INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng 2015; 102:316–331 Published online 24 July 2014 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/nme.4737
An extended finite element method model for carbon sequestration Chris Ladubec, Robert Gracie*,† and James Craig Department of Civil and Environmental Engineering, University of Waterloo, 200 University Ave West, Waterloo, Ontario, Canada
SUMMARY A two-phase flow eXtended Finite Element Method (XFEM) model is presented to analyse the injection and sequestration of carbon dioxide (CO2 ) in deep saline aquifers. XFEM is introduced to accurately approximate near-injection well pressure behaviour with elements significantly larger than the injection well diameter. We present a vertically averaged multiphase flow model that combines XFEM to approximate the pressure field, with a Streamline Upwind/Finite Element Method/Finite Difference Method (SU-FEM-FDM) to approximate the distribution of CO2 in the aquifer. Near-well enrichment functions are presented along with the solution procedure for the coupled problem. Two examples are presented: in the first, CO2 injection into a perfectly horizontal aquifer is modelled with both XFEM and FEM-based methods. The results suggest that XFEM is able to provide low relative errors in the pressure near the well at a reduced computational cost compared with FEM. The impact and selection of the stabilization coefficient of the SU-FEM-FDM is also discussed. In the second example, the XFEM and SU-FEM-FDM model is applied to a more realistic problem of an inclined aquifer to demonstrate the ability of the model to capture the buoyancy-driven migration of CO2 in a deep saline aquifer. Copyright © 2014 John Wiley & Sons, Ltd. Received 12 February 2014; Revised 9 June 2014; Accepted 10 June 2014 KEY WORDS:
multiphase flow; carbon sequestration; extended finite element method; XFEM; streamline upwind
1. INTRODUCTION Carbon sequestration is an operation where carbon dioxide (CO2 ) from point source emitters, such as coal-fired power plants, is injected into deep geological structures for long-term storage. Ideal storage locations are deep saline aquifers that have an impermeable caprock above to prevent the CO2 from leaking into overlaying aquifers. Deep saline aquifers are composed of porous rock, such as sandstone, and initially contain brine (salt water). The resident brine is displaced when the CO2 is injected. At the depths of these aquifers, the temperatures and pressures are such that the CO2 exists in a supercritical state. The supercritical CO2 is less dense than the host brine, and thus, there will be a buoyancy drive causing the CO2 to rise and float on top of the brine. The horizontal dimensions of the aquifers are measured in kilometres, but the diameter of the injection wells is measured in centimetres. To accurately resolve the pressure field in the vicinity of injection or abandoned wells using traditional numerical methods would require a prohibitively fine discretization near the wells.
*Correspondence to: Robert Gracie, Department of Civil and Environmental Engineering, University of Waterloo, 200 University Ave West, Waterloo, Ontario, Canada. † E-mail:
[email protected] ‡ In honour of Professor Ted Belytschko. Copyright © 2014 John Wiley & Sons, Ltd.
