## 1. Introduction

[2] Spontaneous imbibition (SI) (Figure 1) occurs if a wetting fluid (like water or brine) spontaneously enters a porous medium, and displaces a nonwetting fluid (like oil, air, nonaqueous phase liquids (NAPL), or CO_{2}), driven by capillary forces only. It is a process that is of crucial importance for many processes ranging from groundwater contamination by NAPL transport [*Brusseau*, 1992], CO_{2} storage by capillary trapping [*Bickle*, 2009; *Juanes et al.*, 2006; *Pentland et al.*, 2011], steam migration in high-enthalpy geothermal systems [*Li and Horne*, 2009], the mechanical stability and distribution of gas-hydrate bearing sediments [*Clennell et al.*, 1999; *Anderson et al.*, 2009], trapping of CO_{2} in coal seems and generation of methane [*Chaturvedi et al.*, 2009], improved oil recovery from the world's largest remaining oil reserves [*Morrow and Mason*, 2001], evaluating the wettability of a rock [*Jadhunandan and Morrow*, 1991; *Marmur*, 2003] and even many other processes not linked with hydrogeological applications at all [*Alava et al.*, 2004; *Finch-Savage et al.*, 2005].

[3] The advancing displacement front of the wetting phase during SI shows some “roughness” due heterogeneities at the pore scale. While it is well known from statistical physics [*Alava et al.*, 2004] that the averaged position of the rough displacement front and the recovery *R* of the displaced fluid scales with in time, as predicted by the Lucas-Washburn equation [*Lucas*, 1918; *Washburn*, 1921], the properties of this scaling are still unknown.

[4] Scaling groups are used to characterize the influence of key parameters on SI other than time, and are essential in any context where SI needs to be understood and described. For example, they are a central tool for interpretation of laboratory data and upscaling them to field conditions [*Morrow and Mason*, 2001], they lie at the heart of modeling and simulating multiphase flow in different scenarios like flow in heterogeneous, fractured aquifers and reservoirs [*Barenblatt et al.*, 1960; *Warren and Root*, 1963; *Di Donato and Blunt*, 2004], water uptake in plant seeds [*Finch-Savage et al.*, 2005], or are needed as a starting point for evaluating the feasibility of water injection in high-enthalpy geothermal reservoirs [*Li and Horne*, 2009]. Despite this immense practical importance, however, and although the research on SI and scaling groups spans more than 90 years [*Lucas*, 1918; *Washburn*, 1921], not even apparently simple questions—like that of the influence of viscosity ratios on SI—have been resolved satisfactorily because of the strong nonlinearities in the capillary-hydraulic properties [*Morrow and Mason*, 2001; *Marmur*, 2003; *Mason et al.*, 2010].

[5] Scaling groups for SI in realistic porous media (Table 1) have been derived mainly in two ways. Either a curve was fitted against a large body of experimental data and a single parameter was varied to analyze its influence, or simplifying assumptions on the form of the hydraulic diffusivity function in Darcy's equation [*Gummerson et al.*, 1979; *McWhorter and Sunada*, 1990; *Schmid et al.*, 2011] (Table 2) were employed from which new specialized analytical solutions were derived that then served as basis for introducing specific scaling groups. Both approaches yield groups whose applicability is strongly restricted. On the one hand, a general theoretical understanding on why a certain group works and when it would fail is left unanswered. On the other hand, the incorporation of three key aspects into scaling groups remains open, which play a central role in many practical applications.

Author | Dimensionless Time | Proportionality Constant |
---|---|---|

- a
Characteristic values are denoted by . It is now apparent, how previous authors (unknowingly) have derived successively better expressions for *t*by giving approximations to the integral in equation (3). A specific_{d}*t*will give a good scaling if_{d}*c*is the same for the different data sets, and thus*c*can be used to predict the validity of a special scaling group (Figure 4).
| ||

Lucas [1918]; Washburn [1921] | ||

Rapoport [1955], Mattax and Kyte [1962] | ||

Ma et al. [1997] | ||

Zhou et al. [2002] | ||

Tavassoli et al. [2005a] | ||

Li and Horne [2006] | ||

This work | c = 1 |

Author and Year | Assumption |
---|---|

- a
To resolve the influence of capillarity, all of them need to employ additional, nonessential assumptions that restrict their applicability. On contrary, it can be shown [ *Schmid et al.*, 2011] that the solution given in the work of*McWhorter and Sunada*[1990] is general. It can be viewed as the Buckley-Leverett analog for countercurrent SI (see section 2.2). This makes the derivation of further specific solutions unnecessary.
| |

Fokas and Yortsos [1982], Yortsos and Fokas [1983], Philip [1960], Chen [1988], Ruth and Arthur [2011], Wu and Pan [2003] | Specific functional forms for |

Kashchiev and Firoozabadi [2002] | Steady-state, i.e., |

Li et al. [2003] | Piston-like displacement, i.e., |

Barenblatt et al. [1990], Zimmerman and Bodvarsson [1989], Tavassoli et al. [2005b], Tavassoli et al. [2005a], Mirzaei-Paiaman et al. [2011] | Approximate solution for the weak form |

