Advances in Water Resources 78 (2015) 50–59

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Advances in Water Resources

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Ground subsidence due to uniform fluid extraction over a circular regionwithin an aquifer

http://dx.doi.org/10.1016/j.advwatres.2015.01.0150309-1708/� 2015 Elsevier Ltd. All rights reserved.

⇑ Corresponding author.

A.P.S. Selvadurai ⇑, Jueun KimDepartment of Civil Engineering and Applied Mechanics, McGill University, 817 Sherbrooke Street West, Montreal, QC H3A 0C3, Canada

a r t i c l e i n f o

Article history:Received 19 May 2014Received in revised form 26 January 2015Accepted 27 January 2015Available online 7 February 2015

Keywords:Biot poroelasticityFluid extractionGround subsidenceTime-dependent deformations

a b s t r a c t

This paper examines the ground subsidence caused by the uniform extraction of fluids from a disc-shapedregion located within a poroelastic halfspace. Biot’s theory of poroelasticity is used to examine theproblem. The fluid extraction over the circular region at the interior of the poroelastic domain inducestime-dependent axisymmetric surface settlements. The theoretical results are also used to examine theaccuracy of a multi-physics computational scheme that can be used to examine more complex geologicalsettings.

� 2015 Elsevier Ltd. All rights reserved.

1. Introduction

Ground subsidence during water withdrawal from aquifers is aproblem of major importance to environmental geosciences andenvironmental geomechanics. The problem of ground subsidenceis also exacerbated by issues related to climate change and non-re-plenishment of the groundwater regime and extraction of ground-water without considerations of aquifer recharge. The literaturerelated to ground subsidence is vast and can cover a diversity oftopics ranging from sink hole development due to internal erosionby subterranean flows, subsidence due to extraction of coal andother energy and mineral resources, to the conventional situationwhere the depletion of fluid pressures during groundwater extrac-tion, transfers the relief in the pore fluid pressures to the deform-able porous skeleton, which results in additional deformation thattranslates to subsidence. The reader is referred to the articles byPoland and Davis [1], Saxena [2], Poland [3], Barends et al. [4], Gal-loway and Burbey [5] and Galloway et al. [6] for records of theimpact of ground subsidence. The articles by Gambolati [7], Scott[8], Premchitt [9], Harada and Yamanouchi [10], Zekster et al.[11], Calderhead et al. [12] and Zeitoun and Wakshal [13] providefurther references to case studies dealing with ground subsidenceresulting from fluid withdrawal. The fluids being withdrawn neednot always be groundwater; extraction of hydrocarbon fuelsincluding oil and gas can also lead to ground subsidence. For exam-ple, the subsidence of the sea floor from the extraction of oil fromthe Ekofisk Oil Field, south west of Stavanger, Norway is estimated

to be almost 6 m. The reverse can also occur when fluids arepumped into formations. Records of examples of ground heaveduring fluid injection are rare and examples of ground heave dur-ing injection of CO2 into storage formations are documented byVasco et al. [14] and a strategy for anthropogenic uplift of Veniceby injection of brackish water to the underlying geologicalformations is discussed by Gambolati et al. [15] and Teatini et al.[16]. A comprehensive survey of sites worldwide experiencing landuplift caused by fluid injection (water, gas, steam) is given byTeatini et al. [17].

The analysis of problems involving ground subsidence is com-plex not only due to the variability of geological formations inwhich these processes take place but also due to the complexgeomechanical and fluid transport properties of the formations.In this area, the role of modelling is to present plausible approach-es that can be conveniently used to estimate ground subsidence influid-saturated geologic media. Furthermore, the purpose of ele-mentary modelling is to present solutions that can be used as a firstapproximation for the estimation of subsidence and to use suchresults for calibration of computational approaches.

This paper develops mathematical solutions to the problem ofextraction of fluids from a fluid saturated porous medium with adeformable skeleton. The classical Biot [18] theory of poroelasticityaccounts for Hookean elastic deformations of the porous skeletonand Darcy flow through the accessible pore space. The geome-chanical behaviour of the porous skeleton can be much more com-plicated than the elementary linear elastic model assumed in theclassical theory of poroelasticity. The influence of plasticity,damage and viscoelastic effects [9–25] have to be taken into

Fig. 1. Fluid extraction from a poroelastic halfspace.

A.P.S. Selvadurai, J. Kim / Advances in Water Resources 78 (2015) 50–59 51

consideration if irreversible effects are to be accommodated in themodelling of subsidence and/or heave. The linear elastic model,however, is a useful first approximation for modelling the skeletalbehaviour of geologic materials such as over-consolidated non-swelling clays, saturated limestones, sandstones, shale and othersoft rocks [26–31]. The linear theory of poroelasticity [18] (see alsoRice and Cleary [32], Detournay and Cheng [33], Selvadurai [34],Wang [35], Verruijt [36]) has been successfully applied to manyareas in the geosciences including ground water extraction andre-charge problems dealing with both homogeneous and stratifiedfluid-saturated regions [37–43]. (It is also surprising to note thatthe hydrogeology community generally seems to be unaware ofthese latter developments in geomechanics.) Owing to the linearityof the partial differential equations governing classical Biot poroe-lasticity, the solution to problems involving fluid injection into afluid-saturated poroelastic medium can also be recovered fromsolutions to the analogous problem involving fluid extraction.

