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ResearchCite this article: Selvadurai APS, Kim J. 2016Poromechanical behaviour of a surficialgeological barrier during fluid injection into anunderlying poroelastic storage formation.Proc. R. Soc. A 472: 20150418.http://dx.doi.org/10.1098/rspa.2015.0418

Received: 20 June 2015Accepted: 5 February 2016

Subject Areas:civil engineering, environmental engineering,applied mathematics

Keywords:Biot poroelasticity, mechanics of geologicalseals, geological barriers, ground heave,time-dependent deformations

Author for correspondence:A. P. S. Selvaduraie-mail: [email protected]

Electronic supplementary material is availableat http://dx.doi.org/10.1098/rspa.2015.0418 orvia http://rspa.royalsocietypublishing.org.

Poromechanical behaviour ofa surficial geological barrierduring fluid injection into anunderlying poroelastic storageformationA. P. S. Selvadurai and Jueun Kim

Department of Civil Engineering and Applied Mechanics,McGill University, 817 Sherbrooke Street West, Montréal, Quebec,Canada H3A 0C3

A competent low permeability and chemically inertgeological barrier is an essential component of anystrategy for the deep geological disposal of fluidizedhazardous material and greenhouse gases. While theprocesses of injection are important to the assessmentof the sequestration potential of the storage formation,the performance of the caprock is important to thecontainment potential, which can be compromisedby the development of cracks and other defects thatmight be activated during and after injection. Thispaper presents a mathematical modelling approachthat can be used to assess the state of stress in asurficial caprock during injection of a fluid to theinterior of a poroelastic storage formation. Importantinformation related to time-dependent evolution ofthe stress state and displacements of the surficialcaprock with injection rates, and the stress state in thestorage formation can be obtained from the theoreticaldevelopments. Most importantly, numerical resultsillustrate the influence of poromechanics on thedevelopment of adverse stress states in the geologicalbarrier. The results obtained from the mathematicalanalysis illustrate that the surface heave increases asthe hydraulic conductivity of the caprock decreases,whereas the surface heave decreases as the shearmodulus of the caprock increases. The results alsoillustrate the influence of poromechanics on thedevelopment of adverse stress states in the caprock.

1. IntroductionThe injection of fluids into porous formations is regardedas a procedure for managing contaminants and other

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surficial soils

g

fh

h

e

a

d

bc

i

surficial rocks

cap rock

salineaquifer

basementrock

deep salineaquifers

injection well

lake

CO2 plume

Figure 1. Schematic of scCO2 injection into a saline aquifer. (a) Geochemistry and geomechanics of the saline aquifer, (b) plumedevelopment, trapping, pore fabric alterations during scCO2 injection, (c) stability of scCO2-saline pore fluid interfaces in rock,(d) geochemistry and geomechanics of caprock and interface seals, (e) CO2 diffusion and transport through caprock defectsand interface seals, (f ) plume development in surficial rocks, (g) CO2 contamination of groundwater regime, (h) monitoring ofcaprock, (i) movement of saline fluids from deep saline aquifer to the storage horizon [7]. (Online version in colour.)

hazardous waste [1–3]. Deep injection of fluidized forms of greenhouse gases into storageformations has been advocated as a means of mitigating the effects of climate change [4–7].The process of injection into a porous storage formation pre-supposes that the fluid can beinjected without detriment to the fabric of the porous medium and the geological setting canaccommodate hydrodynamic trapping that is provided by a stable geological barrier (figure 1).How the caprock performs is therefore of critical importance to the longevity and effectivenessof the storage strategy. In the context of geological sequestration of greenhouse gases, thefactors controlling the mechanics of the caprock during the injection process has to take intoconsideration a complex set of thermo-hydro-mechanical–chemical (THMC) processes. Thesecan result from the geochemistry of the storage formation, the temperatures of the fluids beinginjected, which can perturb the in situ geothermal gradients, the buoyancy anomalies created bythe injected fluid, the development of migration plumes that correspond to moving boundariesand chemomechanical processes that can lead to deterioration of the fabric of the storageformation. A comprehensive mathematical treatment of all these aspects presents a challengingproblem in environmental geomechanics and such investigations are best addressed throughinvestigations of site-specific situations. If attention is restricted to the investigation of caprockperformance, then the hydromechanical (HM) processes that are created during injection cometo the forefront. For example, the injection rates that will not cause distress to the caprock interms of the potential for fracturing of a given caprock–storage formation can be examined


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by considering the appropriate HM problem. Even in a simplified HM investigation of thecaprock–storage formation interaction, the actual geological setting is important and real-lifeproblems are best examined through the adoption of a computational procedure. Some progresstowards understanding caprock–storage formation interaction during internal injection can bemade by examining the poroelasticity effects of the interaction. The idealization of the storageformation as a poroelastic medium without the influences of chemical interaction is consideredto be appropriate for situations where the materials encountered are competent sandstones.This has been demonstrated by the selection of many storage settings in sandstone formations(e.g. Sleipner, Snøhvit [8], In Salah [9], Nagaoka [10]). When the storage medium has a dominantcarbonate content, the reactive nature of carbon dioxide acidized water can lead to dissolutionof the carbonates causing material erosion and the formation of features such as wormholesthat can compromise the integrity of the storage effort (Selvadurai APS, Couture C-B. 2016Wormhole generation in carbonate zones during reactive flows. McGill University, unpublished).Furthermore, the dissolved carbonates can accumulate in remote regions of the storage formation,which can result in pore clogging and ultimately lead to hydraulic fracture during continuedinjection into the storage formation, thus limiting its trapping potential. In addition, if extensivedissolution takes place in the storage horizon, then this can lead to collapse of the wormholes undergeostatic stresses that can lead to caprock distress owing to loss of geological structural supportprovided by the storage formation.

In this paper, we focus attention on the mathematical modelling of the hydromechanicaleffects that can materialize when a storage formation underlying a caprock layer of uniformthickness is pressurized by the injection of a fluid over a circular region located at a finite distancefrom the base of the caprock. The objective of the study was to determine the stress state inthe caprock during the injection process. If injection is modelled as the rise and maintenanceof the pressure in a specified region in the storage formation [11], then the state of stress inthe caprock can be examined using an elastostatic model of the storage formation where thecaprock is idealized as a structural element with a response that can be modelled as eithera Poisson–Kirchhoff thin plate [5,6] or as a Mindlin–Reissner thick plate [12]. The elastostaticmodel can also provide quantitative estimates for the deflections of the caprock in the long term.The elastostatic model, however, cannot provide an assessment of the time-dependent effectsof injection associated with geologic sequestration activities. In particular, the rates of injectionneed to be controlled in such a way that hydraulic fracturing will not be initiated within thestorage horizon and the injection pressures will not compromise the sealing capacity of thecaprock by initiation and extension of fractures. Because the primary focus of the paper is onthe development of a mathematical solution to study the caprock–storage formation interactionduring injection of fluids to a storage formation, we consider the situation where the caprocklayer is located at the surface of a semi-infinite storage formation. In a practical situation, thecaprock layer may be an embedded stratum (figure 1), but the consideration of a surface layerprovides the opportunity to examine how the absence of confinement influences the performanceof the caprock layer. The particular focus of this study is to provide an analytical result thatcan serve as a useful benchmark for examining the accuracy of computational approachesrather than to provide an analysis of a specific engineering situation associated with geologicalsequestration, which will aid the general applicability of the study. Even within the context ofa simplification that incorporates a surficial caprock layer underlain by a semi-infinite storagehorizon, the analytical solution of the poroelasticity problem can involve extensive mathematicalmanipulations. Neglecting the influence of a poroelastic surficial geological region is a reductionthat makes the benchmark exercise manageable in a theoretical context. In this regard, it shouldbe noted that the case of a more practical situation involving a caprock stratum embeddedbetween a surficial geological medium and a storage horizon, the influence of the surficialgeology can be approximately accounted for by specifying the geostatic loads associated withthe surficial rocks. In such a treatment, the absence of poroelastic effects in the geologicalmedium overlying the caprock will lead to an upper limit of caprock deflections during theinjection process.



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zporoelastic medium

l

y

poroelastic caprock layer circular fluid injection zone

hx

a

Figure 2. Fluid injection at the interior of a halfspace overlain by a caprock layer. (Online version in colour.)

To preserve axial symmetry of the modelling, we consider the case where the fluid injectionis carried out over a circular region located at a finite depth from the boundary of the caprock(figure 2).

