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Coupled Electrokinetic–Hydromechanic Model for \(\text{ CO}_{2}\) Sequestration in Porous Media

Abstract

In this paper, a computational model for the simulation of coupled electrokinetic and hydromechanical flow in a multiphase domain is introduced. Particular emphasis is placed on modeling \(\text{ CO}_{2}\) flow in a deformed, unsaturated geologic formation and its associated streaming potential. The governing field equations are derived based on the averaging theory and solved numerically based on a mixed discretization scheme. The standard Galerkin finite element method is utilized to discretize the deformation and the diffusive dominant field equations, and the extended finite element method, together with the level-set method, is utilized to discretize the advective dominant field equations. The level-set method is employed to trace the \(\text{ CO}_{2}\) plume front, and the extended finite element method is employed to model the high gradient in the saturation field front. This mixed discretization scheme leads to a highly convergent system, giving a stable and effectively mesh-independent model; furthermore, it minimizes the number of degrees of freedom, making the numerical scheme computationally efficient. The capability of the proposed model is evaluated by verification and numerical examples. Effects of the formation stiffness on the \(\text{ CO}_{2}\) flow and the salinity content on the streaming potential are discussed.

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Abbreviations

\(a_J \) :

XFEM extra degree of freedom

\(\mathbf{C}\) :

Electrokinetic coupling coefficient, \(\text{ V} \text{ Pa}^{-1}\)

\(\mathbf{C}_\mathrm{f} \) :

Salt concentration, mole \(\text{ m}^{-3}\)

\(\mathbf{C}_\mathrm{r} \) :

Relative coupling coefficient

\(\mathbf{D}_\mathrm{e} \) :

Stiffness matrix of the solid

\(E\) :

Young’s modulus of elasticity, Pa

\(g\) :

Gravitational acceleration, \(\text{ m} \text{ s}^{-2}\)

\(\mathbf{g}\) :

Gravity acceleration vector, \(\text{ m} \text{ s}^{-2}\)

\(h_\mathrm{e} \) :

Characteristic length of the element, m

\(I\) :

Identity tensor

\(\mathbf{J}\) :

Total electric current density, \(\text{ A} \text{ m}^{-2}\)

\(\mathbf{k}\) :

Intrinsic permeability tensor, \(\text{ m}^{2}\)

\(k_\mathrm{e} \) :

Electro-osmotic permeability, \(\text{ m}^{2}\text{ s}^{-1}\text{ V}^{-1}\)

\(k_{\mathrm{r}\pi }\) :

Relative permeability of \(\pi \) phase

\(K_\pi \) :

Bulk modulus of the phase \(\pi , \text{ Pa}^{-1}\)

\(K_s \) :

Bulk modulus of the grain material, \(\text{ Pa}^{-1}\)

\(\mathbf{L}\) :

Cross coupling coefficients, \(\text{ m}^{2} \text{ V}^{-1} \text{ s}^{-1}\)

\({\hat{\mathbf{L}}}\) :

Displacement-strain operator

m :

A vector equal to \(\left\langle \begin{array}{llllll} 1&1&1&0&0&0 \end{array} \right\rangle ^{T}\)

\(m\) :

Cementation exponent

\(n\) :

Archie’s saturation exponent. Otherwise defined in the text.

\(n_\mathrm{e} \) :

Number of nodes in the element

\(P_\mathrm{c} \) :

Capillary pressure, Pa

\(P_\pi \) :

\(\pi \) Phase pressure, Pa

\(P_\mathrm{b} \) :

Entry pressure, Pa

\(Q_\pi \) :

Imposed mass flux of phase \(\pi \) normal to the boundary, \(\text{ kg} \text{ m}^{-2}\text{ s}\)

\(S_\pi \) :

\(\pi \) phase saturation

\(S_{\mathrm{r}\pi }\) :

Residual saturation of phase \(\pi \)

\(S_\mathrm{e} \) :

