C. J. Van Duijn

The effect of capillary forces on immiscible two-phase flow in heterogeneous porous media

We consider the one-dimensional two-phase flow including capillary effects through a heterogeneous porous medium. The heterogeneity is due to the spatial variation of the absolute permeability and the porosity. Both these quantities are assumed to be piecewise constant. At interfaces where the rock properties are discontinuous, we derive, by a regularisation technique, conditions to match the values of the saturation on both sides. There are two conditions: a flux condition and an extended pressure condition. Applying these conditions we show that trapping of the wetting phase may occur near heterogeneities. To illustrate the behaviour of the saturation we consider a time-dependent diffusion problem without convection, a stationary convection-diffusion problem, and the full time-dependent convection-diffusion problem (numerically). In particular the last two problems explicitly show the trapping behaviour.

Travelling waves in the transport of reactive solutes through porous media: Adsorption and binary ion exchange — Part 2

We study travelling wave solutions for the model developed in Part 1 of this paper. We develop and discuss a condition characterizing their existence. The possibility of finiteness is investigated. We consider the convergence to various limit cases and point out their different qualitative behaviour. Numerical examples are discussed.

Plume development of a nonlinearly adsorbing solute in heterogeneous porous formations

Transport of nonlinearly adsorbing solutes in homogeneous and heterogeneous porous formations is considered. Initially, a fixed amount of solute is assumed to be present in the domain. Transport is characterized in terms of the first and second spatial moments. Nonlinear equilibrium adsorption is described by the Freundlich isotherm, with the Freundlich exponent n, 0 < n < 1. By asymptotic balancing we derive first-order approximations of the limiting behavior of the plume position and plume growth as a function of time for the homogeneous case. In the heterogeneous case we consider random variation of a physical (log conductivity) and a chemical (log adsorption coefficient) parameter, both with an isotropic exponential covariance function. Expected behavior of the relevant spatial moments is obtained by applying a Monte Carlo approach. Individual realizations are solved numerically with a particle-tracking scheme in which nonlinear adsorption is accounted for by a time-dependent retarded velocity. We assess the effects of varying certain transport parameters, such as the degree of physical and chemical heterogeneity, degree of nonlinearity, the adsorption coefficient, and the degree of correlation between hydraulic conductivity and adsorption coefficient. Results of the homogeneous case for a strong degree of nonlinearity show good agreement between the numerical calculations and the limiting analytical expressions. Nonlinear adsorption is shown to have a strong effect on the shape of the plume, especially in longitudinal direction. For the heterogeneous case the analytical expressions predict well the time dependence of the growth rate of the spatial moments. Variation of the transport parameters demonstrates a dominating effect of the degree of nonlinearity on the plume dimensions, which is hardly affected by the degree of heterogeneity, correlation, and the adsorption coefficient. The variance of the mean plume position is affected by all parameters. Although the degree of heterogeneity has a strong impact, the longitudinal and transverse variances are reduced with respect to the linear adsorption case, due to the large size of the plumes.

Multi-phase flow modeling of air sparging

Air injection into groundwater (air sparging) in a homogeneous axially symmetric porous medium is modeled using a two-phase flow approach. A numerical method based on the mixed form of the Richards equation for both phases is presented. Furthermore two analytical approximations are discussed to explain the numerical results. One is a one-dimensional description explaining the occurrence of small air saturations. The other is a closed form approximation for the distribution of the air saturation in the resulting steady state. From the latter we can estimate the maximum radius of influence of air sparging, as a function of the physical parameters. The analytical approximation at steady state and the numerical results are in good agreement.

On the analysis of brine transport in porous media

An analysis is given of brine transport through a porous medium, which incorporates the effect of volume changes due to variations in the salt concentration. Two specific situations are investigated which lead to self-similarity. We develop the existence and uniqueness theory for the corresponding ordinary differential equations, and give a number of qualitative properties of the solutions. In particular, we present an asymptotic expression for the solution in terms of the relative density difference (ρ s −ρ f )/ρ f . Finally, we show some numerical results. It is found that the volume changes have a noticeable effect on the mass transport only when salt concentrations are large.

