Robin-Robin domain decomposition methods for the dual-porosity-conduit system

Abstract

The recently developed dual-porosity-Stokes model describes a complicated dual-porosity-conduit system which uses a dual-porosity/permeability model to govern the flow in porous media coupled with free flow via four physical interface conditions. This system has important applications in unconventional reservoirs especially the multistage fractured horizontal wellbore problems. In this paper, we propose and analyze domain decomposition methods to decouple the large system arisen from the discretization of dual-porosity-Stokes model. Robin boundary conditions are used to decouple the coupling conditions on the interface. Then, Robin-Robin domain decomposition methods are constructed based on the two decoupled sub-problems. Convergence analysis is demonstrated and a geometric convergence order is obtained. Optimized Schwarz methods are proposed for the dual-porosity-Stokes model and optimized Robin parameters are obtained to improve the convergence of proposed algorithms. Three computational experiments are presented to illustrate and validate the accuracy and applicability of proposed algorithms.

Appendix

Proof of Theorem 3.3

Adding (3.29) and (3.30), summing over k from k = 1 to N, then we can obtain:

$$ \begin{array}{@{}rcl@{}} \left\|\varepsilon_{d}^{N+1}\right\|_{L^{2}({\Gamma}_{cd})}^{2} + \left\|\varepsilon_{c}^{N+1}\right\|_{L^{2}({\Gamma}_{cd})}^{2} & =& \left\|{\varepsilon_{d}^{1}}\right\|_{L^{2}({\Gamma}_{cd})}^{2} + \left\|{\varepsilon_{c}^{1}}\right\|_{L^{2}({\Gamma}_{cd})}^{2}\\ &&- 4\gamma\sum\limits_{k=1}^{N}\left( {\vphantom{\frac{\alpha\nu \sqrt{{\mathbf{d}}}}{\sqrt{\text{trace}~\left( \prod\right)}}}}a_{S}\left( {\mathbf{e}}_{u}^{k}, {\mathbf{e}}_{u}^{k}\right) + \frac{1}{\rho} a_{m}\left( {e_{m}^{k}}, {e_{m}^{k}}\right)\right.\\&&+ \frac{1}{\rho} a_{f}\left( {e_{f}^{k}}, {e_{f}^{k}}\right) \\ && + { \frac{1}{\rho} \frac{\sigma k_{m}}{\mu}\left( {e}_{m}^{k}-{e}_{f}^{k},{e_{m}^{k}}-{e_{f}^{k}}\right)_{{\Omega}_{d}} } \\ &&\left.+ \frac{\alpha\nu \sqrt{{\mathbf{d}}}}{\sqrt{\text{trace}~\left( \prod\right)}} \left\| P_{\tau}\mathbf{e}_{u}^{k}\right\|^{2}_{L^{2}({\Gamma}_{cd})} \right). \end{array} $$

(5.1)

By the coercivity of the bilinear forms on the right-hand side of (5.1), we have:

$$ \begin{array}{@{}rcl@{}} 4\gamma C_{0}\sum\limits_{k=1}^{N}\left( \left\|{\mathbf{e}}_{u}^{k}\right\|_{1}^{2}+ { \left\|{e_{m}^{k}}\right\|_{1}^{2} } +\left\|{e_{f}^{k}}\right\|_{1}^{2}\right) &\leq & 4\gamma\sum\limits_{k=1}^{N} \bigg(a_{S}\left( {\mathbf{e}}_{u}^{k}, {\mathbf{e}}_{u}^{k}\right) + \frac{1}{\rho} a_{m}\left( {e_{m}^{k}}, {e_{m}^{k}}\right)\\&&+\frac{1}{\rho} a_{f}\left( {e_{f}^{k}}, {e_{f}^{k}}\right)\\ &&+\frac{1}{\rho} \frac{\sigma k_{m}}{\mu}\left( {e}_{m}^{k}-{e}_{f}^{k},{e_{m}^{k}}-{e_{f}^{k}}\right)_{{\Omega}_{d}} \\ &&+ \frac{\alpha\nu \sqrt{{\mathbf{d}}}}{\sqrt{\text{trace}~\left( \prod\right)}} \left\| P_{\tau}\mathbf{e}_{u}^{k}\right\|^{2}_{L^{2}({\Gamma}_{cd})} \bigg) \\ &\leq & \left\|{\varepsilon_{d}^{1}}\right\|_{L^{2}({\Gamma}_{cd})}^{2} + \left\|{\varepsilon_{c}^{1}}\right\|_{L^{2}({\Gamma}_{cd})}^{2}, \end{array} $$