The modelling adopts a poroelastic formulation for the mechanical behaviour of both thecaprock and the storage formation. The poroelastic formulation of a defect-free geologicalmedium is considered to be a satisfactory model for rocks that are encountered in competentstorage horizons and with caprocks that are free of major defects. This allows the useof Biot’s classical theory of poroelasticity [13] for the study of the problem indicated infigure 2, by assigning different poroelasticity representations for both the storage horizon andthe caprock. The governing equations of poroelasticity for mechanically and hydraulicallyisotropic fluid-saturated porous elastic media are given by several authors including Rice &Cleary [14], Atkinson & Craster [15], Cheng [16], Craster & Atkinson [17], Selvadurai [18–20], Selvadurai & Yue [21] and Selvadurai & Mahyari [22,23]. The mathematical formulationextends previous studies [24,25] that investigated, respectively, the axisymmetric problem of aporoelastic halfspace region with fluid extraction over a circular planar region located at theinterior of a poroelastic halfspace and the three-dimensional problem of line-injection sourceslocated in a poroelastic halfspace region. This study considers the more complex problem wherethe surface of the halfspace region contains a contiguous poroelastic layer with continuity ofdisplacements, tractions and pore fluid pressures and fluxes. The mathematical analysis of theproblem is developed through the application of Laplace and Hankel transform techniques,which reduces the initial boundary value problem to a set of ordinary differential equationsthat can be solved in exact closed form; numerical procedures are then used to invert both theLaplace and Hankel transforms. The paper presents numerical results for the surface deflectionsof the caprock and its stress state at critical locations as a function of the fluid injection rate andother mechanical and fluid flow characteristics of the geologic materials. The emphasis of thepaper is on the study of the poroelastic response of geological media encountered in sequestrationscenarios. The work can be extended to include more advanced nonlinear constitutive modellingof the rocks [26–30] to include thermoporoelastoplastic effects in sequestration activities. In thecase of geological sequestration, however, the geological settings are usually chosen such thatplasticity effects are negligible, but on occasions, such influences are unavoidable. The role ofplasticity can have a notable influence on the mechanical behaviour of rocks in the vicinity ofthe injection location. The results of Selvadurai & Suvorov [30] for the thermoporoelastoplasticityproblem for the internal pressurization of a cavity indicate that plasticity effects can contribute toan increase in displacements in the vicinity of the pressurized cavity.

2. Governing equationsThe fully coupled equations governing the mechanics of a poroelastic medium are given in thereferences cited previously in this section, the relevant final forms of the equations governing



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the displacement and pore pressure fields are presented. Attention is restricted to a state of axialsymmetry in the fluid injection process, which is applied with a prescribed time history over acircular disc-shaped region (figure 2). The dependent variables of the problem are the skeletaldisplacements ur(r, z, t) and uz(r, z, t) in the r- and z-directions, respectively, and the pore fluidpressure p(r, z, t), which are governed by

G(

∇2ur − urr2)

− (2η − 1)∂Θ∂r

= α ∂p∂r

, (2.1)

G∇2uz − (2η − 1)∂Θ∂z

= α ∂p∂z

(2.2)

and β∂p∂t

− γ ∂Θ∂t

= c∇2p. (2.3)

where

α = 3(νu − ν)B(1 − 2ν)(1 + νu) ; β =

(1 − 2νu)(1 − ν)(1 − 2ν)(1 − νu) ; γ =

2GB(1 − ν)(1 + νu)3(1 − 2ν)(1 − νu)

c = 2GB2(1 − ν)(1 + νu)2k

9(νu − ν)(1 − νu)γw ; η =(1 − ν)(1 − 2ν) ; B =

Cm − CsCm − Cs + n(Cf − Cs)

,

⎫⎪⎪⎪⎬⎪⎪⎪⎭ (2.4)and Θ is the volumetric strain. In addition, ∇2 is the axisymmetric form of Laplace’s operatorgiven by

∇2 = ∂2

∂r2+ 1

r∂

∂r+ ∂

2

∂z2. (2.5)

In equations (2.1)–(2.4), G and ν are, respectively, the shear modulus and Poisson’s ratio of theporous skeleton (i. e. the drained elastic parameters); νu is the undrained Poisson’s ratio of thefluid-saturated medium; k is the hydraulic conductivity; γw is the unit weight of the pore fluid;B is Skempton’s pore pressure parameter; Cm is the compressibility of the porous skeleton; Csis the compressibility of the skeletal material; Cf is the compressibility of the pore fluid and nis the porosity. The constitutive parameters have to satisfy certain thermodynamic constraintsto ensure positive definiteness of the strain energy potential; it has been shown [14] that theseconstraints can be expressed in the forms: G > 0; 0 ≤ B ≤ 1; −1 < ν < νu ≤ 0.5. Alternative butequivalent descriptions are described by Cheng [16], Wang [31] and further references are givenin Selvadurai [18,19].

(a) Solution method of the governing equationsSeveral approaches for the solution of problems in poroelasticity have been proposed in theliterature [19,21,32–41]. We use the approach proposed by McNamee & Gibson [34,35] andGibson & McNamee [36] where the solution to the governing coupled partial differentialequations (2.1)–(2.3) can be represented in terms of two scalar functions S(r, z, t) and E(r, z, t),which satisfy the partial differential equations

∇2S = 0 (2.6)

and

c∇4E =(

β + αγ2Gη

)∇2 ∂E

∂t− β

η

∂2S∂z∂t

. (2.7)

The displacements, total stresses and pore fluid pressure can be uniquely represented in terms ofS(r, z, t) and E(r, z, t) as follows

ur = −∂E∂r

+ z∂S∂r

; uz = −∂E∂z

+ z∂S∂z

− S; Θ = ∇2E; p = 2Gα

(∂S∂z

− ηΘ)

. (2.8)



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σrr

2G=(

∂2

∂r2− ∇2

)E − z∂

2S∂r2

+ ∂S∂z

σθθ

2G=(

∂2

r∂r− ∇2

)E − z

r∂S∂r

+ ∂S∂z

σzz

2G=(

∂2

∂z2− ∇2

)E − z∂

2S∂z2

+ ∂S∂z

σrz

2G= ∂

2E∂r∂z

− z ∂2S

∂r∂z.

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(2.9)

The accuracy of the representations (2.8) and (2.9) in terms of S(r, z, t) and E(r, z, t) can be verifiedby back-substitution into the equations (2.1)–(2.3). Analytical solutions for the time-dependentaxisymmetric poroelasticity problem can be obtained by using integral transform techniques.Laplace and zeroth-order Hankel transforms are used to remove, respectively, time-dependencyand dependency on the radial coordinate r

F̄(ξ , z, t) =∫∞

0rJ0(ξr)F(r, z, t) dr (2.10)

and˜̄F(ξ , z, s) = 1

2π i

∫∞0

e−stF̄(ξ , z, t) dt, (2.11)

where ( ) refers to the zeroth-order Hankel transform and (˜ ) refers to the Laplace transform of aparticular function. After successive applications of Laplace and zeroth-order Hankel transforms,the governing PDEs (2.6) and (2.7) can be reduced to the following ODEs for the transformed

variables ˜̄S(ξ , z, s) and ˜̄E(ξ , z, s): (d2

dz2− ξ2

)˜̄S = 0 (2.12)

and (d2

dz2− ξ2

){d2

dz2−[ξ2 + s

c

(β + αγ

2Gη

)]}˜̄E = −βs

ηcd ˜̄Sdz

. (2.13)

3. Initial boundary value problem related to an internally located injection zoneWe consider the problem of a poroelastic halfspace region that is overlain by a deformableporoelastic caprock (figure 2). The pore fluid pressures within the caprock layer and the halfspaceare considered to be sessile and at hydrostatic values. The in situ stress state in the region isassumed to be geostatic with the effective stresses generated purely from the self-weight of thegeological material and the pore fluid pressures. The role of tectonically induced stresses canalso be considered in the analysis provided that the stress states can be represented by equivalentprincipal stress states that preserve axial symmetry. The developments in poroelasticity presentedby (2.1)–(2.9) can therefore be regarded as the representations of displacement, stress and porepressure fields associated purely with the effects of injection. The interface between the poroelasticcaprock layer and the poroelastic halfspace region is assumed to be contiguous thereby allowingcomplete continuity conditions to be prescribed with respect to the displacements, effectivestresses, pore fluid pressures and fluid fluxes. A planar circular area occupying the region 0 ≤ r ≤ aat z = 0 is subjected to fluid influx at a constant total flow rate Q0 (units L3/T). (i.e. This representsthe total volume of fluid that is injected per unit time, over the plane circular region and we note that asthe radius of the injection region a increases the injection intensity measured as volume per unitarea decreases.) The boundary conditions are prescribed in such a way that the injection ratesare dimensionally consistent. The specification of the total volume rate of injection over the circularregion of radius a also makes it possible to compare the results of this study with the point injection



