Fronts in two‐phase porous media flow problems: The effects of hysteresis and dynamic capillarity

Abstract

In this work, we study the behavior of saturation fronts for two-phase flow through a long homogeneous porous column . In particular, the model includes hysteresis and dynamic effects in the capillary pressure and hysteresis in the permeabilities. The analysis uses traveling wave approximation. Entropy solutions are derived for Riemann problems that are arising in this context. These solutions belong to a much broader class compared to the standard Oleinik solutions, where hysteresis and dynamic effects are neglected. The relevant cases are examined and the corresponding solutions are categorized. They include nonmonotone profiles, multiple shocks, and self-developing stable saturation plateaus. Numerical results are presented that illustrate the mathematical analysis. Finally, we discuss the implication of our findings in the context of available experimental results.

Capillary hysteresis and gravity segregation in two phase flow through porous media

“…This behaviour was mentioned briefly in Section 3.5 of [19]. A stationary discontinuity in saturation for the classical problem (with no hysteresis) was also predicted in [2] at the interface between two semi-infinite halves having different h and p c functions.…”

“…The parameter τ is kept small to approximate the results of the model (1.5) accurately. This also ensures that τ does not exceed the threshold required to cause effects such as saturation overshoot [19,21]. We solve (1.3) and (5.1) in a domain (−L, L) for a large L > 0.…”

“…However, there are other more realistic capillary pressure expressions for which the limiting (δ → 0) solution does inherit some properties of the vanishing capillarity. This was studied in detail in [19,31,33,35] for the case where Eq. 1.4 is replaced by the non-equilibrium expression p = p c (S) − τ ∂ t S, where τ > 0 is a relaxation parameter called dynamic capillarity coefficient which attributes to saturation overshoots [17,21,34].…”

“…Travelling wave analysis for hysteresis was conducted for flow of water through soil in [5,20,21,34]. For the two-phase case, travelling waves were studied in [19] for monotone flux functions and vanishing capillarity solutions were derived that differ significantly from the classical ones. These solutions were used to explain the observation of stable and growing saturation plateaus in infiltration experiments.…”

The Origin of Non-thermal Fluctuations in Multiphase Flow in Porous Media

“…Such traveling waves with non-monotonic saturation profiles are known to form under specific conditions for instance for infiltration of water into dry soil (Schneider et al, 2018;Kmec et al, 2019;Mitra et al, 2020) and can be theoretically described with a range of models ranging from generalized Darcy-scale formalisms with dynamic capillary pressure (Schneider et al, 2018) to hybrid models combining Darcy physics and pore scale aspects (Kmec et al, 2019). Mitra et al (2020) provides a mathematical model, where for specific choices of parameters, the solution forms spirals in the S w − p c space. The hysteresis model by Plohr et al (2001) demonstrates that for specific choices of parameter values "loop", i.e., recurring solutions can be found, which implies stable orbits in phase space (Strogatz, 1994).…”