Effective saturation

\({\breve{\mathbf{t}}}_\pi \) :

Intra-phase stress tensor, \(\text{ N} \text{ m}^{-2}\)

\(\mathbf{u}\) :

Displacement vector of solid phase, m

\(\mathbf{v}_\Gamma \) :

Interface velocity, \(\text{ m} \text{ s}^{-1}\)

\(\mathbf{v}_\pi \) :

Mass averaged velocity of phase \({\uppi }, \text{ m} \text{ s}^{-1}\)

\(V\) :

Electrical potential, V

\(\alpha \) :

Biot’s constant

\({\varvec{\varepsilon }}\) :

Total strain of the solid

\({\varvec{\varepsilon }}_0\) :

Initial strain

\(\varepsilon _w \) :

Brine permittivity, \(\text{ F} \text{ m}^{-1}\)

\(\phi \) :

Porosity

\(\Phi \) :

Level set function

\({\varvec{\lambda }}\) :

Pore size distribution index

\(\mu _\pi \) :

Dynamic viscosity of phase \({\uppi }\), Pa s

\(\upsilon \) :

Poisson’s ratio

\(\theta \) :

Time integration parameter

\(\rho _\mathrm{eff} \) :

Effective density in multiphase domain, \(\text{ kg} \text{ m}^{-3}\)

\(\rho _\pi \) :

Intrinsic phase averaged density of phase \(\pi , \text{ kg} \text{ m}^{-3}\)

\({\varvec{\sigma }}\) :

Total stress, \(\text{ N} \text{ m}^{-2}\)

\({\varvec{\sigma _{1}}}^{\prime }\) :

Effective stress, \(\text{ N} \text{ m}^{-2}\)

\({\varvec{\sigma }}^{\prime \prime }\) :

Effective stress with Biot’s constant included, \(\text{ N} \text{ m}^{-2}\)

\(\sigma _\mathrm{e} \) :

Electrical conductivity of bulk formation, \(\text{ S} \text{ m}^{-1}\)

\(\sigma _\pi \) :

Electrical conductivity of phase \(\pi , \text{ S} \text{ m}^{-1}\)

\(\sigma _\mathrm{r} \) :

Relative electric conductivity

\(\tau \) :

Stabilization parameter

\(\zeta \) :

Zeta potential, V

eff:

Effective stress

g:

Gas phase

\(\pi \) :

\(\pi \) phase

r:

Residual saturation

s:

Solid phase

w:

Water phase

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Acknowledgments

This project is financially sponsored by AgentschapNL of the Dutch ministry of economic affairs.

Author information

Correspondence to M. Talebian.

Appendix

Appendix

The matrix coefficients of Eq. 77 are:

