A New Class of Entropy Solutions of the Buckley–Leverett Equation

Abstract

C e n t r u m v o o r W i s k u n d e e n I n f o r m a t i c a MAS Modelling, Analysis and Simulation Modelling, Analysis and SimulationA new class of entropy solutions of the BuckleyLeverett equation ABSTRACT We discuss an extension of the Buckley-Leverett (BL) equation describing two-phase flow in porous media. This extension includes a third order mixed derivatives term and models the dynamic effects in the pressure difference between the two phases. We derive existence conditions for travelling wave solutions of the extended model. This leads to admissible shocks for the original BL equation, which violate the Oleinik entropy condition and are therefore called nonclassical. In this way we obtain non-monotone weak solutions of the BL problem consisting of steady states separated by shocks, confirming results obtained experimentally.

Non-monotonic Travelling Wave Fronts in a System of Fractional Flow Equations from Porous Media

“…Theories of hyperbolic conservation laws treat the boundary condition on one side as a fixed parameter, the boundary condition on the other side as unknown, and the wave velocity c is derived from the Rankine-Hugoniot condition (Duijn et al 2007(Duijn et al , 2013LeVeque 2004;Lax 2006;LeFloch 2002;Cuesta et al 2000;Schaeffer and Shearer 1987). In contrast to this, the present paper treats both boundary resp.…”

“…The traditional method used in the hyperbolic community Duijn et al (2007Duijn et al ( , 2013, LeVeque (2004), LeFloch (2002), Cuesta et al (2000) and Schaeffer and Shearer (1987) also considers the set F, but there the velocity is an output of the Rankine-Hugoniot condition c(u l , v l ; u r , v r ). Traditionally, one starts by fixing one boundary resp.…”

“…3 in a clear and concise way. Furthermore this new approach is compared to the traditional method used, for example, in Duijn et al (2007Duijn et al ( , 2013, LeVeque (2004), LeFloch (2002), Cuesta et al (2000) and Schaeffer and Shearer (1987).…”

“…A strong motivation for these studies were mathematical results excluding the existence of overshoot profiles for the traditional Richards equation from hydrology (Nieber 1996;Otto 1996Otto , 1997Geiger and Durnford 2000;Glass 2001, 2002;Egorov et al 2003;DiCarlo 2005DiCarlo , 2013Duijn et al 2007;Juanes 2008, 2009). It has been conjectured that new theoretical approaches might be unavoidable (Eliassi and Glass 2002;Duijn et al 2007Duijn et al , 2013Juanes 2008, 2009;Hilfer et al 2012;Nieber et al 2005) to reconcile theory and experiment. On the other hand, a recent investigation (Hilfer and Steinle 2014) has uncovered the existence of saturation overshoot profiles within the traditional theory in the hyperbolic limit, while the Richards equation represents a special parabolic limit.…”

Hysteresis models and gravity fingering in porous media

“…For γ = 0, it introduces a term with first time derivative and second spatial derivatives, thus turning the Richards equation into a pseudo-parabolic equation. Even in the limit of a vanishing influence of the capillary pressure, it may change the profile of travelling wave solutions, see [17]. It is well-known that a positive τ can lead to non-monotonic profiles in wetting fingers, [5].…”