# Traveling Wave Solutions in a Generalized Theory for Macroscopic Capillarity

- 929 Downloads
- 2 Citations

## Abstract

One-dimensional traveling wave solutions for imbibition processes into a homogeneous porous medium are found within a recent generalized theory of macroscopic capillarity. The generalized theory is based on the hydrodynamic differences between percolating and nonpercolating fluid parts. The traveling wave solutions are obtained using a dynamical systems approach. An exhaustive study of all smooth traveling wave solutions for primary and secondary imbibition processes is reported here. It is made possible by introducing two novel methods of reduced graphical representation. In the first method the integration constant of the dynamical system is related graphically to the boundary data and the wave velocity. In the second representation the wave velocity is plotted as a function of the boundary data. Each of these two graphical representations provides an exhaustive overview over all one-dimensional and smooth solutions of traveling wave type, that can arise in primary and secondary imbibition. Analogous representations are possible for other systems, solution classes, and processes.

## Keywords

Traveling waves Capillarity Multiphase flow Porous media## 1 Introduction

Almost all accepted and applied theories of multiphase flow in porous media are based on generalized Darcy laws and the concurrent concept of relative permeabilities (Wyckoff and Botset 1936). Despite the fact that (Wyckoff and Botset 1936) strongly emphasized the variation of hydraulically disconnected fluid regions (Jamin 1860), almost all subsequent applications of the relative permeability concept treat the residual nonwetting (or irreducible wetting) saturations as material constants (Bear 1972; Collins 1961; Dullien 1992; Helmig 1997; Wiest 1969; Scheidegger 1957; Marsily 1986). Modern theories of multiphase flow in porous media often resort to microscopic models (e.g., network models) (Bear et al. 1987; Bryant and Blunt 1992; Dias and Payatakes 1986; Dijke and Sorbie 2002; Ferer et al. 2004) in an attempt to derive or estimate macroscopic relative permeabilities from pore scale parameters. It was emphasized in Wyckoff and Botset (1936), however, that the possibility of determining the overall dynamical behavior of nonhomogeneous fluids from a study of microscopic details is remote (Wyckoff and Botset 1936, p. 326).

Experimentally, the volume fraction of stationary, locked, trapped, or nonpercolating fluid phases varies strongly with time and position (Abrams 1975; Avraam and Payatakes 1995; Taber 1969; Wyckoff and Botset 1936). Dispersed droplets, bubbles, or ganglia of one fluid phase obstruct the motion of the other fluid phase. Extensive experimental and theoretical studies of this simple phenomenon exist (Avraam and Payatakes 1999; Jamin 1860). Theoretically, the fundamental difference between trapped percolating and mobile nonpercolating fluid parts was first introduced in Hilfer (1998, 2006a, b, c) and the resulting mathematical model was partially explored further in Hilfer and Doster (2010), Doster and Hilfer (2011), Doster (2011), Doster et al. (2010, 2012).

The objective of this paper is twofold: firstly, the paper finds traveling wave solutions for the model introduced in Hilfer (2006a, b, c). While some approximate analytical and numerical solutions of the theory have been found in Hilfer (2006a, b, c), Hilfer and Doster (2010), Doster et al. (2010), Doster (2011), Doster and Hilfer (2011); Doster et al. (2012) the existence of traveling waves for this system has remained an open question and is analyzed here for the first time. This paper reports the existence of traveling waves for the generalized theory under the assumption that the nonpercolating phases are immobile. The second objective of this paper is to introduce two novel graphical representations of the solution space, that give a complete and exhaustive overview of all traveling wave solutions, that are infinitely often differentiable. Both representations are based on transforming the boundary value problem for the nonlinear partial differential equations to a dynamical system. The first representation is based on the integration constant of the dynamical system related to the boundary data and the wave velocity. The second representation is based on the wave velocity as a function of boundary data.

A word on notation: Throughout the paper, variables with a hat \(\hat{}\) have a physical dimension and all others are dimensionless, e.g. the time \(\hat{t}\) has the dimension of seconds “s”, while \(t\) is dimensionless.