$$\begin{aligned} \mathbf{K}_{11}&= \mathbf{K}_{11}^0 = -\int \limits _\Omega {\mathbf{B}^{T}\mathbf{DB} \text{ d}\Omega }\end{aligned}$$
(86)
$$\begin{aligned} \mathbf{K}_{12}&= \mathbf{K}_{12}^0 =\int \limits _\Omega {\mathbf{B}^{T}\mathbf{m}^{T}\alpha \mathbf{N}\text{ d}\Omega } \end{aligned}$$
(87)
$$\begin{aligned} \mathbf{K}_{13}&= -\int \limits _\Omega {\mathbf{B}^{T}\mathbf{m}^{T}\alpha P_\mathrm{c}^\mathrm{r} \mathbf{N}\text{ d}\Omega } -\int \limits _\Omega {\mathbf{B}^{T}\mathbf{m}^{T}\alpha \frac{\text{ d}P_\mathrm{c} }{\text{ d}S_\mathrm{w} }\mathbf{S}_\mathrm{w}^\mathrm{r} \mathbf{N}\text{ d}\Omega } \nonumber \\&\quad -\int \limits _{\Omega ^{+}} {\mathbf{B}^{T}\mathbf{m}^{T}\alpha \frac{\text{ d}P_\mathrm{c} }{\text{ d}S_\mathrm{w} }\mathbf{a}_J^\mathrm{r} \mathbf{N}^{*}\text{ d}\Omega } +\int \limits _\Omega {\mathbf{N}^T \frac{\partial \rho _\mathrm{eff} }{\partial S_\mathrm{w} } \mathbf{Ng} \text{ d}\Omega } \end{aligned}$$
(88)
$$\begin{aligned} \mathbf{K}_{13}^0&= -\int \limits _\Omega {\mathbf{B}^{T}\mathbf{m}^{T}\alpha P_\mathrm{c}^\mathrm{r} \mathbf{N} \text{ d}\Omega }\end{aligned}$$
(89)
$$\begin{aligned} \mathbf{K}_{14}&= \mathbf{K}_{14}^0 =-\int \limits _{\Omega ^{+}} {\mathbf{B}^{T}\mathbf{m}^{T}\alpha P_\mathrm{c}^\mathrm{r} \mathbf{N}^{*}\text{ d}\Omega } \end{aligned}$$
(90)
$$\begin{aligned} \mathbf{K}_{22}&= \mathbf{K}_{22}^0 =\int \limits _\Omega {(\nabla \mathbf{N})^{T}\mathbf{c}_1^\mathrm{r} (\nabla \mathbf{N}) \text{ d}\Omega } \end{aligned}$$
(91)
$$\begin{aligned} \mathbf{K}_{23}&= \int \limits _\Omega {\mathbf{N}^T \left( {\frac{\partial d_3 }{\partial S_\mathrm{w} } {\dot{\mathbf{u}}}^\mathrm{r}\mathbf{N}} \right) \mathbf{m}^{T}\mathbf{B} } \text{ d}\Omega +\int \limits _\Omega \mathbf{N}^{T}\left( \frac{\partial d_1 }{\partial S_\mathrm{w} }\mathbf{N}{\dot{\mathbf{P}}}_\mathrm{g}^\mathrm{r} \right) \mathbf{N}\text{ d}\Omega \nonumber \\&\quad +\int \limits _\Omega \mathbf{N}^{T}\left( \frac{\partial d_2 }{\partial S_\mathrm{w} }\mathbf{N}{\dot{\mathbf{S}}}_\mathrm{w}^\mathrm{r} \right) \mathbf{N}\text{ d}\Omega +\int \limits _\Omega \mathbf{N}^{T}\left( \frac{\partial d_2 }{\partial S_\mathrm{w} }\mathbf{N}{\dot{\mathbf{a}}}_J^\mathrm{r} \right) \mathbf{N}^{*}\text{ d}\Omega \; \nonumber \\&\quad +\int \limits _\Omega (\nabla \mathbf{N})^{T}\left( \frac{\partial \mathbf{c}_1 }{\partial S_\mathrm{w} }\mathbf{N}{\bar{\mathbf{P}}}_\mathrm{g}^\mathrm{r} \right) (\nabla \mathbf{N}) \text{ d}\Omega -\int \limits _\Omega {(\nabla \mathbf{N})^{T}\left( {\frac{\partial \mathbf{G}_1 }{\partial S_\mathrm{w} }\mathbf{N}} \right) \text{ d}\Omega } \nonumber \\&\quad +\int \limits _\Omega {(\nabla \mathbf{N})^{T}\left( {\frac{\partial \mathbf{c}_2 }{\partial S_\mathrm{w} }\mathbf{N}{\bar{\mathbf{S}}}_\mathrm{w}^\mathrm{r} } \right) (\nabla \mathbf{N}) \text{ d}\Omega } +\int \limits _{\Omega ^{+}} {(\nabla \mathbf{N})^{T}\left( {\frac{\partial \mathbf{c}_2 }{\partial S_\mathrm{w} }\mathbf{Na}_J^\mathrm{r} } \right) (\nabla \mathbf{N}^{*})\text{ d}\Omega } \nonumber \\&\quad +\int \limits _\Omega {(\nabla \mathbf{N})^{T}(\mathbf{c}_2^\mathrm{r} )(\nabla \mathbf{N}) \text{ d}\Omega } +\int \limits _\Omega {(\nabla \mathbf{N})^{T}\left( {\frac{\partial \mathbf{c}_3 }{\partial S_\mathrm{} }\mathbf{N}{\bar{\mathbf{V}}}^\mathrm{r}} \right) (\nabla \mathbf{N}) \text{ d}\Omega }\end{aligned}$$
(92)
$$\begin{aligned} \mathbf{K}_{23}^0&= \int \limits _\Omega {(\nabla \mathbf{N})^{T}\mathbf{c}_2^\mathrm{r} \left( {\nabla \mathbf{N}} \right) \text{ d}\Omega } \end{aligned}$$
(93)
$$\begin{aligned} \mathbf{K}_{24}&= \mathbf{K}_{24}^0 =\int \limits _{\Omega ^{+}} {(\nabla \mathbf{N})^{T}\mathbf{c}_2^\mathrm{r} (\nabla \mathbf{N}^{*})\text{ d}\Omega } \end{aligned}$$
(94)
$$\begin{aligned} \mathbf{K}_{25}&= \mathbf{K}_{25}^0 =\int \limits _\Omega {(\nabla \mathbf{N})^{T}\mathbf{c}_3^\mathrm{r} (\nabla \mathbf{N}) \text{ d}\Omega } \end{aligned}$$
(95)
$$\begin{aligned} \mathbf{K}_{32}&= \mathbf{K}_{32}^0 =\int \limits _\Omega {(\nabla \mathbf{N})^{T}\mathbf{c}_4^\mathrm{r} (\nabla \mathbf{N}) \text{ d}\Omega } \end{aligned}$$
(96)
$$\begin{aligned} \mathbf{K}_{33}&= \int \limits _\Omega \mathbf{N}^{T}\left( \frac{\partial d_6 }{\partial S_\mathrm{w} } \mathbf{N}{\dot{\mathbf{u}}}^\mathrm{r} \right) \mathbf{m}^{T}\mathbf{B}\text{ d}\Omega +\int \limits _\Omega \mathbf{N}^{T}\left( \frac{\partial d_4 }{\partial S_\mathrm{w} }\mathbf{N}{\dot{\mathbf{P}}}_\mathrm{g}^\mathrm{r} \right) \mathbf{N}\text{ d}\Omega \nonumber \\&\quad +\int \limits _\Omega \mathbf{N}^{T}\left( \frac{\partial d_5 }{\partial S_\mathrm{w} }\mathbf{N}{\dot{\mathbf{S}}}_\mathrm{w}^\mathrm{r} \right) \mathbf{N}\text{ d}\Omega +\int \limits _{\Omega ^{+}} \mathbf{N}^{T}\left( \frac{\partial d_5 }{\partial S_\mathrm{w} }\mathbf{N}{\dot{\mathbf{a}}}_J^\mathrm{r} \right) \mathbf{N}^{*}\text{ d}\Omega \; \nonumber \\&\quad +\int \limits _\Omega {(\nabla \mathbf{N})^{T}\left( {\frac{\partial \mathbf{c}_4 }{\partial S_\mathrm{w} }\mathbf{N}{\bar{\mathbf{P}}}_\mathrm{g}^\mathrm{r} } \right) (\nabla \mathbf{N}) \text{ d}\Omega } -\int \limits _\Omega { (\nabla \mathbf{N})^{T}\left( {\frac{\partial \mathbf{G}_2 }{\partial S_\mathrm{w} }\mathbf{N}} \right) \text{ d}\Omega }\end{aligned}$$
(97)
$$\begin{aligned} \mathbf{K}_{42}&= \mathbf{K}_{42}^0 =\int \limits _{\Omega ^{+}} {(\nabla \mathbf{N}^{*})^{T}\mathbf{c}_4^\mathrm{r} (\nabla \mathbf{N})\text{ d}\Omega } \end{aligned}$$
(98)
$$\begin{aligned} \mathbf{K}_{43}&= \int \limits _{\Omega ^{+}} {\mathbf{N}^{*T}\left( \frac{\partial d_6 }{\partial S_\mathrm{w} } \mathbf{N}{\dot{\mathbf{u}}}^\mathrm{r} \right) \mathbf{m}^{T}\mathbf{B}\text{ d}\Omega } +\int \limits _{\Omega ^{+}} \mathbf{N}^{*T}\left( \frac{\partial d_4 }{\partial S_\mathrm{w} }\mathbf{N}{\dot{\mathbf{P}}}_\mathrm{g}^\mathrm{r} \right) \mathbf{N} \text{ d}\Omega \nonumber \\&\quad +\int \limits _{\Omega ^{+}} {\mathbf{N}^{*T}\left( \frac{\partial d_5 }{\partial S_\mathrm{w} }\mathbf{N}{\dot{\mathbf{S}}}_\mathrm{w}^\mathrm{r} \right) \mathbf{N}\text{ d}\Omega } +\int \limits _{\Omega ^{+}} {\mathbf{N}^{*T}\left( \frac{\partial d_5 }{\partial S_\mathrm{w} }\mathbf{N}{\dot{\mathbf{a}}}_J^\mathrm{r} \right) \mathbf{N}^{*}\text{ d}\Omega \;} \nonumber \\&\quad +\int \limits _{\Omega ^{+}} {(\nabla \mathbf{N}^{*})^{T}\left( {\frac{\partial \mathbf{c}_4 }{\partial S_\mathrm{w} }\mathbf{N}{\bar{\mathbf{P}}}_\mathrm{g}^\mathrm{r} } \right) (\nabla \mathbf{N}) \text{ d}\Omega } -\int \limits _{\Omega ^{+}} {(\nabla \mathbf{N}^{*})^{T}\left( {\frac{\partial \mathbf{G}_2 }{\partial S_\mathrm{w} }} \right) \mathbf{N}\text{ d}\Omega } \end{aligned}$$
(99)
$$\begin{aligned} \mathbf{K}_{52}&= \mathbf{K}_{52}^0 =\int \limits _\Omega {(\nabla \mathbf{N})^{T}\mathbf{c}_5^\mathrm{r} (\nabla \mathbf{N}) \text{ d}\Omega } \end{aligned}$$
(100)
$$\begin{aligned} \mathbf{K}_{53}&= \int \limits _\Omega {(\nabla \mathbf{N})^{T}\left( {\frac{\partial c_7 }{\partial S_\mathrm{w} }\mathbf{N}{\bar{\mathbf{V}}}^\mathrm{r}} \right) (\nabla \mathbf{N})\text{ d}\Omega } +\int \limits _\Omega {(\nabla \mathbf{N})^{T}\left( {\frac{\partial \mathbf{c}_5 }{\partial S_\mathrm{w} }\mathbf{N}{\bar{\mathbf{P}}}_\mathrm{g}^\mathrm{r} } \right) (\nabla \mathbf{N}) \text{ d}\Omega } \nonumber \\&\quad +\int \limits _\Omega { (\nabla \mathbf{N})^{T}\mathbf{c}_6^\mathrm{r} (\nabla \mathbf{N}) \text{ d}\Omega } +\int \limits _\Omega {(\nabla \mathbf{N})^{T}\left( {\frac{\partial \mathbf{c}_6 }{\partial S_\mathrm{w} }\mathbf{N}{\bar{\mathbf{S}}}_\mathrm{w}^\mathrm{r} } \right) (\nabla \mathbf{N}) \text{ d}\Omega } \nonumber \\&\quad +\int \limits _{\Omega ^{+}} {(\nabla \mathbf{N})^{T}\left( {\frac{\partial \mathbf{c}_6 }{\partial S_\mathrm{w} }\mathbf{Na}_J^\mathrm{r} } \right) (\nabla \mathbf{N}^{*})\text{ d}\Omega } -\int \limits _\Omega {(\nabla \mathbf{N})^{T}\left( {\frac{\partial \mathbf{G}_3 }{\partial S_\mathrm{w} }} \right) \mathbf{N} \text{ d}\Omega } \end{aligned}$$
(101)
$$\begin{aligned} \mathbf{K}_{53}^0&= \int \limits _\Omega {(\nabla \mathbf{N})^{T}\mathbf{c}_6^\mathrm{r} (\nabla \mathbf{N})\text{ d}\Omega } \end{aligned}$$
(102)
$$\begin{aligned} \mathbf{K}_{54}&= \mathbf{K}_{54}^0 =\int \limits _{\Omega ^{+}} {(\nabla \mathbf{N})^{T}\mathbf{c}_6^\mathrm{r} (\nabla \mathbf{N}^{*}) \text{ d}\Omega } \end{aligned}$$
(103)
$$\begin{aligned} \mathbf{K}_{55}&= \mathbf{K}_{55}^0 =\int \limits _\Omega { (\nabla \mathbf{N})^{T}c_7^\mathrm{r} (\nabla \mathbf{N}) \text{ d}\Omega } \end{aligned}$$
(104)
$$\begin{aligned} \mathbf{C}_{21}&= \mathbf{C}_{21}^0 =\int \limits _\Omega {\mathbf{N}^{T}d_3^\mathrm{r} \mathbf{m}^{T}\mathbf{B}\text{ d}\Omega } \end{aligned}$$
(105)
$$\begin{aligned} \mathbf{C}_{22}&= \mathbf{C}_{22}^0=\int \limits _\Omega {\mathbf{N}^{T}d_1^\mathrm{r} \mathbf{N}\text{ d}\Omega \;} \end{aligned}$$
(106)
$$\begin{aligned} \mathbf{C}_{23}&= \mathbf{C}_{23}^0 =\int \limits _\Omega {\mathbf{N}^{T}d_2^\mathrm{r} \mathbf{N}\text{ d}\Omega \;} \end{aligned}$$
(107)
$$\begin{aligned} \mathbf{C}_{24}&= \mathbf{C}_{24}^0 =\int \limits _{\Omega ^{+}} {\mathbf{N}^{T}d_2^\mathrm{r} \mathbf{N}^{*}\text{ d}\Omega \;} \end{aligned}$$
(108)
$$\begin{aligned} \mathbf{C}_{31}&= \mathbf{C}_{31}^0=\int \limits _\Omega {\mathbf{N}^{T}d_6^\mathrm{r} \mathbf{m}^{T}\mathbf{B}{}\text{ d}\Omega } \end{aligned}$$
(109)
$$\begin{aligned} \mathbf{C}_{32}&= \mathbf{C}_{32}^0 =\int \limits _\Omega {\mathbf{N}^{T}d_4^\mathrm{r} \mathbf{N}\text{ d}\Omega \;} \end{aligned}$$
(110)
$$\begin{aligned} \mathbf{C}_{33}&= \mathbf{C}_{33}^0 =\int \limits _\Omega {\mathbf{N}^{T}d_5^\mathrm{r} \mathbf{N}\text{ d}\Omega \;} \end{aligned}$$
(111)
$$\begin{aligned} \mathbf{C}_{34}&= \mathbf{C}_{34}^0 =\int \limits _\Omega {\mathbf{N}^{T}d_5^\mathrm{r} \mathbf{N}^{*}\text{ d}\Omega \;} \end{aligned}$$
