Numerical simulation for twophase flow in a porous medium
 Souad Gasmi^{1} and
 Fatma Zohra Nouri^{1}Email author
https://doi.org/10.1186/s1366101402566
© Gasmi and Nouri; licensee Springer 2015
Received: 26 August 2014
Accepted: 1 December 2014
Published: 13 January 2015
Abstract
In this paper, we introduce a numerical study of the hydrocarbon system used for petroleum reservoir simulations. This system is a simplified model which describes a twophase flow (oil and gas) with mass transfer in a porous medium, which leads to fluid compressibility. This kind of flow is modeled by a system of parabolic degenerated nonlinear convectiondiffusion equations. Under certain hypotheses, such as the validity of Darcy’s law, incompressibility of the porous medium, compressibility of the fluids, mass transfer between the oil and the gas, and negligible gravity, the global pressure formulation of Chavent (Mathematical Models and Finite Elements for Reservoir Simulation: Single Phase, Multiphase and Multicomponent Flows Through Porous Media, 1986) is formulated. This formulation allows the establishment of theoretical results on the existence and uniqueness of the solution (Gasmi and Nouri in Appl. Math. Sci. 7(42):20552064, 2013). Furthermore, different numerical schemes have been considered by many authors, among others we can refer the reader to (Chen in Finite element methods for the black oil model in petroleum reservoirs, 1994; Chen in Reservoir Simulation: Mathematical Techniques in Oil Recovery, 2007) and (Gagneux et al. in Rev. Mat. Univ. Complut. Madr. 2(1):119148, 1989). Here we make use of a scheme based on the finite volume method and present numerical results for this simplified system.
Keywords
compressible fluids porous medium multiphasic flow finite volume method1 Introduction
The fluids flow within porous media has an important role in various domains, such as geothermal studies, geotechnics (the mechanics of soils), chemical engineering, ground water storage, hydrocarbon exploitation (see references [1] and [2]), etc. In some cases, there are two or more fluids with different characteristics. We are concerned with the simulation of the displacement of a fluid by another one, within a porous medium, while the displacing fluid is immiscible with the fluid being displaced. Different numerical techniques, mainly based on finite elements, have been used by many authors to solve such problems, for example see [3] and [4].
In this paper we introduce a finite volume method for solving the hydrocarbon system model often used for petroleum reservoir simulations. To prevent consistency defects in the scheme, we propose to modify the mesh where the discontinuity occurs, because of the porous media. We propose to design our new scheme, called the ‘diamond meshes scheme’ (DMS), whose convergence is proved, and which can be used to solve the nonlinear discrete equations. Finally, numerical simulations confirm that the DMS is significantly useful for such difficult problems taking into account their physical properties.
In this case, there are only two chemical species, or components, gas and oil, where this last component may exist in both phases (gas and oil), that is to say, the heavy residual component in the oil phase and the light volatile component in the gas phase. In order to reduce confusion, we need to distinguish carefully between the ‘oil component’ and the ‘oil phase’. This model, called a hydrocarbon system, is a simplified compositional model describing twophase flow in a porous medium with mass interchange between them. Therefore it can predict compressibility and mass transfer effects, in the sense that it is assumed that there is mass transfer between the oil and the gas phase. In this model the ‘oil component’ (also called stocktank oil) is the residual liquid at atmospheric pressure left after a differential vaporization, while the ‘gas component’ is the remaining fluid.
2 Mathematical model
One of the fluids wets the porous medium more than the other; we refer to this as the wetting phase fluid and we refer to the other as the nonwetting phase fluid. In an oilgas system, oil is the wetting phase. Let us consider a bounded connex open Ω of \(R^{d}\) (\(d=2\mbox{ or }3\)), describing the porous medium (the reservoir), with a Lipchitz boundary Γ, and let t be the time variable t in \([0,T[\), \(T <\infty\). Let \(C_{Gg}\) be the mass fraction of the gas component in the gas phase, \(C_{Og}\) the mass fraction of the oil component (the light component) in the gas phase, and \(C_{Oo}\) the mass fraction of the oil component (the heavy component) in the oil phase which is equal to 1. While this distribution of the hydrocarbon components between the oil and gas phases plays an important role in a steam drive process, we cannot say that the mass of each phase is conserved because of the possibility of transfer of the oil component between the two phases. Instead, we observe that the total mass of each component must be conserved.
3 Weak formulations
4 Finite volume approximation
In this section we study the approximation of solutions to our model in the discrete finite volume framework. This family of schemes allows very general meshes and deals with the main properties of the physical features of the treated problems. The time interval \([ 0,T [ \) is divided into finite subintervals \([ t_{n},t_{n+1} [ \) of length \(\Delta t_{n}\), \(n=0,\ldots,M\) with \(t_{0}=0\) and \(t_{M}=T\). The space domain (the reservoir Ω) is discretized by a nonstructured stitching \(T_{h}\).
4.1 Notations

Let \(\vert K\vert \) denote the cell K surface, \(N(K)\) the set of triangles having a common side with the cell K.