## 2 Formulation of the Model

In the first part of this section, we give a brief description of the model. For physical details and motivation of constitutive assumptions we refer to the previously published works Hilfer (2006a, b, c). In the second part, problem-specific assumptions are given and a dimensionless fractional flow formulation is derived.

### 2.1 The Generalized Model

A one-dimensional, horizontal, homogeneous, isotropic, and rigid porous column filled with two immiscible Newtonian fluids is considered here. The water phase denoted as \(\mathbb{W }\) with saturation \({S_{\mathbb{W }}}\) consists of percolating \(S_{1}\) and nonpercolating water \(S_{2}\). The oil phase denoted as \(\mathbb{O }\) with saturation \({S_{\mathbb{O }}}\) consists of percolating oil \(S_{3}\) and nonpercolating oil \(S_{4}\). A region occupied by fluid is called percolating if it is path-connected to the boundary of the sample. For a more precise definition see Hilfer (2006b). All saturations are functions of one-dimensional position \(\hat{x}\) and time \(\hat{t}\in \mathbb{R }^{+}\). Throughout the paper we idealize a large but finite system as infinitely extended so that \(\hat{x}\in \mathbb{R }\) holds. Variables with a hat \(\hat{}\) have a physical dimension. The volume fraction of the \(i\)-th phase is \(\phi _{i}=\phi _{i}(\hat{x},\hat{t})=\phi (\hat{x},\hat{t})S_i(\hat{x},\hat{t})\) and the volume fraction of the porous medium is \(\phi _{5}=\phi _{5}(\hat{x},\hat{t})=1-\phi (\hat{x},\hat{t})\) where the porosity \(\phi =\phi (\hat{x},\hat{t})\) is defined as the volume fraction of the pore space.

#### 2.1.1 Balance Laws

#### 2.1.2 Constitutive Assumptions

where \(\hat{R}_{12}=\hat{R}_{34}=0\) was already used because there is no common interface and hence, no direct viscous interaction between these phase pairs.

#### 2.1.3 Reformulation

where the total water flux formed by the percolating and the nonpercolating water flux is denoted by \({\hat{q}_\mathbb{W }}=\hat{q}_{1}+{\hat{q}_2}\). Similarly the oil flux is denoted by \({\hat{q}_\mathbb{O }}=\hat{q}_{3}+{\hat{q}_4}\). Initial and boundary conditions are given below.

#### 2.1.4 Self-Consistence Closure Condition

### 2.2 Problem-Specific Assumptions

#### 2.2.1 Boundary and Initial Conditions

#### 2.2.2 Viscous Drag Domination and Viscous Decoupling

for all pairs \((i,j)\) with \(i\in \{1,2,3,4\}\) and \(j\in \{1,2,3,4,5\}\) such that and \((i,j)\ne (2,5)\) and \((i,j)\ne (4,5)\). The assumptions lead to a constant resistance matrix \(\hat{R}\), to immobile nonpercolating phases and to a mobility matrix \(\hat{\varLambda }\) that has only \(\lambda _{11}\) and \(\lambda _{33}\) as nonzero components. These assumptions may be justified physically by the observation that the motion of contact lines on the internal surface requires to overcome capillary forces and this creates additional resistance that is much higher than the viscous drag.

#### 2.2.3 Model Reduction and Dimensionless Fractional Flow Formulation

\({\eta _{2}}\) | \({\eta _{4}}\) | \(\varPi ^*_\mathrm{a}\) | \(\varPi ^*_\mathrm{b}\) | \(P^*_{2}\) | \(P^*_{4}\) | \(\alpha \) | \(\beta \) | \(\gamma \) | \(\delta \) | \(S_{\mathbb{O }\,\mathrm{im}}\) | \(S_{\mathbb{W }\,\mathrm{dr}}\) | \(\frac{\hat{R}_{11}}{\hat{R}_{33}}\) |
---|---|---|---|---|---|---|---|---|---|---|---|---|

4 | 3 | 1.6 | 0.025 | 2.5 | 0.4 | 0.52 | 0.9 | 1.5 | 3.5 | 0.19 | 0.15 | 2 |

Limiting and initial saturations for primary and secondary imbibition

Parameter | \({S_2^*}\) | \({S_4^*}\) | \({S_\mathbb{W }^*}\) | \(S_{2 0}\) | \(S_{40}\) | \({S_{\mathbb{W }0}}\) |
---|---|---|---|---|---|---|

