Performance studies of the fixed stress split algorithm for immiscible two-phase flow coupled with linear poromechanics

Abstract

In this work, we measure the performance of the fixed stress split algorithm for the immiscible water-oil flow coupled with linear poromechanics. The two-phase flow equations are solved on general hexahedral elements using the multipoint flux mixed finite element method whereas the poromechanics equations are discretized using the conforming Galerkin method. We introduce a rigorous calculation of the update in poroelastic properties during the iterative solution of the coupled system equations. The effects of the coupling parameter on the performance of the fixed stress algorithm is demonstrated in two field studies: the Frio oil reservoir and the Cranfield injection site.

Appendices

Appendix

A Derivation of mass conservation equation for two phase flow model in deformable porous media

When the two phase flow and solid matrix deformation are coupled, the mass conservation equations for flow and solid phase read:

$$ \begin{array}{@{}rcl@{}} &&\frac{\partial (\phi \rho_{\beta}S_{\beta})}{\partial t}+\nabla \cdot (\rho_{\beta}S_{\beta}\phi \mathbf{v}_{f\beta})= q_{\beta}\\&& (mass conservation for fluid phase \beta)\\ &&\frac{\partial ((1-\phi)\rho_{s})}{\partial t}+\nabla \cdot ((1-\phi)\rho_{s} \mathbf{v}_{s})=0\\&& (mass conservation for solid phase) \end{array} $$

where v_fβ is the interstitial velocity of fluid phase β in Eulerian coordinate, ρ_s is the solid phase mass density, and v_s is the solid phase velocity in Eulerian coordinate. The phase Darcy velocity v_β, defined as phase superficial velocity relative to the moving solid skeleton, is connected to v_fβ and v_s as:

$$ \begin{array}{@{}rcl@{}} \mathbf{v}_{\beta} = \phi S_{\beta}(\mathbf{v}_{f\beta}-\mathbf{v}_{s}) \end{array} $$

which can be substituted in the mass conservation of the fluid phase β to obtain

$$ \begin{array}{@{}rcl@{}} \frac{\partial (\phi \rho_{\beta}S_{\beta})}{\partial t}&+&\nabla \cdot (\rho_{\beta}\mathbf{v}_{\beta})+ \rho_{\beta}S_{\beta}\phi \nabla \cdot \mathbf{v}_{s}\\ &+&\mathbf{v}_{s}\cdot \nabla (\rho_{\beta}S_{\beta}\phi)=q_{\beta} \end{array} $$

(A.1)

Similarly, the mass conservation for solid phase can be written as

$$ \begin{array}{@{}rcl@{}} \frac{\partial ((1-\phi)\rho_{s})}{\partial t}&&+(1-\phi)\rho_{s}\nabla \cdot \mathbf{v}_{s} \\ &&+ \mathbf{v}_{s} \cdot \nabla ((1-\phi)\rho_{s}) = 0 \end{array} $$

(A.2)

Using the material time derivative relation:

$$ \begin{array}{@{}rcl@{}} \frac{d(\cdot)}{dt} = \frac{\partial (\cdot)}{dt} + \mathbf{v}_{s} \cdot \nabla (\cdot) \end{array} $$

(A.3)

we obtain (A.1) and (A.2) as

$$ \begin{array}{@{}rcl@{}} \frac{d (\phi \rho_{\beta}S_{\beta})}{d t}+\nabla \cdot (\rho_{\beta}\mathbf{v}_{\beta})+ \rho_{\beta}S_{\beta}\phi \nabla \cdot \mathbf{v}_{s}=q_{\beta} \end{array} $$

(A.4)

$$ \begin{array}{@{}rcl@{}} \frac{d ((1-\phi)\rho_{s})}{d t}+(1-\phi)\rho_{s}\nabla \cdot \mathbf{v}_{s} = 0 \end{array} $$

(A.5)

Substituting the expression for ∇⋅v_s from Eqs. A.5 into A.4, we get

$$ \begin{array}{@{}rcl@{}} \frac{d (\phi \rho_{\beta}S_{\beta})}{dt}+\nabla \cdot (\rho_{\beta}\mathbf{v}_{\beta})- \rho_{\beta}S_{\beta}\phi \left( \frac{\frac{d ((1-\phi)\rho_{s})}{d t}} {(1-\phi)\rho_{s}}\right)=q_{\beta} \end{array} $$

(A.6)