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In this paper, an eXtended Finite Element Method (XFEM) model for accurate and efficient modelling of the pressure field resulting from injecting CO2 into a deep saline aquifer using a vertically averaged multiphase flow formulation is presented. XFEM is then coupled to a Streamline Upwind/Finite Element Method/Finite Difference Method (SU-FEM-FDM) to model the evolution of the average saturation of each phase in the aquifer. This article follows earlier works by the authors on XFEM-based single phase aquifer flow models [1, 2]. The XFEM was comprehensively developed by Belytschko and co-workers over the past 15 years [3–6]. It was first applied to the simulation of the singularities and discontinuities found in linear elastic fracture mechanics [7, 8]. Later applications focussed on dynamic crack propagation [9–11], multiscale analysis [12–14] and material modelling [15–17]. Comparably less attention has been given to the application of XFEM to multiphase flow. Some notable exceptions are the two-dimensional two-phase flow works of Chessa and Belytschko [18] and Cheng and Fries [19], the three-dimensional two-phase flow models of Sauerland and Fries [20] and the analysis of partially saturated porous media of Mohammadnejad and Khoei [21]. One of the major differences between these earlier works and that presented here is that they focus on twodimensional and three-dimensional models, whereas the present model is quasi-three-dimensional, leading to different governing equations, different representation of the interface between fluid phases, and different enrichment functions. Furthermore, the focus here is on the singular behaviour near wells. This article is dedicated to Professor Ted Belytschko for his contributions to the field of computational mechanics, his stewardship over this field and this journal, and his mentorship of young researchers. This article is organized as follows. We present the equations governing conservation of mass in the vertically averaged multiphase flow of CO2 and brine in a porous medium in Section 2. In Sections 3, 4, and 5, we present the government equations, weak form, and discrete equations, respectively. Two numerical examples that demonstrate the capabilities of the outlined model are presented is Section 6 and conclusions are given in Section 7. 2. PROBLEM STATEMENT Consider a saline aquifer, bounded from above and below by impermeable aquicludes, as illustrated in the cross-section shown in Figure 1. CO2 is injected into the saline aquifer displacing the host brine fluid. Because the density of the CO2 is less than that of the brine, it floats on top of the brine and spreads laterally. The injection well acts as a point source, with a locally singular pressure field. Singular pressure fields are also found near hydraulic connections between adjacent aquifers, such as in the vicinity of abandoned wells, faults and fractures [1]. In this paper, we present a computationally efficient methodology to model the injection of CO2 into a carbon sequestration system. Vertically averaged mass balance equations [22] are used to describe the pressure distribution and the average saturation distribution of CO2 in the injection aquifer. The use of a vertically averaged formulation allows us to reduce the dimensionality of the problem from three to two, by integrating the governing equations over the depth of the aquifers [23–26]. Vertical averaging reduces the computational cost of the resulting numerical model by a factor of m, where m is the number of nodes in the vertical direction of the three-dimensional model. This result assumes that the required discretization in the horizontal direction is unchanged by vertical averaging. In practice, it is expected that m 6 n, where n is the number of nodes in each horizontal direction, and so it is expected that vertical averaging will reduce the number of degrees of freedom by as much as a factor of n and by as little as a factor of 2. Reducing the dimensionality of the problem is generally considered a reasonable approximation because the depth of the aquifers is quite small compared with the horizontal dimensions of the aquifer [22]. The governing equations adopted in this paper assume the following: Chemical, thermal and mechanical effects are negligible; The brine, CO2 , and solid matrix are incompressible; Copyright © 2014 John Wiley & Sons, Ltd.
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Figure 1. Defining parameters of carbon sequestration model.
The viscosity and density of brine and CO2 are constant; The interface between the brine and CO2 is sharp, and the capillary pressure is zero; The pressure in the aquifer varies hydrostatically through the depth of the aquifer. The governing equations are the mass balance equations for brine and CO2 that are combined with two equations describing vertically averaged fluxes of each phase using a multiphase extension of Darcy’s law [22]. The mass balance equations for the CO2 and brine phases in the aquifer are: @h C r qO C D qC;i nj ; @t
(1)
@.H h/ C r qO B D 0; @t
(2)
.1 Sres;B /
.1 Sres;B /
where is the porosity of the aquifer, Sres;B is residual saturation of the brine, h.x; t / is the depth of the CO2 , qO C .x; t / is the vertically averaged CO2 flux, qO B .x; t / is the vertically averaged brine flux, qC;i nj .x; t / is the source term to account for the injection of CO2 into the aquifer. CO2 is assumed to be injected from a point source and is defined as qC;i nj D Qi nj ı.x xinj /;
(3)
where Qi nj is the injection rate (e.g., m3 =d , where d is days), and ı./ is the Dirac delta function. The vertically averaged fluxes are given by the multiphase extension of Darcy’s Law [22]: qO C D h
kkrel;C .r pbot B gr H C gr h C C gr ´t op /; C
qO B D .H h/
k .r pbot C B gr ´bot /; B
(4)
(5)
where k is the intrinsic permeability of the aquifer, krel;C is the relative permeability, C is the viscosity of CO2 , B is the viscosity of brine, pbot .x; t / is the pressure at the bottom of the aquifer, C is the density of CO2 , B is the density of the brine, D B C , g is the gravitational constant, ´t op .x/ is the vertical depth of the top of the aquifer, ´bot .x/ is the vertical depth of the bottom of the aquifer and H.x/ is the thickness of the aquifer. Copyright © 2014 John Wiley & Sons, Ltd.