Primary | 0 | \(S_{\mathbb{O }\,\mathrm{im}}\) | \(1-S_{\mathbb{O }\,\mathrm{im}}\) | 0 | 0 | 0 |

Secondary | 0 | \(S_{\mathbb{O }\,\mathrm{im}}\) | \(1-S_{\mathbb{O }\,\mathrm{im}}\) | \(S_{\mathbb{W }\,\mathrm{dr}}\) | 0 | \(S_{\mathbb{W }\,\mathrm{dr}}\) |

## 3 Method of Solution

### 3.1 Traveling Wave Ansatz

are bounded away from their limiting values.

This normalization effectively eliminates the parameter \(y_{0}\) from the discussion. It permits a complete and exhaustive discussion for the special class of all smooth solutions with the help of two reduced graphical representations introduced below in Sect. 5.

### 3.2 Dynamical System Approach

### 3.3 Linearization and Stability

### 3.4 Trajectories and Their Asymptotic Behavior

We now want to analyze the behavior of the trajectories for the limits \(X\searrow {S_{\mathbb{W }0}}\) and \(X\nearrow {S_\mathbb{W }^*}\). This is necessary because the \(Y\)-values of the trajectories go to \(-\infty \) or \(\infty \) where \(X\) goes to its maximum \({S_\mathbb{W }^*}\) or minimum \({S_{\mathbb{W }0}}\) because the capillary function \(\mathrm{D}(X)\) vanishes there.

and differs from their behavior in the traditional theory with Brooks and Corey parameter functions (Brevdo et al. 2001) where they diverge in each limit.

## 4 Results

### 4.1 Phase Portraits

This subsection shows phase portraits and bifurcations for the two main imbibition processes with parameters from Tables 1 and 2 and highlights the differences between the phase portraits found in Brevdo et al. (2001). The notations \({S_{\mathbb{W }}},{S^{\prime }_{\mathbb{W }}}\) instead of \(X,Y\) will be used in this section. Figure 2 shows the parameter functions. A first difference to Brevdo et al. (2001) is that the real and not the effective saturations are used. This permits to explicitly account for differences between primary and secondary imbibitions.

There are four bifurcations at \(c=1.235,1.37,1.738,2.69\). At \(c=1.235\) both separating curves cross each other. At \(c=1.37\) the separating curve \(\mathcal{C }_0\) crosses the trajectory \(\mathcal{N }_2\) and at \(c=1.738\) the separating curve \(\mathcal{C }_1\) crosses the trajectory \(\mathcal{N }_1\). At \(c=2.69\) the trajectories \(\mathcal{N }_1\) and \(\mathcal{N }_2\) coincide and for higher velocities they do not exist.

The phase portraits for secondary imbibition look qualitatively the same. The only differences are the minimal water saturation which is \({S_{\mathbb{W }0}}=S_{\mathbb{O }\,\mathrm{im}}\) instead of \({S_{\mathbb{W }0}}=0\) and the bifurcations happen at velocities \(c=1.52,1.78,1.84,2.9\) instead of \(c=1.235,1.37,1.738, 2.69\).

In contrast to (Brevdo et al. 2001) smooth profiles of class (d) are found for the right boundary condition \({S^\mathrm{{r}}_\mathbb{W }}=0\) with \(\lim _{y\rightarrow \infty }{S_{\mathbb{W }}}(y)=0\). These profiles are produced by the separating curve \(\mathcal{C }_0\) in the phase portrait. For Brooks and Corey parameterizations, profiles where the right boundary condition is \({S^\mathrm{{r}}_\mathbb{W }}=0\) and therefore \(\lim _{y\rightarrow \infty }{S_{\mathbb{W }}}(y)=0\) belong to class (c) and hence are not smooth everywhere. The reason for this difference lies in the different behavior of the capillary function. The derivative of the capillary function \(\mathrm{D}^{\prime }({S_{\mathbb{W }}})\) diverges where for Brooks and Corey it goes to zero as the water saturation reaches its minimum.