In Eq. A.6, $1-\phi = \frac {V_{s}}{V_{b}}$ where V_s = V_b − V_p is the solid grain volume and V_p is the pore volume and V_b is the bulk volume. Use this relation in Eq. A.6 we have:

$$ \begin{array}{@{}rcl@{}} \frac{d (\phi \rho_{\beta}S_{\beta})}{dt}+\nabla \cdot (\rho_{\beta}\mathbf{v}_{\beta})- \rho_{\beta}S_{\beta}\phi \left( \frac{\frac{d (\frac{V_{s}}{V_{b}}\rho_{s})}{d t}} {\frac{V_{s}}{V_{b}}\rho_{s}}\right)=q_{\beta} \end{array} $$

Note that solid grain mass is conserved in a deformable porous media, i.e. ρ_sV_s = constant, implying $\frac {d(\rho _{s} V_{s})}{dt}=0$. Also, $\frac {1}{V_{b}}\frac {dV_{b}}{dt} = \frac {d\epsilon }{dt}$. As a result, we get

$$ \begin{array}{@{}rcl@{}} \frac{d (\phi \rho_{\beta}S_{\beta})}{dt}+\nabla \cdot (\rho_{\beta}\mathbf{v}_{\beta})+ \rho_{\beta}S_{\beta}\phi \frac{d\epsilon}{dt} =q_{\beta} \end{array} $$

(A.7)

Adding and subtracting $\epsilon \frac {d (\phi \rho _{\beta }S_{\beta })}{d t}$ to Eq. A.7, we get

$$ \begin{array}{@{}rcl@{}} \frac{d (\phi^{*} \rho_{\beta}S_{\beta})}{dt}+\nabla \cdot (\rho_{\beta}\mathbf{v}_{\beta})- \epsilon\frac{d (\phi \rho_{\beta}S_{\beta})}{d t} =q_{\beta} \end{array} $$

(A.8)

Knowing the quasi-static nature of deformation i.e. ∥v_s∥ << 1, we get

$$ \begin{array}{@{}rcl@{}} \mathbf{v}_{s} \cdot \nabla (\phi^{*} \rho_{\beta}S_{\beta}) &<<& \frac{\partial (\phi^{*} \rho_{\beta}S_{\beta})}{\partial t}, \quad \mathbf{v}_{s} \cdot \nabla (\phi \rho_{\beta}S_{\beta}) \\&<<& \frac{\partial (\phi \rho_{\beta}S_{\beta})}{\partial t} \end{array} $$

which, in lieu of Eq. A.3, implies that

$$ \begin{array}{@{}rcl@{}} \frac{d (\phi^{*} \rho_{\beta}S_{\beta})}{d t}\approx \frac{\partial (\phi^{*} \rho_{\beta}S_{\beta})}{\partial t}, \quad \frac{d (\phi \rho_{\beta}S_{\beta})}{d t}\approx \frac{\partial (\phi \rho_{\beta}S_{\beta})}{\partial t} \end{array} $$

In lieu of the above, we obtain (A.8) as

$$ \begin{array}{@{}rcl@{}} \frac{\partial (\phi^{*} \rho_{\beta}S_{\beta})}{\partial t}+\nabla \cdot (\rho_{\beta}\mathbf{v}_{\beta})- \epsilon\frac{\partial (\phi \rho_{\beta}S_{\beta})}{\partial t} =q_{\beta} \end{array} $$

(A.9)

Although typically in literature, the last term $\epsilon \frac {\partial (\phi \rho _{\beta }S_{\beta })}{\partial t}$ on the left hand side is ignored in lieu of the small strain deformation assumption 𝜖 << 1, we retain it in this formulation to give due diligence to the fact that the geomechanical feedback from the poromechanics to the flow model is in the form of post-processed volumetric strains.

