Analytical and variational numerical methods for unstable miscible displacement flows in porous media

Abstract

International audienceThe miscible displacement of one fluid by another in a porous medium has received considerable attention in sub-surface, environmental and petroleum engineering applications. When a fluid of higher mobility displaces another of lower mobility, unstable patterns-referred to as viscous fingering-may arise. Their physical and mathematical study has been the object of numerous investigations over the past century. The objective of this paper is to present a review of these contributions with particular emphasis on variational methods. These algorithms are tailored to real field applications thanks to their advanced features: handling of general complex geometries, robustness in the presence of rough tensor coefficients, low sensitivity to mesh orientation in advection dominated scenarios, and provable convergence with fully unstructured grids. This paper is dedicated to the memory of Dr. Jim Douglas Jr., for his seminal contributions to miscible displacement and variational numerical methods

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“…It is recognized that the continuous Galerkin method does not guarantee local mass conservation when applied to Equations and , and the system described by Equation may be subject to spurious oscillations, particularly for an advection‐dominant case. To address this issue, a number of stabilization techniques may be used, including mass lumping, the streamline‐upwind Petrov‐Galerkin method, and enrichment of the Galerkin method . Alternatively, a different method could be employed for spatial discretization, such as the discontinuous Galerkin method .…”

“…Since convergence is not always guaranteed with the Newton‐Raphson procedure, the stability of a solution was checked using the following Courant‐Friedrichs‐Lewy condition, as suggested by Zhu et al: …”

“…where the superior dot denotes a time derivative and each variable (eg, P c ) is interpolated from the vector of nodal variables in the standard manner (ie, P c = ∑ N i P c i = NP c ). It is recognized that the continuous Galerkin method does not guarantee local mass conservation when applied to Equations (21) and (22), and the system described by Equation (23) may be subject to spurious oscillations, 39 particularly for an advection-dominant case. To address this issue, a number of stabilization techniques may be used, including mass lumping, the streamline-upwind Petrov-Galerkin method, and enrichment of the Galerkin method.…”

“…To address this issue, a number of stabilization techniques may be used, including mass lumping, the streamline-upwind Petrov-Galerkin method, and enrichment of the Galerkin method. [39][40][41] Alternatively, a different method could be employed for spatial discretization, such as the discontinuous Galerkin method. 39 In the present work, however, no spurious oscillations were observed in the problems considered, and therefore, it did not prove necessary to employ any of the above stabilization techniques.…”

“…Without loss of generality, bilinear quadrilateral elements were used throughout this study. Since convergence is not always guaranteed with the Newton-Raphson procedure, 39 the stability of a solution was checked using the following Courant-Friedrichs-Lewy condition, as suggested by Zhu et al 12 :…”