Travelling wave solutions for the Richards equation incorporating non-equilibrium effects in the capillarity pressure

Hassanizadeh

2019

Vadose Zone Journal

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.

“…In fact (see van Duijn et al, 2018), we shall distinguish between two classes of possible τ( S ) functions:

\int_{0}^{S_{normalm}} normalτ (S) d S < \infty (Class A)

\int_{0}^{S_{normalm}} normalτ (S) d S < \infty (Class B)

\begin{matrix} left \\ Example: Let f (S) = {(S_{m} - S)}^{ω} for 0 < S < S_{normalm} with - \infty < ω \\ < + \infty . Then \end{matrix}

\int_{0}^{S_{normalm}} f (S) d S = {\begin{array}{l} left & \frac{1}{1 + ω} S_{normalm}^{1 + ω} < \infty & if ω > - 1 \\ left & \infty & if ω \leq - 1 \end{array}

where f is integrable (Class A) if ω > −1 (although f ( S ) has singular behavior as S → S m if −1 < ω < 0) and f is non‐integrable (Class B) if ω ≤ −1.…”

Section: Nondimensionalization Of Equationsmentioning

confidence: 99%

“…Let 0 < S B < S T < S m and distinguish the cases (van Duijn et al, 2018)

\int_{0}^{S_{normalm}} τ (S) d S < \infty (τ in Class A)

\int_{0}^{S_{normalm}} τ (S) d S < \infty (τ in Class B)

…”

Section: Traveling Wave Solutionmentioning

confidence: 99%

“… A. For a given τ( S ) in Class A, introduce the function (van Duijn et al, 2018)

β (S_{normalB}, S_{normalT}) = \int_{S_{normalB}}^{S_{normalm}} G (S; S_{normalB}, S_{normalT}) f (S) d S

and the saturation S β , defined by (van Duijn et al, 2018)

\int_{S_{normalB}}^{S_{normalβ}} G (S; S_{normalB}, S_{normalT}) f (S) d S = 0

The proportion of the function G implies that for each fixed S B ,

β (S_{normalB}, S_{normalT} = S_{normalB}) < 0

β (S_{normalB}, S_{normalT} = S_{normalM}) > 0

and β( S B , S T ) increases monotonically with S T . Hence there exists a unique value of S T = S T * , depending on S B , such that β( S B , S T ) = 0.…”

Section: Traveling Wave Solutionmentioning

confidence: 99%

See 1 more Smart Citation

The Effect of Dynamic Capillarity in Modeling Saturation Overshoot during Infiltration

Zhuang

Hassanizadeh

2019

Vadose Zone Journal

“…In Brokate et al (2012), authors use non-vertical approximations to vertical scanning curves as the original playtype hysteresis model poses difficulties for convergence. Furthermore, in Lamacz et al (2011) and in van Duijn et al (2018), the sign function has been regularized for mathematical analysis, making it resemble non-vertical scanning curves.…”

Section: Extended Playtype Hysteresis Modelmentioning

confidence: 99%

“…10) This type of splitting is well known in the mathematical literature, for instance see Schweizer (2012), van Duijn et al (2018). Let the time interval [0, T ] be divided into N intervals of width t (T = N t) and let w n be the variable w at t = n t, with 1 ≤ n ≤ N .…”

Section: Numerical Schemementioning

confidence: 99%

Hysteresis and Horizontal Redistribution in Porous Media

Mitra

2018

Transp Porous Med

Self Cite

It is well known that multiphase flow in porous media exhibits hysteretic behaviour. This is caused by different fluid-fluid behaviour if the flux reverses. For example, for flow of water in unsaturated soils the process of imbibition and drainage behaves differently. In this paper we study a new model for hysteresis that extends the current playtype hysteresis model in which the scanning curves between drainage and imbibition are vertical. In our approach the scanning curves are non-vertical and can be constructed to approximate experimentally observed scanning curves. Furthermore our approach does not require any book-keeping when the flux reverses at some point in space. Specifically, we consider the problem of horizontal redistribution to illustrate the strength of the new model. We show that all cases of redistribution can be handled, including the unconventional flow cases. For an infinite column, our analysis involves a self-similar transformation of the equations. We also present a numerical approach (L-scheme) for the partial differential equations in a finite domain to recover all redistribution cases of the infinite column provided time is not too large.

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

Mitra

2021

Comput Geosci

Self Cite

We study the gravity driven flow of two fluid phases in a one dimensional homogeneous porous column when history dependence of the pressure difference between the phases (capillary pressure) is taken into account. In the hyperbolic limit, solutions of such systems satisfy the Buckley-Leverett equation with a non-monotone flux function. However, solutions for the hysteretic case do not converge to the classical solutions in the hyperbolic limit in a wide range of situations. In particular, with Riemann data as initial condition, stationary shocks become possible in addition to classical components such as shocks, rarefaction waves and constant states. We derive an admissibility criterion for the stationary shocks and outline all admissible shocks. Depending on the capillary pressure functions, flux function and the Riemann data, two cases are identified a priori for which the solution consists of a stationary shock. In the first case, the shock remains at the point where the initial condition is discontinuous. In the second case, the solution is frozen in time in at least one semi-infinite half. The predictions are verified using numerical results.