Localized linear systems for fully implicit simulation of multiphase multicomponent flow in porous media

Abstract

During the solution of fully implicit reservoir simulation time steps, it is often observed that the computed Newton updates may be very sparse, considering computer precision. This sparsity can be as high as 95% and can vary largely from one iteration to the next. In recent works, a mathematically sound framework was developed to predict the sparsity pattern before the full linear system is solved. The theory is restricted to general, scalar nonlinear advection-diffusion-reaction problems in multidimensional and heterogeneous settings. This theory had been applied to reduce the size of the linear systems that were computed during sequential implicit time steps for two-phase flow. The results confirmed that the linearization computations and the linear solution processes may be localized by as much as 95% while retaining the exact Newton convergence behavior and final solution. Inspired by the great success of that methodology, this work develops algorithmic extensions in order to devise localization algorithms for fully implicit coupled multicomponent problems. We propose, apply, and test a novel algorithm to resolve a system of hyperbolic equations obtained from an Equation of State–based compositional simulator. When applied to various fully implicit flow and multicomponent transport problems, involving six thermodynamic species, on the SPE 10 geological model, the observed reduction in computational effort ranges between six and 49-fold depending on the level of locality present in the system. We apply this algorithm to several injection and depletion scenarios with and without gravity and capillarity in order to investigate the adaptivity and robustness of the proposed method to the underlying heterogeneity and complexity. We demonstrate that the algorithm enables efficient and robust full-resolution fully implicit simulation without the errors introduced by adaptive discretization methods or the stability concerns of semi-implicit approaches.

Appendix: Mathematical Development Including Capillarity

Starting with Eq. 6 and assuming pressure of the first phase to be the primary variable, we obtain

$$ \begin{array}{@{}rcl@{}} &&\Bigg[ \phi \sum\limits_{p} \rho_{p} S_{p} z_{c} \Bigg]^{n+1} - \left[ \phi \sum\limits_{p} \rho_{p} S_{p} z_{c} \right]^{n} \\ && -\overrightarrow{\nabla} \cdot \left[ {\Delta} t \sum\limits_{p} \frac{x_{c,p} \rho_{p} k_{rp}}{\mu_{p}} \mathbf{K} \left( \overrightarrow{\nabla P}_{1} + \overrightarrow{\nabla P}_{cp1} - \overrightarrow{\gamma}_{p} \nabla D \right) \right] \\ &=& {\Delta} t \sum\limits_{p} q_{c,p}, \forall c \in \{1,...,n_{c}\}, \end{array} $$

(33)

where P_cp1 = P_p − P₁ is the capillary pressure between phase p and the primary phase 1. Introducing notation

$$ \begin{array}{@{}rcl@{}} &&\alpha_{c}(x) = \left[ \phi \sum\limits_{p} \rho_{p} S_{p} z_{c} \right]^{n+1} - \left[ \phi \sum\limits_{p} \rho_{p} S_{p} z_{c} \right]^{n} \\ && \qquad\qquad - {\Delta} t \sum\limits_{p} q_{c,p}, \forall c \in \{1,...,n_{c}\}, \end{array} $$

(34)

$$ \begin{array}{@{}rcl@{}} &&\overrightarrow{{\beta}}_{c}(x) = \left[ {\Delta} t \sum\limits_{p} \frac{x_{c,p} \rho_{p} k_{rp}}{\mu_{p}} \mathbf{K} \left( \overrightarrow{\nabla P}_{p} - {\overrightarrow{\gamma}}_{p} \nabla D \right) \right], \\ &&\qquad\qquad \forall c \in \{1,...,n_{c}\}, \end{array} $$

(35)

$$ \begin{array}{@{}rcl@{}} &&{\gamma_{c}^{p}} = {\Delta} t \frac{x_{c,p} \rho_{p} k_{rp}}{\mu_{p}} \mathbf{K}, \forall c \in \{1,...,n_{c}\}, \forall p \in \{1,...,n_{p}\}, \end{array} $$

(36)

$$ \begin{array}{@{}rcl@{}} &&{\xi_{c}^{p}} = P_{c1p}, \forall p \in \{1,...,n_{p}\}, \end{array} $$

(37)

Equation 1 is reduced to

$$ \begin{array}{lll} R_{\infty,c}(x) &= \alpha_{c}(x) - \overrightarrow{\nabla} \cdot \left[ \overrightarrow{\beta}_{c}(x) + \sum\limits_{p} {\gamma_{c}^{p}}(x) \overrightarrow{\nabla \xi}_{c}^{p}(x) \right] \\ &= 0, \forall c \in \{1,...,n_{c}\}. \end{array} $$