Travelling waves during the transport of reactive solute in porous media: combination of Langmuir and Freundlich isotherms

Recently, it has been shown that in the case of nonlinear solute adsorption the displacement may be in the form of a travelling wave. In this paper, we investigate whether a travelling wave type of behaviour can be expected when two different types of sorption sites can be distinguished with different isotherms and kinetics. Illustrations are given for cases where the overall isotherm comprises two contributions that follow the Langmuir and the Freundlich equations, respectively. Boundary conditions are chosen that ensure a decrease in concentration in the direction of flow. Depending on the value of the Freundlich power (p) the travelling wave may exist. For p = 1, the travelling wave always exists, whereas for 1 <p = 2 it depends on the values of the other adsorption parameters and whether a lower bound of the upstream concentration (at x = -8) is exceeded. For p = 2, the existence of the travelling wave requires that the upstream concentration does not exceed an (specified) upper bound. Besides illustrating some waves we show that two different rate functions that have the Freundlich isotherm as their limit for an infinite rate parameter result in qualitatively different travelling waves.

A note on horizontal redistribution with capillary hysteresis.

In a recent paper, Philip analyzed horizontal redistribution with capillary hysteresis. He considered the special case for which one semi-infinite half of a column has an initial condition corresponding to the point where the boundary wetting curve joins the boundary drying curve and the other semi-infinite half of the column has an initial condition corresponding to the point where the boundary drying curve joins the boundary wetting curve. The similarity solution of this problem developed by Philip can be generalized to cases involving any uniform initial conditions of the semi-infinite halves of the column, corresponding to arbitrary histories of the pressure head and the water content. For any such pair of initial conditions for the semi-infinite halves, one of the following three possibilities will apply: (1) no flow between the two halves; (2) conventional flow from the wet half to the dry half; or (3) nonconventional flow from the dry half to the wet half.

Redistribution with air diffusion

Later work has revealed wider implications and applications of the similarity solution for redistribution of soil water with capillary hysteresis [Philip, 1991]. The analysis involves continuity of both water potential and flow velocity across the interface between the draining and wetting regions. Two recent papers, however, claimed that in special circumstances, water potential is discontinuous across the interface and an “extended pressure condition” is needed. The apparent difficulty may be removed by recognizing that in the special case, hydraulics is not enough: volume conservation is maintained by diffusion of dissolved air. The extended pressure condition gives a useful approximation to the distribution of saturation but gets the potential distribution wrong and fails to recognize the air pressure differential between the two regions. Analogous conclusions apply to more general systems such as water/oil.

Simulation of coning in bottom water-driven reservoirs

Waterconing, as a result of oil recovery through a single well, is considered. It is assumed that a sharp transition, an interface, exists between the oil and the water and that the oil region between the cap rock and the bottom water in the lower half space is of infinite radial extent.In this paper, we study the dynamical behavior of the oil-water interface starting from a horizontal position. We give a description in terms of the stream function and extend the vortex theory developed for fresh-salt groundwater flow problems. As a result, we find two singular integral equations: one for the shear flow along the interface and one for the flow component normal to the interface. We also give the algorithm to solve these equations and to determine the time evolution of the cone.The problem is determined by two independent dimensionless quantities: the gravity number (G) [gravity forces/viscous forces] and the mobility ratio (M) [viscous forces in oil/viscous forces in water]. As is to be expected, these computations show that, for the schematization considered here, a stationary cone, and hence, a critical production rate (below which waterfree production is possible) does not exist. Breakthrough times are numerically determined for various values ofG andM.

The Effect of Capillary Forces on Immiscible Two-phase Flow in Strongly Heterogeneous Porous Media

We consider the one-dimensional two-phase flow through a heterogeneous porous medium. The heterogeneity is due to the spatial variation of the absolute permeability and the porosity. Both these quantities are assumed to be piecewise constant. At interfaces where the rock properties are discontinuous, we derive, by a regularisation technique, conditions to match the values of the saturation on both sides. There are two conditions: a flux condition and an extended pressure condition. Applying these conditions we show that trapping of one of the phases may occur near discontinuities in permeability or porosity. To illustrate the behaviour of the saturation we consider a timedependent diffusion problem without convection, a stationary convectiondiffusion problem, and the full time-dependent convection-diffusion problem (numerically). In particular the last two problems explicitly show the trapping behaviour.