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studies, such as the nuclei of fluid influx, that have been developed for both infinite and semi-infinite domains [5,6,12]. We note that the assumption of injection over a circular disc-shapedregion enables the convenient formulation of the initial boundary value problem. In a practicalcontext, if the injected fluids have different physical and mechanical characteristics to the residentfluids, then this will result in the development of a three-dimensional plume, the configuration ofwhich needs to be determined from the solution of a moving boundary problem. Such problemsare more suited to analysis via computational approaches. The assumption of a planar injectionzone and the injection of fluids with identical properties make the resulting initial boundaryvalue problem amenable to the mathematical formulation and solution. From a practical pointof view, the specification of a uniform injection rate over a disc-shaped planar region is perhapsunrealistic, but the solution is intended to provide a fundamental result that can be used, witha superposition technique, to generate other non-uniform axisymmetric injection scenarios. Theinitial boundary value problem governing the fluid injection can be described by referring to thefollowing poroelastic domains: (i) a caprock poroelastic layer region, identified by the superscript( )(C) and occupying the region r ∈ (0, ∞); z ∈ (−h, −(l + h)), (ii) a poroelastic layer region, identifiedby superscript ( )(L) and occupying the region r ∈ (0, ∞); z ∈ (−h, 0) and (iii) a poroelastic halfspaceregion, identified by superscript ( )(H)and occupying the region r ∈ (0, ∞); z ∈ (0, ∞). The followingboundary conditions are applied for each domain.

(i) The boundary conditions applicable to the surface z = −(l + h) are derived from theassumption that the surface of the caprock layer is unloaded and the pore fluid pressure iszero, i.e.

σ(C)zz (r, −(h + l), t) = 0; σ (C)rz (r, −(h + l), t) = 0; p(C)(r, −(h + l), t) = 0. (3.1)

(ii) The continuity conditions at the interface between the poroelastic caprock layer and theporoelastic layer of the halfspace region are derived from the assumption that there is continuityof displacements, tractions, pore fluid pressure and fluid flux. These conditions give

u(C)r (r, −h, t) − u(L)r (r, −h, t) = 0; u(C)z (r, −h, t) − u(L)z (r, −h, t) = 0, (3.2)σ

(C)zz (r, −h, t) − σ (L)zz (r, −h, t) = 0; σ (C)rz (r, −h, t) − σ (L)rz (r, −h, t) = 0, (3.3)

p(C)(r, −h, t) − p(L)(r, −h, t) = 0 (3.4)

andkcγw

(∂p∂z

)(C)z=−h

− ksγw

(∂p∂z

)(L)z=−h

= 0. (3.5)

(iii) The plane of the poroelastic halfspace region containing the injection zone should alsosatisfy the conditions related to the continuity of displacements, tractions and pore fluid pressureapplicable to any plane within the poroelastic layer, whereas the fluid flux should account for thecircular planar injection zone. These conditions give

u(L)r (r, 0, t) − u(H)r (r, 0, t) = 0; u(L)z (r, 0, t) − u(H)z (r, 0, t) = 0, (3.6)σ

(L)zz (r, 0, t) − σ (H)zz (r, 0, t) = 0; σ (L)rz (r, 0, t) − σ (H)rz (r, 0, t) = 0, (3.7)

p(L)(r, 0, t) − p(H)(r, 0, t) = 0 (3.8)

andksγw

(∂p∂z

)(L)z=0

− ksγw

(∂p∂z

)(H)z=0

=⎧⎨⎩

Q0πa2

H(t); 0 ≤ r ≤ a0; a < r ≤ ∞,

(3.9)

where H(t) is the Heaviside step function. For completeness of the formulation of the initialboundary value problem related to the poroelasticity problem, we specify the initial conditions

u(C)(x, 0) = u(L)(x, 0) = u(H)(x, 0) = 0σ (C)(x, 0) = σ (L)(x, 0) = σ (H)(x, 0) = 0p(C)(x, 0) = p(L)(x, 0) = p(H)(x, 0) = 0,

⎫⎪⎬⎪⎭ (3.10)



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where x are the spatial coordinates, u(x, t) is the displacement vector and σ (x, t) is the totalstress tensor. In addition, the solution to the poroelasticity problem should satisfy the regularityconditions applicable to regions where the spatial dimension(s) can extend to infinity.

Because semi-infinite and layer regions that extend to infinity are encountered in the problemformulation, and because the stress states that induce the poroelastic effects are restricted to finiteregions, the solution to the fluid injection problem should also satisfy the regularity conditions

u(C)(x, t) → 0, σ (C)(x, t) → 0, p(C)(x, t) → 0, (3.11)as |x| → ∞, z ∈ (−h, −(l + h)) and for ∀ t ≥ 0 and

u(L)(x, t), u(H)(x, t) → 0, σ (L)(x, t), σ (H)(x, t) → 0, p(L)(x, t), p(H)(x, t) → 0, (3.12)as |x| → ∞, z ∈ (−h, ∞) and for ∀ t ≥ 0.

The uniqueness of solution to the initial boundary value problem posed by (2.1)–(2.3) and theconsistent boundary and regularity conditions (3.1), (3.11) and (3.12) and the initial conditions(3.10) can be assured by the general proof of uniqueness of solution to the linear poroelasticityproblem [42] and the general proof of uniqueness to the problem in classical elasticity [43,44],which is applicable to poroelastic media both at t = 0 and t → ∞.

4. Solution of an initial boundary value problemThe solution procedure commences with the development of solutions of the coupled systemof ordinary differential equations (2.12) and (2.13) applicable to the transformed dependent

variables ˜̄S(ξ , z, s) and ˜̄E(ξ , z, s). The solutions applicable to the poroelastic caprock layer region( )(C), the poroelastic layer region within the halfspace region ( )(L) and the poroelastic halfspaceregion ( )(H) can be expressed in the following forms, ensuring that the regularity conditions (3.11)and (3.12) are satisfied a priori.

(i) For the poroelastic caprock layer occupying the region r ∈ (0, ∞); z ∈ (−h, −(l + h)), thesolutions of (2.12) and (2.13) take the forms

˜̄S(C)(ξ , z, s) = A1e−ξz + B1eξz (4.1)and

˜̄E(C)(ξ , z, s) = C1e−ξz + D1eξz + E1e−ϕcz + F1eϕcz + ΓcA1ze−ξz + ΓcB1zeξz, (4.2)where A1, B1, C1, D1, E1 and F1 are arbitrary constants.

(ii) For the poroelastic layer above the injection zone and occupying the region r ∈ (0, ∞);z ∈ (0, −h), the solutions of (2.12) and (2.13) take the forms

˜̄S(L)(ξ , z, s) = A2e−ξz + B2eξz (4.3)and

˜̄E(L)(ξ , z, s) = C2e−ξz + D2eξz + E2e−ϕsz + F2eϕsz + ΓsA2ze−ξz + ΓsB2zeξz, (4.4)where A2, B2, C2, D2, E2 and F2 are arbitrary constants.

(iii) For the poroelastic halfspace region below the injection zone and occupying the regionr ∈ (0, ∞); z ∈ (0, ∞), the solutions of (2.12) and (2.13) take the forms

˜̄S(H)(ξ , z, s) = A3e−ξz (4.5)and

˜̄E(H)(ξ , z, s) = C3e−ξz + E3e−ϕsz + ΓsA3ze−ξz, (4.6)where A3, C3 and E3 are arbitrary constants. The 15 arbitrary constants can be uniquelydetermined by making use of the boundary conditions and continuity conditions given by the15 equations in (3.1)–(3.9). It is sufficient to note that explicit expressions for the unknownconstants A1, B1 . . . etc., can be obtained through a Mathematica�-based symbolic evaluation.The complete expressions for the unknown constants are given in electronic supplementary



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material, appendix A. The time-dependent displacement fields, stress components and porepressure fields in the poroelastic caprock layer and within the poroelastic halfspace region canbe obtained through a Laplace transform inversion and a Hankel transform inversion of theresulting expressions. At this juncture, it is worth commenting on the poroelastic modelling ofthe problem where a layer of poroelastic caprock is embedded between a geological overburden,which can be modelled as a poroelastic layer of finite thickness and a storage formation, whichis modelled as a poroelastic halfspace. In this case, additional boundary and interface conditionsneed to be prescribed between the poroelastic overburden and the poroelastic caprock, which willgive rise to six additional interface conditions similar to those defined by (3.2)–(3.5), giving riseto 21 arbitrary constants governing the poroelasticity formulation of the idealized analogue of aproblem depicted in figure 1. Quite apart from the complexities associated with the mathematicalsolution of the problem, the illustration of numerical results will have to contend with aninordinate number of material parameter groups that will make the exercise unmanageable andof limited utility as a benchmark problem. For this reason, the approach adopted in the reducedformulation of the benchmark problem is favoured.

