Finite element discretization of Darcy's equations with pressure dependent porosity

Girault, Vivette; Murat, François; Salgado, Abner J.

doi:10.1051/m2an/2010019

Cited by 9 publications

(20 citation statements)

References 30 publications

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“…They showed that the results they obtain are different from the solution to the classical Darcy's equation. Assuming that the "Drag coefficient" is a bounded function of the pressure in the generalized Darcy's equation, Girault et al [27] guaranteed the well-posedness of the system together with the convergence of a finite element method. When the dependence on the pressure is exponential (as in (1.2), the authors propose an interesting splitting algorithm, observed numerically to be robust in the parameter range tested.…”

Section: R a F Tmentioning

confidence: 99%

Flow of a Fluid Through a Porous Solid Due to High Pressure Gradients

2013

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Abstract. It is well known that the viscosity of fluids could vary by several orders of magnitude with pressure. This fact is not usually taken into account in many important applications involving the flow of fluids through a porous media, like the problems of enhanced oil recovery or carbon dioxide sequestration where very high pressure differentials are involved. Another important technical problem where such high pressure differentials are involved is that of extracting unconventional oil deposits such as shale which is becoming ever so important now. In this study, we show that the traditional approach that ignores the variation of the viscosity and drag with the pressure greatly over-predicts the mass flux taking place through the porous structure. While taking the pressure dependence of viscosity and drag leads to a ceiling flux, the traditional approaches lead to a continued increase in the flux with the pressure difference. In this study, we consider a generalization of the classical Brinkman equation that takes the dependence of the viscosity and the Drag coefficient on pressure. To our knowledge, this is the first study to carry out such an analysis.

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Section: R a F Tmentioning

confidence: 99%

Flow of a Fluid Through a Porous Solid Due to High Pressure Gradients

2013

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“…A nonlinear Darcy fluid flow problem, with a permeability dependent upon the pressure was investigated by Azaïez, Ben Belgacem, Bernardi, and Chorfi [2], and Girault, Murat, and Salgado [11]. For a Lipschitz continuous permeability function, bounded above and bounded away from zero, existence of a solution (u, p) ∈ L 2 (Ω) × H 1 (Ω) was established.…”

Section: Introductionmentioning

confidence: 99%

“…In [2] the authors also investigated a spectral numerical approximation scheme for the nonlinear Darcy problem, assuming an axisymmetric domain Ω. A convergence analysis for the finite element discretization of that problem was given in [11].…”

Section: Introductionmentioning

confidence: 99%

Generalized Newtonian fluid flow through a porous medium

Ervin

Lee

Salgado

2016

Journal of Mathematical Analysis and Applications

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We present a model for generalized Newtonian fluid flow through a porous medium. In the model the dependence of the fluid viscosity on the velocity is replaced by a dependence on a smoothed (locally averaged) velocity. With appropriate assumptions on the smoothed velocity, existence of a solution to the model is shown. Two examples of smoothing operators are presented in the appendix. A numerical approximation scheme is presented and an a priori error estimate derived. A numerical example is given illustrating the approximation scheme and the a priori error estimate.

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“…In fact, in [6] the authors present optimal error estimates for a spectral discretization of this system which takes into account the axisymmetry of the domain and of the flow. On the other hand, in [33] the authors consider first the simplified model in which the exponential law defining the porosity is truncated above and below by positive constants, whose corresponding solvability and regularity analysis was previously developed in [6], and propose a primal finite element scheme with polynomial approximations of degrees k − 1 and k for velocity and pressure, respectively. Then, the case of a fully exponential porosity is analyzed in the second part of [33], where, under the heuristic assumption that the resulting model has at least one solution, a suitable change of variables involving only the pressure allows to split the problem into two consecutive linear systems, which are discretized by slight variants of the method from the first part.…”

Section: Introductionmentioning

confidence: 99%

“…Here we also consider the Darcy model with a fully exponential porosity from the second part of [33], assume again the heuristic hypothesis concerning the existence of solution, and apply the same change of variables employed there, but instead of a primal method, we opt for using the dual-mixed approach. As a consequence, the mixed boundary conditions arising from the transformation become readily employable and hence there is no need of additional regularity on the data nor of deriving any other boundary condition.…”

Section: Introductionmentioning

confidence: 99%

A mixed finite element method for Darcy’s equations with pressure dependent porosity

2015

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In this work we develop the a priori and a posteriori error analyses of a mixed finite element method for Darcy's equations with porosity depending exponentially on the pressure. A simple change of variable for this unknown allows to transform the original nonlinear problem into a linear one whose dual-mixed variational formulation falls into the frameworks of the generalized linear saddle point problems and the fixed point equations satisfied by an affine mapping. According to the latter, we are able to show the well-posedness of both the continuous and discrete schemes, as well as the associated Cea estimate, by simply applying a suitable combination of the classical Babuška-Brezzi theory and the Banach fixed point Theorem. In particular, given any integer k ≥ 0, the stability of the Galerkin scheme is guaranteed by employing Raviart-Thomas elements of order k for the velocity, piecewise polynomials of degree k for the pressure, and continuous piecewise polynomials of degree k+1 for an additional Lagrange multiplier given by the trace of the pressure on the Neumann boundary. Note that the two ways of writing the continuous formulation suggest accordingly two different methods for solving the discrete schemes. Next, we derive a reliable and efficient residualbased a posteriori error estimator for this problem. The global inf-sup condition satisfied by the continuous formulation, Helmholtz decompositions, and the local approximation properties of the Raviart-Thomas and Clément interpolation operators are the main tools for proving the reliability. In turn, inverse and discrete inequalities, and the localization technique based on triangle-bubble and edge-bubble functions are utilized to show the efficiency. Finally, several numerical results illustrating the good performance of both methods, confirming the aforementioned properties of the estimator, and showing the behaviour of the associated adaptive algorithm, are reported.

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Finite element discretization of Darcy's equations with pressure dependent porosity

Cited by 9 publications

References 30 publications

Flow of a Fluid Through a Porous Solid Due to High Pressure Gradients

Flow of a Fluid Through a Porous Solid Due to High Pressure Gradients

Generalized Newtonian fluid flow through a porous medium

A mixed finite element method for Darcy’s equations with pressure dependent porosity

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