# A Numerical Study of Two-Phase Flow Models with Dynamic Capillary Pressure and Hysteresis

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## 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.

## Keywords

Castillo–Grone’s mimetic operators Saturation overshoot Pressure overshoot Dynamic capillary pressure Play-type hysteresis## 1 Introduction

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. To describe the non-monotonic behaviour, various extensions to the Richards equation have been investigated. 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). DiCarlo et al. (2008) studied a non-monotonic capillary pressure–saturation relationship and a second-order hyperbolic term, but they mentioned these extensions need a regularization term to produce a unique solution. Cueto-Felgueroso and Juanes (2009) explained the formation of gravity fingers during water infiltration in soil by introducing a fourth-order term to Richards equation. The tip and tail saturations after their phase model have good agreement with the experiments in DiCarlo (2004). Nieber (2003) obtained non-monotonic saturation profiles by supplementing the Richards equation with a non-equilibrium capillary pressure–saturation relationship, as well as including hysteretic effects. Later, 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.

Besides extensions to the Richards equation, other approaches to the characterization of the saturation overshoot have also 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).

Among all the proposed theories and models, the dynamic (or non-equilibrium) capillary pressure relationship proposed by Stauffer (1978), Hassanizadeh and Gray (1993) and Kalaydjian et al. (1992) has received much attention. Results on travelling wave solutions, global existence, phase plane analysis and uniqueness of weak solutions are given in Cuesta et al. (2000), Van Duijn et al. (2007, 2013), Mikelić (2010), Spayd and Shearer (2011) and Cao and Pop (2015). In order to provide accurate simulations, several numerical methods have been proposed in the literature, including a finite difference method with minmod slope limiter in Van Duijn et al. (2007), a cell-centred finite difference method and a locally conservative Eulerian–Lagrangian method in Peszynska and Yi (2008), Godunov-type staggered central schemes in Wang and Kao (2013), two semi-implicit schemes based on equivalent reformulations in Fan and Pop (2013), an adaptive moving mesh method in Zegeling (2015) and the fast explicit operator splitting method in Kao et al. (2015).

In the previous studies of saturation overshoot, the dynamic capillary pressure model with a constant dynamic coefficient usually brought oscillations behind the drainage front (DiCarlo 2005; Sander et al. 2008; van Duijn et al. 2013). 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 \(\tau = \tau _s^0 P'_w(s) (p_0 - p)_+^\gamma \). In this treatment, the Mualem hysteresis model was restricted only to the two-stage wetting–drainage process: trajectories for the wetting stage were located within \(H_w\) (domain above the main wetting curve), while trajectories for the drainage stage were limited to \(H_0\) (main hysteresis loop region). In the wetting stage, \(\tau _s^0 = \tau _w\), and in the drainage stage, \(\tau _s^0 = 0\) (in the numerical simulation, the value was a small constant \(=10^{-3}\)). Recently, the hysteretic dynamic capillary pressure effect has been reported in Sakaki et al. (2010). Mirzaei and Das (2013) show that the dynamic effect in the relationship between capillary pressure and saturation is hysteretic in nature. In this contribution, we consider the hysteresis effects in capillary pressure and study the saturation overshoot and pressure overshoot by adding dynamic capillary pressure, capillary pressure hysteresis and hysteretic dynamic coefficient to a traditional fractional flow equation.

Two-phase flow models in porous media usually consist of coupled, nonlinear partial differential equations. Many reliable discretizations have been proposed for the Richards equation, for instance, the Galerkin finite elements in Arbogast et al. (1993), the multipoint flux approximation in Klausen et al. (2008). The space and time discretizations usually lead to a large system of nonlinear equations, which makes the numerical simulation of two-phase flow a challenging task. Some popular methods used for the Richards equation are the Newton method (Radu et al. 2006), Picard method (Zarba et al. 1990) or L-scheme (Radu et al. 2015); for an extensive review, we refer to List and Radu (2016). The appearance of the dynamic capillary pressure term in the two-phase flow model adds additional difficulty to the numerical treatment. 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) also presented a reliable and efficient semi-implicit scheme for a similar form. In this paper, we present our schemes based on this reformulation. Because of the conservation property and easy implementation of the Castillo–Grone’s mimetic (CGM) operators, we apply the mimetic finite difference method to the elliptic equation in the space direction and then integrate the system in the time direction with the implicit trapezoidal rule.