The results that are of interest to geological sequestration are the time-dependent verticaldisplacement of the surface of the caprock layer and the time-dependent peak radial stress atthe surface of the caprock layer. These results are particularly important for monitoring theinfluence of subsurface injection on caprock heave and estimating the potential for caprockdistress, in terms of the development of tensile fractures in the caprock. The expressions for thetime-dependent axial displacement and time-dependent radial stress are given by

u(C)z (r, z, t) =Q0γwαs2Gsaπks

∫ ζ+i∞ζ−i∞

∫∞0

{(z + h + l)(−ξA1e−ξ (z+h+l) + ξB1eξ (z+h+l))

− [−ξC1e−ξ (z+h+l) + ξD1eξ (z+h+l) − ϕcE1e−ϕc(z+h+l) + ϕcF1eϕc(z+h+l) + ΓcA1e−ξ (z+h+l)− ξΓc(z + h + l)A1e−ξ (z+h+l) + ΓcB1eξ (z+h+l) + ξΓc(z + h + l)B1eξ (z+h+l)

]− (A1e−ξ (z+h+l) + B1eξ (z+h+l))

}J1(ξa)J0(ξr) dξds. (4.7)

and

σ(C)rr (r, z, t) =

Q0γwαs(2Gc)aπks(2Gs)

∫ ζ+i∞ζ−i∞

∫∞0

{[ξ2(z + h + l) − ξ ]A1e−ξ (z+h+l)

+ [ξ2(z + h + l) + ξ ]B1eξ (z+h+l) − [ξ2C1e−ξ (z+h+l) + ξ2D1eξ (z+h+l) + ϕ2c E1e−ϕc(z+h+l)

+ ϕ2c F1eϕc(z+h+l) − 2ξΓcA1e−ξ (z+h+l) + ξ2Γc(z + h + l)A1e−ξ (z+h+l) + 2ξΓcB1eξ (z+h+l)

+ ξ2Γc(z + h + l)B1eξ (z+h+l)]}

J1(ξa)J0(ξr) dξ ds, (4.8)

where ζ is a real constant associated with the Bromwich contour integration encountered inLaplace transform inversion. Other expressions for the displacements, pore fluid pressure andeffective stresses are presented in electronic supplementary material, appendix B. The complexityof the expressions of the type (4.7) and (4.8) makes it difficult to conduct analytical evaluationsof the inversion of the integral transforms. Previous experience [22,25,41,45,46] suggests that theoperations involving Hankel transform inversions and Laplace transform inversions have to becarried out numerically to generate expressions for the displacements and stresses that can beexpressed in terms of r, z and t. In this study, the integral transforms were evaluated using thealgorithms given in Matlab�. In particular, the inverse Laplace transform is evaluated usingthe procedures developed by Crump [47]. The accuracy of the inversion technique is confirmedthrough comparisons with results obtained by McNamee & Gibson [35] for the axisymmetricsurface loading, f , of a halfspace region. Table 1 presents the numerical results evaluated in thisstudy, and the analytical results given in McNamee & Gibson [35]. The error is large when t < 0.1but substantially decreases as t increases. The numerical results have an error of less than 0.1%when t = 10, and, therefore, the procedures are considered satisfactory for evaluating a full rangeof analytical results for the poroelastic problem.



10


...................................................

Table 1. Comparison of the numerical results with the analytical results given in McNamee & Gibson [35].

McNamee & Gibson [35]

ct/a2 2Gfa [uz(0, 0, t) − uz(0, 0, 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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5. COMSOLTM modellingAn objective of the mathematical treatments presented in the paper is also to develop an analyticalresult that can be used to test the accuracy of computational schemes, which can be appliedto examine more complex geologic sequestration settings. In this regard, the computational

solutions were obtained using the finite-element-based multi-physics software COMSOLTM

.This code has been extensively scrutinized, and details of the validation exercises are givenin [24,25,29,30,45,48–51]. Figure 3 shows the representative finite-element discretization modelfor the injection problem that was used to develop the computational simulation. Because theproblem is axisymmetric, only a section of the domain (about z-axis) needs to be modelled. A finitedomain is used instead of the infinite region that was used in the analytical development. Theeffect of the finite boundary is minimized by taking the boundary of the domain sufficientlyremote from the fluid injection region, thereby ensuring that the far-field boundary conditionsare approximately satisfied. An alternative approach is to incorporate infinite elements that canmodel the far field behaviour [46,52,53]. This option is, however, unavailable in the multi-physics

code COMSOLTM

. At the surface, the pore pressure is set to zero, and a flux discontinuity isapplied to the interface between the halfspace layer above the injection site and the halfspaceregion. A symmetry/zero flux boundary condition is applied at the base and the right-hand

side boundary of the domain. Figure 4 shows the mesh configuration used in the COMSOLTM

modelling; mesh refinement is used in the vicinity of the injection region and coarser meshes wereused for the rest of the domain. The mesh consists of 12 570 elements for a/h = 0.1 and increasesto 33 840 for a/h = 10. Each nodal point has three degrees of freedom.

It should be noted that as ks/kc increases, the COMSOLTM

model could not produce resultsthat were comparable to the analytical results. The maximum value for the permeability mismatch

between the caprock layer and storage formation used in the COMSOLTM

model was ks/kc = 100,and therefore, the values of kc and ks were chosen accordingly.

6. ResultsThe results for the vertical surface displacements, pore fluid pressure and effective stressdistributions along the axis of the stratified region are presented in this section. The parameterswith subscript ‘c’ are the caprock variables, and those with subscript ‘s’ are the variables in thestorage formation. For the case where Gc = Gs and kc = ks we used the following variables toevaluate the analytical and computational results:

Gc,s = 20.0 GPa; ν = 0.25; kc = 10−15 m s−1; ks = 10−13 m s−1; γw = 9.81 kN m−3; νu = 0.5.The same variables were used for the case where Gc = Gs, kc = ks, except that Gc was set to 50 GPa.The results are presented for a non-dimensional time, t∗ = ct/h2. The formulation of c is givenin equation (2.4) and is evaluated by using G = Min(Gc, Gs) and k = Min(kc, ks). Throughout thisstudy, the total volume flow rate Q0 and the thickness l of the caprock are fixed. For ease ofpresentation, we neglect the negative sign in the displacement, with the understanding that the



11


...................................................

srz (r, – (h + l), t) = 0

traction free-drained boundary

caprock

r

z

lh region of fluid injection

poroelastichalfspace H0 = 100 a

R0 = 100 a

a

0

sZZ (r, – (h – l), t) = 0p (r, – (h + l), t) = 0

ur (0, z, t) = 0

sr z (0, z, t) = 0

ur (R0, z, t) = 0

uz (r, (H0–h), t = 0

srz (r, (H0–h), t = 0

p (r, (H0–h), t = 0

sr z(R0, z, t) = 0p (R0, z, t) = 0

(∂ p/∂r)(0, z, t) = 0

Figure 3. The axisymmetric region used in the finite-element modelling and the boundary conditions.

Figure 4. Mesh configuration for COMSOLTMmodelling of the injection problem. (Online version in colour.)

surface displacement in the caprock occurs in the negative direction of the z-axis as indicatedin figure 2.

(a) Case when the permeability of the caprock differs from that of the storage formation(i) Surface displacement

The variation in the time-dependent surface displacement is shown in figure 5 for differentvalues of a/h when h/l = 1. The solution at t∗ → ∞ indicates a steady-state response and in the



12


...................................................

a/h = 0.1a/h = 1a/h = 5a/h = 10computational results

0.4

0.3

0.2

0.1

00 10 20

r/ht* = 1

t* = 10

t* •

30

u z(r

,–(h

+l)

,t*)

Q0g

w/2

Gsk s

Figure 5. Time-dependent surface displacement for different radii of circular injection region. (Online version in colour.)

numerical calculation, it is effectively reached when t∗ = 1000. As indicated in equation (4.7), fora constant Q0, the injection intensity will decrease as a increases. As is evident from figure 5, thesurface heave decreases as a/h increases. The corresponding computational results are denotedby circles. The discrepancy between the analytical and computational solutions is approximately11% around the central location.

