van Cj Hans Duijn

Infiltration in porous media with dynamic capillary pressure: travelling waves

textabstractWe consider a model for non-static groundwater flow where the saturation-pressure relation is extended by a dynamic term. This approach together with a convective term due to gravity, results in a pseudo-parabolic Burgers type equation. We give a rigorous study of global travelling wave solutions, with emphasis on the role played by the dynamic term and the appearance of fronts.

Transverse dispersion from an originally sharp fresh-salt interface caused by shear flow

In this paper the influence of transversal dispersion and molecular diffusion on the distribution of salt in a plane flow through a homogeneous porous medium is studied. Since the dispersion depends on the velocity and the velocity on the distribution of salt (through the specific weight) this is a nonlinear phenomenon. In particular for the flow situation considered, this leads to a differential equation which has the character of nonlinear diffusion. n nThe initial situation (at t = 0) is chosen such that the fresh- and salt water are separated by an interface, and each fluid has a constant specific weight γ1 and γ2, respectively. For this initial situation, the solution of the nonlinear diffusion equation has the form of a similarity solution, depending only on ζ√t, where ζ denotes the local coordinate normal to the original interface plane and t denotes time. n nProperties of this similarity solution are discussed. In particular it is shown how to obtain this solution numerically. The interpretation of these mathematical results in terms of their hydrological significance is given for a number of worked out examples. These examples describe the distribution of salt, as a function of ζ and t, for various flow conditions at the boundaries ζ = ± ∞. Also examples are given where the molecular diffusion can be disregarded with respect to the transversal dispersion.

A Mathematical Model for Hysteretic Two-Phase Flow in Porous Media

We develop a mathematical model for hysteretic two-phase flow (of oil and water) in waterwet porous media. To account for relative permeability hysteresis, an irreversible trapping-coalescence process is described. According to this process, oil ganglia are created (during imbibition) and released (during drainage) at different rates, leading to history-dependent saturations of trapped and connected oil. As a result, the relative permeability to oil, modelled as a unique function of the connected oil saturation, is subject to saturation history. A saturation history is reflected by history parameters, that is by both the saturation state (of connected and trapped oil) at the most recent flow reversal and the most recent water saturation at which the flow was a primary drainage. Disregarding capillary diffusion, the flow is described by a hyperbolic equation with the connected oil saturation as unknown. This equation contains functional relationships which depend on the flow mode (drainage or imbibition) and the history parameters. The solution consists of continuous waves (expansion waves and constant states), shock waves (possibly connecting different modes) and stationary discontinuities (connecting different saturation histories). The entropy condition for travelling waves is generalized to include admissible shock waves which coincide with flow reversals. It turns out that saturation history generally has a strong influence on both the type and the speed of the waves from which the solution is constructed.We develop a mathematical model for hysteretic two-phase flow (of oil and water) in waterwet porous media. To account for relative permeability hysteresis, an irreversible trapping-coalescence process is described. According to this process, oil ganglia are created (during imbibition) and released (during drainage) at different rates, leading to history-dependent saturations of trapped and connected oil. As a result, the relative permeability to oil, modelled as a unique function of the connected oil saturation, is subject to saturation history. A saturation history is reflected by history parameters, that is by both the saturation state (of connected and trapped oil) at the most recent flow reversal and the most recent water saturation at which the flow was a primary drainage. Disregarding capillary diffusion, the flow is described by a hyperbolic equation with the connected oil saturation as unknown. This equation contains functional relationships which depend on the flow mode (drainage or imbibition) and the history parameters. The solution consists of continuous waves (expansion waves and constant states), shock waves (possibly connecting different modes) and stationary discontinuities (connecting different saturation histories). The entropy condition for travelling waves is generalized to include admissible shock waves which coincide with flow reversals. It turns out that saturation history generally has a strong influence on both the type and the speed of the waves from which the solution is constructed.

A stationary flow of fresh and salt groundwater in a coastal aquifer

Consider a horizontally situated homogeneous aquifer of constant thickness h. In this aquifer, fresh and salt water are present and separated by an abrupt interface Γ. Let the flow be two dimensional in the vertical (x,z) plane

Large Time Profiles in Reactive Solute Transport

In this paper we consider the large time structure of reactive solute plumes in two dimensional, macroscopically homogeneous, flow domains. The reactions between the dissolved chemicals and the porous matrix are equilibrium adsorption reactions, given by an isotherm of Freundlich type. We also incorporate the effect of partial and full decay. We use the method of asymptotic balancing to obtain, to leading order, the large time behaviour of the solute concentration and the relevant moments (mass, centre of mass,variance). The method of balancing is based on certain conjectures about the form of the temporal decay and partial spreading of the solute. These conjectures are verified numerically.