(38)

Following the derivation in the previous section, the infinite-dimensional iteration, in three (Cartesian) spatial dimensions, can be written as

$$ \begin{array}{@{}rcl@{}} &&\sum\limits_{d} \frac{\partial \alpha_{c}}{\partial z_{d}} \delta_{\infty, d} \\ &-& \sum\limits_{i} \frac{\partial}{\partial x_{i}} \left[ \sum\limits_{d} \frac{\partial {\beta^{i}_{c}}}{\partial z_{d}} \delta_{\infty,d} \right] \\ &-& \sum\limits_{i} \frac{\partial}{\partial x_{i}} \Bigg[ \sum\limits_{p} \frac{\partial {\xi_{c}^{p}}}{\partial x_{i}} \sum\limits_{d} \frac{\partial {\gamma_{c}^{p}}}{\partial z_{d}} \delta_{\infty, d}\Bigg] \\ &-& \sum\limits_{i} \frac{\partial}{\partial x_{i}} \Bigg[ \sum\limits_{p} {\gamma_{c}^{p}} \frac{\partial}{\partial x_{i}}\left( \sum\limits_{d} \frac{\partial {\xi_{c}^{p}}}{\partial z_{d}} \delta_{\infty,d} \right) \Bigg] \\ &=& (-)R_{\infty,c} \forall c \in \{1,...,n_{c}\}, i = {1,2,3}. \end{array} $$

(39)

Expanding the last term and grouping the second, third, and fourth term gives

$$ \begin{array}{@{}rcl@{}} &&\sum\limits_{d} \frac{\partial \alpha_{c}}{\partial z_{d}} \delta_{\infty, d} \\ &- &\sum\limits_{i} \frac{\partial}{\partial x_{i}} \Bigg[ \sum\limits_{d} \frac{\partial {\beta^{i}_{c}}}{\partial z_{d}} \delta_{\infty,d} + \sum\limits_{p} \frac{\partial {\xi_{c}^{p}}}{\partial x_{i}} \sum\limits_{d} \frac{\partial {\gamma_{c}^{p}}}{\partial z_{d}} \delta_{\infty, d} \end{array} $$

(40)

$$ \begin{array}{@{}rcl@{}} &+& \sum\limits_{p} {\gamma_{c}^{p}} \sum\limits_{d} \frac{\partial^{2} {\xi_{c}^{p}}}{\partial x_{i} \partial z_{d}} \delta_{\infty,d}\Bigg] \\ &-& \sum\limits_{i} \frac{\partial}{\partial x_{i}} \Bigg[ \sum\limits_{p} {\gamma_{c}^{p}} \sum\limits_{d} \frac{\partial {\xi_{c}^{p}}}{\partial z_{d}} \frac{\partial \delta_{\infty,d}}{\partial x_{i}} \Bigg] \\ &= &(-)R_{\infty,c} \forall c \in \{1,...,n_{c}\}, i = {1,2,3}. \end{array} $$

Expanding the last term once again gives

$$ \begin{array}{@{}rcl@{}} &&\sum\limits_{d} \frac{\partial \alpha_{c}}{\partial z_{d}} \delta_{\infty, d} \\ &-& \sum\limits_{i} \frac{\partial}{\partial x_{i}} \Bigg[ \sum\limits_{d} \frac{\partial {\beta^{i}_{c}}}{\partial z_{d}} \delta_{\infty,d} + \sum\limits_{d} \sum\limits_{p} \frac{\partial {\gamma_{c}^{p}}}{\partial z_{d}} \frac{\partial {\xi_{c}^{p}}}{\partial x_{i}} \delta_{\infty, d} \\ &+& \sum\limits_{d} \sum\limits_{p} {\gamma_{c}^{p}} \frac{\partial^{2} {\xi_{c}^{p}}}{\partial x_{i} \partial z_{d}} \delta_{\infty,d}\Bigg] \\ &-& \sum\limits_{i} \sum\limits_{d} \frac{\partial}{\partial x_{i}} \Bigg[ \sum\limits_{p} {\gamma_{c}^{p}} \frac{\partial {\xi_{c}^{p}}}{\partial z_{d}} \Bigg] \frac{\partial \delta_{\infty,d}}{\partial x_{i}}\\ &-& \sum\limits_{i} \sum\limits_{d} \Bigg[ \sum\limits_{p} {\gamma_{c}^{p}} \frac{\partial {\xi_{c}^{p}}}{\partial z_{d}} \Bigg] \frac{\partial^{2} \delta_{\infty,d}}{\partial {x_{i}^{2}}} \\ &=& (-)R_{\infty,c} \forall c \in \{1,...,n_{c}\}, i = {1,2,3}. \end{array} $$