The rest of the paper is organized as follows. In Sect. 2, we first derive the traditional equation for one-dimensional two-phase flow, and then we present the extended models in Sects. 2.1, 2.2 and 2.3 by incorporating the dynamic capillary pressure term and hysteresis effects. Section 3.1 is devoted to presenting the CGM operators. In Sect. 3.2, we apply the CGM operators in the space direction and an implicit trapezoidal integrator in the time direction to discretize the system. In Sect. 4, numerical experiments are carried out to show the effectiveness and reliability of the extended models. Section 5 summarizes the conclusions.

## 2 Mathematical Models

*x*-direction; for each phase, the mass conservation law is represented by the equation

*K*is the intrinsic permeability of the porous medium,

*g*is the gravitational acceleration constant, \(k_{r \alpha }, \mu _\alpha , \lambda _\alpha = \frac{k_{r \alpha } K}{\mu _\alpha }\) and \(p_\alpha \) are the relative permeability function, viscosity, mobility and pressure of phase \(\alpha \), respectively.

*q*is the flux value used in DiCarlo (2004, 2007) and \(S_{w}^{\mathrm{R}}\) is the initial water saturation at \(x_{\mathrm{R}}\).

### 2.1 Dynamic Capillary Pressure Model

### 2.2 Play-Type Capillary Pressure Hysteresis Model

- 1.
Set \(p_c^{n}= p_c^{n-1} - \beta (S_w^n - S_w^{n-1})\).

- 2.
If \(p_c^{n} < P_{c}^{\mathrm{im}}(S_w^n)\), set \(p_c^{n} = P_c^{\mathrm{im}}(S_w^n)\).

- 3.
If \(p_c^{n} > P_c^{\mathrm{dr}}(S_w^n)\), set \(p_c^{n} = P_c^{\mathrm{dr}}(S_w^n)\).

### 2.3 Hysteretic Dynamic Capillary Pressure Model with Play-Type Capillary Pressure Hysteresis

Sakaki et al. (2010) conducted a series of experiments to measure values of the dynamic capillary coefficient \(\tau \) for a porous medium. Their result suggests that \(\tau \) is hysteretic. Mirzaei and Das (2013) investigated the hysteretic behaviour between \(\tau \) and \(S_w\). The experiments demonstrate that the value of \(\tau \) for imbibition is generally larger as compared to the \(\tau \) value for drainage at the same saturation. Thus, it is reasonable to introduce hysteresis in the \(\tau \)–\(S_w\) relationship. We assume in the imbibition process \(\tau = \tau ^{\mathrm{im}}\); in the hysteresis process, \(\tau \) decreases from \(\tau ^{\mathrm{im}}\) to \(\tau ^{\mathrm{dr}}\); and at the tail, the dynamic coefficient is \(\tau ^{\mathrm{dr}}\). To the best of our knowledge, the ratio \({\tau ^{\mathrm{dr}}}{/}{\tau ^{\mathrm{im}}}\) has not been investigated in the literature. In this work, we follow Van Duijn et al. (2007) to show the influence of \(\tau \) by using the travelling ansatz.

*s*determined by the Rankine–Hugoniot condition

From the analysis above, we can conclude that when \(\tau < \tau _s\), saturation oscillation will not appear at the drainage front, which means \(\partial S_w/ \partial t \rightarrow 0^-\). Therefore, the phase pressure difference \(p_n - p_w\) will tend to \(P_c^{\mathrm{dr}}\) at equilibrium.