The inclusion of coupling between the fluid flow process andthe skeletal deformations through the Biot poroelasticity modelis an improvement to the conventional modelling of the groundsubsidence or ground heave problem, which treats the geologicformation as an elastic solid and the primary effect of fluid injec-tion/extraction is modelled as a distribution of either centres ofcompression (simulating fluid extraction) or centres of dilation(simulating fluid injection). Examples of such studies are givenby Geertsma [44], Segall [45], Segall and Fitzgerald [46], Du andOlson [47] and more recently by Selvadurai [48–50].

The applications of the coupled Biot poroelasticity approach toproblems of fluid withdrawal have largely focused on the investi-gation of the point sink problem. The point sink fluid withdrawalproblem for a poroelastic solid of infinite extent has been investi-gated by Derski [51] and subsequently by Cleary [52] and Rudnicki[53]. It has also been investigated in the articles [37–43] cited pre-viously. In a practical application, however, the region over whichthe fluid extraction takes place is generally three-dimensional withan arrangement of extraction wells configured to control thesubsidence. An alternative is to restrict the state of deformationin the poroelastic medium to only displacements along a specifieddirection, which results in a simplification of the general three-dimensional equations of poroelasticity to simpler decoupledforms [54]. Since the governing equations of classical poroelasticityare linear, it is always possible to obtain effects of generaliseddistributions of injections of varying strength that are distributedin both space and time. The objective of this paper is to proceedbeyond the point sink problem and to develop certain solutionsthat can account for the dimensions of the region in which fluidis extracted at a specified rate. The canonical problem associatedwith the extraction from a poroelastic halfspace is to considerthe situation where these processes take place over a circulardisc-shaped region of finite radius (a) and zero thickness locatedat a finite depth (h) below the surface of the poroelastic halfspacepreserving axial symmetry (Fig. 1). To the authors’ knowledge thisproblem has not been treated in the literature in the earth sciencesor geomechanics; it is also useful to compare the solutionsobtained for the disc-shaped region, which has a finite arealdimension and a finite depth of location, with the results for thepoint sink problem that has dimensions only for the location ofthe sink point. The solution to the problem is obtained from therigorous mathematical analysis of the initial boundary value prob-lem, which, in view of the axial symmetry and time-dependency ofthe problem, uses Hankel and Laplace transform techniques. Thisprocedure can be used to develop numerical results for the prob-lem of extraction over the circular disc-shaped region. The analysisprovides both a comparison of results from a disc-shaped region ofextraction with the point sink problem and an assessment of theaccuracy of the multi-physics finite element code COMSOL™.

2. Governing equations

The fully coupled equations governing the mechanics of aporoelastic medium are given in several references (see e.g. Riceand Cleary [32]; Detournay and Cheng [33]; Selvadurai [34,55],Selvadurai and Yue [56]; Yue and Selvadurai [57]) and in thissection the relevant final forms of the equations governing thedisplacement and pore pressure fields are summarised. We alsorestrict attention to a state of axial symmetry in the fluid extrac-tion process which is applied with a prescribed time history overa circular disc-shaped region (Fig. 1). The dependent variables inthe problem are the skeletal displacements ur(r, z, t) and uz(r, z, t)in the r and z directions, respectively, and the pore fluid pressurep(r, z, t), which are governed by the system of coupled partialdifferential equations

G r2ur �ur

r2

� �� ð2g� 1Þ @H

@r¼ a

@p@r

ð1Þ

Gr2uz � ð2g� 1Þ @H@z¼ a

@p@z

ð2Þ

b@p@t� c

@H@t¼ cr2p ð3Þ

where

a¼ 3ðmu�mÞBð1�2mÞð1þmuÞ

; b¼ð1�2muÞð1�mÞð1�2mÞð1�muÞ

; c¼2GBð1�mÞð1þmuÞ3ð1�2mÞð1�muÞ

c¼2GB2ð1�mÞð1þmuÞ2k9ðmu�mÞð1�muÞcw

; g¼ ð1�mÞð1�2mÞ ; B¼ Cm�Cs

Cm�CsþnðCf �CsÞð4Þ

and H is the volumetric strain; r2 is the axisymmetric form ofLaplace’s operator given by

r2 ¼ @2

@r2 þ1r@

@rþ @2

@z2 ð5Þ

In Eqs. (1)–(4), G and m are, respectively, the shear modulus andPoisson’s ratio of the porous skeleton (i.e. the drained elastic para-meters); mu is the undrained Poisson’s ratio of the fluid-saturatedmedium; k is the hydraulic conductivity; cw is the unit weight ofpore fluid; B is Skempton’s pore pressure parameter; Cm is the com-pressibility of the porous skeleton; Cs is the compressibility of theskeletal material; Cf is the compressibility of the pore fluid and nis the porosity. Certain thermodynamic constraints need to be sat-isfied [32] to ensure positive definiteness of the strain energypotential; it can be shown that these constraints can be expressedin the forms: G > 0; 0 6 B 6 1; �1 < m < mu 6 0:5. Alternative but

