Model adaptation for non-linear elliptic equations in mixed form: existence of solutions and numerical strategies

Abstract

Depending on the physical and geometrical properties of a given porous medium, fluid flow can behave differently, going from a slow Darcian regime to more complicated Brinkman or even Forchheimer regimes for high velocity. The main problem is to determine where in the medium one regime is more adequate than others. In order to determine the low-speed and high-speed regions, this work proposes an adaptive strategy which is based on selecting the appropriate constitutive law linking velocity and pressure according to a threshold criterion on the magnitude of the fluid velocity itself. Both theoretical and numerical aspects are considered and investigated, showing the potentiality of the proposed approach. From the analytical viewpoint, we show existence of weak solutions to such model under reasonable hypotheses on the constitutive laws. To this end, we use a variational approach identifying solutions with minimizers of an underlying energy functional. From the numerical viewpoint, we propose a one-dimensional algorithm which tracks the transition zone between the low- and high-speed regions. By running numerical experiments using this algorithm, we illustrate some interesting behaviors of our adaptive model on academic cases and on small networks of intersecting fractures.

Well-posedness and variational numerical scheme for an adaptive model in highly heterogeneous porous media

“…(2.3) Additional theoretical arguments require that η be monotonously nondecreasing and coercive in its second variable [13,14], which intuitively implies that the viscosity perceived by the fluid cannot decrease with its speed.…”

“…To locate the linear and nonlinear regions, we propose to use the adaptive model introduced in [13,14]. There, the authors derive a model where the constitutive law linking velocity and pressure gradient switches from linear to nonlinear according to a fixed threshold on the seepage-flux magnitude: when the magnitude is below the threshold, the linear law is solved, and when it is above, the nonlinear law is solved instead.…”

“…As a result, a law which naturally adapts itself given the flux speed is obtained. Although well-posedness of the adaptive model is proved in [13,14], the adaptive law is discontinuous in speed, leading to a jump of pressure gradient at the transition zones between the two laws. Consequently, the adaptive model is rather unsuited for numerical simulation.…”

“…The paper is organized as follows: in Section 2, we summarize the various nonlinear laws considered by our model, we derive an expression for the flux threshold depending on the prescribed tolerance and we recall the adaptive and regularized models of [13,14]; in Section 3, we show the validity of the approach by experimenting on four different test cases in two and three space dimensions, the first two cases being inspired by an application in landfill management [32] and the remaining two cases being taken from the SPE10 benchmark reservoir scenario [8]. The codes used to obtain our results are written in Python and can be found at [12], while the underlying discretization and equation resolution are performed using the open source Python simulation tool PorePy [1,17].…”

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An adaptive model for the description of flows in highly heterogeneous porous media is developed in [13,14]. There, depending on the magnitude of the fluid's velocity, the constitutive law linking velocity and pressure gradient is selected between two possible options, one better adapted to slow motion and the other to fast motion.

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