A note on transport of a pulse of nonlinearly reactive solute in a heterogeneous formation

Saturated flow takes place in geological formations of spatially variable permeability which is regarded as a stationary random space function of given statistical moments. The flow is assumed to be uniform in the mean and the Eulerian velocity field has stationary fluctuations. Water carries solutes that react according to the nonlinear equilibrium Freundlich isotherm. Neglecting pore scale dispersion (high Peclet number), we study the behavior of an initially finite pulse injection of constant concentration.Mean flux-averaged concentration is derived in a simple manner by using the previously determined solution of transport in a homogeneous one-dimensional medium and the Lagrangian methodology developed by Cvetkovic and Dagan [5] to model reactive transport in a three-dimensional flow field.The mean breakthrough curves are computed and the combined effect of reactive parameters and heterogeneity upon reduction of the concentration peak is investigated. Furthermore, with the aid of temporal moments, we determine equivalent reaction and macrodispersion coefficients pertinent to a homogeneous medium.

Brine transport in porous media: On the use of Von Mises and similarity transformations

In this paper we use a Von Mises transformation to study brine transport in porous media. The model involves mass balance equations for fluid and salt, Darcys law and an equation of state, relating the salt mass fraction to the fluid density. Application of the Von Mises transformation recasts the model equations into a single nonlinear diffusion equation. A further reduction is possible if the problem admits similarity. This yields a formulation in terms of a boundary value problem for an ordinary differential equation which can be treated by semi‐analytical means. Three specific similarity problems are considered in detail: (i) one‐dimensional, stable displacement of fresh water and brine in a porous column, (ii) flow of fresh water along the surface of a salt rock, (iii) mixing of parallel layers of brine and fresh water.

Mathematical analysis of the influence of power-law fluid rheology on a capillary diffusion zone

Abstract Non-Newtonian fluids are used in current oil recovery processes. These fluids do not satisfy the linear Darcy Law for flow through porous media. A generalization is needed to model the flow processes involved. Furthermore, when two immiscible fluids are present in a porous medium, capillary pressure will cause a transition zone to develop between them. This transition zone may lead to early breakthrough of water into an oil well. In this paper, we study the effect of the non-Newtonian behaviour of fluids on a capillary transition zone. A general framework is set up for modelling processes involving two-phase flow of non-Newtonian, immiscible and incompressible fluids in a porous medium. The equations are applied to the one-dimensional diffusion process of power-law fluids. The model allows for general capillary pressure and relative permeability functions. The mathematical model consists of a degenerate diffusion equation, giving rise to a free boundary formulation. The free boundaries represent the endpoint of the diffusion zone. Qualitative properties, and some analytical solutions, can be obtained for the saturation profile. Two numerical methods are presented. One is applicable if the rheology of both fluids is modelled with equal powers. The other is applicable to the general situation. Both methods make use of the qualitative, analytical properties of the solutions, which clearly improved the results obtained by standard methods. One-dimensional results can be used to interpret the general multi-dimensional flow behaviour of non-Newtonian fluids when capillarity is considered. They can also be used to test numerical algorithms developed for multi-dimensional displacement processes.

Regularity of the free boundary degenerate parabolic equation

On etudie le probleme aux valeurs initiales: u t =(D(u)φ)u x )) x dans Q=R×(0,∞), u(x,0)=u 0 (x) pour x∈R, ou D∈C 1 [0,1] est concave et satisfait D>0 dans (0,1) et D(0)=D(1)=0, et φ∈C 1 [−1,1] est telle que φ>0 dans (−1,1) et φ(−1)=φ(+1)=0

SIMILARITY SOLUTIONS FOR GRAVITY-DOMINATED SPREADING OF A LENS OF ORGANIC CONTAMINANT

Similarity solutions are developed for gravity-dominated spreading at the water table of a lens of organic liquid largely immiscible with water. Different solutions deal with different mechanisms by which the lens volume decreases with time: dissolution or evaporation, trapping as water invades the region beneath the central part of the lens, and uniform degradation. In the last case, the solution is a special case of the Barenblatt-Pattle solution. The same is true for the limiting case of a lens of constant volume. In all cases information is presented on variation of lens radius and thickness with time. All solutions except that involving trapping can be carried over with minor modifications to the case of a lens simultaneously spreading and translating along a slightly inclined water table.