52 A.P.S. Selvadurai, J. Kim / Advances in Water Resources 78 (2015) 50–59

equivalent descriptions are also available and these are describedby Detournay and Cheng [33], Wang [35] and further referencesare given in Selvadurai [34,55,58].

2.1. Methods of solution of the governing equations

We adopt the approach proposed by McNamee and Gibson [59]and Kanok-Nukulchai and Chau [41] where the partial differentialEqs. (1)–(3) governing poroelastic behaviour for axisymmetricstates of deformation can be represented in terms of two scalarfunctions S(r, z, t) and E(r, z, t) which satisfy the partial differentialequations

r2S ¼ 0 ð6Þ

cr4E ¼ bþ ac2Gg

� �r2 @E

@t� b

g@2S@z@t

ð7Þ

The displacements, total stresses, and pore fluid pressure can beuniquely represented in terms of S(r, z, t) and E(r, z, t) as follows:

ur ¼ �@E@rþ z

@S@r

; uz ¼ �@E@zþ z

@S@z� S; H ¼ r2E;

p ¼ 2Ga

@S@z� gH

� �ð8Þ

rrr

2G¼ @2

@r2 �r2

!E� z

@2S@r2 þ

@S@z

;

rzz

2G¼ @2

@z2 �r2

!E� z

@2S@z2 þ

@S@z

;rrz

2G¼ @2E@r@z

� z@2S@r@z

ð9Þ

The accuracy of the representations in terms of S(r, z, t) andE(r, z, t) can be verified by back-substitution into the Eqs. (1)–(3).The analytical solutions for the time-dependent axisymmetricporoelasticity problem can be obtained by using integral transformtechniques. Laplace and zeroth-order Hankel transforms are usedto remove, respectively, time-dependency and dependency onthe radial coordinate: i.e.

�Fðn; z; tÞ ¼Z 1

0rJ0ðnrÞFðr; z; tÞdr ð10Þ

~�Fðn; z; sÞ ¼ 12pi

Z 1

0e�st�Fðn; z; tÞdt ð11Þ

where ð�Þ refers to the zeroth-order Hankel transform and ðeÞ refersto the Laplace transform of a particular function. After successiveapplications of Laplace and zeroth-order Hankel transforms, thePDEs governing Eqs. (6) and (7) can be reduced to the following

ODEs for the transformed variables ~�Sðn; z; sÞ and ~�Eðn; z; sÞ: i.e.

d2

dz2 � n2

!~�S ¼ 0 ð12Þ

d2

dz2 � n2

!d2

dz2 � n2 þ sc

bþ ac2Gg

� �� �( )~�E ¼ � bs

gcd~�Sdz

ð13Þ

3. Fluid extraction from an internally located circular zone

We consider the problem of a poroelastic halfspace region thatis subjected to fluid extraction at a flow rate of Q0 (units L3/T),which represents the total volume of fluid that is extracted per unittime, over the plane circular region of radius a, located at a finitedepth h (Fig. 1) (i.e. As the radius of the extraction region aincreases the extraction per unit area decreases.) We reiterate

the assumption that the circular disc-shaped extraction regionhas zero thickness. The boundary conditions are prescribed in sucha way that the extraction rates are dimensionally consistent. Thespecification of the total volume rate of extraction over the circularregion of radius a also makes it possible to compare the results ofthe present study with the point extraction studies presented in theliterature. In practise, the specification of a uniform extraction rateover a region is perhaps unrealistic, but the solution is intended toprovide a fundamental result that can be used, with judicioussuperposition, to generate other non-uniform axisymmetricextraction scenarios. The boundary conditions and initial condi-tions governing the fluid extraction problem can be described byconsidering (i) a layer region, superscript ()L, occupying the regionr e (0,1); z e (0, �h) and (ii) a halfspace region, superscript ()H,occupying the region r e (0,1); z e (0,1). The boundary condi-tions applicable to the surface z = �h are

rðLÞzz ðr;�h; tÞ ¼ 0; rðLÞrz ðr;�h; tÞ ¼ 0; pðLÞðr;�h; tÞ ¼ 0 ð14Þ

and at the interface z = 0 the continuity conditions are

uðLÞr ðr;0; tÞ � uðHÞr ðr;0; tÞ ¼ 0; uðLÞz ðr;0; tÞ � uðHÞz ðr;0; tÞ ¼ 0 ð15Þ

rðLÞzz ðr;0; tÞ � rðHÞzz ðr;0; tÞ ¼ 0; rðLÞrz ðr;0; tÞ � rðHÞrz ðr;0; tÞ ¼ 0 ð16Þ

