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Abstract. Traditional two-phase flow models use an algebraic relationship between capillary pressure and saturation. This relationship is based on measurements made under static conditions. However, this static relationship is then used to model dynamic conditions, and evidence suggests that the assumption of equilibrium between capillary pressure and saturation may not be be justified. Extended capillary pressure-saturation relationships have been proposed that include an additional term accounting for dynamic effects. In the present work we study some of the underlying pore-scale physical mechanisms that give rise to this so-called dynamic effect. The study is carried out with the aid of a simple bundle-of-tubes model wherein the pore space of a porous medium is represented by a set of parallel tubes. We perform virtual two-phase flow experiments in which a wetting fluid is displaced by a non-wetting fluid. The dynamics of fluid-fluid interfaces are taken into account. From these experiments, we extract information about the overall system dynamics, and determine coefficients that are relevant to the dynamic capillary pressure description. We find dynamic coefficients in the range of 10 2 − 10 3 kg m −1 s −1 , which is in the lower range of experimental observations. We then analyze certain behavior of the system in terms of dimensionless groups, and we observe scale dependency in the dynamic coefficient. Based on these results, we then speculate about possible scale effects and the significance of the dynamic term.

ABSTRACTwhere P n and P w are the average pressures of nonwetting and wetting phases, respectively; P c is capillary pressure, Capillary pressure plays a central role in the description of waterand S is the wetting phase saturation. A schematic depicflow in unsaturated soils. While capillarity is ubiquitous in unsaturated tion of P c vs. S curves is given in Fig. 1. 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 his-C apillarity plays a central role in the description of tory of flow is known as capillary pressure hysteresis; multiphase (and unsaturated) flow in porous media. analyses, the theoretical basis and practical implications of capillaritythis 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, how-P n Ϫ P w ϭ P c ϭ f(S) [1] ever, are less pronounced than in the case of capillary pressure. It must be noted that the dynamic effect con-S.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

[1] Micromodels have been increasingly employed in various ways in porous media research, to study the pore-scale behavior of fluids. Micromodels have proven to be a valuable tool by allowing the observation of flow and transport at the micron scale in chemical, biological, and physical applications. They have helped to improve our insight of flow and transport phenomena at both microscale and macroscale. Up to now, most micromodels that have been used to study the role of interfaces in two-phase flow were small, square, or nearly square domains. In this work, an elongated PDMS micromodel, bearing a flow network with dimensions 5Â30 mm 2 was manufactured. The pore network was designed such that the REV size was around 5Â7 mm 2 . So, our flow network was considered to be nearly four times the REV size. Using such micromodels, we established that the inclusion of interfacial area between the wetting and the nonwetting fluids models the hysteretic relationship between capillary pressure and saturation in porous media. In this paper, we first present the procedure for manufacturing PDMS micromodels with the use of soft lithography. Then, we describe an innovative and novel optical setup that allows the real-time visualization of elongated samples. Finally, we present the results obtained by quasi-static, two-phase flow experiments.

The transport of viruses in unsaturated porous media has been a subject of great interest in recent years because of the enhanced removal of these microorganisms compared with saturated conditions. We studied the transport of bacteriophages MS2 and ϕX174, used as surrogate pathogenic viruses, at various water contents and solution chemistries in terms of pH and ionic strength (IS). The objective was to explore the interaction of viruses with the solid–water interfaces (SWI) and air–water interfaces (AWI) for a range of conditions. The experimental data were fitted with a transport model to determine the adsorption (attachment and detachment rate) parameters. Our results show that the retention of viruses in the soil column increases as water saturation decreases when the chemical conditions are favorable for adsorption (pH 7 and relatively high IS). Our analysis indicates that the enhanced retention of ϕX174 viruses at lower water contents is caused by increased attachment to the SWI and that retention by the AWI is not significant. Results obtained from a first series of experiments (pH 9 and low IS) showed insignificant attachment of MS2 viruses to both the SWI and the AWI. The MS2 breakthrough data for a second series of experiments (pH 7 and high IS) did not allow us to separate out the role of the AWI. Although attachment of MS2 viruses to the AWI cannot be ruled out in our experiments, we suspect that the increased retention of this phage under unsaturated condition was also due to enhanced attachment to the SWI. Increased attachment to the SWI under unsaturated conditions is attributed to increased mass transfer of viruses to the SWI due to a reduced diffusion length at lower water contents. Our results demonstrate that if there is any attachment to the AWI, it is reversible. When unfavorable conditions occur for attachment to the SWI, the attached viruses may be detached by moving solid–water–air contact lines (SWA).

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In previous work by the authors a rigorous procedure for the derivation of global watershed-scale balance laws for mass, momentum, energy and entropy has been pursued. To complement these, a set of constitutive relations for the closure of the mass and momentum balance equations has also been derived, based on the exploitation of the second law of thermodynamics. In this paper these governing equations, including the constitutive relations, are rederived for the simpler case of the stream channel network of a natural watershed. The derived constitutive relationships for mass and force exchanges amongst channel reaches are physically consistent and thermodynamically admissible insofar as they respect physical constraints and keep the total entropy production of the system always positive. Next, the resulting system of coupled nonlinear ordinary di¬erential equations are simultaneously solved for a natural watershed under realistic conditions. The numerical model presented permits the estimation of space-time elds of average velocity, storage and discharge within all reaches of the network tree during run-o¬ events. The network response, as well as space-time elds of velocity and discharge, are computed for a number of rainfall events of di¬erent magnitude and di¬erent levels of network discretization. The nonlinearity of the response and the e¬ects of di¬erent discretizations of the network are analysed in terms of computational experiments.