Handy [1960], Chen et al. [1995], Sanchez Bujanos et al. [1998], Rangel-German and Kovscek [2002] | Existence of an equivalent constant capillary diffusion coefficient |

Ruth et al. [2007] | Self-similarity behaves according to to specific functional form |

Cil and Reis [1996], Reis and Cil [1993] | Linear capillary pressure, i.e., |

Rasmussen and Civan [1998], Civan and Rasmussen [2001] | Asymptotic approximation of laplace transformation for S_{w} |

Zimmerman and Bodvarsson [1991] | Piecewise linear S profile_{w} |

[6] First, the mobility of the fluids is governed by the relative permeability of one phase to the other and is weighted by their respective viscosities. It is unclear [*Morrow and Mason*, 2001; *Mason et al.*, 2010] how this weighting should depend on the viscosity ratio such that in the limit of an inviscid nonwetting phase the Lucas-Washburn equation is obtained [*Lucas*, 1918; *Washburn*, 1921], and how a single relative permeability value should be chosen such that it characterizes the strong nonlinear dependence on the wetting phase volume fraction over the whole saturation range (Figure 2).

[7] Second, if the porous medium initially contains a wetting phase, competition occurs between the low capillary pressure force and the high phase mobilities (Figure 2) [*Parrish and Leopold*, 1977; *Morrow and Mason*, 2001]. So far, this effect has only been characterized for cases where the ratio of nonwetting to wetting phase viscosity is close to one, and if the capillary pressure and the wetting behavior can be characterized by a single value [*Li and Horne*, 2006] which is unlikely in realistic porous media [*Valvatne and Blunt*, 2004] (Figure 2).

[8] Third, capillary pressure curves and the phase mobilities not only depend on the fluids, but also on the geometry of the pore structure, and thus are different for different materials [*Valvatne and Blunt*, 2004] (Figure 2). Up to now, however, scaling groups try to characterize the influence of capillary pressure and wetting by some single value that is representative of the entire porous medium [*Tavassoli et al.*, 2005a; *Li and Horne*, 2006; *Marmur*, 2003].

[9] In addition to these three practical issues, the very theoretical framework for describing SI has been the center of debate in physics and engineering. It has been proposed that the classical Darcy approach [*Bear*, 1972] is unsuitable for SI and should be replaced by a model that incorporates dynamic changes in capillary pressure (for recent overviews, see *Hall* [2007], *Bottero et al.* [2011], *Goel and O'Carroll* [2011], *Manthey et al.* [2008]).

[10] In the following, for the first time all three practical aspects will be accounted for. We also discuss the validity of the classical Darcy description for describing SI.

[11] The remainder of our paper is structured as follows: First, the problem formulation and an exact analytical solution for countercurrent imbibition are introduced. Until recently [*Schmid et al.*, 2011], the derivations of analytical solutions for capillary dominated two-phase flow has been the matter of intensive research (Table 2). We rigorously derive our scaling group from the only known general analytical solution for imbibition [*Schmid et al.*, 2011; *McWhorter and Sunada*, 1990], which can be viewed as the capillary counterpart to the Buckley-Leverett solution for viscous dominated flow [*Buckley and Leverett*, 1942]. No assumptions other than those needed for Darcys model are made. No fitting parameters are introduced. In section 3.1 we show the validity of our scaling group by correlating 42 published imbibition studies that vary all key parameters, namely material and capillary-hydraulic properties, viscosity ratios, initial water saturation, and characteristic lengths (Table 3). We then show that our group is a “master equation” for scaling SI, which contains many of the previously defined groups as special cases (Table 1), and demonstrate how the generality of our approach allows the prediction of the validity range of specialized groups (Table 1). This is the first predictive theory for evaluating scaling groups. We also will give strong evidence that the classical Darcy description for SI is appropriate. The paper is finished with some conclusions.

Sample^{a} | L [cm]_{c} | k [mD] | φ [−] | μ_{w} [Pa · s] | μ_{n} [Pa · s] | σ [mN/m] | S_{0} [−] | A |
---|---|---|---|---|---|---|---|---|

- a
The porous material in the work of *Zhang et al.*[1996]*Bourbiaux and Kalaydjian*[1990],*Fischer et al.*[2006],*Babadagli and Hatiboglu*[2007] was a Berea sandstone, the materials reported in the work of*Hamon and Vidal*[1986] were performed on a synthetic porous material, and*Zhou et al.*[2002] used a diatomite rock was used. For all the experiments, the wetting-phase was water. For all experiments reported for the first five groups of samples, the nonwetting phase was oil; for the ones reported in the work of*Babadagli and Hatiboglu*[2007] the nonwetting phase was air.
| ||||||||