## 3 Numerical Scheme

### 3.1 Castillo–Grone’s Mimetic Operators

To discretize Eq. (30) in the space direction, we adopt the mimetic finite difference method which satisfies the discrete version of continuum conservation law. Castillo and Grone (2003) developed a set of mimetic operators knows as Castillo–Grone’s mimetic (CGM) operators. CGM operators have been used in many fields, such as seismic studies (Rojas et al. 2008), electrodynamics (Runyan 2011) and image processing (Bazan et al. 2011). Numerical results in these fields validate the high efficiency and reliability of the CGM operators.

The main features of CGM operators are that they preserve symmetry properties of the continuum and have overall high-order accuracy. The CGM operators can be implemented as efficient as the standard finite difference schemes. Here, we briefly describe the CGM operators in one dimension as applied in this work. The CGM 2-D operators can be obtained by the Kronecker products of block matrices. For more details, see Castillo and Miranda (2013) and references therein.

*f*and

*v*are two smooth real-valued functions defined in interval \(\varOmega = [x_{\mathrm{L}}, x_{\mathrm{R}}]\). Let \(L = x_{\mathrm{R}} - x_{\mathrm{L}}\), and the step size \(\Delta x = L{/}N\), then \(\varOmega \) can be partitioned into

*N*equal-sized cells \([x_i, x_{i+1}]\), where \(0 \le i \le N-1\). The cell centres can be indexed as \(x_{i+1/2} = \frac{1}{2} (x_i + x_{i+1})\) for \(0 \le i \le N-1\). The cell nodes \(x_i\) and cell centres \(x_{i+1/2}\) build up the uniform staggered grid. We discretize

*v*at the cell nodes:

*f*at the cell centres and the boundary nodes:

*I*is the \((N+2) \times (N+2)\) identity matrix, and

*Q*and

*P*are the weight matrices for \(\hat{D}\) and

*G*, respectively. The matrix \(\hat{B} = Q \hat{D} + G^\mathrm{T} P\) embodies the global conservation requirement for the discrete conservation law and is called the boundary operator. Castillo and Yasuda (2005) presented the second-order divergence mimetic operator as

*Q*is the \((N+2) \times (N+2) \) identity matrix. The second-order CGM gradient operator reads as

*D*and

*G*, the boundary operator \(\hat{B}\) can be written as

### 3.2 Mimetic Discretizations for the Two-Phase Flow Equations

*p*at the centres as \(\bar{S}_w = [S_{w0}, S_{w\frac{1}{2}}, \ldots , S_{w\frac{N-1}{2}}, S_{wN}]\) and \(\bar{p} = [p_{0}, p_{\frac{1}{2}}, \ldots , p_{\frac{N-1}{2}}, p_{N}]\). Then, we use linear interpolation to get \(S_w\) at the nodes

*A*,

*B*and vector \(b(\bar{S}_w^{n+1})\) represent the boundary conditions. As a result of the flux boundary condition at \(x_{\mathrm{L}}\) and the Dirichlet boundary condition at \(x_{\mathrm{R}}\), the only nonzero element in

*A*is \(A_{N+2,N+2} = 1\), and

*B*differs from \(\hat{B}\) in the last three rows,

*l*as an iteration counter, the algorithm for each time step is as follows:

- 1.
Set \(\bar{S}_w^{n+1,0} = \bar{S}_w^n, \bar{p}^{n+1,0} = \bar{p}^n, P_c(\bar{S}_w^{n+1, 0}) = P_c(\bar{S}_w^{n})\) [or \(P_c^{\mathrm{hyst}}(\bar{S}_w^{n+1,0})=P_c^{\mathrm{hyst}}(\bar{S}_w^n\))], \(l = 0\).

