Dynamic Effect in the Capillary Pressure–Saturation Relationship and its Impacts on Unsaturated Flow

where Pn and Pw are the average pressures of nonwetting and wetting phases, respectively; Pc is capillary pressure, Capillary pressure plays a central role in the description of water and S is the wetting phase saturation. A schematic depicflow in unsaturated soils. While capillarity is ubiquitous in unsaturated tion of Pc vs. S curves is given in Fig. 1. analyses, the theoretical basis and practical implications of capillarity This simple model is implicitly assumed to account for in soils remain poorly understood. In most traditional treatments of capillary pressure, it is defined as the difference between pressures all effects and processes that influence the equilibrium of phases, in this case air and water, and is assumed to be a function distribution of fluids, such as surface tension, presence of saturation. Recent theories have indicated that capillary pressure of fluid–fluid interfaces, wettability of solid surfaces, should be given a more general thermodynamic definition, and its grain size distribution, and microscale heterogeneities. functional dependence should be generalized to include dynamic efAll of these effects are essentially lumped into the Pc–S fects. Experimental evidence has slowly accumulated in the past derelationship. Moreover, this relationship and graphs cades to support a more general description of capillary pressure that similar to those depicted in Fig. 1 are obtained experiincludes dynamic effects. A review of these experiments shows that mentally under equilibrium conditions. Thus, to obtain the coefficient arising in the theoretical analysis can be estimated from a drainage (or imbibition) curve, one starts with a wet the reported data. The calculated values range from 104 to 107 kg (or dry) soil sample, then the capillary pressure is in(m s) 1. In addition, recently developed pore-scale models that simulate interface dynamics within a network of pores can also be used creased (or decreased) incrementally, and at each step to estimate the appropriate dynamic coefficients. Analyses of experithe water content is measured after equilibrium is ments reported in the literature, and of simulations based on porereached. The time to equilibrium after each imposed scale models, indicate a range of dynamic coefficients that spans about pressure increment ranges from a few hours to many three orders of magnitude. To examine whether these coefficients days, depending on the soil type and saturation degree have any practical effects on larger-scale problems, continuum-scale (see, e.g., Elrick, 1963; Stephens, 1995, p. 189). The simulators may be constructed in which the dynamic effects are intypical time needed to construct a complete capillary cluded. These simulators may then be run to determine the range of pressure–saturation curve is weeks or longer. Now, the coefficients for which discernable effects occur. Results from such question arises whether such curves adequately describe simulations indicate that measured values of dynamic coefficients are the relationship between Pn–Pw and S in drainage or within one order of magnitude of those values that produce significant effects in field simulations. This indicates that dynamic effects may imbibition events with a time scale in the order of hours. be important for some field situations, and numerical simulators for In fact, there is ample theoretical and experimental eviunsaturated flow should generally include the additional term(s) assodence that this simple relationship is not unique, but it ciated with dynamic capillary pressure. depends on the flow dynamics—it depends on both the history and the rate of change of saturation. The dependence of capillary pressure–saturation curves on the hisC plays a central role in the description of tory of flow is known as capillary pressure hysteresis; multiphase (and unsaturated) flow in porous media. this is a well-known effect and has been the subject of In quantitative modeling of multiphase flow, a relationextensive investigations. The dependence of capillary ship is needed to describe capillary pressure as a funccurves on the rate of change of saturation is due to dytion of other medium properties. Although the underlynamic effects. It is much less known and is not quantified ing processes that determine the distribution of fluid properly. The latter effect is the subject of this study. phases in porous media are extremely complicated, the Another important parameter in the description of main theoretical and practical tool currently used to unsaturated flow is relative permeability, which is also quantify the capillary pressure function is an empirical considered to be a function of saturation. There are some relationship between capillary pressure and saturation indications that the relative permeability–saturation rein the form (see, e.g., Bear and Verruijt, 1987): lationship also shows hysteresis effects and may depend on the rate of change of saturation. These effects, howPn Pw Pc f(S) [1] ever, are less pronounced than in the case of capillary pressure. It must be noted that the dynamic effect conS.M. Hassanizadeh, Section for Hydrology and Ecology; Faculty of sidered in this paper is different from the flow-rate deCivil Engineering and Geosciences, Delft University of Technology; pendence of the relative permeability coefficient. It is P.O. Box 5048, 2600GA Delft, The Netherlands; Michael A. Celia, known that relative permeability shows some depenEnvironmental Engineering and Water Resources Program, Departdence on capillary number (basically on the rate of fluid ment of Civil and Environmental Engineering, Princeton University; flow). This dependence is more significant at high capilPrinceton, NJ 08544, USA; Helge K. Dahle, Department of Mathematics, University of Bergen, Johannes Brunsgate 12, 5008 Bergen, lary numbers and is present under steady-state condiNorway. Received 26 Nov. 2001. *Corresponding author (majid@citg. tions as well (see, e.g., Blom and Hagoort, 1997; Ramatudelft.nl). krishnan and Wasan, 1986). However, the dynamic effect, related to the rate of change of saturation, is a transient Published in Vadose Zone Journal 1:38–57 (2002).