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3. STRONG FORM The solution approach we take is to manipulate (1) and (2) to arrive at two equations: one that is solved for pressure (pressure equation) and the second that is solved for the average saturation distribution (saturation equation). We first obtain the strong form of the pressure equation and saturation equation from (1) and (2) combined with (4) and (5). The pressure equation is obtained by adding the mass balance equations for each phase. The transient term disappears, and the pressure equation becomes an elliptical steady state equation, that is, a Poisson-type equation. This leads to an instantaneous propagation of changes in pressure through the domain . The strong form of the pressure equation is to find pbot .x/ such that ! C kkREL r h .r pbot C gr h B gr H C C gr ´t op / C (6) k C r .H h/ .r pbot C B gr ´bot / B D qC;i nj ; x 2 ; and pbot .x; t / D p bot .x; t / on p ;
(7)
qO C .x; t / D qC .x; t / on q ;
(8)
qO B .x; t / D qB .x; t / on q ;
(9)
where p [ q is the boundary of and p \ q D ;. The saturation equation is obtained directly from the mass balance equation for the brine, (2) and (5). The strong form of the saturation equation is to find h.x; t / such that @.H h/ k .1 Sres;B / .r pbot C B gr´bot / r .H h/ @t B (10) D 0; x 2 , t 2 Œ0; tend ; h.x; t / D h.x; t / on h ;
(11)
h.x; 0/ D ho .x/ on ;
(12)
where h is the boundary of . 4. WEAK FORM In this section, the weak forms of the pressure equation and the saturation equation are presented. The weak form of the pressure equation for the pressure at the bottom of the aquifer (6) is to find pbot .x/ 2 W such that Z Z k k T .r w/ .H h/ r pbot d C .r w/T .H h/ B gr ´bot d ; B B Z Z kkrel;C kkrel;C C .r w/T h gr hd .r w/T h B gr Hd ; (13) C C Z I kkrel;C C .r w/T h C gr ´t op d w T qO TB nd ; C Copyright © 2014 John Wiley & Sons, Ltd.
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Z
kkrel;C C .r w/ h r pbot d C Z w T qC;i nj d ; 8w 2 W0 : D T
I
w T qO TC nd ;
The function spaces are given as W D ¹pbot .x/jpbot .x/ 2 H 1 nwel l ; pbot .x/ D p bot on p º; W0 D ¹w.x/jw.x/ 2 H 1 nwel l ; w.x/ D 0 on p º;
(14)
where wel l is the domain of the well. The weak form of the saturation equation is to find h.x; t / 2 U such that Z Z @.H h/ k v T .1 SRE S;B / .r v/T .H h/ .r pbot C B gr´bot /d ; d C @t B I (15) T T v qO B nd ; 8v 2 U0 ; D
where U and U0 are the appropriate function spaces for h.x; t / and v.x; t /, respectively. 5. DISCRETE EQUATIONS AND APPROXIMATION In this section, the discretization of the weak forms of the pressure and the saturation equations is presented. The pressure is discretized in space using the XFEM, and the CO2 saturation is discretized using a Streamline Upwind Finite Element Method (SU-FEM) in space and a Finite Difference Method (FDM) in time. 5.1. XFEM approximation of the pressure equation The solution to the pressure equation (6) is known to be singular at the injection well. To better approximate the pressure field, an XFEM approximation is adopted. The key to the XFEM is that the FEM approximation is enriched in the vicinity of the injection wells with functions that capture the asymptotic behaviour near the well. In contrast, an FEM model would require a fine mesh around the wells in order to obtain accurate solutions. As mentioned, XFEM was previously used to model single phase porous media flow in an aquifer [1, 2]. For the multiphase XFEM model presented here, we investigate the use of the same enrichment function. The enrichment function for the pressure near well ˛ is ² log.r˛ .x//; r˛ > rw ; (16) ˛ .x/ D log.rw /; r˛ 6 rw ; where r˛ is the distance to the centre of the well number, ˛, and rw is the radius of the well. The enrichment function is illustrated in Figure 2. The distance to the centre of well ˛ is given by r˛ .x/ D jjx xi˛nj jj;
(17)
where xi˛nj are the spatial coordinates of the injection wells. The XFEM approximation of pressure is given by p.x/ D
X
ni nj
NI .x/pI C
I 2N
X
˛D1
w˛ .x/
˛ .x/
X
NJ .x/a˛J ; x 2 ;
(18)
J 2S˛
where N is the set of all nodes, pI is the pressure at node I, ni nj is the number of injection wells, S˛ is the set of nodes in the enriched domain of well ˛ (Figure 3), w˛ .x/ is a weighting function Copyright © 2014 John Wiley & Sons, Ltd.