pðLÞðr;0; tÞ � pðHÞðr;0; tÞ ¼ 0 ð17Þ

When the interface is subjected to a fluid efflux over the pre-scribed circular region, a pressure gradient discontinuity occurs,which leads to the boundary condition

kcw

@p@z

� �ðLÞz¼0� k

cw

@p@z

� �ðHÞz¼0¼ Q 0

pa2 HðtÞ 0 6 r 6 a ð18Þ

kcw

@p@z

� �ðLÞz¼0� k

cw

@p@z

� �ðHÞz¼0¼ 0 a < r <1 ð19Þ

where H(t) is the Heaviside step function. The initial conditions gov-erning the fluid extraction problem are

rðLÞðx;0Þ ¼ rðHÞðx;0Þ ¼ uðLÞðx;0Þ ¼ uðHÞðx;0Þ ¼ 0 ð20Þ

pðLÞðx;0Þ ¼ pðHÞðx;0Þ ¼ 0 ð21Þ

In addition, the solution should satisfy the regularity conditionssuch that

rðiÞðx; tÞ; pðiÞðx; tÞ; uðiÞðx; tÞ ! 0; ði ¼ L;HÞ as jxj ! 1;t 2 ð0;1Þ ð22Þ

4. Final results for the poroelastic problem

The generalised solutions for the ODEs (12) and (13) for~�Sðn; z; sÞ and ~�Eðn; z; sÞ applicable to the poroelastic layer takethe forms

~�SðLÞðn; z; sÞ ¼ A1e�nz þ B1enz ð23Þ

~�EðLÞðn; z; sÞ ¼ C1e�nz þ D1enz þ E1e�uz þ F1euz þ CA1ze�nz

þ CB1zenz ð24Þ

and the solutions applicable to the semi-infinite region, which areselected to satisfy the regularity conditions, take the forms

~�SðHÞðn; z; sÞ ¼ A2e�nz ð25Þ

~�EðHÞðn; z; sÞ ¼ C2e�nz þ E2e�uz þ CA2ze�nz ð26Þ

Table 1Comparison of the numerical results with the analytical results in McNamee andGibson [62].

ct/a2 McNamee and Gibson [62]2G/fa(w � wt=0)

Present study Error (%)

0.01 0.1062 0.1128 6.2%0.1 0.35 0.3528 0.8%1 0.7239 0.7288 0.68%10 0.9102 0.9105 0.03%

A.P.S. Selvadurai, J. Kim / Advances in Water Resources 78 (2015) 50–59 53

where

u ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin2 þ s

cbþ ac

2Gg

� �s; C ¼ bG

2gbGþ acð27Þ

Altogether 9 arbitrary functions of n and s are encountered inthe formulation and these can be determined by satisfying theboundary conditions (14)–(19). The final expressions for the dis-placements, uz(r, z, t) and ur(r, z, t), and the pore water pressurecaused by the fluid extraction, p(r, z, t), take the forms

uðLÞz ðr;z;tÞQ 0cwa=2Gpak

¼Z fþi1

f�i1

Z 1

0J1ðanÞJ0ðrnÞ

e�nh�e�uh

gCðu�nÞ2�gðu2�n2Þþnðu�nÞ

!(

�"

nCðzþhÞ�C�nðzþhÞð Þe�nz�nh

þð2gnC�nÞgðu2�n2Þ

ue�uz�uh�ne�nz�nh #

þn e�nz�nhþenzþnh

e�nh

2gnðu2�n2Þþu e�uz�uhþeuzþuh

�e�uh

2guðu2�n2Þ

)est

sdnds ð28Þ

uðHÞz ðr;z;tÞQ 0cwa=2Gpak

¼Z fþi1

f�i1

Z 1

0J1ðanÞJ0ðrnÞ

e�nh�e�uh

gCðu�nÞ2�gðu2�n2Þþnðu�nÞ

!(

�"

nCðzþhÞ�C�nðzþhÞð Þe�nz�nh

þ 2gnC�nð Þgðu2�n2Þ

ue�uz�uh�ne�nz�nh #

þn e�nz�nhþe�nzþnh

e�nh

2gn u2�n2 þu e�uz�uhþe�uzþuh �e�uh

2guðu2�n2Þ

)est

sdnds ð29Þ

uðLÞr ðr;z;tÞQ0cwa=2Gpak

¼Z fþi1

f�i1

Z 1

0J1ðanÞJ0ðrnÞ

e�nh�e�uh

gCðu�nÞ2�gðu2�n2Þþnðu�nÞ

!(

�"

