [31] Coordinates sed to integrate the vertical displacements reslting from the dilatational and niaxial strain nclei over the circlar areas (in plan) of the reservoir are shown in Figre 5. As indicated in Figre 5, the sorce is positioned at r on the r axis, and we mst integrate the vertical displacement cased by the sorce over the circlar area; an arbitrary point is located at (r, J) with the ncles at r = as reference. Integrating a fnction, say g, over the area A of the circle in Figre 5 is accomplished by carrying ot the operation Z gðr;# ÞdA ¼ 2 Z =2 Z r grdr þ Z rl grdr d#; ða1þ where r and r l are the maximm lengths of the pper and lower chord segments indicated in Figre 5. The lengths r and r l of the pper and lower chord segments are fond from Figre 5 to be given by the expressions r ¼ a cos # þ r cos f r l ¼ a cos # r cos f; ða2þ

8 ETG 6-8 WALSH: SUBSIDENCE ABOVE A PLANAR RESERVOIR Applying the reciprocal theorem, as described in the text, we find Z F z p þ 2 z p h ¼ p T D z hda; ða7þ where z p and r p are the axial and radial components of strain at r prodced by the interfacial tractions p T and the integration over the circlar area A of the reservoir is that given by eqation (A1). Carrying ot the integration in eqation (A7), we find Figre 5. Coordinate system sed for integrating over the srface of the reservoir as in eqation (A1) and (A9). where, from the Law of Sines, a sin # ¼ r sin f: [32] Here, we are interested in the work done by p T (as in Figre 1c) against the displacements prodced by the strain sorces. The procedre is similar to that sed in the text to find the state of strain at the center of the reservoir except that the integration of the vertical displacement field is replaced here by eqation (A1). Expressions for vertical and radial displacements for varios strain nclei given by Okada s [1992, Table 2] consist of terms which have in the denominator either R 1 or R 2 where R 2 1 ¼ r2 þ ðz cþ 2 R 2 2 ¼ r2 þ ðz þ cþ 2 : ða3þ One finds pon carrying ot the integration in eqation (A1) that the integrands of terms containing R 1 dominate other terms in the vicinity of the reservoir (where z c is relatively small). As a conseqence, we need consider only the terms containing R 1 here; Okada s [1992] catalog gives w D ¼ Fh 1 2n z c 8pm " 1 n R 3 1 w T ¼ Fh # 1 2n z c þ 3 ðz cþ 3 : ða4þ 8pm 1 n 1 n R 3 1 Definitions of the symbols in eqation (A4) can be fond in the text. The radial components of displacement for these sorces contain only R 2 and can be neglected. [33] Compaction z D reslting from the dilatational sorce is given by D z h ¼ w Dðr; c h=2;# Þ w D ðr; c þ h=2;# Þ: ða5þ From eqations (A4) and (A5) we find D Fh 1 2n h z h ¼ h i 8pm 1 n 3=2 : ða6þ r 2 þ ðh=2þ 2 R 5 1 r z þ 2r r ¼ pt 2m 1 2n ; ða8þ 1 n qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where terms of the order h= r 2 þ ðh=2þ2 and h= r 2 l þðh=2þ2 in the denominator have been neglected. Note in eqation (A8) that the hydrostatic component of compaction is niform across the reservoir, having the same vale as that given by eqation (6) in the text. The tensile component of compaction, which is fond by repeating the calclation sing Okada s tensile sorce, is fond also to be niform with the vale given by eqation (8). A2. Srface Sbsidence [34] The analysis is similar to that for reservoir except that the axiliary stress system needed in addition to that in Figre 1c is a vertical force F, as shown in Figre 3. The vertical and radial displacements in the half-space indced by F are given by eqation (13) in the text and the vertical compaction in the half-space at the horizon of the reservoir is given by eqation (14). Applying