Selvadurai & Kim [24] investigated the ground subsidence due to fluid extraction from acircular disc-shaped region, which can also be used as a solution for the fluid injection problemowing to the linearity of the fundamental equations governing Biot’s theory of poroelasticity.Figure 6 compares the heave at the interface between the caprock and the storage halfspace,z = −h, with the surface heave when the caprock is absent [24]. The surface heave is normalized bythe steady-state value at r = 0. It is observed that without the caprock, the surface displacementoccurs only around the injection region and decreases quickly as r is remote from the injectionregion. However, in the presence of the caprock, the surface displacement occurs over a broaderregion and decreases gradually as the point of interest is remote from the injection region. Inthe case without the caprock, surface flattening and widening of the centre were observed in theinjection region at t∗ = 1 and 10 when a/h = 5 and 10. However, when the caprock is present,no flattening of the surface is observed for the cases a/h = 5 and 10; only widening of the centreis observed in the present results. When the present results are compared with those obtainedfor the case without the caprock, it was found that the ratio of the surface heave to the steady-state displacement is larger when the caprock is present. When the caprock is present, the surfacedisplacement when a/h = 10 attains in excess of 60% of the steady-state response at t∗ = 10 and30% at t∗ = 1; however, the surface displacement for the case a/h = 10 at t∗ = 10 was only 40% ofthe steady-state displacement without the caprock.

Figure 7 illustrates the surface heave for different depths of location of the injection zone, i.e.a/h = 0.1 and h/l = 1 to 10. It can be seen from figure 7 that varying the depth of the injection regioninfluences the surface heave in a similar manner to varying the injection size. As h/l increases, thesurface heave decreases. This observation is analogous to a Saint Venant-type effect in classicalelasticity, where remoteness of the injection zone ensures that the caprock displacements arereduced. In the limit as h/l becomes large (e.g. h/l > 10), the injection pressures have only amarginal influence on the caprock displacements. The corresponding computational results areshown as circles in figure 7. The agreement between analytical and computational results isgenerally good; however, there is a discrepancy between the two solutions.

In this study, the maximum discrepancy is 20% at the central location.Figure 8 compares the surface heave for different ks/kc at t∗ = 0 and t∗ → ∞. The results

obtained for the case ks/kc = 105 are larger than those obtained for ks/kc = 102. No distinct change



13


...................................................

t* = 1uz (r, –h, t*)

Uz (0, –h, •)

uz (r, –h, t*)

Uz (0, –h, •)

1.0

0.8

0.6

0.4

0.2

0r/h

–50 50

1.0

0.8

0.6

0.4

0.2

0r/h

–50 50

t* = 10t* •

t* = 1uz (r, –h, t*)

Uz (0, –h, •)

1.0

0.8

0.6

0.4

0.2

0r/h

–100 –50 50 100

t* = 10t* •

t* = 1uz (r, –h, t*)

Uz (0, –h, •)

1.0

0.8

0.6

0.4

0.2

0r/h

–150 –100 –50 50 100 150

t* = 10t* •

t* = 1t* = 10t* •

(b)(a)

(c) (d )

Figure 6. Comparisons of the normalized surface heave at the interface (blue line) with the results given in Selvadurai &Kim [24] (red lines): (a) a/h= 0.1; (b) a/h= 1; (c) a/h= 5; (d) a/h= 10. (Online version in colour.)

h = 0h/l = 1h/l = 2h/l = 5h/l = 10computational results

0.4

0.5

0.3

0.2

0.1

00

1020

30 t* = 1

t* = 10

t* •

r/h

u z(r

,–(h

+l)

,t*)

Q0g

w/2

Gsk s

Figure 7. Time-dependent surface displacement for different depths of injection. (Online version in colour.)



14


...................................................

0.50

0.45

0.40

0.35

0.30

0.25

0.20

0.15

0.10

0.05

0

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.10 50 100 150 200

a/h = 0.1a/h = 1a/h = 5a/h = 10

r/h0 50 100 150 200

r/h

(b)(a)

u z(r

,–(h

+l)

,t*)

Q0g

w/2

Gsk s

a/h = 0.1a/h = 1a/h = 5a/h = 10

Figure 8. Comparison of the surface displacement for different ks/kc; blue line: ks/kc = 105; red line: ks/kc = 102: (a) t∗ = 1;(b) t∗ → ∞. (Online version in colour.)

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

01 102

ks/kc

104

u z(0

,0,•

)

Q0g

w/2

Gsk s

Figure 9. Maximum surface heave for different ks/kc. (Results applicable for a/h= 0.10). (Online version in colour.)

in the shape of the surface heave is observed. The maximum surface heave for different valuesof ks/kc is shown in figure 9; as ks/kc increases, the surface displacement increases. A large ks/kcindicates a low hydraulic conductivity in the caprock layer. When the hydraulic conductivity islow, the pore fluid pressure tends to diffuse slowly from the vicinity of the injection zone, whichcontributes to a greater surface heave.

(ii) Pore fluid pressure

Figure 10 presents the changes in the pore fluid pressure owing to fluid injection into theporoelastic medium when a/h = 0.1 and h/l = 1. The pore fluid pressure is at its maximum atthe injection site, which is located at the interface between the layer (L) and the halfspace, anddecreases very rapidly as the position z/h is remote from the injection location. However, in thecaprock where the hydraulic conductivity is low compared with the layer (L) and the halfspace,the change in the pore fluid pressure is very small as z moves away from the injection site.



15


...................................................

–2.0

–1.5

–1.0

–0.5

0

0.5

1.00 0.5 1.0

halfspace

layer

caprock t* = 0.01t* = 0.1

z/h

t* •computational results

1.5 2.0p (0, z, t*)Q0gw / ks h

Figure 10. Changes in the pore fluid pressure. (Online version in colour.)

–2.0

–1.5

–1.0

–0.5

0

0.5

1.01.2 –0.8 –0.4 0

halfspace

layer

caprockt* = 0.01t* = 0.1t* •computational results

z/h

s¢rr (0, z, t*)Q0gw /ks h

Figure 11. Changes in effective stress in the r-direction. (Online version in colour.)

The corresponding computational results are presented as circles and they show good agreementwith the analytical solutions except at locations close to the injection region.

(iii) Effective stresses and caprock distress

It was observed that the pore fluid pressure increased owing to fluid injection, and therefore, theeffective stresses are expected to decrease in the injection region. The change in the effective stress(σ ′ij)

Q0 is evaluated from the result

(σ ′ij)Q0 = (σij)Q0 − α(p)Q0δij, (6.1)

where (σij)Q0 and (p)Q0 are the changes in the total stress and pore fluid pressure owing to fluidinjection. Figures 11 and 12 show the effective stresses in the r- and z-directions, respectively.



16


...................................................

•

z/h

–2.0

–1.5

–1.0

–0.5

0

0.5

1.0–1.5 –1.0 –0.5 0

halfspace

layer

caprockt* = 0.01t* = 0.1t* computational results

s¢zz (0, z, t*)Q0gw / ks h

Figure 12. Changes in effective stress in the z-direction. (Online version in colour.)

The negative values indicate a decrease in the effective stresses in both directions owing to fluidinjection. In addition, in the caprock layer, it was observed that negative effective stress (tension)developed in the r-direction. At the surface, traction free boundary conditions are invoked, andtherefore, only radial stresses are present. The possibility of tensile failure can be checked bycomparing this effective stress with the tensile strength (T) of the caprock medium

σ ′rr > T. (6.2)

A typical caprock material is shale (e.g. Sleipner, In Salah [8–10,54–56]) and we supposed thatthe caprock considered in this study consists of shales. The tensile strength for the shale is inthe range of 0.2 to 2.0 MPa [56]. In this study, when kc = 10−12 m s−1, ks = 10−10 m s−1 a = 50 andh = 500, the effective stress at the surface of the caprock layer in the r-direction is approximately4.289 × Q0 × 102 MPa (figure 11). For tensile failure to occur at the surface, the following conditionneeds to be satisfied

σ ′rr = 4.289 × Q0 × 102 ≥ T, (6.3)where T is the tensile strength of the shale caprock, which is estimated at 2 MPa. Possible tensilefailure can occur when Q0 > 0.466 × 10−2 m3 s−1 (≈0.1456 megatonnes per year (Mt yr−1)).

