The Effect of Dynamic Capillarity in Modeling Saturation Overshoot during Infiltration

Abstract

Core Ideas We provide detailed analysis of various forms of the dynamic capillarity coefficient. We focus on the region of saturation overshoot at larger saturation. The results are evaluated with the aid of experimental observations. Gravity‐driven fingering has been observed during downward infiltration of water into dry sand. Moreover, the water saturation profile within each finger is non‐monotonic, with a saturation overshoot at the finger tip. As reported in the literature, these effects can be simulated by an extended form of the Richards equation, where a dynamic capillarity term is included. The coefficient of proportionality is called the dynamic capillarity coefficient. The dynamic capillarity coefficient may depend on saturation. However, there is no consensus on the form of this dependence. We provide a detailed traveling wave analysis of four distinctly different functional forms of the dynamic capillarity coefficient. In some forms, the coefficient increases with increasing saturation, and in some forms, it decreases. For each form, we have found an explicit expression for the maximum value of saturation in the overshoot region. In current formulations of dynamic capillarity, if the value of the capillarity coefficient is large, the value of saturation in the overshoot region may exceed unity, which is obviously nonphysical. So, we have been able to ensure boundedness of saturation regardless of the value of the dynamic capillarity coefficient by extending the capillary pressure–saturation relationship. Finally, we provide a qualitative comparison of the results of traveling wave analysis with experimental observations.

Transient Two-Phase Flow in Porous Media: A Literature Review and Engineering Application in Geotechnics

“…They found the conventional model with hysteresis only performed well for the initial condition of very dry, while the interfacial area model can simulate for all experimental settings by fitting out the parameter in the interfacial production term in thermodynamic-based theory. Later, with the prior experimental findings on the log-linear relationship between saturation and the dynamic coefficient [47], Zhuang, van Duijn and Hassanizadeh [48] modelled saturation overshoot during the infiltration through dry sand in consideration of saturation-dependent dynamic coefficient. Not only was analytical analysis given in detail regarding parameter studies for saturation overshoot, but Zhuang, van Duijn and Hassanizadeh [48] also validated the thermodynamic-based theory with prior experimental works from DiCarlo [201] and Fritz [202] on primary imbibition exclusively.…”

“…Later, with the prior experimental findings on the log-linear relationship between saturation and the dynamic coefficient [47], Zhuang, van Duijn and Hassanizadeh [48] modelled saturation overshoot during the infiltration through dry sand in consideration of saturation-dependent dynamic coefficient. Not only was analytical analysis given in detail regarding parameter studies for saturation overshoot, but Zhuang, van Duijn and Hassanizadeh [48] also validated the thermodynamic-based theory with prior experimental works from DiCarlo [201] and Fritz [202] on primary imbibition exclusively. The most updated work from Zhuang, Hassanizadeh, et al [203] modelled spontaneous imbibition in the thin fibrous layer.…”

“…where τ = dynamic coefficient, S = saturation, t = time, and the rest notations were already given in Equations ( 47) and (48). The formulation of Equation ( 49), when χ = S e = (S − S r )/ (1 − S r ), was accepted for sandy soil in other hydro-mechanical coupling studies [253,254].…”

“…In order to simulate the transient two-phase flow in porous media under nonequilibrium conditions, several works developed a series of novel theories from various physical bases [22,[43][44][45]. By far, the investigations of dynamic effects have been still continually conducted using experimental [46], theoretical [47], and numerical methods [48] in the continuum scale. Diamantopoulos and Durner [49] comprehensively reviewed the so-defined dynamic nonequilibrium effects from a soil science and hydrology perspective, merely focusing on the air-water system in natural soil.…”

An efficient numerical scheme for fully coupled flow and reactive transport in variably saturated porous media including dynamic capillary effects

“…Typically, the capillary pressure is assumed to be a nonlinear decreasing function depending on the water saturation. However, numerous studies are showing that such formulation is often too simplistic and that dynamic effects, due to the changes in time of the water phase, should also be included [2,3,5,11,13]. Based on this, we consider here the system:…”