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Figure 2. Illustration of the enrichment function.
Figure 3. Enriched domain. S˛ is the set of nodes within the enrichment radius of well ˛.
that blends the enriched and unenriched parts of the domain ([1, 27]), a˛J are the enriched degrees of freedom and NI .x/ are the standard finite element basis functions. The pressure approximation can be written in matrix form as p.x/ D Npbot C Na;
(19)
where N is the matrix of standard FEM shape functions and N is the matrix of the enriched shape functions. pbot and a are vectors containing the standard FEM pressure degrees of freedom and the enriched degrees of freedom, respectively. pTbot D ¹p1 ; p2 ; : : :; pnn º;
(20)
aT D ¹a1 ; a2 ; : : :; amm º;
(21)
where nn and mm are the number of nodes in the mesh and the number of enriched nodes, respectively. The blending weight function is defined as X NI .x/wI ; (22) w.x/ D I 2N
where wI is one for an enriched node within the enrichment radius, re nr , and wI is zero for an enriched node outside the enrichment radius. The XFEM discrete pressure equations are ³ ² ³ ² Fp pbot (23) D Kaquif er u Fp ; Fp D Fp1 C Fp2 C Fp3 C Fp4 ; Copyright © 2014 John Wiley & Sons, Ltd.
(24)
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Fp D Fp1 C Fp2 C Fp3 C Fp4 ; " Kaquif er D
K K T K K;
(25)
# (26)
where Kp is the stiffness matrix, Fp1 is the injection vector, Fp2 is the boundary flux vector, Fp3 is the buoyancy vector and Fp4 is the aquifer slope vector. Kp and Kp are enriched stiffness matrices, Fp1 is the enriched injection vector, Fp2 is the enriched boundary flux vector, Fp3 is the enriched buoyancy vector and Fp4 is the enriched aquifer slope vector. The components of the matrices and vectors are Z Kp;IJ D
Z K p;IJ D
Z K p;IJ D
T BI
T BI
H h hkREL;C kBJ d ; 8I; J 2 N ; C C B
(27)
H h hkREL;C kBJ d ; 8I 2 N ; 8J 2 S˛ ; C C B
(28)
H h hkREL;C kBJ d ; 8I 2 S˛ ; 8J 2 Sˇ ; C C B
(29)
BTI
Z Fp1;I D
Z
T
F p1;I D
I
T
F p2;I D
Z Fp3;I D
Z F p3;I D
F p4;I
(30)
(31)
(32)
NI .qO B C qO C / nd ; 8I 2 S˛ ;
(33)
kkrel;C ghr hd ; 8I 2 N ; C
(34)
BTI
T kkrel;C
BI
C
ghr hd ; 8I 2 S˛ ;
(35)
kkrel;C k h D .B gr H C gr ´t op / .H h/ B gr ´bot d ; 8I 2 N ; C B (36) Z kkrel;C k T D BI h .B gr H C gr ´t op / .H h/ B gr ´bot d ; 8I 2 S˛ ; C B (37) Z
Fp4;I
NI qC;i nj d ; 8I 2 S˛ ;
NTI .qO B C qO C / nd ; 8I 2 N ;
Fp2;I D I
NTI qC;i nj d ; 8I 2 N ;
BTI
where B and B are matrices of the derivatives of the standard and enriched shape functions, respectively and n is a unit normal vector to the Neumann boundary. 5.1.1. Numerical integration. Numerical integration of the unenriched elements in the pressure equation is performed using 2 2 Gauss quadrature. Enriched elements that do not contain a well are integrated using 44 Gauss quadrature. Enriched elements containing wells are integrated using Copyright © 2014 John Wiley & Sons, Ltd.