In a practical geological sequestration scenario, however, the caprock is most likely to beoverlain by either surficial soils or rocks similar to the situation illustrated in figure 1. We considerthe case where the caprock is overlain by a surficial soil, such as clay of saturated unit weight,γs = 20 kN m−3. The surficial soil layer is assumed to contribute little to the poromechanicalinteraction between the caprock and the storage formation but provides only a surcharge loadingat the surface of the caprock layer by virtue of the self-weight. The self-weight of the surficial soilswill introduce an additional effective radial compressive stress in the caprock, which is given by

σ ′rr(induced by soil layer) =ν

1 − ν (γs − γw)D (6.4)

where D is the thickness of the surficial soil layer. Table 2 presents an estimate of how this surficialsoil layer, by virtue of the additional self weight-induced loading, can contribute to the increase inthe allowable injection rate that can be attained without the development of fracture at the upperboundary of the caprock layer. Table 3 provides a summary of the typical annual CO2 injectionrates for the selected greenhouse gas storage sites.

The corresponding computational results for the r- and z-direction effective stresses arepresented as circles in figures 11 and 12, respectively. The agreement is good for the effective



17


...................................................

Table 2. Maximum allowable volume flow rate to ensure the integrity of the caprock.

D Q0 (m3 s−1) Q0 (Mt yr−1)0.5 km 0.862 × 10−2 0.269

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.0 km 1.258 × 10−2 0.393. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.5 km 1.654 × 10−2 0.517. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Table 3. Annual injection rates for selected storage site (see Hosa et al. [56]).

depth of storage annual CO2 injection

project name formation (m) rate (Mt yr−1)In Salah 1850 1.00

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Weyburn 1418 2.70. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Snøhvit 2550 0.75. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Sleipner 1000 1.00. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

stress in the z-direction; however, the COMSOLTM

model underestimates the effective stress inthe r-direction. In particular, the effective stress in the r-direction obtained by COMSOL

TMat

the surface is less than 40% of the analytical result. It must be emphasized that the analyticalresults presented in the paper are accurate and represent the solution to the initial boundary valueproblem defined by (3.1)–(3.12). The accuracy of the analytical technique has also been establishedthrough comparisons with known analytical results given by McNamee & Gibson [35], albeitfor the surface loading of a poroelastic halfspace region. The issue of the discrepancy betweenthe analytical results and the computational results obtained, particularly for the radial stress,can arise from a number of approximations including the representation of the halfspaceregion by a domain of finite extent and the application of specific boundary conditions at thefar-field boundary of the computational domain as opposed to regularity conditions applicableto the semi-infinite domains. The mesh discretization in the vicinity of the injection zone canalso contribute to the discrepancy. The trends in all computationally derived effective stressdistributions are consistent with the results obtained from the analytical approach. We also notethat the results presented for the poroelastic halfspace without a surficial layer is not meant toreflect any practical engineering situation but are included to illustrate the moderating influenceof the surficial barrier in mitigating the boundary displacements of the storage formation.

(iv) Case when both elastic modulus and permeability of the caprock differ from those inthe storage formation

The surface displacement was also investigated for the case where the caprock has a differentshear modulus and hydraulic conductivity than those in the layer and halfspace. Figure 13shows the non-dimensional surface displacement for different cases considered in this studywhen a/h = 0.1 and h/l = 1. It is observed that the magnitude of the surface heave is greaterwhen Gc = Gs and kc = ks compared with the cases when Gc = Gs and kc = ks. The surface heaveobtained when Gc = Gs and kc = ks has a maximum at the centre and quickly converges to 0as r/h moves away from the centre, whereas the results obtained from the other cases show amore gradual decrease. When the hydraulic conductivities differ, i.e. kc = ks, the magnitude ofthe surface displacement obtained for the case where Gc = Gs is larger than that obtained whenGc = Gs.

Figure 14 presents the maximum surface heave for different Gs/Gc. Two cases, (i) kc = ks and(ii) kc = ks, are considered, and both cases show exactly the same trend except for the magnitudeof the surface heave. In both cases, the surface heave increases as Gs/Gc increases.



18


...................................................

0.40

0.35

0.30

0.25

0.20

0.15

0.10

0.05

0 5 10 15r/h

20 25 30

t* = 1t* = 10t* •

u z(r

,–(h

+l)

,t*)

Q0g

w/2

Gsk s

Figure 13. Time-dependent surface displacement. Blue lines: Gc = Gs, kc = ks; red lines: Gc = Gs, kc = ks; green lines: Gc =Gs, kc = ks. (Online version in colour.)

0.36

0.34

0.32

0.30

0.28

0.26

0.24

0.008

0.075

0.070

0.065

0.060

0.055

0.050

0.045

0.04010–1 1 10–1 1

Gs/Gc Gs/Gc

kc = kskc π ks

(b)(a)

u z(r

,0,•

)

Q0g

w/2

Gsk s

Figure 14. Maximum surface heave for different Gs/Gc; (a) kc = ks, (i) kc = ks. (Online version in colour.)

7. Concluding remarksThe paper develops analytical solutions for the poroelastic modelling of the interaction betweena poroelastic caprock and a poroelastic storage aquifer, for the case where steady injection takesplace over a planar circular injection zone. The mathematical analysis can be performed to



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develop exact closed results in integral representations that can be evaluated using standardalgorithms available for Laplace and Hankel transform inversions. Two cases were examined: inthe first case, the elastic shear modulus for the caprock is the same as that of the storage horizon,but the hydraulic conductivities are different (Gc = Gs, kc = ks). In the second case, both the shearmodulus and the hydraulic conductivities are different (Gc = Gs, kc = ks). We investigated howthe radius and the depth of the planar injection region influence the surface displacement forthe first case. When the total flow rate to the injection zone is constant, as expected, an increasein the radius of the injection region leads to a reduction in the surface heave. Similarly, surfaceheave decreased as the injection depth increased. The zone of the storage formation above theinjection area acts as a space for diffusion of the injected fluid, so that an increase in depth ofthe injection zone results in a decreased surface heave. In this study, it has been shown thatboth the shape and magnitude of the surface heave change significantly when there is a caprocklayer present. As the hydraulic conductivity of the caprock decreases, the magnitude of thesurface heave increased. We observe that in the hypothetical case where injection takes placeinto a poroelastic formation without a caprock, both the magnitude and profile of the surfaceheave can be distinctly different from the case where a caprock layer with different poroelasticityproperties is present. The stiffening action of the caprock diffuses the spatial distribution ofheave. These effects have also been observed in the purely elastic modelling of caprock–storagehorizon interaction [5–7,12], the surface heave becomes flattened and localized to the region ofthe injection region for the case when the caprock is not present [24]. It was also found thatthe ratio of the surface heave to the steady-state displacement was larger when a caprock waspresent. Owing to the fluid injection, it was observed that the pore fluid pressure increased inthe area surrounding the injection region and this caused a decrease in the effective stresses(both in the r- and z-directions). Because of the low hydraulic conductivities, the changes inthe pore fluid pressure and the effective stresses were very small in the caprock compared withthose within the halfspace storage region. One important observation is that adverse stresses,which lead to tensile failure, are developed at the caprock surface during fluid injection. For thecase considered in this study, which is applicable to a shale similar to that encountered at thegeological sequestration site at In Salah, it is observed that tensile failure can occur when Q0 >0.1456 Mt yr−1. Assuming the presence of a surficial geological zone of thickness 1.5 km, whichmerely provides a surcharge load, the injection rate can be increased up to 0.517 Mt yr−1, withoutdistress to the caprock. The injection rates obtained through theoretical modelling of the caprock–storage formation interaction in this study are comparable with those associated with typical CO2sequestration projects.

For the second case when Gc = Gs and kc = ks, the caprock experienced less surface heavecompared with those obtained in the first case. As Gc increased, the surface heave decreasedand the rate of decrease was larger when kc = ks. The magnitude of the surface heave decreasedby 45% when kc = ks, whereas the magnitude of the heave decreased by 27% when kc = ks. Theanalytical solutions were used to calibrate the results obtained using the finite-element-based

code COMSOLTM

. The computational results compare favourably with the exact analyticalsolutions.