(112)
$$\begin{aligned} \mathbf{C}_{41}&= \mathbf{C}_{41}^0 =\int \limits _{\Omega ^{+}} {\mathbf{N}^{*T}d_6^\mathrm{r} \mathbf{m}^{T}\mathbf{B}\text{ d}\Omega } \end{aligned}$$
(113)
$$\begin{aligned} \mathbf{C}_{42}&= \mathbf{C}_{42}^0 =\int \limits _{\Omega ^{+}} {\mathbf{N}^{*T}d_4^\mathrm{r} \mathbf{N}\text{ d}\Omega \;}\end{aligned}$$
(114)
$$\begin{aligned} \mathbf{C}_{43}&= \mathbf{C}_{43}^0 =\int \limits _{\Omega ^{+}} {\mathbf{N}^{*T}d_5^\mathrm{r} \mathbf{N}\text{ d}\Omega \;} \end{aligned}$$
(115)
$$\begin{aligned} \mathbf{C}_{44}&= \mathbf{C}_{44}^0 =\int \limits _{\Omega ^{+}} {\mathbf{N}^{*T}d_5^\mathrm{r} \mathbf{N}^{*}\text{ d}\Omega }\end{aligned}$$
(116)
$$\begin{aligned} \mathbf{f}_1&= -\int \limits _\Omega {\mathbf{N}^T \rho ^{r}\mathbf{g} \text{ d}\Omega } - \int \limits _{\Gamma _\mathrm{u}^\mathrm{q} }\, {\mathbf{N}^T \, {\bar{\mathbf{t}}} \,\text{ d}\Gamma } \end{aligned}$$
(117)
$$\begin{aligned} \mathbf{f}_2&= \int \limits _\Omega {(\nabla \mathbf{N})^{T}\mathbf{G}_1 ^\mathrm{r}\text{ d}\Omega } -\int \limits _{\Gamma _\mathrm{w}^\mathrm{q} } {\mathbf{N}^{T}\frac{Q_\mathrm{w} }{\rho _\mathrm{w} } \text{ d}\Gamma }\end{aligned}$$
(118)
$$\begin{aligned} \mathbf{f}_3&= -\int \limits _{\Gamma _\mathrm{g}^\mathrm{q} } {\mathbf{N}^{T}\frac{Q_\mathrm{g} }{\rho _\mathrm{g} } \text{ d}\Gamma } +\int \limits _\Omega {(\nabla \mathbf{N})^{T}\mathbf{G}_2^\mathrm{r} \text{ d}\Omega } \end{aligned}$$
(119)
$$\begin{aligned} \mathbf{f}_4&= -\int \limits _{\Gamma _\mathrm{g}^\mathrm{q} } {\mathbf{N}^{T}\frac{Q_\mathrm{g} }{\rho _\mathrm{g} } \text{ d}\Gamma } +\int \limits _{\Omega ^{+}} { (\nabla \mathbf{N}^{*})^{T}\mathbf{G}_2^\mathrm{r} \text{ d}\Omega } \end{aligned}$$
(120)
$$\begin{aligned} \mathbf{f}_{5}&= \int \limits _\Omega {(\nabla \mathbf{N})^{T}\mathbf{G}_{3}^\mathrm{r} \text{ d}\Omega } -\int \limits _{\Gamma _\mathrm{e}^\mathrm{q} } {\mathbf{N}^{T}Q_\mathrm{e} \text{ d}\Gamma } \end{aligned}$$
(121)

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Talebian, M., Al-Khoury, R. & Sluys, L.J. Coupled Electrokinetic–Hydromechanic Model for \(\text{ CO}_{2}\) Sequestration in Porous Media. Transp Porous Med 98, 287–321 (2013). https://doi.org/10.1007/s11242-013-0145-y

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Keywords

  • Multiphase flow
  • \(\text{ CO}_{2}\) sequestration
  • Streaming potential
  • XFEM
  • Level-set