### 4.2 Saturation Profiles

This subsection shows saturation profiles for primary and secondary imbibition processes with parameters from Tables 1 and 2 and highlights the differences to the saturation profiles found in Brevdo et al. (2001).

Figures | \(c\) | \(c_0\) | \({S^\mathrm{{\ell }}_\mathbb{W }}\) | \({S^\mathrm{{r}}_\mathbb{W }}\) | \(S^\mathrm{{\ell }}_{2}\) | \(S^\mathrm{{r}}_{2}\) | \(S^\mathrm{{\ell }}_{4}\) | \(S^\mathrm{{r}}_{4}\) | Profile class |
---|---|---|---|---|---|---|---|---|---|

5a | 1 | -0.3 | 0.81 | 0 | 0 | 0 | 0.19 | 0 | (a) |

5e | 1 | -0.3 | 0.81 | 0.15 | 0 | 0.15 | 0.19 | 0 | (a) |

5b | 1 | -0.25 | 0.81 | 0.72 | 0 | 0 | 0.19 | 0.19 | (b) |

5f | 1 | -0.25 | 0.81 | 0.72 | 0 | 0 | 0.19 | 0.19 | (b) |

5c | 1 | -0.1 | 0.53 | 0 | 0 | 0 | 0.18 | 0 | (c) |

5g | 1 | -0.1 | 0.54 | 0.15 | 0.01 | 0.15 | 0.18 | 0 | (c) |

5d | 1 | 0.1 | 0.40 | 0.11 | 0 | 0 | 0.17 | 0.07 | (d) |

5h | 1 | 0.2 | 0.35 | 0.21 | 0.04 | 0.10 | 0.12 | 0.05 | (d) |

6a | 1.37 | 0 | 0.66 | 0 | 0 | 0 | 0.19 | 0 | (d) |

6b | 1.78 | 0.26 | 0.64 | 0.15 | 0 | 0.15 | 0.19 | 0 | (d) |

Figure 5b contains a profile of class (b) for primary imbibition. A water front imbibes a medium filled with maximal amount of nonpercolating oil and to \(72\,\%\) with percolating water. The water saturation increases to its maximal value \(1-S_{\mathbb{O }\,\mathrm{im}}\) and the nonpercolating phases do not change. Figure 5f contains a profile of class (b) for secondary imbibition. There is almost no difference between primary and secondary imbibition because the nonpercolating phases are constant and identical for \({S_{\mathbb{W }}}\in (0.7,0.81)\), see Fig. 1a.

Figure 5c contains a profile of class (c) for primary imbibition. A water front imbibes a medium completely filled with percolating oil. The water saturation increases to \(53\,\%\) and nonpercolating oil is produced up to a value of \(18\,\%\). Figure 5g contains a profile of class (c) for secondary imbibition. A water front imbibes a medium filled with maximal percolating oil and maximal nonpercolating water. The water saturation increases to \(54\,\%\), nonpercolating oil is produced up to a value of \(18\,\%\) and almost all nonpercolating water becomes percolating.

Figure 5d contains a profile of class (d) for primary imbibition. A water front imbibes a medium filled with \(82\,\%\) percolating oil, \(7\,\%\) nonpercolating oil and \(11\,\%\) percolating water. The water saturation increases to \(40\,\%\) and nonpercolating oil is produced up to a value of \(17\,\%\). Figure 5h contains a profile of class (d) for secondary imbibition. A water front imbibes a medium filled with \(74\,\%\) percolating oil, \(5\,\%\) nonpercolating oil, \(11\,\%\) percolating water and \(10\,\%\) nonpercolating water. The water saturation increases to \(35\,\%\), nonpercolating oil is produced up to a value of \(12\,\%\) and up to \(4\,\%\) all nonpercolating water becomes percolating.

## 5 Two Reduced Graphical Representations of all Differentiable Traveling Waves

From here on, the discussion is restricted to traveling waves of class (d). It is the only class with profiles that are everywhere smooth. Because of this property, many authors Cueto-Felgueroso and Juanes (2009), Gilding and Kersner (2001), Volpert et al. (1994) limit their studies to class (d) beforehand. We follow this practice here. The limitation to smooth solutions allows to effectively eliminate the parameter \(y_{0}\) from the discussion by setting \(y_{0}=\infty \) and changing the normalization from Eqs. (45) to (46).

