A cell-centered multipoint flux approximation method with a diamond stencil coupled with a higher order finite volume method for the simulation of oil–water displacements in heterogeneous and anisotropic petroleum reservoirs

Contreras, Fernando Raul Licapa; Lyra, Paulo Roberto Maciel; Souza, M. R. A.; Carvalho, Darlan Karlo Elisiário de

doi:10.1016/j.compfluid.2015.11.013

Cited by 36 publications

(20 citation statements)

References 24 publications

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“…This initial problem was adapted from . It consists in a 1D linear displacement of oil by water in a homogeneous petroleum reservoir initially fully saturated by oil.…”

Section: Numerical Resultsmentioning

confidence: 99%

“…Finally, we also note that, even for these two distorted mesh configurations, the use of the MIMPES strategy with the HOFV‐E method provides a solution that is quite close to the IMPES solution with a relative error on the cumulative oil of 0.037%, using only 15.91% of the CPU time of the classical IMPES method for the distorted aligned mesh configuration, and 0.029% and only 26.01% of the CPU time, for the distorted and non‐aligned mesh configuration, showing a clear improvement in computational efficiency, with a relatively good accuracy, even though, for the non‐aligned mesh configuration, stable but non‐monotone solutions were obtained for the saturation field, with a minimum saturation value of

S_{w}^{\min} = prefix- 0.13585 \times 10^{prefix- 3}

throughout the simulation. A possible and relatively simple solution for this major drawback has been recently proposed in .Example Discontinuous and Highly Anisotropic one‐fourth of Five‐Spot Problem.…”

Section: Numerical Resultsmentioning

confidence: 99%

“…Further details about the described finite volume formulation, including its performance for non‐homogeneous and non‐isotropic media and the imposition of general Neumann and Cauchy boundary conditions, can be found in . Other locally conservative approaches that arguably produce more accurate results for velocities, particularly for heterogeneous media with high permeability contrasts, include the mixed finite element method and the multipoint flux approximation methods .…”

Section: Numerical Formulationmentioning

confidence: 99%

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A higher resolution edge‐based finite volume method for the simulation of the oil–water displacement in heterogeneous and anisotropic porous media using a modified IMPES method

Silva¹,

Lyra²,

Willmersdorf³

et al. 2016

Numerical Methods in Fluids

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In this article, we present a higher-order finite volume method with a 'Modified Implicit Pressure Explicit Saturation' (MIMPES) formulation to model the 2D incompressible and immiscible two-phase flow of oil and water in heterogeneous and anisotropic porous media. We used a median-dual vertex-centered finite volume method with an edge-based data structure to discretize both, the elliptic pressure and the hyperbolic saturation equations. In the classical IMPES approach, first, the pressure equation is solved implicitly from an initial saturation distribution; then, the velocity field is computed explicitly from the pressure field, and finally, the saturation equation is solved explicitly. This saturation field is then used to re-compute the pressure field, and the process follows until the end of the simulation is reached. Because of the explicit solution of the saturation equation, severe time restrictions are imposed on the simulation. In order to circumvent this problem, an edge-based implementation of the MIMPES method of Hurtado and co-workers was developed. In the MIMPES approach, the pressure equation is solved, and the velocity field is computed less frequently than the saturation field, using the fact that, usually, the velocity field varies slowly throughout the simulation. The solution of the pressure equation is performed using a modification of Crumpton's two-step approach, which was designed to handle material discontinuity properly. The saturation equation is solved explicitly using an edge-based implementation of a modified second-order monotonic upstream scheme for conservation laws type method. Some examples are presented in order to validate the proposed formulation. Our results match quite well with others found in literature.In Eq. (2),K denotes the absolute permeability tensor of the rock, and λ i = k i /μ i is the scalar phase mobility, with k i being the phase relative permeability and μ i the phase viscosity. Henceforth, we will assume incompressible medium and fluids. We will also neglect the capillary pressure and assume that P = P w = P o , where (w) and (o) stand, respectively, for the wetting (water) and the non-R. S. DA SILVA ET AL.

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“…This initial problem was adapted from . It consists in a 1D linear displacement of oil by water in a homogeneous petroleum reservoir initially fully saturated by oil.…”

Section: Numerical Resultsmentioning

confidence: 99%

S_{w}^{\min} = prefix- 0.13585 \times 10^{prefix- 3}

Section: Numerical Resultsmentioning

confidence: 99%

Section: Numerical Formulationmentioning

confidence: 99%

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A higher resolution edge‐based finite volume method for the simulation of the oil–water displacement in heterogeneous and anisotropic porous media using a modified IMPES method