(41)

The final equation grouped by the derivatives of $\delta _{\infty ,d}$ is written as

$$ \begin{array}{@{}rcl@{}} &\sum\limits_{d} \Bigg[ \frac{\partial \alpha_{c}}{\partial z_{d}} - \sum\limits_{i} \frac{\partial}{\partial x_{i}} \Bigg(\frac{\partial {\beta^{i}_{c}}}{\partial z_{d}} + \sum\limits_{p} \frac{\partial {\gamma_{c}^{p}}}{\partial z_{d}} \frac{\partial {\xi_{c}^{p}}}{\partial x_{i}} \\ &+ \sum\limits_{p} {\gamma_{c}^{p}} \frac{\partial^{2} {\xi_{c}^{p}}}{\partial x_{i} \partial z_{d}}\Bigg) \Bigg] \delta_{\infty,d} \\ &- \sum\limits_{d} \sum\limits_{i} \Bigg[ \frac{\partial {\beta^{i}_{c}}}{\partial z_{d}} + \sum\limits_{p} \frac{\partial {\gamma_{c}^{p}}}{\partial z_{d}} \frac{\partial {\xi_{c}^{p}}}{\partial x_{i}} + \sum\limits_{p} {\gamma_{c}^{p}} \frac{\partial^{2} {\xi_{c}^{p}}}{\partial x_{i} \partial z_{d}}\\ &+ \frac{\partial}{\partial x_{i}} \left( \sum\limits_{p} {\gamma_{c}^{p}} \frac{\partial {\xi_{c}^{p}}}{\partial z_{d}} \right)\Bigg] \frac{\partial \delta_{\infty,d}}{\partial x_{i}} \\ &- \sum\limits_{d} \Bigg[ \sum\limits_{p} {\gamma_{c}^{p}} \frac{\partial {\xi_{c}^{p}}}{\partial z_{d}} \Bigg] \sum\limits_{i} \frac{\partial^{2} \delta_{\infty,d}}{\partial {x_{i}^{2}}} \\ &= (-)R_{\infty,c} \forall c \in \{1,...,n_{c}\}, i = {1,2,3}. \end{array} $$

(42)

Substituting

$$(A)_{c,d}:=\frac{\partial \alpha_{c}}{\partial z_{d}}, (B^{i})_{c,d}:=\frac{\partial {\beta^{i}_{c}}}{\partial z_{d}}, ({\Gamma}^{p})_{c,c}:={\gamma^{p}_{c}}, $$

$$({{\Gamma}^{p}}^{\prime})_{c,d}:=\frac{\partial {\gamma^{p}_{c}}}{\partial z_{d}}, ({\Xi}^{p})_{c,c}:={\xi^{p}_{c}}, \text{and } ({{\Xi}^{p}}^{\prime})_{c,d}:=\frac{\partial {\xi^{p}_{c}}}{\partial z_{d}},$$

for all c,d ∈{1,...,n_c} in Eq. 10, gives

$$ \begin{array}{@{}rcl@{}} &\Bigg[ A - \sum\limits_{i} \frac{\partial}{\partial x_{i}} \left( B^{i} + \sum\limits_{p} {{\Gamma}^{p}}^{\prime} \frac{\partial {\Xi}^{p}}{\partial x_{i}} + \sum\limits_{p} {\Gamma}^{p} \frac{\partial {{\Xi}^{p}}^{\prime}}{\partial x_{i}} \right)\Bigg] \delta_{\infty}\\ &- \sum\limits_{i} \Bigg[ B^{i} + \sum\limits_{p} {{\Gamma}^{p}}^{\prime} \frac{\partial {\Xi}^{p}}{\partial x_{i}} + \sum\limits_{p} {\Gamma}^{p} \frac{\partial {{\Xi}^{p}}^{\prime}}{\partial x_{i}} \end{array} $$

(43)

$$ \begin{array}{@{}rcl@{}} &+ \frac{\partial }{\partial x_{i}} \sum\limits_{p} {\Gamma}^{p} {{\Xi}^{p}}^{\prime} \Bigg] \frac{\partial \delta_{\infty}}{\partial x_{i}} \\ & -\sum\limits_{p} {\Gamma}^{p} {{\Xi}^{p}}^{\prime} \sum\limits_{i} \frac{\partial^{2} \delta_{\infty}}{\partial {x_{i}^{2}}} = (-)R_{\infty}, \forall p \in \{1,...,n_{p}\}. \end{array} $$

(44)

