Global Pressure Equation Research Articles

Modeling and simulation of surfactant–polymer flooding using a new hybrid method

Chemical enhanced oil recovery by surfactant–polymer (SP) flooding has been studied in two space dimensions. A new global pressure for incompressible, immiscible, multicomponent two-phase porous media flow has been derived in the context of SP flooding. This has been used to formulate a system of flow equations that incorporates the effect of capillary pressure and also the effect of polymer and surfactant on viscosity, interfacial tension and relative permeabilities of the two phases. The coupled system of equations for pressure, water saturation, polymer concentration and surfactant concentration has been solved using a new hybrid method in which the elliptic global pressure equation is solved using a discontinuous finite element method and the transport equations for water saturation and concentrations of the components are solved by a Modified Method Of Characteristics (MMOC) in the multicomponent setting. Numerical simulations have been performed to validate the method, both qualitatively and quantitatively, and to evaluate the relative performance of the various flooding schemes for several different heterogeneous reservoirs.

Modeling compositional compressible two-phase flow in porous media by the concept of the global pressure

We derive a new formulation for the compositional compressible two-phase flow in porous media. We consider a liquid–gas system with two components: water and hydrogen. The formulation considers gravity, capillary effects, and diffusivity of each component. The main feature of this formulation is the introduction of the global pressure variable that partially decouples the system equations. To formulate the final system, and in order to avoid primary unknowns changing between one-phase and two-phase zones, a second persistent variable is introduced: the total hydrogen mass density. The derived system is written in terms of the global pressure and the total hydrogen mass density. The system is capable of modeling the flows in both one and two-phase zones with no changes of the primary unknowns. The mathematical structure is well defined: the system consists of two nonlinear parabolic equations, the global pressure equation, and the total hydrogen mass density equation. The derived formulation is fully equivalent to the original one. Numerical simulations show ability of this new formulation to model efficiently the phase appearance and disappearance.

A convergence of a MFE-FV method for immiscible compressible flow in heterogeneous porous media

Abstract This paper deals with the development and analysis of a numerical method for a coupled system describing immiscible compressible two-phase flow through heterogeneous porous media. The system is modelled in a fractional flow formulation which consists of a parabolic equation (the global pressure equation) coupled with a nonlinear degenerated diffusion-convection one (the saturation equation). A mixed finite element (MFE) method is used to discretize the pressure equation and is combined with a conservative finite volume (FV) method on unstructured grids for the saturation equation. It is shown that the FV scheme satisfies a discrete maximum principle. We derive L ∞ and BV estimates under an appropriate CFL condition. Then we prove the convergence of the approximate solution to a weak solution of the coupled system. Numerical results for water-gas flow through engineered and geological barriers for a geological repository of radioactive waste are presented to illustrate the performance of the method in two space dimensions.

An existence result for a coupled system modeling a fully equivalent global pressure formulation for immiscible compressible two-phase flow in porous media

A new model describing immiscible, compressible two-phase flow, such as water–gas, through heterogeneous porous media is considered. The main feature of this model is the introduction of a new global pressure and the full equivalence to the original equations. The resulting equations are written in a fractional flow formulation and lead to a coupled system which consists of a nonlinear parabolic equation (the global pressure equation) and a nonlinear diffusion–convection one (the saturation equation). Under some realistic assumptions on the data, we show an existence result with the help of appropriate regularizations and a time discretization. We use suitable test functions to get a priori estimates. In order to pass to the limit in nonlinear terms, we also obtain compactness results which are nontrivial due to the degeneracy of the system.