- 2.
Update \(\bar{S}_w^{n+1,l+1} = \bar{S}_w^n + \frac{\Delta t }{2} \left[ \frac{P_c(\bar{S}_w^n) - p^n)}{\tau } + \frac{P_c(\bar{S}_w^{n+1,l}) - \bar{p}^{n+1,l}}{\tau }\right] \), solve (42) for \(\bar{p}^{n+1,l+1}\), and update \(P_c(\bar{S}_w^{n+1,l+1})\) [or \(P_c^{\mathrm{hyst}}(\bar{S}_w^{n+1,l+1})\)].

- 3.
\(l = l+1\).

- 4.
Repeat steps 2 and 3 until \(|\bar{S}_w^{n+1,l+1} - \bar{S}_w^{n+1, l}| < \mathrm {tol}\).

- 5.
Set \(\bar{S}_w^{n+1} = \bar{S}_w^{n+1,l+1}, p^{n+1} = p^{n+1,l+1}\).

### Remark

The application of MFD to partial differential equations constitutes an active filed of research (Lipnikov et al. 2014). Formal analysis of MFD for the Richards equation can be achieved by combining the convergence results in Brezzi et al. (2005), with the equivalence between MFD and multipoint flux approximation (MPFA) established in Stephansen (2012) and the convergence proof of MPFA for Richards equation in Klausen et al. (2008). However, difficulties arise when establishing convergence for (30), because of the nonlinearity and the reformulation. In Sect. 3, we present numerical results to demonstrate the convergence of the method.

## 4 Numerical Experiments

Physical parameters for 20 / 30 sand

Sand | \(\kappa ~\mathrm {[m~s^{-1}]}\) | \(\phi \) (–) | Drainage | Imbibition | ||||
---|---|---|---|---|---|---|---|---|

\(S_{wr}\) (–) | \(\lambda \) (–) | \(p_d~\text {(Pa)}\) | \(S_{wr}\) (–) | \(\lambda \) (–) | \(p_d ~\text {(Pa)}\) | |||

20 / 30 | \(2.5\text {e}{-}03\) | 0.35 | 0 | 5.57 | 850 | 0 | 5 | 490 |

Constants and Brooks–Corey models

Density \(\mathrm {(kg~m^{-3})}\) | \(\rho _w = 998.21\) | \(\rho _n = 1.2754\) |

Viscosity \(\mathrm {(kg~m^{-1} s^{-1})}\) | \(\mu _w = 1.002\text {e}{-}03\) | \(\mu _n = 1.82\text {e}{-}05\) |

Mobility \(\mathrm {(m~s~kg^{-1})}\) | \(\lambda _w = \frac{K k_{rw}}{\mu _w}\) | \(\lambda _n = \frac{K k_{rn}}{\mu _n}\) |

Constants | \(g = 9.81 ~\mathrm {(m~s^{-2})}\) | \(K = \frac{\kappa \mu _w}{\rho _w g}~\mathrm {(m^2)}\) |

Capillary pressure | Relative permeability | |
---|---|---|

Brooks–Corey model | \(S_e = \frac{S_w - S_{wr}}{1 - S_{wr}}\) | \(k_{rw} = S_e^{\frac{2+ 3 \lambda }{\lambda }}\) |

\(p_c = p_d S_e^{-\frac{1}{\lambda }},~~\mathrm {for}~p_c > p_d\) | \(k_{rn} = (1-S_e)^2 \big (1-S_e^{\frac{2+\lambda }{\lambda }}\big )\) |

| Model 1 | Model 2 | Model 3 | |||
---|---|---|---|---|---|---|

\(L^{2}\) error | \(L^2\) order | \(L^2\) error | \(L^2\) order | \(L^2\) error | \(L^2\) order | |

64 | 4.1726e−04 | – | 3.6657e−04 | – | 3.6661e−04 | – |

128 | 1.0224e−04 | 2.0289 | 7.9465−05 | 2.2057 | 7.9475e−05 | 2.2057 |

256 | 2.4289e−05 | 2.0736 | 1.7996e−05 | 2.1426 | 1.7998e−05 | 2.1427 |

512 | 4.8183e−06 | 2.3337 | 4.0298e−06 | 2.1589 | 4.0300e−06 | 2.1590 |

\(\Delta t\) | Model 1 | Model 2 | Model 3 | |||
---|---|---|---|---|---|---|

\(L^{2}\) error | \(L^2\) order | \(L^{2}\) error | \(L^2\) order | \(L^2\) error | \(L^2\) order | |