B Mixed Finite Element Formulation for the Flow Model

Let ${\mathscr{T}}_{h}$ be the finite element partition of Ω consisting of elements E. The problem statement is : Find $\tilde {\mathbf {z}}_{\beta _{h}},\mathbf {z}_{\beta _{h}}\in \mathbf {V}_{h}$ and $p_{\beta _{h}},S_{w},S_{o}\in W_{h}$ such that

$$ \begin{array}{@{}rcl@{}} \left.\begin{array}{c} \sum\limits_{E\in \mathscr{T}_{h}}\int\limits_{E}\mathbf{K}^{-1}\tilde{\mathbf{z}} {o_{h}}\cdot \mathbf{v}=\sum\limits_{E\in \mathscr{T}_{h}}\int\limits_{E} \big[ p_{o_{h}}\nabla \cdot \mathbf{v}+\rho_{o} \mathbf{g}\cdot \mathbf{v}\big]\\ \sum\limits_{E\in \mathscr{T}_{h}}\int\limits_{E}\mathbf{K}^{-1}\tilde{\mathbf{z}} {w_{h}}{}\cdot{} \mathbf{v} = \sum\limits_{E\in \mathscr{T}_{h}}\int\limits_{E}\big[p_{o_{h}}\nabla {}\cdot{} \mathbf{v} - p_{c_{h}}\nabla {}\cdot{} \mathbf{v} + \rho_{w} \mathbf{g{}}\cdot{} \mathbf{v}\big]\\ \sum\limits_{E\in \mathscr{T}_{h}}\int\limits_{E}\big[\phi^{*} \rho_{o}S_{o}-\epsilon\phi \rho_{\beta}S_{o}+{\Delta} t\nabla \cdot \mathbf{z}_{o_{h}}\big]w \\=\sum\limits_{E\in \mathscr{T}_{h}}\int\limits_{E}\big[(\phi^{*} \rho_{o}S_{o})^{n}-\epsilon(\phi \rho_{o}S_{o})^{n}+{\Delta} tq_{o}\big]w\\ \sum\limits_{E\in \mathscr{T}_{h}}\int\limits_{E}\big[\phi^{*} \rho_{w}S_{w}-\epsilon\phi \rho_{w}S_{w}+{\Delta} t\nabla \cdot \mathbf{z}_{w_{h}}\big]w \\=\sum\limits_{E\in \mathscr{T}_{h}}\int\limits_{E}\big[(\phi^{*} \rho_{w}S_{w})^{n}-\epsilon(\phi \rho_{w}S_{w})^{n}+{\Delta} tq_{w}\big]w\\ \sum\limits_{E\in \mathscr{T}_{h}}\int\limits_{E}\mathbf{z}_{o_{h}}\cdot \mathbf{v}=\sum\limits_{E\in \mathscr{T}_{h}}\int\limits_{E}\lambda_{o}\tilde{\mathbf{z}}_{o_{h}}\cdot \mathbf{v}\\ \sum\limits_{E\in \mathscr{T}_{h}}\int\limits_{E}\mathbf{z}_{w_{h}}\cdot \mathbf{v}=\sum\limits_{E\in \mathscr{T}_{h}}\int\limits_{E}\lambda_{w}\tilde{\mathbf{z}}_{w_{h}}\cdot \mathbf{v}\\ \sum\limits_{E\in \mathscr{T}_{h}}\int\limits_{E}\big[S_{w}+ S_{o} \big]w= \sum\limits_{E\in \mathscr{T}_{h}}\int\limits_{E}w \end{array}\right. \end{array} $$

where the terms (⋅)ⁿ are evaluated at the previous time step, the finite dimensional spaces W_h and V_h are given as

$$ \begin{array}{@{}rcl@{}} &&W_{h}\equiv \big\{w:w\vert_{E}\in \mathbb{P}_{0}(E) \forall E\in \mathscr{T}_{h}\big\}\\ &&\mathbf{V}_{h}\equiv \big\{\mathbf{v}:\mathbf{v}\vert_{E}\leftrightarrow \hat{\mathbf{v}}\vert_{\hat{E}}:\hat{\mathbf{v}}\vert_{\hat{E}}\in \hat{\mathbf{V}}(\hat{E}) \forall E\in \mathscr{T}_{h},\\&&\qquad\quad\mathbf{v} \cdot \mathbf{n}=0 \text{on} {{\varGamma}_{N}^{f}}\big\} \end{array} $$

and the details of $\hat {\mathbf {V}}(\hat {E})$ are presented next. The notation $\mathbf {v}\vert _{E}\leftrightarrow \hat {\mathbf {v}}\vert _{\hat {E}}$ means that the space of velocities v on E correspond to the space of velocities $\hat {\mathbf {v}}$ on $\hat {E}$. Let $\mathbf {V}^{*}_{h}\times W_{h}$ be the lowest order BDDF₁ MFE spaces on hexahedra [63]. With $\mathbf {x}\equiv (\hat {x},\hat {y},\hat {z})\in \hat {E}$, these spaces are defined on $\hat {E}$ as