Silva¹,

Lyra²,

Willmersdorf³

et al. 2016

Numerical Methods in Fluids

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show abstract

“…The objective of this work is to simulate the two-phase fluid flow of oil and water in fractured porous media. Aiming to obtain a locally conservative formulation (condition that, in general, is not met by classical finite element methods) which is capable to deal with full permeability tensors (condition that, in general, is not met by finite difference methods), in the present work, we adopt the so-called MPFA-D (multipoint flux approximation with a diamond stencil), 21,22 because it is a FPS formulation which is more robust than a MPFA method with a triangle pressure support (TPS), particularly for highly anisotropic porous media. 23 Thus, we have used the MPFA-D together with the HyG approach to accurately handle the two-phase fluid flow in fractured porous media.…”

Section: Introductionmentioning

confidence: 99%

A multipoint flux approximation with diamond stencil finite volume scheme for the two‐dimensional simulation of fluid flows in naturally fractured reservoirs using a hybrid‐grid method

Cavalcante¹,

Contreras²,

Lyra

et al. 2020

Numerical Methods in Fluids

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Summary Two‐phase flows of oil and water in naturally fractured reservoirs can be described by a system of nonlinear partial differential equations that comprises of an elliptic pressure equation and hyperbolic saturation equation coupled through the total velocity field. Modeling this problem is a great challenge, due to the complexity of the depositional environments, including inclined layers and fractures with different sizes and shapes, and random spatial distribution. In this work, to solve the pressure equation, we adopted a cell‐centered finite‐volume method with a multipoint flux approximation that uses the “diamond stencil” (MPFA‐D) coupled with a hybrid‐grid method (HyG) to deal with the fractures. The classical first‐order upwind method was used to solve the saturation equation, in its explicit and implicit versions. The MPFA‐D is a very robust and flexible formulation that is capable of handling highly heterogeneous and anisotropic domains using general polygonal meshes. In the strategy developed in this work, the mesh that discretize the domain must fit the spatial position of the fractures, so that they are associated to the control surfaces—as (n − 1)D cells—therefore, the calculation of the fluxes in these control surfaces is dependent on the pressures on fractures and on the adjacent volumes. In HyG, the fractures are expanded to nD in the computational domain. The proposed formulation presented quite remarkable results when compared with similar formulations using classical full pressure support and triangle pressure support methods, or even the with MPFA‐D itself when the fractures are treated as nD geometric entities.

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“…Discontinuous and high-order approximations, upwinding, and adaptive meshes were the most successful techniques to deal with coupled equations. 47,48 In the finite element method context, streamline upwind Petrov-Galerkin, Galerkin least squares, and algebraic subgrid scale with high-order elements together with shock-capturing techniques have been the most successful techniques. 27,33,38,49 In general and within all these methodologies, the case of systems of equations has been traditionally tackled using techniques previously successful in the case of a single equation.…”

Section: Introductionmentioning

confidence: 99%

A stabilization technique for coupled convection‐diffusion‐reaction equations

Hernández

Massart

Peerlings

et al. 2018

Numerical Meth Engineering

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Partial differential equations having diffusive, convective, and reactive terms appear in the modeling of a large variety of processes in several branches of science. Often, several species or components interact with each other, rendering strongly coupled systems of convection-diffusion-reaction equations. Exact solutions are available in extremely few cases lacking practical interest due to the simplifications made to render such equations amenable by analytical tools. Then, numerical approximation remains the best strategy for solving these problems. The properties of these systems of equations, particularly the lack of sufficient physical diffusion, cause most traditional numerical methods to fail, with the appearance of violent and nonphysical oscillations, even for the single equation case. For systems of equations, the situation is even harder due to the lack of fundamental principles guiding numerical discretization. Therefore, strategies must be developed in order to obtain physically meaningful and numerically stable approximations. Such stabilization techniques have been extensively developed for the single equation case in contrast to the multiple equations case. This paper presents a perturbation-based stabilization technique for coupled systems of one-dimensional convection-diffusion-reaction equations. Its characteristics are discussed, providing evidence of its versatility and effectiveness through a thorough assessment. KEYWORDS convection-diffusion equation, differential equations, finite element methods, stability Int J Numer Methods Eng. 2018;116:43-65.wileyonlinelibrary.com/journal/nme

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A cell-centered multipoint flux approximation method with a diamond stencil coupled with a higher order finite volume method for the simulation of oil–water displacements in heterogeneous and anisotropic petroleum reservoirs

Cited by 36 publications

References 24 publications

A higher resolution edge‐based finite volume method for the simulation of the oil–water displacement in heterogeneous and anisotropic porous media using a modified IMPES method

A higher resolution edge‐based finite volume method for the simulation of the oil–water displacement in heterogeneous and anisotropic porous media using a modified IMPES method

A multipoint flux approximation with diamond stencil finite volume scheme for the two‐dimensional simulation of fluid flows in naturally fractured reservoirs using a hybrid‐grid method

A stabilization technique for coupled convection‐diffusion‐reaction equations

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