This section discusses two reduced representations of all smooth traveling waves. The reduced representations enable us to plot all global information of smooth traveling waves—the wave velocity \(c\) and the boundary conditions \(({S^\mathrm{{\ell }}_\mathbb{W }},{S^\mathrm{{r}}_\mathbb{W }})\)—in one single figure. Without this representation many phase portraits are necessary to illustrate all global information. This exacerbates a quantitative comparison of the solution space \((c,{S^\mathrm{{\ell }}_\mathbb{W }},{S^\mathrm{{r}}_\mathbb{W }})\) of traveling waves for different models or model parameters. With the reduced representations, the difference between two models can be assessed through the comparison of two figures. Important information such as the maximal traveling wave height or the maximal velocity can be very easily read off one of these figures.

In this article, these representations are mainly used to illustrate the differences between secondary and primary imbibition of one parameter set of the generalized model assuming immobile nonpercolating phases. But the reduced representations can be applied to other models with flow functions, even if they are non-monotonic. Moreover, these representations can be used for the Buckley–Leverett limit and therefore for shock waves because traveling waves and shock waves are connected through the vanishing viscosity limit (Duijn et al. 2007). The ideas for these representations are partially borrowed from the standard analysis of shock waves of the Buckley–Leverett limit.

### 5.1 Representation 1: The Integration Constant \(c_0\)

The first reduced representation uses the established picture (Welge 1952) that a shock wave in the Buckley–Leverett limit can be represented as a straight line in the flow function figure. This line will be called traveling wave line in this article. Its slope represents the wave velocity. The \(y\) intercept is the negative of the integration constant of the traveling wave formulation. Its intersections with the flow function are the possible boundary values. Segments of the traveling wave line having higher values than the flow function are imbibition shocks and segments of the traveling wave line having lower values than the flow function are drainage shocks. The integration constant can now be written as a function of one possible boundary condition and the velocity. The sign of the derivative of the integration constant function in respect to the boundary condition tells us whether the boundary condition is left- or right-sided. Plotting the integration constant function and using the sign of its derivative, represents all global information.

#### 5.1.1 Method

#### 5.1.2 Results

Important information about the solution space of traveling waves, such as the maximal velocity or maximal saturation, can be easily identified in Fig. 8. The maximal velocity of a traveling wave is \(c^\mathrm{max}=\max _{{S_{\mathbb{W }}}\in (0,1)}f^{\prime }({S_{\mathbb{W }}})\) and its corresponding saturation \({S_{\mathbb{W }}^\mathrm{m}}\) is the only inflection point of \(f({S_{\mathbb{W }}})\). The maximal left-sided boundary saturation is \({S_{\mathbb{W }}^\mathrm{BL}}=\{{S_{\mathbb{W }}}|c^\mathrm{BL}=f({S_{\mathbb{W }}})/({S_{\mathbb{W }}}-{S_{\mathbb{W }0}}) \text{ maximal }\}\). The saturation \({S_{\mathbb{W }}^\mathrm{BL}}\) is the left-sided limit of a maximal Buckley–Leverett shock (Welge 1952). One can see in Fig. 8 that higher velocities are possible for the secondary imbibition. The saturations \({S_{\mathbb{W }}^\mathrm{m}}\) for the fastest wave are however identical. The maximal saturation \({S_{\mathbb{W }}^\mathrm{BL}}\) is \(2\,\%\) lower for a secondary imbibition although there is already \(15\,\%\) initially present.

- (i)
Traveling waves connecting C and D with boundary saturations \(({S^\mathrm{{\ell }}_\mathbb{W }},{S^\mathrm{{r}}_\mathbb{W }})\in \{({S^\mathrm{{\ell }}_\mathbb{W }},0)|{S^\mathrm{{\ell }}_\mathbb{W }}\in (0,{S_{\mathbb{W }}^\mathrm{BL}})\}\) and velocities \(c=f^{\prime }({S^\mathrm{{\ell }}_\mathbb{W }})/{S^\mathrm{{\ell }}_\mathbb{W }}\). These are the maximal imbibition waves in a completely dry porous medium.