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an iterative bisection scheme as described in [1]. The subcells are then integrated using 3 3 Gauss quadrature if they are located outside the well radius and using 1 1 Gauss quadrature if they are located within the well radius. The number of subcells was determined so that the integration errors are sufficiently small to not affect the results. 5.2. Stabilized Galerkin FEM—Crank–Nicolson discretization of the saturation equation An SU-FEM-FDM approximation is used to discretize the saturation equation. Space is discretized using an SU-FEM approach. SU is implemented as an added artificial diffusion term that counteracts the negative diffusion that occurs because of the Galerkin FEM discretization of the hyperbolic saturation equation (i.e. the advection equation). A Crank–Nicolson (trapezoidal) FDM is used to discretize the time domain. The average saturation of the brine is approximated in space by X h.x; t / D NI .x/hI .t /: (38) I 2N
The semi-discrete FEM saturation equations are given by ° ± ŒCs hP B C ŒKs ¹hB º D ¹Fs º; Z Cs D
(39)
Ne T .1 SRE S;B /Ne d ;
(40)
k .r pbot C B gr´bot /Ne d ; B
(41)
Z Kadv D
Be T
I Fs D
Ne T qO B nd ;
(42)
where ¹hB º is the unknown vector of brine depth in the aquifer and hB .x; t / D H.x/ h.x; t /, Cs is the mass matrix, Kadv is the advection matrix , Fs is the boundary flux vector. Ne are the shape functions for each element, and Be are the derivatives of the shape functions. The XFEM pressure approximation is utilized in the advection matrix, Kadv above. The streamline upwind (SU) method is used to stabilize the saturation equation, which is a pure advection equation. SU is implemented using artificial diffusion as described in [28] and [29]. Artificial diffusion acts to offset the negative diffusion that is created because of the Galerkin FEM approach. The artificial diffusion to the system, which for the pure advection case simplifies to Z k H Be T
jr pbot C B gr´bot jBe de ; (43) KSU D e .1 S / res;B B where is a parameter to control the amount of diffusion applied and is meant to replace the constant 1 multiplier of p.15/ that was used in [28]. The amount of artificial diffusion is element dependent because it is a function of the pressure gradient in each cell. This approach in not consistent with respect to the transient term and the boundary flux vector; therefore, must be selected carefully to avoid overly diffuse solutions [28]. Improved stabilization approaches are desirable and should be the subject of future research. In the saturation equation, time is discretized using the FDM. This can be written as .ŒCs C t ŒKs n / ¹hB ºnC1 D .ŒCs t .1 /ŒKs n / ¹hB ºn C t ¹Fs ºnC1 C .1 /¹Fs ºn ;
(44)
where ŒKs D ŒKadv C ŒKSU . In the aforementioned equation, D 0:5 gives the Crank–Nicolson method. Copyright © 2014 John Wiley & Sons, Ltd.
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Figure 4. Solution strategy. Pressure is solved using (23), and saturation is solved using (44).
5.3. Solution procedure The system of equations that we obtain from the discretization of the pressure and saturation equations are strongly coupled. A sequential solution strategy similar to IMplicit Pressure Explicit Saturation (IMPES) [30], which is common in the reservoir simulation community is adopted. In this solution scheme, initial conditions for the saturation are specified and the initial pressure distribution are computed. Then for each timestep, the saturation equation is solved using the pressure distribution from the previous timestep. The pressure distribution is then updated using the saturation distribution from the current timestep. This solution strategy is shown in Figure 4. 6. NUMERICAL EXAMPLES The solution approach described in this paper is demonstrated using two examples. 6.1. Example 1 In this example, CO2 is injected into a brine filled aquifer. The setup of this problem is shown in Figure 5. The system properties are given in Table I. The aquifer is bounded above and below by Copyright © 2014 John Wiley & Sons, Ltd.
