An efficient and physically consistent numerical method for the Maxwell–Stefan–Darcy model of two‐phase flow in porous media

Abstract

Numerical modeling of two-phase flow in porous media has extensive applications in subsurface flow and petroleum industry. A comprehensive Maxwell-Stefan-Darcy (MSD) two-phase flow model has been developedrecently, which takes into consideration the friction between two phases by a thermodynamically consistent way. In this article, we for the first time propose an efficient energy stable numerical method for the MSD model, which can preserve multiple important physical properties of the model. First, the proposed scheme can preserve the original energy dissipation law. This is achieved through a newly-developed energy factorization approach that leads to linear semi-implicit discrete chemical potentials. Second, the scheme preserves the famous Onsager's reciprocal principle and the local mass conservation law for both phases by introducing different upwind strategies for two phase saturations and applying the cell-centered finite volume method to the original formulation of the model. Third, by introducing two auxiliary phase velocities, the scheme has ability to guarantee the positivity of both saturations under proper conditions. Another distinct feature of the scheme is that the resulting discrete system is totally linear, well-posed and unbiased for each phase. Numerical results are also provided to show the excellent performance of the proposed scheme. K E Y W O R D S energy stability, Maxwell-Stefan-Darcy model, Onsager's reciprocal principle, two-phase flow in porous media 546

An energy stable, conservative and bounds‐preserving numerical method for thermodynamically consistent modeling of incompressible two‐phase flow in porous media with rock compressibility

“…EF is originally proposed in Reference 46 and has been successfully applied to the phase-field models 58,59 and two-phase flow in porous media. 42 For S k 𝛼 > 0 and S k+1 𝛼 > 0, using the concavity of ln(S 𝛼 ), we can deduce that…”

“…[5][6][7] To guarantee full mass conservations of both phases, the phase mobilities should be first treated by some discretization strategies, and then total mobility should be obtained through summing the discrete phase mobilities of two phases together. 6,7,41,42 On the basis of the transport property of two-phase flow models, the upwind strategies have been employed as approximations of phase mobilities on the grid-cell boundaries, which can lead to the schemes that ensure the mass conservation law for each phase. 6,7,41,42,[49][50][51] Interestingly but not surprisingly, with the help of upwind mobility strategies, the semi-implicit linear schemes are able to guarantee the positivity of saturations under the Courant-Friedrichs-Lewy (CFL) condition, 6,41,42 while the implicit nonlinear schemes never suffer from this restriction to preserve the positivity.…”

“…6,7,41,42 On the basis of the transport property of two-phase flow models, the upwind strategies have been employed as approximations of phase mobilities on the grid-cell boundaries, which can lead to the schemes that ensure the mass conservation law for each phase. 6,7,41,42,[49][50][51] Interestingly but not surprisingly, with the help of upwind mobility strategies, the semi-implicit linear schemes are able to guarantee the positivity of saturations under the Courant-Friedrichs-Lewy (CFL) condition, 6,41,42 while the implicit nonlinear schemes never suffer from this restriction to preserve the positivity. [49][50][51] In this paper, the fully discrete scheme is constructed combining the locally conservative cell-centered finite difference (CCFD) method with the upwind mobilities.…”

“…Thermodynamically consistent mathematical models, that is, models obeying the second law of thermodynamics, have been increasingly important in many fields, such as phase‐field problems 34‐38 and multicomponent flow with realistic equations of state 32,33,39 . In recent years, thermodynamically consistent models and numerical simulation of incompressible and immiscible two‐phase flows in porous media have been intensively studied in References 22,40‐42. In References 22,40, the two‐phase interfacial free energy and chemical potentials have been introduced to account for capillarity effect, and as a result, the models naturally obey the second law of thermodynamics, but rock compressibility has been never considered.…”

“…Nevertheless, due to the lack of thermodynamically consistent models of two‐phase flows in incompressible or compressible porous media, energy stable methods have been scarcely studied in the past. Very recently, in References 41,42, the energy stable numerical methods were just proposed to approximate the thermodynamically consistent two‐phase flow models established in References 22,40, and meanwhile, an energy stable and mass conservative numerical scheme was proposed for the single‐phase gas flow in porous media with rock compressibility in Reference 43. To our best knowledge, there is no such a scheme developed for two‐phase flow with rock compressibility.…”