(5.2)

where C₀ is given in (2.22). Hence, $\left \|{\mathbf {e}}_{u}^{k}\right \|_{1}$, $\left \|{e_{m}^{k}}\right \|_{1}$, and $\left \|{e_{f}^{k}}\right \|_{1}$ tend to zero, respectively. Passing k to the limit in the error (3.16) will result in:

$$ \begin{array}{@{}rcl@{}} \left\langle\varepsilon_{d}^{k}, \psi_{f}\right\rangle \rightarrow 0 ~~\text{as}~~k\rightarrow \infty, ~~ \forall \psi_{f}\in H^{\frac12}({\Gamma}_{cd}) \end{array} $$

(5.3)

which implies the convergence of ${\varepsilon _{d}^{k}}$ in $H^{-\frac 12}({\Gamma }_{cd})$. Through the convergence of ${\varepsilon ^{k}_{d}}$ and ${e_{f}^{k}}$ together with the error (3.19), we deduce the convergence of ${\varepsilon _{c}^{k}}$ in $H^{-\frac 12}({\Gamma }_{cd})$. The convergence of the pressure ${e_{p}^{k}}$ follows from the inf-sup condition (2.21) and (3.17). Then, the theorem is obtained. □

Proof of Theorem 3.4

It is easy to show the following coercivity properties:

$$ \begin{array}{@{}rcl@{}} &&a_{m}\left( {e_{m}^{k}}, {e_{m}^{k}}\right){+}a_{f}\left( {e_{f}^{k}}, {e_{f}^{k}}\right) {+}\frac{\sigma k_{m}}{\mu}\left( {e}_{m}^{k}{-}{e}_{f}^{k},{e_{m}^{k}}{-}{e_{f}^{k}}\right)_{{\Omega}_{d}} {\ge} C_{1} \frac{k_{f}}{\mu} \left\|{e_{f}^{k}}\right\|_{1}^{2}, \end{array} $$

(5.4)

$$ \begin{array}{@{}rcl@{}} && a_{S}\left( \mathbf{e}^{k}_{u},\mathbf{e}^{k}_{u}\right)\geq 2\nu C_{2}\left\|\mathbf{e}^{k}_{u}\right\|_{1}^{2}, \end{array} $$

(5.5)

where the constants C₁ and C₂, also the constants C₃, C₄, and C₅ used below, are given in (2.13)–(2.16). Since (3.31) and (3.32) are satisfied, then from (5.4) and (5.5) and trace inequalities (2.15) and (2.16), we deduce:

$$ \begin{array}{@{}rcl@{}} &&\frac{1}{\rho^{2}} \left( 1-\left( \frac{\gamma_{c}}{\gamma_{d}}\right)^{2}\right) \left\|{e_{f}^{k}}\right\|_{L^{2}({\Gamma}_{cd})}^{2} - \frac{2\gamma_{c}}{\rho}\left( 1+\frac{\gamma_{c}}{\gamma_{d}}\right)\left( a_{m}\left( {e_{m}^{k}}, {e_{m}^{k}}\right)+a_{f}\left( {e_{f}^{k}}, {e_{f}^{k}}\right) \right.\\ &&\left.+\frac{\sigma k_{m}}{\mu}\left( {e}_{m}^{k}-{e}_{f}^{k},{e_{m}^{k}}-{e_{f}^{k}}\right)_{{\Omega}_{d}} \right) \\ &\leq & \left( \frac{1}{\rho^{2}}\left( 1-\left( \frac{\gamma_{c}}{\gamma_{d}}\right)^{2}\right)C_{4} - \frac{2\gamma_{c}}{\rho}\left( 1+\frac{\gamma_{c}}{\gamma_{d}}\right) C_{1}\frac{k_{f}}{\mu} \right)\left\|{e_{f}^{k}}\right\|_{1}^{2}\leq 0, \end{array} $$