Ethics. The research conducted is unrelated ethical requirements indicated in the author guidelines.Data accessibility. The data files, computer codes and subroutines can be accessed at the following sites:On page: https://www.mcgill.ca/civil/people/selvadurai/research-repositoryFile 1: https://www.mcgill.ca/civil/files/civil/instruction_for_source_file.pdfFile 2: https://www.mcgill.ca/civil/files/civil/surficial_caprock_model.mph_.txtAuthors’ contributions. The conceptual development of the paper and the solution approach was developed byA.P.S.S. The mathematical formulation of the problem was prepared by A.P.S.S. and J.K. The computationswere performed by J.K.; the accuracy of the computations was verified by A.P.S.S. The paper was written byA.P.S.S. and J.K.; the responses to the reviewer’s questions was written by A.P.S.S. and J.K.Competing interests. No competing interests.Funding. The work performed was supported by an NSERC Discovery Grant and the James McGillProfessorship Grant awarded to A.P.S.S.


https://www.mcgill.ca/civil/people/selvadurai/research-repositoryhttps://www.mcgill.ca/civil/files/civil/instruction_for_source_file.pdfhttps://www.mcgill.ca/civil/files/civil/surficial_caprock_model.mph_.txthttp://rspa.royalsocietypublishing.org/

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Acknowledgements. The work described in this paper was supported by a NSERC Discovery Grant, the JamesMcGill Research Chairs support and the Carbon Management Canada Grant awarded to the first author. J.K.acknowledges the partial research support provided by the Brace Center for Water Resources Management atMcGill University and a McGill Engineering Doctoral Award. The authors are indebted to the reviewers fortheir constructive comments on the manuscript.Disclaimer. The use of the computational code COMSOLTM is for demonstration purposes only. The authorsneither advocate nor recommend the use of the code without conducting suitable validation procedures totest the accuracy of the code in a rigorous fashion.

References1. Selvadurai APS. 2002 The advective transport of a chemical from a cavity in a porous medium.

Comput. Geotech. 29, 525–546. (doi:10.1016/S0266-352X(02)00007-1)2. Selvadurai APS. 2003 Contaminant migration from an axisymmetric source in a porous

medium. Water Resour. Res. 39, 1204. (doi:10.1029/2002WR001442)3. Selvadurai APS. 2006 Gravity-driven advective transport during deep geological disposal of

contaminants. Geophys. Res. Lett. 33, L08408. (doi:10.1029/2006GL025944)4. Rutqvist J. 2012 The geomechanics of CO2 storage in deep sedimentary formations. Geotech.

Geol. Eng. 30, 515–551. (doi:10.1007/s10706-011-9491-0)5. Selvadurai APS. 2009 Heave of a surficial rock layer due to pressures generated by injected

fluids. Geophys. Res. Lett. 36, L14302. (doi:10.1029/2009GL038187)6. Selvadurai APS. 2012 A geophysical application of an elastostatic contact problem. Math.

Mech. Solids 18, 192–203. (doi:10.1177/1081286512462304)7. Selvadurai APS. 2013 Caprock breach: a potential threat to secure geologic sequestration

of CO2. In Geomechanics in CO2 storage facilities (eds G Pijaudier-Cabot, J-M Pereira), Ch. 5,pp. 75–93. Hoboken, NJ: John Wiley & Sons Inc.

8. Eiken O, Ringrose P, Hermanrud C, Nazarian B, Torp TA, Høier L. 2011 Lessons learnedfrom 14 years of CCS operations: Sleipner, In Salah and Snøhvit. Energy Proc. 4, 5541–5548.(doi:10.1016/j.egypro.2011.02.541)

9. Vasco DW, Ferretti A, Novali F. 2008 Reservoir monitoring and characterization using saellitegeodetic data: Interferometric synthetic aperture radar observations from the Krechba field,Algeria. Geophysics 73, WA113–WA122. (doi:10.1190/1.2981184)

10. Sato K, Mito S, Horie T, Ohkuma H, Saito H, Watanabe J, Yoshimura T. 2011 Monitoring andsimulation studies for assessing macro- and meso-scale migration of CO2 sequestered in anonshore aquifer: experiences from Nagaoka pilot site, Japan. Int. J. Greenhouse Gas Control 5,125–137. (doi:10.1016/j.ijggc.2010.03.003)

11. Geertsma J. 1973 Land subsidence above compacting oil and gas reservoirs. J. Petrol. Technol.25, 734–744. (doi:10.2118/3730-PA)

12. Selvadurai APS. 2014 Mechanics of contact between bi-material elastic solids perturbed by aflexible interface. IMA J. Appl. Math. 79, 739–752. (doi:10.1093/imamat/hxu001)

13. Biot MA. 1941 General theory of three-dimensional consolidation. J. Appl. Phys. 12, 155–164.(doi:10.1063/1.1712886)

14. Rice JR, Cleary MP. 1976 Some basic stress diffusion solutions for fluid-saturated elasticporous media with compressible constituents. Rev. Geophys. 14, 227–241. (doi:10.1029/RG014i002p00227)

15. Atkinson C, Craster RV. 1991 Plane strain fracture in poroelastic media. Proc. R. Soc. Lond. A434, 605–633. (doi:10.1098/rspa.1991.0116)

16. Cheng AH-D. 2015 Poroelasticity. Berlin, Germany: Springer.17. Craster RV, Atkinson C. 1996 Theoretical aspects of fracture in porous elastic media. In

Mechanics of poroelastic media (ed. APS Selvadurai), pp. 23–45. Dordrecht, The Netherlands:Kluwer Academic.

18. Selvadurai APS (ed.). 1996 Mechanics of poroelastic media. Dordrecht, The Netherlands: KluwerAcademic.

19. Selvadurai APS. 2007 The analytical method in geomechanics. Appl. Mech. Rev. 60, 87–106.(doi:10.1115/1.2730845)

20. Selvadurai APS. 2015 Contact problems for a finitely deformed incompressible elastic halfspace. Contin. Mech. Thermodyn. 27, 287–304. (doi:10.1007/s00161-014-0376-3)


http://dx.doi.org/doi:10.1016/S0266-352X(02)00007-1http://dx.doi.org/doi:10.1029/2002WR001442http://dx.doi.org/doi:10.1029/2006GL025944http://dx.doi.org/doi:10.1007/s10706-011-9491-0http://dx.doi.org/doi:10.1029/2009GL038187http://dx.doi.org/doi:10.1177/1081286512462304http://dx.doi.org/doi:10.1016/j.egypro.2011.02.541http://dx.doi.org/doi:10.1190/1.2981184http://dx.doi.org/doi:10.1016/j.ijggc.2010.03.003http://dx.doi.org/doi:10.2118/3730-PAhttp://dx.doi.org/doi:10.1093/imamat/hxu001http://dx.doi.org/doi:10.1063/1.1712886http://dx.doi.org/doi:10.1029/RG014i002p00227http://dx.doi.org/doi:10.1029/RG014i002p00227http://dx.doi.org/doi:10.1098/rspa.1991.0116http://dx.doi.org/doi:10.1115/1.2730845http://dx.doi.org/doi:10.1007/s00161-014-0376-3http://rspa.royalsocietypublishing.org/

21


...................................................

21. Selvadurai APS, Yue ZQ. 1994 On the indentation of a poroelastic layer. Int. J. Numer. Anal.Methods Geomech. 18, 161–175. (doi:10.1002/nag.1610180303)

22. Selvadurai APS, Mahyari AT. 1997 Computational modeling of the indentation of a crackedporoelastic half-space. Int. J. Fract. 86, 59–74. (doi:10.1023/A:1007372823282)

23. Selvadurai APS, Mahyari AT. 1998 Computational modelling of steady crack extension inporoelastic media. Int. J. Solids Struct. 35, 4869–4885. (doi:10.1016/S0020-7683(98)00098-5)

24. Selvadurai APS, Kim J. 2015 Ground subsidence due to uniform fluid extraction over a circularregion within an aquifer. Adv. Water Resour. 78, 50–59. (doi:10.1016/j.advwatres.2015.01.015)

25. Kim J, Selvadurai APS. 2015 Ground heave due to line injection sources. Geomech. EnergyEnviron. 2, 1–14. (doi:10.1016/j.gete.2015.03.001)

26. Davis RO, Selvadurai APS. 2002 Plasticity and geomechanics. Cambridge, UK: CambridgeUniversity Press.

27. Laloui L, Cekerevac C, François B. 2005 Constitutive modelling of the thermo-plasticbehaviour of soils. Eur. J. Civil Eng. 9, 635–650.

28. Gens A. 2010 Soil-environment interactions in geotechnical engineering. Geotechnique 60, 3–74.(doi:10.1680/geot.9.P.109)

29. Selvadurai APS, Suvorov AP. 2012 Boundary heating of poroelastic and poro-elastoplasticspheres. Proc. R. Soc. A 468, 2779–2806. (doi:10.1098/rspa.2012.0035)

30. Selvadurai APS, Suvorov AP. 2014 Thermo-poromechanics of a fluid-filled cavity in a fluid-saturated porous geomaterial. Proc. R. Soc. A 470, 20130634. (doi:10.1098/rspa.2013.0634)

31. Wang HF. 2000 Theory of linear poroelasticity: with applications to geomechanics and hydrogeology.Princeton, NJ: Princeton University Press.