Assuming that the residual can be projected onto a discrete Banach space as shown in Eq. 17, Eq. 44 can be written out for each control volume as

$$ \left\{\begin{array}{lll} &\Bigg[ A - \sum\limits_{i} \frac{\partial}{\partial x_{i}} \left( B^{i} + \sum\limits_{p} {{\Gamma}^{p}}^{\prime} \frac{\partial {\Xi}^{p}}{\partial x_{i}} + \sum\limits_{p} {\Gamma}^{p} \frac{\partial {{\Xi}^{p}}^{\prime}}{\partial x_{i}} \right)\Bigg] \delta_{\infty,{\Omega}} \\ & \qquad- \sum\limits_{i} \Bigg[ B^{i} + \sum\limits_{p} {{\Gamma}^{p}}^{\prime}\frac{\partial {\Xi}^{p}}{\partial x_{i}} + \sum\limits_{p} {\Gamma}^{p} \frac{\partial {{\Xi}^{p}}^{\prime}}{\partial x_{i}} \\ & \qquad+ \frac{\partial }{\partial x_{i}} \sum\limits_{p} {\Gamma}^{p} {{\Xi}^{p}}^{\prime} \Bigg] \frac{\partial \delta_{\infty,{\Omega}}}{\partial x_{i}} \\ & \qquad-\sum\limits_{p} {\Gamma}^{p} {{\Xi}^{p}}^{\prime} \sum\limits_{i} \frac{\partial^{2} \delta_{\infty,{\Omega}}}{\partial {x_{i}^{2}}} = 0, \forall x \not \in {\Omega},\\ &\Bigg[ A - \sum\limits_{i} \frac{\partial}{\partial x_{i}} \left( B^{i} + \sum\limits_{p} {{\Gamma}^{p}}^{\prime}\frac{\partial {\Xi}^{p}}{\partial x_{i}} + \sum\limits_{p} {\Gamma}^{p} \frac{\partial {{\Xi}^{p}}^{\prime}}{\partial x_{i}} \right)\Bigg] \delta_{\Omega} \\ & \qquad = (-)R_{h,{\Omega}}, \qquad\qquad\qquad\qquad \forall x \in {\Omega}. \end{array}\right. $$

(45)

The boundary conditions used to solve the continuous Newton iteration in this case are (1) $\delta _{\infty , {\Omega }} = \delta _{\Omega } \forall x\in {\Omega }$, given by the second sub-equation in Eq. 45 and (2) $\delta _{\infty , {\Omega }}$ is bounded at $x = \infty$. Similar to the derivation presented in the previous section, the next step towards solving (45) is to homogenize the variable coefficients. After the first step of homogenization, we obtain

$$ \left\{\begin{array}{lll} &\Bigg[ A^{*} - \sum\limits_{i} \frac{\partial}{\partial x_{i}} \Bigg(B^{*} + \left\{ \sum\limits_{p} {{\Gamma}^{p}}^{\prime} \frac{\partial {\Xi}^{p}}{\partial x_{i}} \right\}^{*} \\ &\qquad+ \left\{ \sum\limits_{p} {\Gamma}^{p} \frac{\partial {{\Xi}^{p}}^{\prime}}{\partial x_{i}} \right\}^{*} \Bigg)\Bigg] \delta^{*}_{\infty,{\Omega}} -\\ &\qquad\sum\limits_{i} \Bigg(B^{*} + \left\{ \sum\limits_{p} {{\Gamma}^{p}}^{\prime} \frac{\partial {\Xi}^{p}}{\partial x_{i}}\right\}^{*} + \left\{ \sum\limits_{p} {\Gamma}^{p} \frac{\partial {{\Xi}^{p}}^{\prime}}{\partial x_{i}} \right\}^{*} \\ &\qquad+ \left\{ \frac{\partial }{\partial x_{i}} \sum\limits_{p} {\Gamma}^{p} {{\Xi}^{p}}^{\prime} \right\}^{*} \Bigg) \frac{\partial \delta^{*}_{\infty,{\Omega}}}{\partial x_{i}} \\ &\qquad - \left\{\sum\limits_{p} {\Gamma}^{p} {{\Xi}^{p}}^{\prime} \right\}^{*} \sum\limits_{i} \frac{\partial^{2} \delta^{*}_{\infty,{\Omega}}}{\partial {x^{2}_{i}}} = 0, \qquad \forall x \not \in {\Omega},\\ &\Bigg(A - \sum\limits_{i} \frac{\partial}{\partial x_{i}} \Bigg[ B^{i} + \sum\limits_{p} {{\Gamma}^{p}}^{\prime}\frac{\partial {\Xi}^{p}}{\partial x_{i}} \\ &\qquad + \sum\limits_{p} {\Gamma}^{p} \frac{\partial {{\Xi}^{p}}^{\prime}}{\partial x_{i}} \Bigg]\Bigg) \delta^{*}_{\Omega} = (-)R_{h,{\Omega}}, \qquad \forall x \in {\Omega}. \end{array}\right. $$