(5.6)

and

$$ \begin{array}{@{}rcl@{}} &&\left( {\gamma_{d}^{2}} - {\gamma_{c}^{2}}\right) \left\|{\mathbf{e}}_{u}^{k}\cdot \mathbf{n}_{cd}\right\|_{L^{2}({\Gamma}_{cd})}^{2}-2(\gamma_{c}+\gamma_{d}) a_{S}\left( {\mathbf{e}}_{u}^{k}, {\mathbf{e}}_{u}^{k}\right)\leq \left( ({\gamma_{d}^{2}}-{\gamma_{c}^{2}})C_{5}-4\nu C_{2}(\gamma_{c}+\gamma_{d}) \right)\\ && \left\|{\mathbf{e}}_{u}^{k}\right\|^{2}_{1}\leq0, \end{array} $$

(5.7)

where $\left \|{\mathbf {e}}_{u}^{k}\cdot \mathbf {n}_{cd}\right \|_{L^{2}({\Gamma }_{cd})}$ is bounded from above by $\left \|{\mathbf {e}}_{u}^{k}\right \|_{L^{2}({\Gamma }_{cd})}$ and then by (2.16). Plugging (5.6) and (5.7) into (3.21) and (3.22), respectively, we can obtain:

$$ \begin{array}{@{}rcl@{}} \left\|\varepsilon_{c}^{k+1}\right\|_{L^{2}({\Gamma}_{cd})} \leq \frac{\gamma_{c}}{\gamma_{d}} \left\|\varepsilon_{c}^{k-1}\right\|_{L^{2}({\Gamma}_{cd})} ~~~\text{and}~~~ \left\|\varepsilon_{d}^{k+1}\right\|_{L^{2}({\Gamma}_{cd})} \leq \frac{\gamma_{c}}{\gamma_{d}} \left\|\varepsilon_{d}^{k-1}\right\|_{L^{2}({\Gamma}_{cd})}. \end{array} $$

(5.8)

Above recurrence formula implies the geometric convergence rate $\sqrt { \gamma _{c}/\gamma _{d}}$ for ${\varepsilon _{c}^{k}}$ and ${\varepsilon _{d}^{k}}$, as follows:

$$ \begin{array}{@{}rcl@{}} \left\|{\varepsilon_{d}^{k}}\right\|_{L^{2}({\Gamma}_{cd})} \leq \max \left\{ \sqrt{\frac{\gamma_{d}}{\gamma_{c}}} \left\|{\varepsilon_{d}^{1}}\right\|_{L^{2}({\Gamma}_{cd})}, \left\|{\varepsilon_{d}^{0}}\right\|_{L^{2}({\Gamma}_{cd})} \right\} \left( \sqrt{\frac{\gamma_{c}}{\gamma_{d}}} \right)^{k}, \end{array} $$

(5.9)

$$ \begin{array}{@{}rcl@{}} \left\|{\varepsilon_{c}^{k}}\right\|_{L^{2}({\Gamma}_{cd})} \leq \max \left\{ \sqrt{\frac{\gamma_{d}}{\gamma_{c}}} \left\|{\varepsilon_{c}^{1}}\right\|_{L^{2}({\Gamma}_{cd})}, \left\|{\varepsilon_{c}^{0}}\right\|_{L^{2}({\Gamma}_{cd})} \right\} \left( \sqrt{\frac{\gamma_{c}}{\gamma_{d}}} \right)^{k}. \end{array} $$

(5.10)