nCðzþhÞþ1�nðzþhÞð Þe�nz�nh

þnð2gnC�nÞgðu2�n2Þ

e�uz�uh�e�nz�nh #

�n enzþnh�e�nz�nh

e�nh

2gnðu2�n2Þ�n euzþuh�e�uz�uh �e�uh

2guðu2�n2Þ

)est

sdnds ð30Þ

uðHÞr ðr;z;tÞQ0cwa=2Gpak

¼Z fþi1

f�i1

Z 1

0J1ðanÞJ0ðrnÞ

ðe�nh�e�uhÞgCðu�nÞ2�gðu2�n2Þþnðu�nÞ

!(� nCðzþhÞþ1�nðzþhÞð Þe�nz�nh�þnð2gnC�nÞ

gðu2�n2Þe�uz�uh�e�nz�nh #

�n e�nzþnh�e�nz�nh

e�nh

2gnðu2�n2Þ�n e�uzþuh�e�uz�uh �e�uh

2guðu2�n2Þ

)est

sdnds ð31Þ

pðLÞðr;z;tÞQ 0cwa=pak

¼Z fþi1

f�i1

Z 1

0J1ðanÞJ0ðrnÞ �

e�uz�2uh�euz

2u

� ���ð2gCn�nÞ e�nh�e�uh

e�nz�nh�e�uz�uh

gCðu�nÞ2�gðu2�n2Þþnðu�nÞ

" #)est

sdnds ð32Þ

pðHÞðr;z;tÞQ 0cwa=pak

¼Z fþi1

f�i1

Z 1

0J1ðanÞJ0ðrnÞ �

e�uz�2uh�e�uz

2y

� ���ð2gCn�nÞ e�nh�e�uh

e�nz�nh�e�uz�uh

gCðu�nÞ2�gðu2�n2Þþnðu�nÞ

" #)est

sdnds ð33Þ

where f is a real number related to the Bromwich contour integra-tion associated with Laplace transform inversion [60]. The expres-sion for the total stresses in the layer and halfspace region aregiven in Appendix A. Both operations involving Hankel transforminversion and Laplace transform inversion have to be carried outto generate expressions for the displacements and stresses that

can be expressed in terms of r, z and t. To the author’s knowledge,the transformed expressions of the type (28)–(33) and those givenin Appendix A cannot be inverted exactly to generate explicit ana-lytical results for the displacements, pore fluid pressures and totalstresses. Therefore numerical techniques need to be used to per-form Hankel and Laplace transform inversions. In this study, theintegral transforms are evaluated using the algorithms given inMATLAB

�. In particular, the inverse Laplace transform is evaluated

using the procedures developed by Crump [61]. The accuracy ofthe inversion technique is confirmed through comparisons withresults obtained by McNamee and Gibson [62] for the axisymmetricsurface loading, f, of a halfspace region. The numerical results andthe analytical results given by [62] are presented in Table 1. Theerror substantially decreases when t P 0:1 (see also Table 1 inYue and Selvadurai [63]) and becomes less than 0.1% whenct/a2 = 10. The procedures are therefore considered satisfactory forthe purpose of developing a full range of analytical results for theproblem of fluid extraction over a circular region located at theinterior of a fluid-saturated halfspace region. Specific numericalresults will be presented in a subsequent section.

5. Numerical solutions and validation of COMSOL™

An objective of developing an analytical solution to the Biotporoelasticity modelling of the fluid extraction over a circularregion is also to use the result to calibrate the accuracy of compu-tational schemes that can be used to examine extraction scenarioswith more complicated geometries and geological features that donot lend themselves to convenient solution via analytical methods.In this study we also model the axisymmetric problem of fluidextraction over a circular region within a poroelastic halfspaceusing the multi-physics based finite element software, COMSOL™.Fig. 2 shows the domain considered in the finite element simula-tion of the poroelastic halfspace region. Since the problem isaxisymmetric, the entire domain can be simulated by a two-di-mensional equivalent referred to the cylindrical polar coordinatesystem (r, h, z), with appropriate boundary conditions specifiedon the axis of symmetry. Ideally the poroelastic semi-infinitedomain should be modelled using infinite elements [64]. In thisstudy the region modelled has a finite geometry and suitableboundary conditions can be prescribed for the displacements, trac-tions and pore fluid pressures. The dimensions of the finite elementdomain should be sufficiently large to ensure that the far-fieldboundary conditions, which are the regularity conditions indicatedby (22), can be accurately accommodated by the boundary condi-tions indicated in Fig. 2.

The finite element mesh used in the modelling is shown inFig. 3. The mesh consists of triangular elements with 3 degrees offreedom at a nodal point. These elements have been used quitesuccessfully in previous investigations where fluid flow, poroelas-ticity, poroelasto-plasticity and thermo-poroelasticity problemshave been examined [24,25,30,65–68]. In the final computationsperformed, the finite element mesh consisted of 5147 elements(in the case of a/h = 10, the number of elements was increased to15,000) with 3 degrees of freedom at a nodal point. The COMSOL™

Fig. 2. The geometry of the region and the boundary conditions used in the finite element modelling.