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Figure 5. Set up of Example 1.
Table I. System properties for Example 1. Property B C B C Sres;B kx ky krel;C qi nj rw
Value
Units
5.11e4 6.11e5 1099 400 0 0.15 1e15 1e15 1 1600 0.15
N s/m2 N s/m2 kg/m3 kg/m3 — — m2 m2 — m3 /d m
impermeable caprock layers (aquicludes). Initially, the aquifer is filled with brine. The water table begins at the top boundary of the aquifer. CO2 is injected at a constant rate of 1600 m3 /d. As the injection progresses, the CO2 plume spreads throughout the aquifer. Dirichlet boundary conditions of hydrostatic pressure are applied to the pressure equation along the whole boundary of the domain. An initial condition of h.x; 0/ D 0 throughout the domain is used (i.e. brine only). Dirichlet boundary conditions of h.x; t / D 0 are imposed at all boundaries, and as such, the domain must be large enough so that the CO2 plume remains far enough away from the domain boundaries. The simulation is conducted for a period of 3 days. Figure 6 compares the pressure distributions for two XFEM and two FEM simulations with different mesh densities. The pressure is plotted in the radial direction from the centre of the well along section A–A, as shown in Figure 5. Near the injection well, the XFEM solutions give higher pressures. The pressure near the injection well increases along with refinement of the FEM and XFEM meshes. At further distances away from the well, XFEM and FEM give similar pressure fields. Copyright © 2014 John Wiley & Sons, Ltd.
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Figure 6. Comparison of XFEM and FEM pressure distributions along a radius centred at injection well and along line A–A using bilinear quadrilateral elements.
Figure 7. Comparison of XFEM and FEM average CO2 saturation distributions along a radius centred at injection well and along line A–A using bilinear quadrilateral elements.
Figure 7 shows the average CO2 saturation distribution along a radius that follows section A–A (Figure 5). The XFEM and FEM pressure approximations give similar results. The depth of CO2 at the injection well converges to the depth of the aquifer as element size is reduced. A study of the relative error in the pressure at the well is shown in Figure 8. The relative error is obtained by comparison with the pressure at the well using a fine FEM mesh (3:6 105 elements) with the injection at a node. This compares with a maximum of 2:3104 elements used in the study. For the coarsest meshes, the FEM with the injection at a node provides the least relative error. This relative error decreases rather slowly. XFEM reaches a 1% relative error much sooner than both FEM approaches. Therefore, to achieve a relative error of less than 1% it is computationally more efficient to use XFEM. XFEM can achieve low relative errors even when the injection takes place mid-element. FEM performs poorly when the injection takes place mid-element. The magnitudes of the slopes of the relative error in the pressure at the well versus number of element plots in Figure 8 are 0.19, 0.28 and 0.91 for the FEM with injection mid-element, FEM with injection at a node, and XFEM with injection mid-element, respectively. The slopes were computed using a best fit to the data. Copyright © 2014 John Wiley & Sons, Ltd.
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Figure 8. Comparison of relative error in pressure at the injection well for XFEM and FEM.
Figure 9. Effect of non-dimensional stabilization parameter on average CO2 saturation profile along a radius centred at injection well and along line A–A.
It is noted that Figure 8 plots the convergence of a point value and not of a norm. The authors are not aware of any theorectical derivation of the convergence rate of any norm for the current set of coupled governing equations, which could be used to more rigourously evaluate the convergence of the method presented. It is further noted that the meshes used to generate Figure 8 are not locally refined near the well (point of singularity). This is standard practice in the literature for demonstrating convergence of XFEM in the modelling of cracks [31, 32], dislocations [33], and well leakage [1]. Increased accuracy of both FEM and XFEM can be expected with non-uniform meshes, refined at the well. The effect of the stabilization ( ) on the average saturation profile is studied using XFEM in Figure 9. At low values of , the saturation profile is non-physical. The approximation becomes smoother as is increased. One side effect of increased damping is the reduction of the depth of CO2 at the injection well. Care must be taken in the selection of to avoid overdamping. The effect of on the pressure is shown in Figure 10. Once is large enough to eliminate the non-physical values, increasing increases the pressure near the well and reduces the pressure further away from the well. Therefore, a careful selection of is important. We have not investigated the behaviour of the presented method in the limit as rw approaches zero. However, typical well radii encountered in the field are of similar magnitude to the radius Copyright © 2014 John Wiley & Sons, Ltd.