0.016 | 1.0078e−08 | – | 5.5162e−05 | – | 6.0156e−05 | – |

0.008 | 2.5009e−09 | 2.0107 | 2.5972e−05 | 1.0867 | 2.8378e−05 | 1.0839 |

0.004 | 5.9543e−10 | 2.0704 | 1.1182e−05 | 1.2158 | 1.2230e−05 | 1.2144 |

0.002 | 1.1908e−11 | 2.3219 | 3.7358e−06 | 1.5816 | 4.0881e−06 | 1.5809 |

Setting \(S_w^{\mathrm{L}} = 0.025, S_w^{\mathrm{R}} = 0.003, q = 1.32\text {e}{-}04, \tau = 4.0\text {e03}\), the saturation and pressure profiles obtained by Models 1, 2 and 3 are presented in Fig. 2. The imbibition fronts obtained by the three models are similar, but the saturations and pressures at the plateaus and behind drainage fronts are different. For Model 1, the value of the plateau saturation is constant and behind the drainage front, oscillations appear and the phases pressure difference follows the imbibition capillary pressure. For Model 2, the plateau saturations decrease a little from the imbibition front to the drainage front and behind the drainage front, there is a slight oscillation in the saturation profile and the phase pressure difference moves to the imbibition capillary pressure. For Model 3, because of the hysteresis in the capillary pressure and the dynamic coefficient, no oscillation appears in the saturation profile. As a result, the pressure keeps constant and follows the drainage capillary pressure. The phases pressure difference–saturation curves are presented in Fig. 2d. The result obtained by Model 3 shows similar behaviour as the measured data in Fig. 6 in DiCarlo (2007).

*q*to the three models, we have to know the end times for the simulations. In DiCarlo (2004), the end times for the experiments are not given, but the inner diameter of the tube is given as \(d = 1.27\text {e}{-}02 (\mathrm{m})\), then the total volume of water injected into each tube can be calculated by \(\hbox {Volume} = \phi \pi \big (\frac{d}{2}\big )^2 \sum _{k = 0}^{N-1} (x_{k+1} - x_{k}) \frac{S_w(x_k) + S_w(x_{k+1})}{2}\), where \(x_k\) is the sample point in DiCarlo (2004). Thus, we can calculate the end times using \(T_{\mathrm{end}} = \frac{\mathrm{Volume}}{q \pi (d/2)^2} \text {(s)}\).

Parameters for different fluxes

Volume of water \(\mathrm {(m^{-3})} \) | \(1.26\text {e}{-}05\) | \(1.17\text {e}{-}05\) | \(6.85\text {e}{-}06\) | \(4.56\text {e}{-}06\) | \(2.06\text {e}{-}06\) | \(2.18\text {e}{-}06\) |

\(q \mathrm {~(m~s^{-1})}\) | \(2.0\text {e}{-}03\) | \(1.32\text {e}{-}03\) | \(1.32\text {e}{-}04\) | \(1.32\text {e}{-}05\) | \(1.32\text {e}{-}06\) | \(1.32\text {e}{-}07\) |

\(\tau (\tau ^{\mathrm{im}})\) (Pa s) | 40 | 1.0e02 | 4.0e03 | 9.0e04 | 4.0e05 | 1.0e06 |

\(\tau ^{\mathrm{dr}}\) (Pa s) | 8 | 20 | 800 | 1.8e04 | \(8.0\text {e04}\) | \(2.0\text {e05}\) |

\(\tau _s\) | – | 163.4 | 328.5 | 4991 | \(6.483\text {e04}\) | \(6.083\text {e05}\) |