Zhang et al. [1996] | ||||||||

AA01 | 0.5364 | 510.8 | 0.218 | 0.03782 | 50.62 | 0 | ||

AA02 | 0.8029 | 498.5 | 0.219 | 0.03782 | 50.62 | 0 | ||

AA03 | 0.9723 | 519.8 | 0.222 | 0.03782 | 50.62 | 0 | ||

AA04 | 1.089 | 521.7 | 0.224 | 0.03782 | 50.62 | 0 | ||

AA05 | 1.1837 | 505.5 | 0.215 | 0.03782 | 50.62 | 0 | ||

AA06 | 1.3059 | 501.6 | 0.218 | 0.03782 | 50.62 | 0 | ||

BC21 | 6.092 | 481.9 | 0.213 | 0.00398 | 47.38 | 0 | ||

BC13 | 4.998 | 503.6 | 0.209 | 0.03782 | 47.38 | 0 | ||

BC22 | 5.687 | 496.8 | 0.208 | 0.1563 | 51.77 | 0 | ||

BD15 | 1.3506 | 523.8 | 0.214 | 0.00398 | 47.38 | 0 | ||

BD14 | 1.3506 | 518.9 | 0.218 | 0.03782 | 50.62 | 0 | ||

BD18 | 1.3506 | 509.7 | 0.218 | 0.1563 | 51.77 | 0 | ||

BA3 | 13.87 | 907.1 | 0.214 | 0.03782 | 50.62 | 0 | ||

Hamon and Vidal [1986] | ||||||||

A10 | 9.7 | 4000 | 0.472 | 0.001 | 0.0115 | 49.0 | 0.189 | |

A10-20 | 19.7 | 3430 | 0.453 | 0.001 | 0.0115 | 49.0 | 0.187 | |

A10-30 | 30.0 | 3830 | 0.453 | 0.001 | 0.0115 | 49.0 | 0.151 | |

A10-40 | 40.0 | 3550 | 0.478 | 0.001 | 0.0115 | 49.0 | 0.172 | |

A10-85 | 84.7 | 3000 | 0.478 | 0.001 | 0.0115 | 49.0 | 0.164 | |

A10-VI-20 | 19.8 | 3200 | 0.456 | 0.001 | 0.0115 | 49.0 | 0.164 | |

A10-X-20 | 20.0 | 2300 | 0.458 | 0.001 | 0.0115 | 49.0 | 0.132 | |

Zhou et al. [2002] | ||||||||

Z-2 | 9.5 | 6.1 | 0.72 | 0.001 | 51.4 | 0 | ||

Z-3 | 9.5 | 7.9 | 0.77 | 0.001 | 45.7 | 0 | ||

Z-4 | 9.5 | 2.5 | 0.78 | 0.001 | 51.4 | 0 | ||

Z-5 | 9.5 | 6.0 | 0.68 | 0.001 | 51.4 | 0 | ||

Bourbiaux and Kalaydjian [1990] | ||||||||

GVB-3 | 29.0 | 124.0 | 0.233 | 0.0012 | 0.0015 | 35.0 | 0.4 | |

GVB-4 | 14.5 | 118.0 | 0.233 | 0.0012 | 0.0015 | 35.0 | 0.411 | |

Fischer et al. [2006] | ||||||||

EV6-22 | 7.18 | 109.2 | 0.18 | 0.495 | 0.0039 | 28.9 | 0 | |

EV6-18 | 7.62 | 140.0 | 0.181 | 0.001 | 0.063 | 51.3 | 0 | |

EV6-21 | 7.7 | 107.3 | 0.187 | 0.0278 | 0.0039 | 34.3 | 0 | |

EV6-13 | 7.75 | 113.2 | 0.187 | 0.001 | 0.0039 | 50.5 | 0 | |

EV6-14 | 7.66 | 127.2 | 0.178 | 0.0041 | 0.0039 | 41.2 | 0 | |

EV6-20 | 7.52 | 132.9 | 0.181 | 0.0041 | 0.0633 | 41.7 | 0 | |

EV6-16 | 7.78 | 136.8 | 0.181 | 0.0278 | 0.0633 | 34.8 | 0 | |

EV6-23 | 7.36 | 132.1 | 0.179 | 0.0977 | 0.0039 | 31.3 | 0 | |

EV6-15 | 7.3 | 107.0 | 0.183 | 0.4946 | 0.0633 | 29.8 | 0 | |

EV6-17 | 7.54 | 128.1 | 0.19 | 0.0977 | 0.0633 | 32.1 | 0 | |

Babadagli and Hatiboglu [2007] | ||||||||

F-11 | 10.16 | 500.0 | 0.21 | 0.001 | 72.9 | 0 | ||

F-12 | 15.24 | 500.0 | 0.21 | 0.001 | 72.9 | 0 | ||

F-14 | 10.16 | 500.0 | 0.21 | 0.001 | 72.9 | 0 | ||

F-16 | 5.08 | 500.0 | 0.21 | 0.001 | 72.9 | 0 | ||

F-16 | 10.16 | 500.0 | 0.21 | 0.001 | 72.9 | 0 | ||

F-18 | 15.24 | 500.0 | 0.21 | 0.001 | 72.9 | 0 |