Analysis of a finite volume element scheme for solving the model two-phase nonequilibrium flow problem
DOI:
https://doi.org/10.26577/JMMCS.2022.v114.i2.012Keywords:
Finite volume element method, nonequilibrium fluid flow, dynamic capillary pressure, dual mesh, a priori estimate, convergence, stability, computational experimentAbstract
The paper proposes a hybrid numerical method for solving a model problem of two-phase nonequilibrium flow of an incompressible fluid in a porous medium. This problem is relevant in the modern theory of the motion of multiphase fluids in porous media and has many applications. The studied model is based on the assumption that the relative phase permeabilities and capillary pressure depend not only on saturation, but also on its time derivative. The saturation equation in this problem refers to the type of convection-diffusion with a predominance of convection, which also includes a third-order term to account for the nonequilibrium effects. Due to the hyperbolic nature of the equation, its solution is accompanied by a number of difficulties that lead to the need for an appropriate choice of the solution method. In contrast to previous works, this paper uses a finite volume element method for solving the problem, the construction of which is based on integral balance equations, and an approximate solution is chosen from the finite element space. To discretize the problem, two different dual grids are used based on the main triangulation. In this paper, a number of a priori estimates are obtained which yields the unconditional stability of the scheme as well as its convergence with the second order. The advantages of the approach used include the local conservatism of the scheme, as well as the comparative simplicity of the software implementation of the method. These results are confirmed by a numerical test carried out on the example of a model problem.
References
[2] Al-Bayati D., Saeedi A., Ktao I., Myers M., White C., and Xie Q., "Insight into Influence of Crossflow in layered Sandstone porous media during Miscible and Immiscible CO2 WAG Flooding", InterPore 2019 Valencia: Book of Abstracts (2019): 66.
[3] Allen G.H. and Sahimi M., "Flow, Transport, and Reaction in Porous Media: Percolation Scaling, Critical-Path Analysis, and Effective Medium Approximation" (AGU Publications, 2017).
[4] Hassanizadeh S.M. and Graw W.G., "Toward an improved description of the physics of twophase flow", Advances in Water Resources, 16 (1993): 53-67.
[5] O'Carroll D.M., Phelan T.J., and Abriola L.M., "Exploring dynamic effects in capillary pressure in multistep outflow experiments", Water Resour Res., 41 (2005): W11419.
[6] Cuesta C., van Duijn C.J., and Hulshof J., "Infiltration in porous media with dynamic capillary pressure: travelling waves", Eur. J. Appl. Math., 11 (2000): 381-397.
[7] DiCarlo D.A., Juanes R., LaForce T., and Witelski T.P., "Nonmonotonic travelling wave solutions of infiltration in porous media", Water Resour. Res., 44 (2008): W02406.
[8] Barenblatt G.I., "Filtratsiya dvuh nesmeshivayuschihsya zhidkostei v odnorodnoi poristoi srede [Filtration of two nonmixing fluids in a homogeneous porous medium]", Mehanika gazov i zhidkostei, 5 (1971): 57-64 (in Russian).
[9] Cao X. and Mitra K., "Error estimates for a mixed finite element discretization of a two-phase porous media flow model with dynamic capillarity", Journal of Computational and Applied Mathematics, 353 (2019): 164-178.
[10] Nicaise S. and Bbekkouche F., "A posteriori error estimates for a fully discrete approximation of sobolev equations", Confluentes Math., 11 (2019): 3-28.
[11] Fan Y., and Pop I.S., "A class of pseudo-parabolic equations: existence, uniqueness of weak solutions, and error estimates for the Euler-implicit discretization", Math. Meth. Appl. Sci., 34 (2011): 2329-2339.
[12] Bouadjila K., Mokrane A., Saad A.S., and Saad M., "Numerical analysis of a finite volume scheme for two incompressible phase flow with dynamic capillary pressure", Computers and Mathematics with Applications, 75:10 (2018): 3614-3631.
[13] Luo Z.D., "A Stabilized Crank-Nicolson Mixed Finite Volume Element Formulation for the Non-stationary Incompressible Boussinesq Equations", Journal of Scientific Computing, 66 (2016): 555-576.
[14] Kumar S., "Finite volume element methods for incompressible miscible displacement problems in porous media (Ph.D. thesis)" (Department of mathematics indian institute of technology, Bombay, 2008).
[15] Vasilevsky Yu.V. and Kapyrin I.V., "Dve shemy rasshheplenija dlja nestacionarnoj zadachi konvekcii-diffuzii na tetrajedralnyh setkah [Two splitting schemes for the nonstationary convection-diffusion problem on tetrahedral meshes ]", Zhurnal vychislitel'noj matematiki i matematicheskoj fiziki, 48 (2008): 1429-1447 (in Russian).
[16] Luo Z., Li H., and Sun P., "A fully discrete stabilized mixed finite volume element formulation for the non-stationary conduction–convection problem", Journal of Mathematical Analysis and Applications, 404 (2013): 71-85.
[17] Ferraris S., Bevilacqua I., Canone D., Pognant D., and Previati M., "The Finite Volume Formulation for 2D Second-Order Elliptic Problems with Discontinuous Diffusion/Dispersion Coefficients", Mathematical Problems in Engineering, 187634 (2012): 1-24.
[18] Nikitin K.D., "Nelinejnyj metod konechnyh obyomov dlja zadach dvuhfaznoj filtracii [Nonlinear finite volume method for two-phase filtration problems]", Matematicheskoe modelirovanie, 22 (2010): 131-147 (in Russian).
[19] Lin Y., Liu J., and Yang M., "Finite volume element methods: an overview on recent developments", International Journal of Numerical analysis and modeling, Series B., 4 (2013): 14-34.
[20] Sayevand K. and Arjang F., "Finite volume element method and its stability analysis for analyzing the behavior of sub-diffusion problems", Applied Mathematics and Computation, 290 (2016): 224-239.
[21] Voller V. R., "Basic control volume finite element methods for fluids and solids" (World Scientific Publishing, 2009).
[22] Omariyeva D., Temirbekov N., Madiyarov M., "Stabilized finite element method for solving the saturation equation in the two-phase non-equilibrium flow problem", Joint Issue of Bulletin of the National Engineering Academy of the Republic of Kazakhstan and Computational Technologies, 3:1 (2020): 209-216.