(46)

The coefficient of $\delta ^{*}_{\Omega }$ in the second part of Eq. 46 is not homogenized as it is constant and evaluated at the centroid of the control volume (see assumption 3 in the next section). Finally, radially symmetric solutions, $\tilde \delta ^{*}_{\infty , {\Omega }}(r)$, can be obtained by homogenizing the coefficient matrices containing vector entries. Homogenizing the coefficient of $\frac {\partial \delta ^{*}_{\infty ,{\Omega }}}{\partial x_{i}}$ in Eq. 46 and writing

$$ \begin{array}{@{}rcl@{}} \tilde A := &A^{*} - \sum\limits_{i} \frac{\partial}{\partial x_{i}} \Bigg(B^{*} + \left\{ \sum\limits_{p} {{\Gamma}^{p}}^{\prime} \frac{\partial {\Xi}^{p}}{\partial x_{i}} \right\}^{*} \\ &+ \left\{ \sum\limits_{p} {\Gamma}^{p} \frac{\partial {{\Xi}^{p}}^{\prime}}{\partial x_{i}} \right\}^{*} \Bigg) \end{array} $$

(47)

$$ \begin{array}{@{}rcl@{}} \tilde C := &\min\limits_{i} \Bigg(B^{*} + \left\{ \sum\limits_{p} {{\Gamma}^{p}}^{\prime} \frac{\partial {\Xi}^{p}}{\partial x_{i}}\right\}^{*} + \left\{ \sum\limits_{p} {\Gamma}^{p} \frac{\partial {{\Xi}^{p}}^{\prime}}{\partial x_{i}} \right\}^{*} \\ &+ \left\{ \frac{\partial }{\partial x_{i}} \sum\limits_{p} {\Gamma}^{p} {{\Xi}^{p}}^{\prime} \right\}^{*} \Bigg) \end{array} $$

(48)

$$ \begin{array}{@{}rcl@{}} \tilde D := &\left\{\sum\limits_{p} {\Gamma}^{p} {{\Xi}^{p}}^{\prime} \right\}^{*} \end{array} $$

(49)

we get

$$ \tilde A \tilde\delta^{*}_{\infty,{\Omega}}(r) - \tilde C \frac{\partial \tilde\delta^{*}_{\infty,{\Omega}}(r)}{\partial r} - \tilde D \frac{\partial^{2} \tilde\delta^{*}_{\infty,{\Omega}}(r) }{\partial r^{2}}= 0. $$

(50)

The solution of this equation depends on the sign of ${4 \tilde A \tilde D + \tilde C^{2}}$. Real and distinct roots are obtained for ${4 \tilde A \tilde D + \tilde C^{2}} > 0$. Cases when ${4 \tilde A \tilde D + \tilde C^{2}} < 0$, complex conjugates are obtained and ${4 \tilde A \tilde D + \tilde C^{2}} = 0$ results in repeated real roots. In this work, it is assumed that ${4 \tilde A \tilde D + \tilde C^{2}} > 0$. This condition is enforced by taking the norm of the eigenvalues while computing the local coefficient matrices. Further research is required to study the behaviour of the solutions obtained when this condition is not satisfied or when the local matrices have complex eigenvalues. The solution then becomes

$$ \tilde\delta^{*}_{\infty,{\Omega}}(r) = \exp\left[ - \frac{\sqrt{4 \tilde A \tilde D + \tilde C^{2}} + \tilde C}{2\tilde D} (r - r_{\Omega}) \right] \tilde\delta^{*}_{\Omega}. $$

(51)

Finally, following the derivation in the previous section, the conservative estimate, $\hat \delta ^{*}_{\infty , {\Omega }}$, such that $\left \Vert {\hat \delta ^{*}_{\infty , {\Omega }}}\right \Vert \ge \left \Vert {\tilde \delta ^{*}_{\infty , {\Omega }}}\right \Vert$, is given by

$$ \hat\delta^{*}_{\infty,{\Omega}}(r) = \tilde\delta^{*}_{\Omega} \exp \left[ -\lambda_{\min} (r - r_{\Omega})\right], $$

(52)

where $\lambda _{\min \limits }$ is the norm of the minimum eigenvalue of $\frac {\sqrt {4 \tilde A \tilde D + \tilde C^{2}} + \tilde C}{2\tilde D}$ in Eq. 51. The radius of influence is calculated by Eq. 25.