32. Mandel J. 1953 Consolidation des sols etude mathematique. Geotechnique 7, 287–299. (doi:10.1680/geot.1953.3.7.287)

33. De Josselin De Jong G. 1957 Application of stress functions to consolidation problems. Proc.4th Int. Conf. on Soil Mechanics and Foundation Engineering, vol. 1, pp. 320–323. London, UK:Butterworths Scientific Publishers.

34. McNamee J, Gibson RE. 1960 Displacement functions and linear transforms applied todiffusion through porous elastic media. Q. J. Mech. Appl. Math. 13, 98–111. (doi:10.1093/qjmam/13.1.98)

35. McNamee J, Gibson RE. 1960 Plane strain and axially symmetric problems of consolidation ofa semi-infinite clay stratum. Q. J. Mech. Appl. Math. 13, 210–227. (doi:10.1093/qjmam/13.2.210)

36. Gibson RE, McNamee J. 1963 A three-dimensional problem of the consolidation of a semi-infinite clay stratum. Q. J. Mech. Appl. Math. 16, 115–127. (doi:10.1093/qjmam/16.1.115)

37. Freudenthal AM, Spillers WR. 1962 Solutions for the infinite layer and the half-spacefor quasi-static consolidating elastic and viscoelastic media. J. Appl. Phys. 33, 2661–2668.(doi:10.1063/1.1702529)

38. Rajapakse RKND, Senjuntichai T. 1993 Fundamental solutions for a poroelastic half-spacewith compressible constituents. J. Appl. Mech. 60, 847–856. (doi:10.1115/1.2900993)

39. Yue ZQ, Selvadurai APS. 1994 On the asymmetric indentation of a consolidating poroelastichalf space. Appl. Math. Model. 18, 170–185. (doi:10.1016/0307-904X(94)90080-9)

40. Yue ZQ, Selvadurai APS. 1995 On the mechanics of a rigid disc inclusion embedded ina fluid saturated poroelastic medium. Int. J. Eng. Sci. 33, 1633–1662. (doi:10.1016/0020-7225(95)00031-R)

41. Lan Q, Selvadurai APS. 1996 Interacting indentors on a poroelastic half-space. Z. Angew. Math.Phys. 47, 695–716. (doi:10.1007/BF00915270)

42. Altay GA, Dökmeci MC. 1998 A uniqueness theorem in Biot’s poroelsticity theory. Z. Angew.Math. Phys. 49, 838–846. (doi:10.1007/PL00001489)

43. Knops RJ, Payne LE. 1971 Uniqueness theorems in linear elasticity. Berlin, Germany: Springer.44. Selvadurai APS. 2000 Partial differential equations in mechanics, 2: the biharmonic equation,

Poisson’s equation. Berlin, Germany: Springer-Verlag.45. Selvadurai APS, Najari M. 2015 Laboratory-scale hydraulic pulse testing: influence of air

fraction in the fluid-filled cavity in the estimation of permeability. Geotechnique 65, 126–134.(doi:10.1680/geot.14.P.174)

46. Karpurapu R, Selvadurai APS, Tanoesoedibjo RES. 1990 Consolidation analysis ofsymmetrically loaded strip footings on a poroelastic layer. Comput. Geotech. 9, 171–184.(doi:10.1016/0266-352X(90)90012-K)

47. Crump K. 1976 Numerical inversion of Laplace transforms using a Fourier seriesapproximation. J. Assoc. Comput. Mach. 23, 89–96. (doi:10.1145/321921.321931)


http://dx.doi.org/doi:10.1002/nag.1610180303http://dx.doi.org/doi:10.1023/A:1007372823282http://dx.doi.org/doi:10.1016/S0020-7683(98)00098-5http://dx.doi.org/doi:10.1016/j.advwatres.2015.01.015http://dx.doi.org/doi:10.1016/j.gete.2015.03.001http://dx.doi.org/doi:10.1680/geot.9.P.109http://dx.doi.org/doi:10.1098/rspa.2012.0035http://dx.doi.org/doi:10.1098/rspa.2013.0634http://dx.doi.org/doi:10.1680/geot.1953.3.7.287http://dx.doi.org/doi:10.1680/geot.1953.3.7.287http://dx.doi.org/doi:10.1093/qjmam/13.1.98http://dx.doi.org/doi:10.1093/qjmam/13.1.98http://dx.doi.org/doi:10.1093/qjmam/13.2.210http://dx.doi.org/doi:10.1093/qjmam/16.1.115http://dx.doi.org/doi:10.1063/1.1702529http://dx.doi.org/doi:10.1115/1.2900993http://dx.doi.org/doi:10.1016/0307-904X(94)90080-9http://dx.doi.org/doi:10.1016/0020-7225(95)00031-Rhttp://dx.doi.org/doi:10.1016/0020-7225(95)00031-Rhttp://dx.doi.org/doi:10.1007/BF00915270http://dx.doi.org/doi:10.1007/PL00001489http://dx.doi.org/doi:10.1680/geot.14.P.174http://dx.doi.org/doi:10.1016/0266-352X(90)90012-Khttp://dx.doi.org/doi:10.1145/321921.321931http://rspa.royalsocietypublishing.org/

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...................................................

48. Selvadurai APS, Selvadurai PA. 2010 Surface permeability tests: experiments and modellingfor estimating effective permeability. Proc. R. Soc. A 466, 2819–2846. (doi:10.1098/rspa.2009.0475)

49. Selvadurai PA, Selvadurai AP. 2014 On the effective permeability of a heterogeneousporous medium: the role of the geometric mean. Philos. Mag. 94, 2318–2338. (doi:10.1080/14786435.2014.913111)

50. Selvadurai APS, Najari M. 2013 On the interpretation of hydraulic pulse tests on rockspecimens. Adv. Water Resour. 53, 139–149. (doi:10.1016/j.advwatres.2012.11.008)

51. Selvadurai APS, Suvorov AP, Selvadurai PA. 2015 Thermo-hydro-mechanical processesin fractured rock formations during glacial advance. Geosci. Model. Dev. 8, 2167–2185.(doi:10.5194/gmd-8-2167-2015)

52. Zienkiewicz OC, Emson C, Bettess P. 1983 A novel boundary infinite element. Int. J. Numer.Methods Eng. 19, 393–404. (doi:10.1002/nme.1620190307)

53. Selvadurai APS, Karpurapu R. 1989 Composite infinite element for modeling unboundedsaturated soil media. J. Geotech. Eng. ASCE 115, 1633–1646. (doi:10.1061/(ASCE)0733-9410(1989)115:11(1633))

54. Rutqvist J, Vasco DW, Myer L. 2009 Coupled reservoir-geomechanical analysis of CO2injection at In Salah, Algeria. Energy Proc. 1, 1847–1854. (doi:10.1016/j.egypro.2009.01.241)

55. Gor YG, Elliot TR, Prévost JH. 2013 Effects of thermal stresses on caprock integrity duringCO2 storage. Int. J. Greenhouse Gas Control 12, 300–309. (doi:10.1016/j.ijggc.2012.11.020)

56. Hosa A, Esentia M, Stewart J, Haszeldine S. 2011 Injection of CO2 into saline formation;Benchmarking worldwide projects. Chem. Eng. Res. Des. 89, 1855–1864. (doi:10.1016/j.cherd.2011.04.003)


http://dx.doi.org/doi:10.1098/rspa.2009.0475http://dx.doi.org/doi:10.1098/rspa.2009.0475http://dx.doi.org/doi:10.1080/14786435.2014.913111http://dx.doi.org/doi:10.1080/14786435.2014.913111http://dx.doi.org/doi:10.1016/j.advwatres.2012.11.008http://dx.doi.org/doi:10.5194/gmd-8-2167-2015http://dx.doi.org/doi:10.1002/nme.1620190307http://dx.doi.org/doi:10.1061/(ASCE)0733-9410(1989)115:11(1633)http://dx.doi.org/doi:10.1061/(ASCE)0733-9410(1989)115:11(1633)http://dx.doi.org/doi:10.1016/j.egypro.2009.01.241http://dx.doi.org/doi:10.1016/j.ijggc.2012.11.020http://dx.doi.org/doi:10.1016/j.cherd.2011.04.003http://dx.doi.org/doi:10.1016/j.cherd.2011.04.003http://rspa.royalsocietypublishing.org/

IntroductionGoverning equationsSolution method of the governing equations

Initial boundary value problem related to an internally located injection zoneSolution of an initial boundary value problemCOMSOL™ modellingResultsCase when the permeability of the caprock differs from that of the storage formation

Concluding remarksReferences

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