$$ \begin{array}{@{}rcl@{}} \hat{\mathbf{V}}^{*}(\hat{E})&=&(\mathbb{P}_{1}(\hat{E}))^{3} +r_{0} curl(0,0,\hat{x}\hat{y}\hat{z})^{T} \\&&+r_{1} curl(0,0,\hat{x}\hat{y}^{2})^{T} +s_{0} curl(\hat{x}\hat{y}\hat{z},0,0)^{T}\\ &&+s_{1} curl(\hat{y}\hat{z}^{2},0,0)^{T} +t_{0} curl(0,\hat{x}\hat{y}\hat{z},0)^{T} \\&&+t_{1} curl(0,\hat{x}^{2}\hat{z},0)^{T}\\ \hat{W}(\hat{E})&=&\mathbb{P}_{0}(\hat{E}) \end{array} $$

with the following properties

$$ \begin{array}{@{}rcl@{}} &&\hat{\nabla} \cdot \hat{\mathbf{V}}^{*}(\hat{E})=\hat{W}(\hat{E}), \qquad \text{and} \qquad \forall \hat{\mathbf{v}}\in \hat{\mathbf{V}}^{*}(\hat{E}),\\&&\forall \hat{e}\subset \partial \hat{E}, \hat{\mathbf{v}}\cdot \hat{\mathbf{n}}_{\hat{e}}\in \mathbb{P}_{1}(\hat{e}) \end{array} $$

where $\hat {e}$ represents a face of $\hat {E}$ and $\hat {\mathbf {n}}_{\hat {e}}$ the unit outward normal to $\hat {e}$. The multipoint flux approximation procedure requires on each face one velocity degree of freedom to be associated with each vertex thus requiring four degrees of freedom per face. Since $\mathbf {V}^{*}_{h}$ has only three degrees of freedom per face, it is augmented with one degree of freedom per face resulting in addition of six degrees of freedom per element. Since the properties of constant divergence, linear independence of the shape functions and continuity of the normal component across the element faces are to be preserved, six curl terms are added (see [64]) to $\mathbf {V}^{*}_{h}$. Let V_h × W_h be the enhanced BDDF₁ spaces on hexahedra. On $\hat {E}$, these spaces are

$$ \begin{array}{@{}rcl@{}} \hat{\mathbf{V}}(\hat{E})&=&\hat{\mathbf{V}}^{*}(\hat{E}) +r_{2} curl(0,0,\hat{x}^{2}\hat{z})^{T} \\&&+r_{3} curl(0,0,\hat{x}^{2}\hat{y}\hat{z})^{T} +s_{2} curl(\hat{x}\hat{y}^{2},0,0)^{T}\\ &&+s_{3} curl(\hat{x}\hat{y}^{2}\hat{z}^{2},0,0)^{T} +t_{2} curl(0,\hat{y}\hat{z}^{2},0)^{T} \\&&+t_{3} curl(0,\hat{x}\hat{y}\hat{z}^{2},0)^{T}\\ \hat{W}(\hat{E})&=&\mathbb{P}_{0}(\hat{E}) \end{array} $$

with the following properties

$$ \begin{array}{@{}rcl@{}} &&\hat{\nabla} \cdot \hat{\mathbf{V}}(\hat{E})=\hat{W}(\hat{E}), \qquad \text{and} \qquad \forall \hat{\mathbf{v}}\in \hat{\mathbf{V}}(\hat{E}),\\&&\forall \hat{e}\subset \partial \hat{E}, \hat{\mathbf{v}}\cdot \hat{\mathbf{n}}_{\hat{e}}\in \mathbb{Q}_{1}(\hat{e}) \end{array} $$

Since $dim \mathbb {Q}_{1}(\hat {e})=4$, the dimension of $\hat {\mathbf {V}}(\hat {E})$ is 24 as shown in Fig. 10.