Above two inequalities are true whether k is even or odd. For the sake of simplification, define:

$$\tilde{C}=\max \left\{ \sqrt{\frac{\gamma_{d}}{\gamma_{c}}} \left\|{\varepsilon_{c}^{1}}\right\|_{L^{2}({\Gamma}_{cd})}, \sqrt{\frac{\gamma_{d}}{\gamma_{c}}} \left\|{\varepsilon_{d}^{1}}\right\|_{L^{2}({\Gamma}_{cd})}, \left\|{\varepsilon_{c}^{0}}\right\|_{L^{2}({\Gamma}_{cd})}, \left\|{\varepsilon_{d}^{0}}\right\|_{L^{2}({\Gamma}_{cd})} \right\},$$

thus (5.9) and (5.10) become:

$$ \begin{array}{@{}rcl@{}} \left\|{\varepsilon_{d}^{k}}\right\|_{L^{2}({\Gamma}_{cd})} \leq \tilde{C} \left( \sqrt{\frac{\gamma_{c}}{\gamma_{d}}} \right)^{k} ~~\text{and}~~ \left\|{\varepsilon_{c}^{k}}\right\|_{L^{2}({\Gamma}_{cd})} \leq \tilde{C} \left( \sqrt{\frac{\gamma_{c}}{\gamma_{d}}} \right)^{k}. \end{array} $$

(5.11)

Next, we derive the convergence rate for ${e_{m}^{k}}$, ${e_{f}^{k}}$, ${\mathbf {e}}_{u}^{k}$ and ${e_{p}^{k}}$ via the error equations (3.16)–(3.18) and (5.11). Setting $\psi _{m}={e_{m}^{k}}$ and $\psi _{f}={e_{f}^{k}}$ in (3.16), we have:

$$ \begin{array}{@{}rcl@{}} \frac{1}{\gamma_{d} }\left\langle {\varepsilon_{d}^{k}}, {e_{f}^{k}}\right\rangle &= &a_{m}\left( {e_{m}^{k}}, {e_{m}^{k}}\right)+a_{f}\left( {e_{f}^{k}}, {e_{f}^{k}}\right) +\frac{\sigma k_{m}}{\mu}\left( {e}_{m}^{k}-{e}_{f}^{k},{e_{m}^{k}}-{e_{f}^{k}}\right)_{{\Omega}_{d}} \\&&+ \frac{1}{\gamma_{d}\rho}\left\langle {e_{f}^{k}}, {e_{f}^{k}}\right\rangle. \end{array} $$

(5.12)

By (5.12), Poincaré inequality (2.13), Cauchy-Schwarz inequality, and trace inequality (2.15), we deduce

$$ \begin{array}{@{}rcl@{}} &&\frac{k_{f}}{\mu} C_{1} \left\|{e^{k}_{f}}\right\|^{2}_{1} {\leq} a_{f}\left( {e^{k}_{f}},{e^{k}_{f}}\right) {\leq} \frac{1}{\gamma_{d}} \left\langle {\varepsilon^{k}_{d}},{e^{k}_{f}} \right\rangle {\leq} \frac{\sqrt{C_{4}}}{\gamma_{d}}\left\|{\varepsilon^{k}_{d}}\right\|_{L^{2}({\Gamma}_{cd})}\left\|{e^{k}_{f}}\right\|_{1} , \end{array} $$

(5.13)

$$ \begin{array}{@{}rcl@{}} &&\frac{k_{m}}{\mu} C_{1} \left\|{e^{k}_{m}}\right\|^{2}_{1} {\leq} a_{m}\left( {e^{k}_{m}},{e^{k}_{m}}\right) {\leq} \frac{1}{\gamma_{d}} \left\langle {\varepsilon^{k}_{d}},{e^{k}_{f}} \right\rangle {\leq} \frac{\sqrt{C_{4}}}{\gamma_{d}}\left\|{\varepsilon^{k}_{d}}\right\|_{L^{2}({\Gamma}_{cd})}\left\|{e^{k}_{f}}\right\|_{1}, \end{array} $$

(5.14)

where the middle inequality follows from (5.12). Substituting the first estimate in (5.11) into (5.13), then we can obtain the convergence rate of ${p^{k}_{f}}$:

$$ \begin{array}{@{}rcl@{}} &&\left\|{e^{k}_{f}}\right\|_{1} \leq \frac{\mu \sqrt{C_{4}}\tilde{C}}{\gamma_{d} k_{f} C_{1}} \left( \sqrt{\frac{\gamma_{c}}{\gamma_{d}}} \right)^{k}. \end{array} $$