Fig. 3. Mesh configuration for the finite element modelling of the fluid extraction problem.

54 A.P.S. Selvadurai, J. Kim / Advances in Water Resources 78 (2015) 50–59

multi-physics finite element code has an automatic mesh gen-eration routine and mesh refinement can be attained at locationswhere the boundary conditions change abruptly.

The following material parameters were used in thecomputations.

G ¼ 20:0 GPa; m ¼ 0:25; k ¼ 10�5 m=s; cw ¼ 9:81 kN=m3; mu ¼ 0:5

5.1. Surface displacement

Fig. 4 compares the analytical solutions given in this study forthe cases of fluid extraction at a point extraction and over a circulardisc region (a/h = 0.1) and those obtained by Kanok-Nukulchai andChau [41]. Uz(0, �h,1), the displacement corresponding to thenon-dimensional time t⁄(=ct/h2) =1 at r = 0, indicates a steady-state response and the analytical expression is given as

uzðr;�h;1Þa=h¼0:1

aQ 0cw=2Gpak¼Z 1

0J1ðanÞJ0ðrnÞ

he�nh

ð�2gþ 1Þn

" #dn ð34Þ

In the computation, the limit of infinite time is effectivelyreached when t⁄ � 1000. The results developed in this study arenormalised with respect to the steady-state response at r = 0 andare consistent with the results given by Kanok-Nukulchai and Chau[41].

The normalised surface displacements for different values of a/hat different times, t⁄, are shown in Fig. 5. All the values are nor-malised by the maximum settlement for the case, a/h = 0.1, whichapproximately models the point sink solution. When a/h is small,the results show a narrow inverted-bell shape, and, as a/h increas-es, the surface displacement becomes flattened in the extractionregion. The results obtained are the same for both cases of a/h = 0.1 and a/h = 0.001. This indicates that the results for the circu-lar extraction region converge to those for a point source problemas the relative radius of embedment a/h decreases, which, in thiscalculation, was a=h 6 0:1 (Fig. 5). The maximum surfacedisplacement decreases as a/h increases. The maximum settlementfor the case a/h = 10 is approximately 18% of that for the casewhere a/h = 0.1. Also the time required to reach a certain degree

Fig. 4. Comparison of the analytical solutions of the point source and circularregion source problems obtained in this study with results given by Kanok-Nukulchai and Chau [41].

Fig. 5. Surface displacements for different circular disc sizes.

Fig. 7. Maximum non-dimensionalized surface settlement for different a/h.

Fig. 6. Non-dimensionalized surface displacement at different times t⁄.

A.P.S. Selvadurai, J. Kim / Advances in Water Resources 78 (2015) 50–59 55

of surface settlement increases as a/h increases. The surface dis-placement reaches approximately 20% of the maximum settlementat the non-dimensional time t⁄ = 10 when a/h = 5; however, whena/h = 10, the time required to reach 20% of the maximum settle-ment is t⁄?1. The corresponding computational results obtainedin this study are shown as circles. The analytical and the computa-tional results show good agreement.

Fig. 6 shows the surface displacements normalised with respectto the volume flow rate, Q0, for t⁄ = 1, t⁄ = 10, and t⁄?1. Asexpected from the result (28), the surface displacements and thevolume rates show a near-linear relationship. If the volume ratedoubles, the surface displacement also doubles. The correspondingcomputational results are depicted by circles. The computationalresults slightly overestimate the displacement value at t⁄ = 1; how-ever, the agreement between the analytical solutions and the com-putational solutions is satisfactory.

The non-dimensional values of the maximum surfacesettlement for different values of a/h are shown in Fig. 7. The

surface displacement does not change until it reaches up to a/h = 0.1. It is consistent with the previous observation that the sur-face settlement converges to the point sink solution whena=h 6 0:1. The surface settlement starts to decrease as a/h exceeds0.1 and converges to zero when a/h > 102.

Fig. 8 shows the surface displacement for different Poisson’sratios. For the case mu = 0.5, the surface settlement is larger whenm is smaller. The surface settlement for one extreme case (m = 0,mu = 0.01) is also presented in Fig. 8. It was observed in this studythat the surface settlement for the case m = 0 is larger than whenm – 0.

5.2. Change in pore pressure by fluid extraction

Fig. 9 shows the variation of pore pressure along the z-axis dueto the fluid extraction as t⁄?1. As a/h increases, the change inpore pressure decreases. Unlike the surface displacement, the porepressure obtained in the case a=h 6 0:1 is not the same as that for apoint source. In particular, there is a limited change in the pore

Fig. 8. Surface settlement for different values of Poisson’s ratio when a/h = 0.1. Fig. 9. Changes in pore pressure at t⁄?1 for different a/h and the comparisonwith the point sink solutions.

56 A.P.S. Selvadurai, J. Kim / Advances in Water Resources 78 (2015) 50–59

pressure for the case of a circular injection region whereas the porepressure becomes singular for a point source (Fig. 9). The corre-sponding computational solutions are denoted by circles and theyshow good agreement with the analytical solutions with a maxi-mum discrepancy of 1.0%.