Fig. 10

Degrees of freedom and basis functions for the enhanced BDDF₁ velocity space on hexahedra

C Derivation of mass conservation equation for the popular Biot system

The mass conservation equation of the popular ‘Biot’ set of equations is pertinent to linearized single phase flow coupled with linear poromechanics, and is a special case of Eq. A.9 with β = 1, S_β = 1, and the last term $\epsilon \frac {\partial (\phi \rho _{\beta }S_{\beta })}{\partial t}$ on the left hand side ignored. We rewrite (A.9) as

$$ \begin{array}{@{}rcl@{}} \frac{\partial (\phi^{*} \rho)}{\partial t}-\nabla \cdot (\rho\boldsymbol{\kappa}(\nabla p-\rho \mathbf{g}))=q \end{array} $$

(C.1)

We first comment that the difference between the slightly compressible single phase flow model and the linearized flow model lies in the expression for the density variation as a function of pore pressure. The former models the density variation as $\rho =\rho _{0} e^{c(p-p_{0})}$ which when expanded as a Taylor series, is written as ρ = ρ₀(1 + c(p − p₀) + c²(p − p₀)^/2! + ⋯ ) which, after noting $c=\mathcal {O}(10^{-6})$, can be approximated as

$$ \begin{array}{@{}rcl@{}} \rho\approx \rho_{0} (1+c(p-p_{0})) \end{array} $$

(C.2)

after truncating the expansion to the first order term. We also note that while the slightly compressible single phase flow might require a nonlinear solve with Newton iterations within every coupling iteration, the linearized flow model requires only one solve within every coupling iteration. Furthermore, the Lagrangian porosity variation, instead of having two different expressions for the flow solve and the poromechanics solve as we saw in modules Sections 2.2.1 and 2.2.2 respectively, has only one expression given by

$$ \begin{array}{@{}rcl@{}} \phi^{*}= \phi_{0}+\alpha \epsilon+\frac{1}{N}(p-p_{0}) \end{array} $$

(C.3)

with $\frac {1}{N}\equiv \frac {(\alpha -\phi _{0})(1-\alpha )}{K_{b}}$ in lieu of Eq. 2.8. We substitute (C.2) and (C.3) in (C.1), and divide by ρ₀ to get

$$ \begin{array}{@{}rcl@{}} &&\frac{\partial (\phi_{0}+\alpha \epsilon+\frac{1}{N}(p-p_{0}))}{\partial t}+(\phi_{0}+\alpha \epsilon)c\frac{\partial (p-p_{0})}{\partial t}\\&&+c(p-p_{0})\frac{\partial (\phi_{0}+\alpha\epsilon)}{\partial t}\\ &&+\frac{\partial (\frac{1}{N}(p-p_{0})c(p-p_{0}))}{\partial t}-\nabla\\&& \cdot (\boldsymbol{\kappa}(1+c(p-p_{0}))(\nabla p-\rho \mathbf{g}))=\frac{q}{\rho_{0}} \end{array} $$

which, in lieu of $\epsilon =\mathcal {O}(10^{-3})$ resulting in α𝜖 ≪ ϕ₀, and $\frac {\partial \phi _{0}}{\partial t}=\frac {\partial p_{0}}{\partial t}=0$, is written as

$$ \begin{array}{@{}rcl@{}} &&\!\!\!\!\frac{\partial (\alpha \epsilon+\frac{1}{N}p+\phi_{0} cp)}{\partial t}+\frac{2c}{N}(p-p_{0})\frac{\partial p}{\partial t}+\alpha c(p-p_{0})\frac{\partial \epsilon}{\partial t}\\ &&\!\!\!\!-\nabla \cdot (\boldsymbol{\kappa}c(p-p_{0})(\nabla p-\rho \mathbf{g}))-\nabla \cdot (\boldsymbol{\kappa}(\nabla p-\rho \mathbf{g}))=\frac{q}{\rho_{0}} \end{array} $$

We neglect the nonlinear terms $\frac {2c}{N}(p-p_{0})\frac {\partial p}{\partial t}$, $\alpha c(p-p_{0})\frac {\partial \epsilon }{\partial t}$ and ∇⋅ (κc(p − p₀)(∇p − ρg)) to obtain

$$ \begin{array}{@{}rcl@{}} &\frac{\partial \zeta}{\partial t}-\nabla \cdot (\boldsymbol{\kappa}(\nabla p-\rho \mathbf{g}))=\frac{q}{\rho_{0}} \end{array} $$

where $\zeta = \alpha \epsilon +\frac {1}{N}p+\phi _{0} cp \equiv \alpha \epsilon + \frac {1}{M}p$ is refered to as the fluid content and $\frac {1}{M}=\frac {1}{N}+\phi _{0} c \equiv \phi _{0} c+\frac {(\alpha -\phi _{0})(1-\alpha )}{K_{b}}$