(5.15)

Using (5.11), (5.14), and (5.15), we can obtain the convergence rate of ${p^{k}_{m}}$:

$$ \begin{array}{@{}rcl@{}} && \left\|{e^{k}_{m}}\right\|_{1} \leq \frac{\mu\sqrt{C_{4}} \tilde{C}}{\gamma_{d} \sqrt{k_{m}k_{f}}C_{1}} \left( \sqrt{\frac{\gamma_{c}}{\gamma_{d}}} \right)^{k}. \end{array} $$

(5.16)

From (3.17) and the trace inequality (2.16), we have:

$$ \begin{array}{@{}rcl@{}} &&2\nu C_{2} \left\|\mathbf{e}^{k}_{u}\right\|_{1}^{2} \leq a_{S}\left( \mathbf{e}^{k}_{u},\mathbf{e}^{k}_{u}\right) \leq \left\langle {\varepsilon^{k}_{c}},\mathbf{e}^{k}_{u}\cdot \mathbf{n}_{cd} \right\rangle \leq \left\|{\varepsilon_{c}^{k}}\right\|_{L^{2}({\Gamma}_{cd})} \left\|\mathbf{e}^{k}_{u}\right\|_{{L}^{2}({\Gamma}_{cd})}\\ &&\qquad \leq C_{5}\left\|{\varepsilon_{c}^{k}}\right\|_{L^{2}({\Gamma}_{cd})} \left\|\mathbf{e}^{k}_{u}\right\|_{1} \end{array} $$

(5.17)

and the convergence rate of u^k is

$$ \begin{array}{@{}rcl@{}} \left\|\mathbf{e}^{k}_{u}\right\|_{1} \leq \frac{C_{5}\tilde{C}}{{2}\nu C_{2}} \left( \sqrt{\frac{\gamma_{c}}{\gamma_{d}}} \right)^{k}. \end{array} $$

(5.18)

Furthermore, rewriting (3.17) and estimating the upper bound of the bilinear form b_S(⋅,⋅), as follows:

$$ \begin{array}{@{}rcl@{}} b_{S}\left( \mathbf{v},{e_{p}^{k}}\right) &=& \left\langle {\varepsilon_{c}^{k}},\mathbf{v}\cdot \mathbf{n}_{cd}\right\rangle -a_{S}\left( {\mathbf{e}}_{u}^{k}, \mathbf{v}\right) - \gamma_{c}\left\langle {\mathbf{e}}_{u}^{k}\cdot \mathbf{n}_{cd}, \mathbf{v}\cdot \mathbf{n}_{cd}\right\rangle -\frac{\alpha\nu \sqrt{{\mathbf{d}}}}{\sqrt{\text{trace}~\left( \prod\right)}}\left\langle P_{\tau}{\mathbf{e}}_{u}^{k},P_{\tau}\mathbf{v}\right\rangle \\ &\leq& \left( \sqrt{C}_{5} \left\| {\varepsilon_{c}^{k}} \right\| +\left( 2\nu C_{3}+\gamma_{c}C_{5}+\frac{\alpha \nu }{\sqrt{k_{f}}}C_{5}\right)\left\|\mathbf{e}_{u}^{k} \right\|_{1} \right) \left\|\mathbf{v}\right\|_{1} \end{array} $$

(5.19)

for all v ∈V₀. Due to the inf-sup condition (2.21) combined with the upper bound (5.19), we deduce:

$$ \begin{array}{@{}rcl@{}} \left\|{e_{p}^{k}}\right\|_{0}\leq \frac{\tilde{C}}{\beta}\left( \sqrt{C_{5}} +\frac{C_{3}C_{5}}{C_{2}} +\frac{\gamma_{c} {C_{5}^{2}}}{{2}\nu C_{2}} +\frac{\alpha {C_{5}^{2}}}{2\sqrt{k_{f}}C_{2}} \right) \left( \sqrt{\frac{\gamma_{c}}{\gamma_{d}}}\right)^{k}. \end{array} $$

(5.20)

Hence, we have proved the convergence theorem for Case 2. □