\(\Delta t\) (s) | 7.3e−04 | 1.0e−03 | 5.6e−03 | 3.2e−02 | 1.8e−01 | 1 |

\(T_{\mathrm{end}} \text {~(s)}\) \(\big (\frac{\mathrm{Volume}}{q \pi (d/2)^2}\big )\) | 49.7 | 70.0 | 409.7 | \(2.727\text {e03}\) | \(1.2320\text {e04}\) | \(1.3037\text {e05}\) |

\(T_{\mathrm{end}} \text {~(s)}\) (Model 1) | 54.0 | 216.0 | 425.0 | \(2.380\text {e03}\) | \(9.000\text {e03}\) | \(3.120\text {e04}\) |

\(T_{\mathrm{end}} \text {~(s)}\) (Model 2) | 54.0 | 216.0 | 460.0 | \(2.660\text {e03}\) | \(1.000\text {e04}\) | \(3.120\text {e04}\) |

\(T_{\mathrm{end}} \text {~(s)}\) (Model 3) | 54.0 | 216.0 | 460.0 | \(2.660\text {e03}\) | \(1.000\text {e04}\) | \(3.120\text {e04}\) |

*q*using a log–log diagram. Realizing the near log–log relationship between \(\tau \) and

*q*, we set \(\tau = 40\) for \(q = 2.0\text {e}{-}03, \tau = 2.0\text {e06}\) for \(q = 1.32\text {e}{-}07\). The number of nodes used in space is \(N = 256\), and the values of \(\tau (\tau ^{\mathrm{im}}), \tau ^{\mathrm{dr}}, \tau _s\), time steps as well as the end times are presented in Table 5. As can be seen, the end times for simulations are near to the calculated times except when \(q = 1.32\text {e}{-}07\), we will explain this later. For all three models, when the flux

*q*is \(1.32\text {e}{-}03\), the imbibition front moves quickly, while the change in hysteretic capillary pressure is slow. In order to show the overshoot saturation phenomenon, we have to enlarge the interval to [0, 1]. In Fig. 5, we compare the numerical solutions with the experiments.

Figure 5a–c shows that all three models can obtain saturation overshoots for \(q = 1.32\text {e}{-}06, q = 1.32\text {e}{-}05, q = 1.32\text {e}{-}04, q = 1.32\text {e}{-}03\). From Fig. 5a, we can see oscillations behind the drainage fronts when \(q = 1.32\text {e}{-}06, q = 1.32\text {e}{-}05\) and \(q = 1.32\text {e}{-}04\), while Fig. 5b only presents weaker oscillations for \(q = 1.32\text {e}{-}04\) and \(1.32\text {e}{-}05\) and there is no oscillation behind the drainage fronts in Fig. 5c. Although Table 5 shows that for \(q = 1.32\text {e}{-}04, 1.32\text {e}{-}05\) and \(.32\text {e}{-}06\), the \(\tau ^{\mathrm{dr}}\) values are slightly larger than \(\tau _s\), no oscillation appears behind the drainage fronts. This may be caused by the hysteresis in the capillary pressure, because in Fig. 5b the oscillations are weaker than Fig. 5a even without hysteresis in dynamic capillary coefficient.

In Fig. 5d, we plot the relationship between phases pressure difference and saturation obtained by Model 3 for all fluxes. At the imbibition front, the phases pressure difference is smaller than the equilibrium imbibition pressure. Behind the front, the pressure–saturation follows the equilibrium drainage pressure and finally stops at the tail saturation. This figure shows similar pressure–saturation behaviour as the experiments presented in Fig. 6 in DiCarlo (2007). As can be seen, at the lowest flux, significant pressure overshoot can still be obtained by Model 3. This phenomenon was also observed in the experiment in DiCarlo (2007). Selker et al. (1992) shows that saturation overshoot is associated with pressure overshoot. Thus, we guess even that at low flux, Model 3 can still produce saturation overshoot. Let the space interval be [0, 1] and \(T_{\mathrm{end}} = 96{,}000\), and the saturation and pressure profiles computed by Model 3 are presented in Fig. 5f. It shows that at \(q = 1.32\text {e}{-}07\), very small saturation overshoot appears.

