Abstract
We explore and develop a Proper Orthogonal Decomposition (POD)-based deflation method for the solution of ill-conditioned linear systems, appearing in simulations of two-phase flow through highly heterogeneous porous media. We accelerate the convergence of a Preconditioned Conjugate Gradient (PCG) method achieving speed-ups of factors up to five. The up-front extra computational cost of the proposed method depends on the number of deflation vectors. The POD-based deflation method is tested for a particular problem and linear solver; nevertheless, it can be applied to various transient problems, and combined with multiple solvers, e.g., Krylov subspace and multigrid methods.
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Funding
This study was funded by the Mexican Insitute of Petroleum (IMP), the ‘Consejo Nacional de Ciencia y Tecnología (CONACYT)’, and the ‘Secretaría de Energía (SENER)’, through the programs: ‘Programa de Captación de Talento, Reclutamiento, Evaluación y Selección de Recursos Humanos (PC-TRES)’, and ‘Formación de Recursos Humanos Especializados para el Sector Hidrocarburos (CONACYT-SENER Hidrocarburos, 2015-2016, CUARTO PERIODO)’.
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Appendices
Appendix A: Two-phase flow through porous media
For the simulation of two-phase (oil-water) flow through a porous medium, we can consider the phases as separated, i.e., they are immiscible and there is no mass transfer between them. We usually consider one of the fluids as the wetting phase (w), which is more attracted to the mineral particles than the other phase, known as the non-wetting phase (nw). In the case of a water-oil system, water is considered the wetting phase.
The saturation of a phase \(\left (S_{\alpha }\right )\) is the fraction of void space filled with that phase in the medium. When modeling two-phase flow, the permeability of each phase, α, will be affected by the presence of the other phase. Therefore, an effective permeability, Kα, has to be used instead of the absolute permeability K, and relative permeability values are taken into account.
As in the single-phase case, the governing equations for two-phase flow in a porous medium are the mass conservation principle and Darcy’s law. The mass balance equation for a phase α is given by:
and the Darcy’s law reads:
where ρα, μα, qα and pα are the density, viscosity, sources and pressure of each phase, g is the gravity constant, and d is the depth of the reservoir.
Combining Darcy’s law (21), the mass balance (20) and using the phase mobilities, the system reads:
which is a parabolic equation for pressures and saturations, and where
The previously-mentioned equations can be separated into a pressure equation and a saturation or transport equation via the fractional flow formulation [27], approach used in this work. For an immiscible, incompressible flow, the pressure equation becomes elliptic and the transport equation becomes hyperbolic. With this formulation, the pressure and transport equations are solved in separate steps in a sequential procedure, for more details see [14]. This approach is used throughout this work, therefore, we present a brief description of the method in Appendix B.
Appendix B: Fractional flow formulation
In the case of incompressible flow, the porosity ϕ and the densities ρα do not depend on the pressure. Considering a two-phase system with a wetting (w) and a non wetting phase (nw). Therefore, (20) for each phase reduces to:
To solve them, the fractional flow formulation takes the total Darcy’s velocity as the sum of the velocity in both phases:
Summing up the continuity equations for the two phases we obtain:
where q = qnw + qw is the total source term. Defining the total mobility as λ = λnw + λw (see Eq. 23), and using Darcy’s law, (26) becomes:
which is an equation for the pressure of the non wetting phase. This equation depends on the saturation via the capillary pressure pc and the total mobility λ.
Multiplying each phase velocity by the relative mobility of the other phase and subtracting the result we get:
Therefore, for the wetting phase velocity, vw, we have:
We introduce the fractional flow function,
which, together with the previously computed velocity vw, transforms the transport Eq. (20), idem for the rest of the referenced equations in Appendix A to:
where Δρ = ρw − ρnw.
With this approach, the system is expressed in terms of the non wetting phase pressure, (27), and the saturation of the wetting phase, (31). In the pressure equation, the coupling to saturation is present via the phase mobilities, and the derivative of the capillary function. For the saturation, we have an indirect coupling with the pressure through the total Darcy velocity, (25). With this scheme, the equations are solved for the pressure using the previously computed saturation, and the saturation is updated by substituting the computed pressure.
Appendix C: Discretization methods
In this work, we use the sequential scheme to simulate two-phase flow. With this approach, an unknown is fixed, e.g., the saturation of the wetting phase (Sw), and the resulting elliptic equation is solved for the pressure of the non-wetting phase (pnw). Once pnw is computed, we update the total velocity, v, and we solve the parabolic transport equation for Sw (more details in [14]).
The resulting system depends on space and time. The space derivatives are discretized using finite differences scheme; for the temporal discretization, we use the backward Euler method, details can be found in [14]. In the examples presented in Section 3, the discretization is performed with the Matlab Reservoir Simulation Toolbox, MRST [30].
Appendix D: Complexity
This appendix is devoted to the computation of the number of operations necessary to implement the methods studied throughout this work. For the implementation of the POD-based deflation method, first, we need to compute the snapshots with the ICCG method. Once the snapshots are computed, we obtain the eigenvalues (Λ) and eigenvectors (V) of the covariance matrix \(\textbf {R}\in \mathbb {R}^{n\times n}\) by computing the SVD of \(\textbf {R}^{T}= \textbf {x}s\textbf {x}s^{T} \in \mathbb {R}^{p\times p}\). The eigenvalues of both matrices are the same and the eigenvectors of R can be computed from:
where V are the eigenvectors of RT (see [17, 40]). The eigenvectors corresponding to the largest eigenvalues are selected as the POD basis.
The operational cost of the ICCG method, the DICCG method, and the SVD process is presented in Table 10 (see [17]). For the computation of the number of flops of the DICCG method, we assume that the matrix Z is already given, i.e. it does not change during the iteration process. The flops are computed for sparse matrices with m the number of non-zero diagonals, and p the number of deflation vectors.
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Diaz Cortés, G.B., Vuik, C. & Jansen, JD. Accelerating the solution of linear systems appearing in two-phase reservoir simulation by the use of POD-based deflation methods. Comput Geosci 25, 1621–1645 (2021). https://doi.org/10.1007/s10596-021-10041-6
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DOI: https://doi.org/10.1007/s10596-021-10041-6