-
rspa.royalsocietypublishing.org
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
2016 The Author(s) Published by the Royal Society. All rights
reserved.
on May 5, 2016http://rspa.royalsocietypublishing.org/Downloaded
from
http://crossmark.crossref.org/dialog/?doi=10.1098/rspa.2015.0418&domain=pdf&date_stamp=2016-03-09mailto:[email protected]://rspa.royalsocietypublishing.org/
-
2
rspa.royalsocietypublishing.orgProc.R.Soc.A472:20150418
...................................................
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
on May 5, 2016http://rspa.royalsocietypublishing.org/Downloaded
from
http://rspa.royalsocietypublishing.org/
-
3
rspa.royalsocietypublishing.orgProc.R.Soc.A472:20150418
...................................................
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.
on May 5, 2016http://rspa.royalsocietypublishing.org/Downloaded
from
http://rspa.royalsocietypublishing.org/
-
4
rspa.royalsocietypublishing.orgProc.R.Soc.A472:20150418
...................................................
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
on May 5, 2016http://rspa.royalsocietypublishing.org/Downloaded
from
http://rspa.royalsocietypublishing.org/
-
5
rspa.royalsocietypublishing.orgProc.R.Soc.A472:20150418
...................................................
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)
on May 5, 2016http://rspa.royalsocietypublishing.org/Downloaded
from
http://rspa.royalsocietypublishing.org/
-
6
rspa.royalsocietypublishing.orgProc.R.Soc.A472:20150418
...................................................
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
on May 5, 2016http://rspa.royalsocietypublishing.org/Downloaded
from
http://rspa.royalsocietypublishing.org/
-
7
rspa.royalsocietypublishing.orgProc.R.Soc.A472:20150418
...................................................
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)
on May 5, 2016http://rspa.royalsocietypublishing.org/Downloaded
from
http://rspa.royalsocietypublishing.org/
-
8
rspa.royalsocietypublishing.orgProc.R.Soc.A472:20150418
...................................................
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
on May 5, 2016http://rspa.royalsocietypublishing.org/Downloaded
from
http://rspa.royalsocietypublishing.org/
-
9
rspa.royalsocietypublishing.orgProc.R.Soc.A472:20150418
...................................................
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.
on May 5, 2016http://rspa.royalsocietypublishing.org/Downloaded
from
http://rspa.royalsocietypublishing.org/
-
10
rspa.royalsocietypublishing.orgProc.R.Soc.A472:20150418
...................................................
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
on May 5, 2016http://rspa.royalsocietypublishing.org/Downloaded
from
http://rspa.royalsocietypublishing.org/
-
11
rspa.royalsocietypublishing.orgProc.R.Soc.A472:20150418
...................................................
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
on May 5, 2016http://rspa.royalsocietypublishing.org/Downloaded
from
http://rspa.royalsocietypublishing.org/
-
12
rspa.royalsocietypublishing.orgProc.R.Soc.A472:20150418
...................................................
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
on May 5, 2016http://rspa.royalsocietypublishing.org/Downloaded
from
http://rspa.royalsocietypublishing.org/
-
13
rspa.royalsocietypublishing.orgProc.R.Soc.A472:20150418
...................................................
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.)
on May 5, 2016http://rspa.royalsocietypublishing.org/Downloaded
from
http://rspa.royalsocietypublishing.org/
-
14
rspa.royalsocietypublishing.orgProc.R.Soc.A472:20150418
...................................................
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.
on May 5, 2016http://rspa.royalsocietypublishing.org/Downloaded
from
http://rspa.royalsocietypublishing.org/
-
15
rspa.royalsocietypublishing.orgProc.R.Soc.A472:20150418
...................................................
–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.
on May 5, 2016http://rspa.royalsocietypublishing.org/Downloaded
from
http://rspa.royalsocietypublishing.org/
-
16
rspa.royalsocietypublishing.orgProc.R.Soc.A472:20150418
...................................................
•
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
on May 5, 2016http://rspa.royalsocietypublishing.org/Downloaded
from
http://rspa.royalsocietypublishing.org/
-
17
rspa.royalsocietypublishing.orgProc.R.Soc.A472:20150418
...................................................
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.
on May 5, 2016http://rspa.royalsocietypublishing.org/Downloaded
from
http://rspa.royalsocietypublishing.org/
-
18
rspa.royalsocietypublishing.orgProc.R.Soc.A472:20150418
...................................................
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
on May 5, 2016http://rspa.royalsocietypublishing.org/Downloaded
from
http://rspa.royalsocietypublishing.org/
-
19
rspa.royalsocietypublishing.orgProc.R.Soc.A472:20150418
...................................................
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.
on May 5, 2016http://rspa.royalsocietypublishing.org/Downloaded
from
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/
-
20
rspa.royalsocietypublishing.orgProc.R.Soc.A472:20150418
...................................................
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)
on May 5, 2016http://rspa.royalsocietypublishing.org/Downloaded
from
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
rspa.royalsocietypublishing.orgProc.R.Soc.A472:20150418
...................................................
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)
on May 5, 2016http://rspa.royalsocietypublishing.org/Downloaded
from
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/
-
22
rspa.royalsocietypublishing.orgProc.R.Soc.A472:20150418
...................................................
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)
on May 5, 2016http://rspa.royalsocietypublishing.org/Downloaded
from
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