Abstract
Saturation overshoot and pressure overshoot are studied by incorporating dynamic capillary pressure, capillary pressure hysteresis and hysteretic dynamic coefficient with a traditional fractional flow equation in one-dimensional space. Using the method of lines, the discretizations are constructed by applying the Castillo–Grone’s mimetic operators in the space direction and a semi-implicit integrator in the time direction. Convergence tests and conservation properties of the schemes are presented. Computed profiles capture both the saturation overshoot and pressure overshoot phenomena. Comparisons between numerical results and experiments illustrate the effectiveness and different features of the models.
Highlights
Water infiltrating into initially dry sandy porous media has been shown to produce saturation overshoot and pressure overshoot in Selker et al (1992), Shiozawa and Fujimaki (2004), DiCarlo (2004, 2007). Eliassi and Glass (2001) and Egorov et al (2003) have demonstrated that the traditional Richards equation is unable to describe saturation overshoot
Eliassi and Glass (2001) studied three additional forms referred to as hypodiffusive form, hyperbolic form and mixed form; saturation overshoot is obtained by using the hypodiffusive form in Eliassi and Glass (2003)
Chapwanya and Stockie (2010) studied gravity-driven fingering instabilities based on the work of Nieber (2003), and their results demonstrate that the non-equilibrium Richards equation is capable of reproducing realistic fingering flows for a wide range of physically relevant parameters
Summary
Water infiltrating into initially dry sandy porous media has been shown to produce saturation overshoot and pressure overshoot in Selker et al (1992), Shiozawa and Fujimaki (2004), DiCarlo (2004, 2007). Eliassi and Glass (2001) and Egorov et al (2003) have demonstrated that the traditional Richards equation is unable to describe saturation overshoot. Besides extensions to the Richards equation, other approaches to the characterization of the saturation overshoot have been proposed, such as the generalized theory by introducing percolating and non-percolating fluid phases into the traditional mathematical model (Hilfer and Besserer 2000; Hilfer et al 2012; Doster et al 2010), fractional flow approach (DiCarlo et al 2012) and moment analysis (Xiong et al 2012). The hysteretic non-equilibrium model proposed in Beliaev and Hassanizadeh (2001) postulates that the dynamic capillary effects are significant only outside the main hysteresis loop Following this idea, Nieber (2003) adopted a saturation- and pressure-dependent dynamic coefficient τ = τs0 Pw(s)( p0 − p)γ+. In Sander et al (2008), a reformulation of the non-equilibrium two-phase flow equation, which consists of an elliptic equation and an ODE, is shown to be effective for numerical simulations, Fan and Pop (2013) presented a reliable and efficient semi-implicit scheme for a similar form.
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