Fig. 10. Comparison of the surface subsidence solution with Geertsma’s [44]results.

6. Comparison with Geertsma’s solution

Geertsma [44] presented a solution to the elastic analysis of theproblem of a disc-shaped reservoir undergoing a uniform pressuredecline Dp over a region 0 < r < a, located at a depth h. The analy-tical solution for the vertical displacement at the surface of thehalfspace given by Geertsma [40] can be re-written as

½uzðr;0Þ�Geertsma

p0ð1� 2mÞa=2G¼ u�z ¼ �2a

H0

a

� �Z 1

0J1ðnÞJ0ðnqÞe�nddn ð35Þ

where H0 is the thickness of the pressurized zone, p0 is the pressure,a is the Biot coefficient given in (4) and

q ¼ ðr=aÞ; d ¼ ðh=aÞ ð36Þ

For comparison, we examined the problem of the pressurizationof the poroelastic halfspace by a circular disc-shaped region. Thegoverning equations and the boundary conditions are the sameexcept for the pressure gradient jump conditions (18) and (19);in this case, the boundary conditions (18) and (19) become

pðLÞðr;0; tÞ ¼ pðHÞðr;0; tÞ ¼ p0HðtÞ 0 6 r 6 a ð37Þ

pðLÞðr;0; tÞ ¼ pðHÞðr;0; tÞ ¼ 0 a < r <1 ð38Þ

Again, the circular disc-shaped pressurization region is assumedto have zero thickness. The analytical solution for the axial dis-placement at the surface of the poroelastic halfspace region takesthe form

uðLÞz ðr;�h;tÞp0a=2G

¼Z fþi1

f�i1

Z 1

0J1ðanÞJ0ðrnÞ 2guð1�e�nhþuhÞ�

gðu�nÞ e�uh�euhð Þ gCð½

�

Since the solution by Geertsma [44] (see also [48–50]) is basedon an elasticity formulation, the results are applicable only for thelimiting cases t = 0 and t ?1. The results for t = 0 will be inappli-cable since the undrained medium has m = 0.5 at t = 0 and theresults (35) will give zero displacement. For the solution obtainedin this study, the limiting case t ?1 is obtained by using the finalvalue theorem of the Laplace transform and this is given as

½uzðr;�h;0Þ�Present Study

p0ð1�2mÞa=2G¼u�z ¼

Z 1

0J1ðnÞJ0ðnqÞ

2deð�ndÞ

ðeð�2ndÞ �1Þ

� �dn ð40Þ

The non-dimensional surface displacement (35) developed byGeertsma [44] and the equivalent result (40) developed in thispaper, cannot be directly compared in view of the arbitrariness

1ðu�nÞ�gðuþnÞþnÞþ2gu 1�e�nhþuhð Þð2gCn�nÞ e�uh�e�nhð Þ�

est

sdnds

ð39Þ

A.P.S. Selvadurai, J. Kim / Advances in Water Resources 78 (2015) 50–59 57

that is introduced through the multiplier (2aH0/a). It is, howev-er, instructive to compare the displacement profiles obtainedfrom the result given by (35) with the displacement profile gen-erated through (40) by setting, d ¼ ðh=aÞ ¼ 1, and setting themultiplier 2aH0/a � 2.75, such that the displacements derivedfrom both solutions (35) and (40) are identical at the centrallocation at the surface, (0,0). Fig. 10 illustrates the deflectionprofiles obtained from both solutions given by (35) and (40).

7. Conclusions

The paper develops analytical solutions for the transient poroelas-ticity problem related to fluid extraction over a circular disc-shapedregion located at the interior of a poroelastic halfspace. The solutionexamines the coupled influences of skeleton deformations and porefluid pressures. The analytical solutions are used to calibrate theresults obtained from the finite element-based multi-physics codeCOMSOL™. The computational results compare favourably with theanalytical solutions and the maximum discrepancy between thetwo sets of results is less than 1% (excluding the results in Fig. 6).

For the problem considered, we observe that for a constant flowrate at a fixed depth of extraction, the surface displacement andthe pore fluid pressure depend on the size of the withdrawalregion. Both the surface displacement and the pore fluid pressuredecrease as the radius of the disc-shaped fluid extraction regionincreases. We also observe that the solution for the surface dis-placement converges to the solution obtained for the problem ofa point sink when the ratio between the radius of the extractionregion and its depth of location is less than or equal to 0.1. Howev-er, unlike the displacement solution, the convergence to the pointsink solution was not observed for the pore fluid pressure. This isdue to the singular nature of the point extraction solution.

8. Disclaimer

The use of the computational code COMSOL Multiphysics™ is onlyfor demonstration purposes only. The authors neither advocate norrecommend the use of the code without conducting suitable valida-tion procedures to test the accuracy of the code in a rigorous fashion.