Let \(e_{K,L}\) be the common side of the triangles K and L and \(\overrightarrow{\eta}_{K,L}\) be the normal oriented from K towards L.

\(\overrightarrow{\eta}_{e_{i}}\) is the external normal corresponding to the part of \(e_{i}\) at the boundary Γ.

Let \(S_{h}\) be the set of sides of the stitching \(T_{h}\) and \(S_{h}^{\ast} \) be the set of the interior sides.

For a given side e, let us denote by S and N the extremities e, by W and E the two triangles where \(e=\bar{W}\cap\bar{E}\); by \(\chi _{e}\) the diamond cell associated with e given by connecting the centers of gravities of the cells W and E with the extremities N and S of e.

\(( ( \varepsilon_{_{i}} ) _{i=1,4} ) \) are the four segments forming the border of \(\chi_{e}\).

\(\overrightarrow{\eta}_{\varepsilon}=\frac{1}{\vert \varepsilon _{_{i}}\vert } ( \mu_{x_{i}},\mu_{y_{i}} ) \) is the normal on \(\varepsilon_{_{i}}\) outgoing of \(\chi_{e}\).

For a given node, \(V ( N ) \) is the set of triangles with this node in common.
For the numerical resolution of problem (22)(25), the first two equations will be discretized separately. For more details on finite volume methods, see for example [6] and [7].
4.2 Discretization of the first equation
4.3 Discretization of the second equation
5 Numerical experiments
The problem given in (22)(25) was said not to be typical of a hydrocarbon system simulation, but it can be designed to test the resolution capabilities of the numerical method for problems involving sharp fronts.
The numerical tests are done on a homogeneous \(isotropic \) reservoir. The physical setting in 2D was as follows. A horizontal rectangular reservoir \(\Omega=\,]0,300[\,\times\,]0,60[\), discretized with a structured mesh made up of \(3{,}000\) cells, with an intrinsic permeability of 0.001 was initially filled with a mixture of gas and oil, their residual saturations are 0.02 and 0.03, respectively. The initial fluid (oil) distribution was taken to be uniform on the whole reservoir surface, and it had an associated initial pressure \(P_{0}=2{,}000~\mbox{bar}\). The porosity is \(\Phi=0.02\). The mobilities and the capillary pressure are given by \(P_{c}=(s^{2}1)/2\), \(\lambda _{o}=0.5\), \(\lambda_{g}= 0.2\), and \(\mu_{w}=1\), and \(\mu_{o}=3\), the time step is \(\Delta t=10~\mbox{s}\). These tests were carried out using the free and opensource simulator for flow and transport processes in porous media, based on the Distributed and Unified Numerics Environment DUNE (see www.dumux.org).
5.1 Results and discussion
6 Conclusion
In this paper we proposed a new scheme based on the finite volume method for solving the displacement of a fluid by another one, within a porous medium, while the displacing fluid is immiscible with the fluid being displaced. This gives the hydrocarbon system model often used for petroleum reservoir simulations. To prevent consistency defects in the scheme, we suggested to modify the mesh where the discontinuity occurs, because of the porous media. This convergent scheme, called the ‘diamond meshes scheme’ (DMS) was designed, to solve the associated nonlinear discrete equations. Finally, the numerical results, which are linked to our theoretical ones in [5], confirm that the DMS is significantly useful for such difficult problems (see Figures 13).
Declarations
Acknowledgements
This work was financially supported by the national research project (PNR08/23/997, 20112013). The authors are very grateful to the referees for their valuable and helpful comments, remarks and suggestions.
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
References
 Chavent, G, Jaffre, J: Mathematical Models and Finite Elements for Reservoir Simulation: Single Phase, Multiphase and Multicomponent Flows Through Porous Media, pp. 140. Elsevier, Amsterdam (1986) MATHGoogle Scholar
 Gagneux, G, Lefevere, AM, MadauneTort, M: Analyse mathématique de modèles variationnels en simulation pétrolière: le cas du modèle blackoil pseudocompositionnel standard isotherme. Rev. Mat. Univ. Complut. Madr. 2(1), 119148 (1989) MATHMathSciNetGoogle Scholar
 Chen, Z: Finite element methods for the black oil model in petroleum reservoirs, pp. 128. I.M.A. Preprint Series #1238 (1994) Google Scholar
 Chen, Z: Reservoir Simulation: Mathematical Techniques in Oil Recovery. SIAM, Philadelphia (2007) View ArticleGoogle Scholar
 Gasmi, S, Nouri, FZ: A study of a biphasic flow problem in porous media. Appl. Math. Sci. 7(42), 20552064 (2013) MathSciNetGoogle Scholar
 Afif, M, Amaziane, B: On convergence of finite volume schemes for onedimensional twophase flow in porous media. J. Comput. Appl. Math. 145, 3148 (2002) View ArticleMATHMathSciNetGoogle Scholar
 Boyer, F, Hubert, F: Finite volume method for 2D linear and nonlinear elliptic problems with discontinuities. SIAM J. Numer. Anal. 43(6), 30323070 (2008) View ArticleMathSciNetGoogle Scholar