- (ii)
Traveling waves connecting A and B with boundary saturations \(({S^\mathrm{{\ell }}_\mathbb{W }},{S^\mathrm{{r}}_\mathbb{W }})\in \{({S^\mathrm{{\ell }}_\mathbb{W }},{S^\mathrm{{r}}_\mathbb{W }})|{S^\mathrm{{\ell }}_\mathbb{W }}\in ({S_{\mathbb{W }}^\mathrm{m}},{S_{\mathbb{W }}^\mathrm{BL}}) \cup {S^\mathrm{{r}}_\mathbb{W }}\in (0,{S_{\mathbb{W }}^\mathrm{m}})\}\) and velocities \(c=(f({S^\mathrm{{\ell }}_\mathbb{W }})-f({S^\mathrm{{r}}_\mathbb{W }}))/({S^\mathrm{{\ell }}_\mathbb{W }}-{S^\mathrm{{r}}_\mathbb{W }})\). These are imbibition waves with the maximal saturation difference into a porous medium with an initial saturation \({S^\mathrm{{r}}_\mathbb{W }}\). This is the Welge construction (Welge 1952) for given initial saturation \({S^\mathrm{{r}}_\mathbb{W }}\).

- (iii)
Line E represents traveling waves with \({S^\mathrm{{\ell }}_\mathbb{W }}={S^\mathrm{{r}}_\mathbb{W }}\in (0,{S_{\mathbb{W }}^\mathrm{m}})\) and velocities \(c=f^{\prime }({S^\mathrm{{\ell }}_\mathbb{W }})\). These are the waves with constant saturations.

### 5.2 Representation 2: The shock speed \(c\)

The second representation uses the Rankine–Hugoniot condition to plot the functionality between the velocity and the two boundary conditions. In addition, the Rankine–Hugoniot condition can be defined as a function of the nonpercolating boundary saturations because the nonpercolating phases are bijective functions of the water saturation for fixed \((S_{2 0},S_{40},{S_{\mathbb{W }0}})\). This leads to some interesting insights about nonpercolating phases and their influence on the velocity of a traveling wave.

#### 5.2.1 Method

where the \(S_2^{-1}\) and \(S_4^{-1}\) denote inverse functions with respect to the water saturation \({S_{\mathbb{W }}}\) because the nonpercolating phases are bijective functions of the water saturation for fixed \((S_{2 0},S_{40},{S_{\mathbb{W }0}})\).

#### 5.2.2 Results

Equation (63) allows to study the behavior of the nonpercolating phases. Figure 10c does not show any color contours because there is no initial nonpercolating water \(S^\mathrm{{r}}_{2}=0\) in a primary imbibition process and during an imbibition process nonpercolating water is not produced \(S^\mathrm{{\ell }}_{2}=0\). For the secondary imbibition in Fig. 10d the velocities increase if the nonpercolating water saturations behind the front are minimal. The velocities increase for given nonpercolating water saturations behind the front with decreasing initial nonpercolating water saturations. Consequently, we observe that larger differences in nonpercolating water saturation lead to smaller velocities. Higher ratios between nonpercolating and percolating water lead to slower velocities. From a physical point of view this is obvious because nonpercolating phases are assumed to be immobile and these fluid parts can only be mobilized by coalescence with the percolating phase. Therefore, the mass exchange reduces the velocity. A small amount of nonpercolating water, which remains immobile, increases the maximal velocity because the maximal velocity is \(30\,\%\) higher for secondary imbibition as compared to primary imbibition. We assume that this immobile nonpercolating water occupies pore volumes which have very low conductance leading to a higher overall conductance for the porous medium.

In Fig. 10e, f, the behavior of the nonpercolating oil does not differ qualitatively between primary and secondary imbibition. It also shows similarities to the behavior of the water saturation which displays the correlation between water and nonpercolating oil due to the mass exchange term. An increasing water saturation leads to an increasing nonpercolating oil saturation due to break up. The velocities increase for given initial nonpercolating oil saturations if the nonpercolating saturations behind the front increase. For given nonpercolating oil saturations behind the front velocities increase if the initial nonpercolating saturations increase. A high presence of immobile nonpercolating oil occupying pore volume which has low conductance leads to a higher overall conductance for the porous medium.