Saturations at \(x = 0\) obtained by Eq. (48) and numerical simulations

\(q \mathrm {(m s^{-1})}\) | Brooks–Corey model | ||||
---|---|---|---|---|---|

\(\theta \) (experiment) | \(\theta \) (analytical) | \(\theta \) (Model 1) | \(\theta \) (Model 2) | \(\theta \) (Model 3) | |

2.0e−03 | 0.3500 | 0.3279 | 0.3289 | 0.3491 | 0.3494 |

1.32e−03 | 0.2665 | 0.2902 | 0.2902 | 0.2902 | 0.2902 |

1.32e−04 | 0.1250 | 0.1474 | 0.1474 | 0.1421 | 0.1474 |

1.32e−05 | 0.0790 | 0.0749 | 0.0749 | 0.0777 | 0.0749 |

1.32e−06 | 0.0500 | 0.0380 | 0.0380 | 0.0377 | 0.0380 |

1.32e−07 | 0.0450 | 0.0193 | 0.0193 | 0.0211 | 0.0198 |

Since Fig. 5c shows more realistic profiles than Fig. 5a, b, we will focus on this model and apply more fluxes to test its effectiveness. In Fig. 6a, we plot \(\tau , T_{\mathrm{end}}, \Delta t\) and *q* used in Fig. 5 with solid triangles. Then, we use logarithmic interpolation to get the values of \(\tau , T_{\mathrm{end}}\) and \(\Delta t\) for intermediate fluxes, these parameters are plotted using open triangles. Figure 6b plots the values of \(\tau \) as functions of tip and tail saturations. For both imbibition and drainage, the values of \(\tau \) increase as water saturations decrease. This trend seems to agree with the measured data in Manthey et al. (2005), Das and Mirzaei (2012) and Mirzaei and Das (2013).

In Fig. 6c, the computed tip and tail saturations from Model 3 are compared with the measured data in DiCarlo (2004). As the flux increases, the tip saturation increases very fast for flux value in interval \([1.0\text {e}{-}06, 1.0\text {e}{-}04]\), while the tip saturation increases slowly when flux *q* is above \(1.32\text {e}{-}04\). For tail saturations, both the experimental data and computed results follow the analytical curve given by Eq. (48) when flux is bigger than \(1.0\text {e}{-}06\).

Figure 6d plots the tip length versus flux obtained by Model 3. For flux values between \(1.0\text {e}{-}05\) and \(1.0\text {e}{-}03\), the tip length increases monotonically with the flux. This trend matches Fig. 12 in DiCarlo (2004).

Figure 6e presents the phases pressure differences at the imbibition front and the tail obtained by Model 3. For intermediate fluxes, the phases pressure differences at the tail follow the equilibrium drainage capillary pressure, while the phases pressure differences at the imbibition front are below the equilibrium imbibition capillary pressure as a result of the dynamic capillary pressure effect.

## 5 Conclusion

In this study, we applied the Castillo–Grone’s mimetic operators and the implicit trapezoidal rule to solve two-phase flow models including dynamic capillary pressure (with constant and hysteretic coefficient) and capillary pressure hysteresis in porous media. Numerical simulations show that the second-order mimetic operators mimic the Green–Gauss–Stokes theorem with high accuracy for all three models. The hysteretic dynamic capillary pressure model with capillary pressure hysteresis produces realistic saturation overshoot and pressure overshoot phenomena as observed in DiCarlo (2004, 2007).

## Notes

### Acknowledgements

The research of H. Zhang was funded by the China Scholarship Council (No. 201503170430). We thank Prof. S. Majid Hassanizadeh for useful discussions. Moreover, the valuable suggestions and comments from the anonymous reviewers are highly appreciated.

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