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Figure 10. Effect of non-dimensional stabilization parameter on pressure along a radius centred at injection well and along line A–A.
Figure 11. Set up of Example 2.
rw D 0:15 m used in our examples. The method presented is insensitive to values of rw typical of actual oil and gas wells. 6.2. Example 2 In this example, CO2 is injected into a sloping aquifer. The problem is illustrated in Figure 11 and the system properties are described in Table II. The aquifer is bounded above and below by impermeable layers. The top boundary of the aquifer slopes upwards to the right, and the bottom Copyright © 2014 John Wiley & Sons, Ltd.
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Table II. System properties for Example 2. Property B C B C Sres;B kx ky krel;C qi nj rw
Value
Units
5.11e4 6.11e5 1099 400 0 0.15 1e15 1e15 1 1600 0.15
N s/m2 N s/m2 kg/m3 kg/m3 — — m2 m2 — m3 /d m
Figure 12. XFEM average CO2 saturation distribution after 60 days of injection.
boundary slopes downwards to the right. Therefore, the depth of the aquifer is a function of the x and y coordinates. Dirichlet boundary conditions are specified for the pressure equations as the hydrostatic pressure caused by the brine that exists between the top of the water table and the bottom of the aquifer. For the saturation equation, Dirichlet boundary conditions are specified such that at the boundaries of the domain, the full depth of the aquifer is filled with brine. Initial conditions of a completely brine-filled aquifer are applied. The average CO2 saturation profile is shown in Figure 12. Because the CO2 has a lower density than the host brine fluid, the CO2 should preferentially flow to the right, following the up-sloping top boundary of the aquifer. Figure 12 shows the average saturation after 60 days of injection for the system shown in Figure 11 compared with the system shown in Figure 5. The average saturation is shown along axis b, defined in Figure 11. Comparing the evolution of the CO2 plume in the horizontal aquifer case and the sloping aquifer, we can see that for the sloping aquifer, the CO2 preferentially flows up the slope. This preferential flow is caused by the buoyant drive resulting from the lower density of the CO2 compared with brine. Thus, the current formulation adequately captures the updip effect of CO2 migration. Figure 13 shows the pressure distribution along axis b, defined in Figure 11 after 60 days of injection. The pressure to the left of the well is larger than to the right of the well because of the sloping aquifer. The larger pressure on the left drives the flow updip to the right. During the early Copyright © 2014 John Wiley & Sons, Ltd.
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C. LADUBEC, R. GRACIE AND J. CRAIG
Figure 13. XFEM pressure distribution after 60 days of injection.
stages of injection, the saturation of CO2 is significantly impacted by the sloping aquifer geometry because of buoyancy forces. 7. CONCLUSIONS A computationally efficient model of carbon sequestration, where carbon dioxide (CO2 ) is injected into deep saline aquifers, is presented. An effective computational scheme is obtained by combining the efficiency of a vertically averaged formulation and through the enrichment of the pressure approximation using the XFEM. The XFEM-based formulation is used to improve the approximation of the singular pressure field in the vicinity of an injection well. The XFEM pressure approximation is combined with a SU-FEM-FDM framework to approximate the average CO2 saturation through the thickness of the aquifer. Using the XFEM-SU-FEM-FDM framework, we examined two examples. In the first, it was shown that XFEM is able to achieve low relative errors in the pressure near injection wells at a lower computational cost, when compared with an FEM approximation. The SU stabilization parameter, , must be selected carefully to avoid over diffuse saturation distributions. In the second example, the XFEM-SU-FEM-FD simulator is demonstrated to be able to capture the important effect of buoyancy-driven flow of CO2 in a sloping aquifer. ACKNOWLEDGEMENTS
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