the reciprocal theorem gives expressions for W r and W z, the two components of the work done by p T in Figre 1c against the radial and axial displacements in Figre 3. where W r ¼ 2hp T F 4pm Fð; r Þwð; r Þ ¼ W r þ W z Z p=2 ("! # c 1 R ð1 2nÞþ c R 3 þ c 1 ð1 2nÞþ r2 l c ) R 1 R 3 d# l W z ¼ hp T F Z ( p=2 Z r 3c 3 2nc 4pm R5 R 3 2rdr Z rl 3c 3 ) 2nc þ R5 R 3 2rdr d# ða9þ and r and r l are defined in Figre 5, and R 2 = r 2 + c 2, R l 2 = r l 2 + c 2. [35] Carrying ot the integration over r in eqation (A9) gives, after algebraic redction, the expression W r þ W z ¼ hp T F Z p=2 1 2n pm ð Þ 2 c c d#: R R l ða1þ

9 WALSH: SUBSIDENCE ABOVE A PLANAR RESERVOIR ETG 6-9 Note that eqation (A1) can be rewritten in the form W r þw z ¼ hp T F Z p=2 pm ð1 2nÞ Z r Z c rl R 3 rdrþ c R 3 rdr d#: ða11þ The calclation in eqation (A11) is simply the integration of the fnction (c/r 3 ) over the circlar planar area of the reservoir. The fnction (c/r 3 ) is identical to the Green s fnction sed by Geertsma and by Segall to describe the vertical displacement of a point on the srface cased by a ncles of dilatational strain in the reservoir; therefore, the integrations are identical and all of their nmerical reslts can be applied here. [36] Acknowledgments. Discssions with Herbert Wang and Tengfong Wong were invalable early in the project when I was attempting to rationalize my approach, and probing qestions later by Robert Meij, Pal Segall, Art McGarr, and Florian Lehner were invalable when I was trying to rationalize my reslts. John Rdnicki s commnication noted in the text was a major contribtion and one which had a fndamental inflence on the conclsions. I particlarly thank Robert Meij and John Rdnicki for their interest and perseverance. The National Science Fondation (grant EAR ) spported my recent work on this project. References Eshelby, J. D., The determination of the elastic field of an ellipsoidal inclsion, and other related problems, Proc. R. Soc. London, Ser. A, 241, , Fng, Y. C., Fondations of Solid Mechanics, vol. 1, Prentice-Hall, 525 pp., Old Tappan, N. J., Geertsma, J., Problems of rock mechanics in petrolem prodction engineering, paper presented at Congress of International Society of Rock Mechanics, Lisbon, Geertsma, J., A basic theory of sbsidence de to reservoir compaction: The homogeneos case, Verh. K. Ned. Geol. Mijnbowkd. Genoot., 28, 43 62, 1973a. Geertsma, J., Land sbsidence above compacting oil and gas reservoirs, J. Pet. Technol., 25, , 1973b. McGarr, A., On a possible connection between three major earthqakes in California and oil prodction, Bll. Seismol. Soc. Am., 81, , Mindlin, R. D., and D. H. Cheng, Ncleii of strain in the semi-infinite solid, J. Appl. Phys., 21, , 195. Okada, Y., Internal deformation de to shear and tensile falts in a halfspace, Bll. Seismol. Soc. Am., 82, , Segall, P., Earthqakes triggered by flid extraction, Geology, 17, , Segall, P., Indced stresses de to flid extraction from axisymmetric reservoirs, Pre Appl. Geophys., 139, , Segall, P., J.-R. Grasso, and A. Mossop, Poroelastic stressing and indced seismicity near the Lacq gas field, sothwestern France, J. Geophys. Res., 99, 15,423 15,438, Seo, K., and T. Mra, The elastic field in a half space de to ellipsoidal inclsions with niform dilatational eigenstrains, J. Appl. Mech., 46, , Timoshenko, S., and J. N. Goodier, Theory of Elasticity, 56 pp., McGraw- Hill, New York, Wong, T.-F., and J. B. Walsh, Deformation-indced gravity changes in volcanic regions, Geophys. J. Int., 16, , Yerkes, R. F., and R. O. Castle, Seismicity and falting attribtable to flid extraction, Eng. Geol., 1, , J. B. Walsh, Box 22, Adamsville, RI 281, USA.