Acknowledgements

The work described in this paper was supported by a NSERCDiscovery Grant awarded to the first author. The second authoracknowledges the partial research support provided by a MEDAAward. The authors are grateful to the referees for their construc-tive comments and recommendations.

Appendix A

The analytical solutions for the displacements and pore fluid pres-sure are given in Section 4 (Eqs. (28)–(33)). This Appendix containsthe remaining analytical results for the problem examined in this paper.

rðLÞrr ðr;z;tÞQ 0cwa=pak

¼Z fþi1

f�i1

Z 1

0J1ðanÞJ0ðrnÞ�

("2nC�n2CðzþhÞ

þn2ðzþhÞ�2n e�nh�e�uh

gCðu�nÞ2�gðu2�n2Þþnðu�nÞ

!e�nz�nh

þ 2gCn�n

gðu2�n2Þn2e�nz�nh�u2e�uz�uh e�nh�e�uh

gCðu�nÞ2�gðu2�n2Þþnðu�nÞ

!þ

u2 e�uz�2uh�euz 2guðu2�n2Þ

�n2 e�nz�2nh�enz 2gnðu2�n2Þ

#)est

sdnds ðA1Þ

rðHÞrr ðr;z;tÞQ0cwa=pak

¼Z fþi1

f�i1

Z 1

0J1ðanÞJ0ðrnÞ�

��2nC�n2CðzþhÞ

þn2ðzþhÞ�2n e�nh�e�uh

gCðu�nÞ2�gðu2�n2Þþnðu�nÞ

!e�nz�nh

þ 2gCn�n

gðu2�n2Þ

�n2e�nz�nh�u2e�uz�uh

�e�nh�e�uh

gCðu�nÞ2�gðu2�n2Þþnðu�nÞ

!þ

u2 e�uz�2uh�e�uz 2guðu2�n2Þ

�n2 e�nz�2nh�e�nz 2gnðu2�n2Þ

#)est

sdnds ðA2Þ

rðLÞzz ðr;z; tÞQ0cwa=pak

¼Z fþi1

f�i1

Z 1

0J1ðanÞJ0ðrnÞ�

��ðn2C�n2ÞðzþhÞ

e�nh�e�uh

gCðu�nÞ2�gðu2�n2Þþnðu�nÞ

!e�nz�nh

þ2gCn3�n3

gðu2�n2Þe�nh�e�uh

gCðu�nÞ2�gðu2�n2Þþnðu�nÞ

!e�uz�uh�e�nz�nh

þ n

2gðu2�n2Þe�nz�2nh�enz

þ n2

2guðu2�n2Þeuz�e�uz�2uh #)est

sdnds ðA3Þ

rðHÞzz ðr;z;tÞQ0cwa=pak

¼Z fþi1

f�i1

Z 1

0J1ðanÞJ0ðrnÞ�

��ðn2C�n2ÞðzþhÞ

e�nh�e�uh

gCðu�nÞ2�gðu2�n2Þþnðu�nÞ

!e�nz�nh

þ2gCn3�n3

gðu2�n2Þe�nh�e�uh

gCðu�nÞ2�gðu2�n2Þþnðu�nÞ

!e�uz�uh�e�nz�nh

þ n

2gðu2�n2Þe�nz�2nh�e�nz

þ n2

2guðu2�n2Þe�uz�e�uz�2uh #)est

sdnds ðA4Þ

rðLÞrz ðr;z;tÞQ0cwa=pak

¼Z fþi1

f�i1

Z 1

0J1ðanÞJ0ðrnÞ�

��ðn2C�n2ÞðzþhÞ�Cþn� �

e�nh�e�uh

gCðu�nÞ2�gðu2�n2Þþnðu�nÞ

!e�nz�nh

þ 2gCn�n

gðu2�n2Þ

�nye�uz�uh�n2e�nz�nh

�e�nh�e�uh

gCðu�nÞ2�gðu2�n2Þþnðu�nÞ

!�

nu e�uz�2uh�euz 2guðu2�n2Þ

þn2 e�nz�2nh�enz 2gnðu2�n2Þ

#)est

sdnds ðA5Þ

rðHÞrz ðr;z;tÞQ0cwa=pak

¼Z fþi1

f�i1

Z 1

0J1ðanÞJ0ðrnÞ�

��ðn2C�n2ÞðzþhÞ�Cþn

e�nh�e�uh

gCðu�nÞ2�gðu2�n2Þþnðu�nÞ

!e�nz�nh

þ 2gCn�n

gðu2�n2Þ

�nye�uz�uh�n2e�nz�nh

�e�nh�e�uh

gCðu�nÞ2�gðu2�n2Þþnðu�nÞ

!

�nu e�uz�2uh�euz 2guðu2�n2Þ

þn2 e�nz�2nh�enz 2gnðu2�n2Þ

#)est

sdnds ðA6Þ

58 A.P.S. Selvadurai, J. Kim / Advances in Water Resources 78 (2015) 50–59

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