## 6 Conclusion

This paper has computed traveling wave solutions of a recent generalized theory for macroscopic capillarity when it can be assumed that the nonpercolating phases are immobile. Only primary and secondary imbibition processes were considered here, but drainage processes can be studied along the same lines. The solutions have been compared to traveling wave solutions of the traditional theory and significant differences were found, because the generalized theory accounts explicitly for nonpercolating and immobile fluid parts. Breakup and coalescence slow down the front, but depending on the parameters a small amount of nonpercolating immobile fluids can also lead to higher velocity. Methodically, the complete analysis of smooth solutions was based on two novel graphical representations of the solution space that may be generalized to other solution classes. The reduced representation combines the wave speed, the boundary data and the integration constant. It simplifies the comparison between different models and provides important information, such as the maximal velocity or maximal saturation at the inlet. It can also be used to discuss shock fronts and the hyperbolic limit.

## Notes

### Acknowledgments

The authors gratefully acknowledge financial support from the Deutsche Forschungsgemeinschaft (DFG) within the International Research and Training Group on Nonlinearities and Upscaling in Porous Media (the NUPUS project) and fruitful discussions with Paul Zegeling.

## References

- Abrams, A.: Influence of fluid viscosity, interfacial-tension, and flow velocity on residual oil saturation feft by waterflood. Soc. Petroleum Eng. J.
**15**(5), 437–447 (1975)Google Scholar - Avraam, D., Payatakes, A.: Generalized relative permeability coefficients during steady-state two-phase flow in porous media, and correlation with the flow mechanisms. Transp. Porous Media
**20**, 135–168 (1995)CrossRefGoogle Scholar - Avraam, D., Payatakes, A.: Flow mechanisms, relative permeabilities, and coupling effects in steady-state two-phase flow through porous media, the case of strong wettability. Ind. Eng. Chem. Res.
**38**(3), 778–786 (1999)CrossRefGoogle Scholar - Bear, J.: Dynamics of Fluids in Porous Media. Dover Publications, New York (1972)Google Scholar
- Bear, J., Braester, C., Menier, P.C.: Effective and relative permeabilities of anisotropic porous media. Transp. Porous Media
**2**, 301–316 (1987)Google Scholar - Brevdo, L., Helmig, R., Haragus-Courcelle, M., Kirchgässner, K.: Permanent fronts in two-phase flows in a porous medium. Transp. Porous Media
**44**, 507–537 (2001)CrossRefGoogle Scholar - Bryant, S.L., Blunt, M.J.: Prediction of relative permeability in simple porous media. Phys. Rev. A
**46**(4), 2004–2011 (1992)CrossRefGoogle Scholar - Collins, R.: Flow of Fluids Through Porous Materials. Reinhold Publishing Corporation, New York (1961)Google Scholar
- Cueto-Felgueroso, L., Juanes, R.: Stability analysis of a phase-field model of gravity-driven unsaturated flow through porous media. Phys. Rev. E (Statistical, Nonlinear, and Soft Matter Physics)
**79**(3), 036301 (2009)Google Scholar - De Marsily, G.: Quantitative Hydrogeology-Groundwater Hydrology for Engineers. Academic Press, Orlando (1986)Google Scholar
- de Wiest, R.: Flow Through Porous Media. Academic Press, New York (1969)Google Scholar
- Dias, M.M., Payatakes, A.C.: Network models for two-phase flow in porous media part 1. Immiscible microdisplacement of non-wetting fluids. J. Fluid Mech.