Modeling and Numerical Simulations of Immiscible Compressible Two-Phase Flow in Porous Media by the Concept of Global Pressure

A new formulation is presented for the modeling of immiscible compressible two-phase flow in porous media taking into account gravity, capillary effects, and heterogeneity. The formulation is intended for the numerical simulation of multidimensional flows and is fully equivalent to the original equations, contrary to the one introduced in Chavent and Jaffre (Mathematical Models and Finite Elements for Reservoir Simulation, 1986). The main feature of this formulation is the introduction of a global pressure. The resulting equations are written in a fractional flow formulation and lead to a coupled system which consists of a nonlinear parabolic (the global pressure equation) and a nonlinear diffusion–convection one (the saturation equation) which can be efficiently solved numerically. A finite volume method is used to solve the global pressure equation and the saturation equation for the water and gas phase in the context of gas migration through engineered and geological barriers for a deep repository for radioactive waste. Numerical results for the one-dimensional problem are presented. The accuracy of the fully equivalent fractional flow model is demonstrated through comparison with the simplified model already developed in Chavent and Jaffre (Mathematical Models and Finite Elements for Reservoir Simulation, 1986).

Accurate velocity reconstruction for Discontinuous Galerkin approximations of two-phase porous media flows

We consider Discontinuous Galerkin approximations of two-phase, immiscible porous media flows in the global pressure/fractional flow formulation with capillary pressure. A sequential approach is used with a backward Euler step for the saturation equation, equal-order interpolation for the pressure and the saturation, and without any limiters. An accurate total velocity field is recovered from the global pressure equation to be used in the saturation equation. Numerical experiments show the advantages of the proposed reconstruction. To cite this article: A. Ern et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).

A new formulation of immiscible compressible two-phase flow in porous media

A new formulation is proposed to describe immiscible compressible two-phase flow in porous media. The main feature of this formulation is the introduction of a global pressure. The resulting equations are written in a fractional flow formulation and lead to a coupled system which consists of a nonlinear parabolic (the global pressure equation) and a nonlinear diffusion–convection one (the saturation equation) which can be efficiently solved numerically. To cite this article: B. Amaziane, M. Jurak, C. R. Mecanique 336 (2008).

Numerical solution of two-phase flow for the advection-dominated and non-linear case

This work presents a highly efficient numerical scheme for solving immiscible, advection-dominated two-phase flow in heterogeneous porous media. The pressure equation is decoupled from the saturation equation using an IMPES approach, while the advective terms are decoupled from the capillary diffusive terms in the saturation equation through sequential operator splitting. The parabolic and hyperbolic equations are approximated in time by implicit and explicit schemes, respectively. Damped Newton linearization is applied to the implicit non-linear diffusive step. Mixed hybrid finite elements are applied to the global pressure equation and to the regularized capillary diffusion term. For both linear systems arising from the approximation procedure, an AMG preconditioned conjugate gradient solver is used. A finite volume scheme with slope limiter is applied to the advective step. Numerical comparison with standard preconditioners demonstrates the reliability of the proposed AMG-preconditioner. Benchmark examples illustrate the robustness of the method.

A Finite Volume Scheme for Two-Phase Immiscible Flow in Porous Media

In this paper, we prove the convergence of a numerical method for solving two-phase immiscible, incompressible flow in porous media. The method combines an upwind time implicit finite volume scheme for the saturation equation (hyperbolic-parabolic type) and a centered finite volume scheme for the Chavent global pressure equation (elliptic type). The capillary pressure is not neglected, and we study the case when the diffusion term in the saturation equation is weakly degenerated. Estimates on the approximate solution are proven; then by using compactness theorems we obtain a limit when the size of the discretization goes to zero, and we prove that this limit is the unique weak solution of the problem that we study.

Practical implementation of the fractional flow approach to multi-phase flow simulation

Fractional flow formulations of the multi-phase flow equations exhibit several attractive attributes for numerical simulations. The governing equations are a saturation equation having an advection diffusion form, for which characteristic methods are suited, and a global pressure equation whose form is elliptic. The fractional flow approach to the governing equations is compared with other approaches and the implication of equation form for numerical methods discussed. The fractional flow equations are solved with a modified method of characteristics for the saturation equation and a finite element method for the pressure equation. An iterative algorithm for determination of the general boundary conditions is implemented. Comparisons are made with a numerical method based on the two-pressure formulation of the governing equations. While the fractional flow approach is attractive for model problems, the performance of numerical methods based on these equations is relatively poor when the method is applied to general boundary conditions. We expect similar difficulties with the fractional flow approach for more general problems involving heterogenous material properties and multiple spatial dimensions.