**164**(1), 305–336 (1986)CrossRefGoogle Scholar - Doster, F.: Die bedeutung perkolierender und nichtperkolierender phasen bei mehrphasenströmungen in porösen medien auf laborskala. Ph.D. thesis, Universität Stuttgart, Holzgartenstr. 16, 70174 Stuttgart (2011)Google Scholar
- Doster, F., Hilfer, R.: Generalized Buckley–Leverett theory for two phase flow in porous media. New J. Phys.
**13**, 123,030 (2011)CrossRefGoogle Scholar - Doster, F., Hönig, O., Hilfer, R.: Horizontal flow and capillarity-driven redistribution in porous media. Phys. Rev. E
**86**(1), 016,317 (2012)CrossRefGoogle Scholar - Doster, F., Zegeling, P.A., Hilfer, R.: Numerical solutions of a generalized theory for macroscopic capillarity. Phys. Rev. E
**81**(3), 036307 (2010)Google Scholar - Dullien, F.: Porous Media: Fluid Transport and Pore Structure, 2nd edn. Academic Press, San Diego (1992)Google Scholar
- Ferer, M., Ji, C., Bromhal, G., Cook, J., Ahmadi, G., Smith, D.: Crossover from capillary fingering to viscous fingering for immiscible unstable flow: experiment and modeling. Phys. Rev. E
**70**(1), 16,303 (2004)CrossRefGoogle Scholar - Gilding, B., Kersner, R. Travelling waves in nonlinear diffusion–convection-reaction. Memorandum 1585, Department of Applied Mathematics, University of Twente, Enschede (2001)Google Scholar
- Helmig, R.: Multiphase Flow and Transport Processes in the Subsurface. Springer, Berlin (1997)CrossRefGoogle Scholar
- Hilfer, R.: Macroscopic equations of motion for two-phase flow in porous media. Phys. Rev. A
**58**, 2090 (1998)Google Scholar - Hilfer, R.: Capillary pressure, hysteresis and residual saturation in porous media. Physica A
**359**, 119 (2006)CrossRefGoogle Scholar - Hilfer, R.: Macroscopic capillarity and hysteresis for flow in porous media. Phys. Rev. E
**73**, 016,307 (2006)CrossRefGoogle Scholar - Hilfer, R.: Macroscopic capillarity without a constitutive capillary pressure function. Physica A
**371**, 209–225 (2006)CrossRefGoogle Scholar - Hilfer, R., Doster, F.: Percolation as a basic concept for macroscopic capillarity. Transp. Porous Media
**82**(3), 507–519 (2010)Google Scholar - Hilfer, R., Oeren, P.: Dimensional analysis of pore scale and field scale immiscible displacement. Transp. Porous Media
**22**, 53–72 (1996)CrossRefGoogle Scholar - Jamin, J.: Notes about equilibrium and flow of fluids in porous body. Acad. Sci.
**50**, 172 (1860)Google Scholar - Perko, L.: Differential Equations and Dynamical Systems. Springer, New York (1993)Google Scholar
- Scheidegger, A.E.: The Physics of Flow through Porous Media. University of Toronto Press, Toronto (1957)Google Scholar
- Smoller, J.: Shock Waves and Reaction–Diffusion Equations, 2nd edn. Springer, New York (1983)CrossRefGoogle Scholar
- Taber, J.: Dynamic and static forces required to remove a discontinuous oil phase from porous media containing both oil and water. Soc. Petroleum Eng. J.
**9**(1), 3 (1969)Google Scholar - van Duijn, C.J., Peletier, L.A., Pop, I.S.: A new class of entropy solutions of the Buckley–Leverett equation. SIAM J. Math. Anal.
**39**(2), 507–536 (2007)CrossRefGoogle Scholar - van Dijke, M., Sorbie, K.: Pore-scale network model for three-phase flow in mixed-wet porous media. Phys. Rev. E
**66**(4), 46,302 (2002)Google Scholar - Volpert, A.I., Volpert, V.A., Volpert, V.A.: Traveling Wave Solutions of Parabolic Systems. Translations of Mathematical Monographs, p. 448. American Mathematical Society, Providence (1994)Google Scholar
- Welge, H.J.: A simplified method for computing oil recovery by gas or water drive. AIME Trans.
**195**, 99–108 (1952)Google Scholar - Wyckoff, R.D., Botset, H.G.: The flow of gas–liquid mixtures through unconsolidated sands. Physics
**7**(9), 325–345 (1936)Google Scholar

## Copyright information

**Open Access**This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.