Category: Two-Phase Flow

Macroscopic capillarity without a constitutive capillary pressure function

Post author By R Hilfer
Post date January 20, 2006

R. Hilfer

Physica A 371, 209 (2006)
https://doi.org/10.1016/j.physa.2006.04.051

submitted on
Friday, January 20, 2006

This paper challenges the foundations of the macroscopic capillary pressure concept. The capillary pressure function, as it is traditionally assumed in the constitutive theory of two-phase immiscible displacement in porous media, relates the pressure difference between nonwetting and wetting fluid to the saturation of the wetting fluid. The traditional capillary pressure function neglects the fundamental difference between percolating and nonpercolating fluid regions as first emphasized in R. Hilfer [Macroscopic equations of motion for two phase flow in porous media, Phys. Rev. E 58 (1998) 2090]. The theoretical approach proposed here starts from residual saturations as the volume fractions of nonpercolating phases. The resulting equations of motion open the possibility to describe flow processes where drainage and imbibition occur simultaneously. The theory predicts hysteresis and process dependence of capillary phenomena. The traditional theory is recovered as a special case in the residual decoupling approximation. Explicit calculations are presented for quasistatic equilibrium profiles near hydrostatic equilibrium. The results are found to agree with experiment. r 2006 Elsevier B.V. All rights reserved.

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Simulating the Saturation Front Using a Fractional Diffusion Model

Post author By R Hilfer
Post date June 30, 2005

E. Gerolymatou, I. Vardoulakis, R. Hilfer

in: Proceedings of the GRACM05 International Congress on Computational Mechanics, Limassol 2005
edited by: G. Georgiou, P. Papanastasiou, M. Papadrakakis
GRACM, Athens, 653 (2005)

submitted on
Thursday, June 30, 2005

In this paper the possibility of making use of fractional derivatives for the simulation of the flow of water through porous media and in particular through soils is considered. The Richards equation, which is a non-linear diffusion equation, will be taken as a basis and is used for the comparison of results. Fractional derivatives differ from derivatives of integer order in that they entail the whole history of the function in a weighted form and not only its local behavior, meaning that a different numerical approach is required. Previous work on the topic will be examined and a consistent approach based on fractional time evolutions will be presented.

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Porous Media Two-Phase Flow

Macroscopic Capillarity and Hysteresis for Flow in Porous Media

Post author By R Hilfer
Post date May 27, 2005

R. Hilfer

Physical Review E 73, 016307 (2006)
https://doi.org/10.1103/PhysRevE.73.016307

submitted on
Friday, May 27, 2005

A macroscopic theory for capillarity in porous media is presented, challenging the established view that capillary pressure and relative permeability are constitutive parameter functions. The capillary pressure function in the present theory is not an input parameter but an outcome. The theoretical approach is based on introducing the residual saturations explicitly as state variables [as in Phys. Rev. E 58, 2090 (1998)]. Capillary pressure and relative permeability functions are predicted to exist for special cases. They exhibit hysteresis and process dependence as known from experiment.

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Porous Media Two-Phase Flow

Capillary Pressure, Hysteresis and Residual Saturation in Porous Media

Post author By R Hilfer
Post date May 20, 2005

R. Hilfer

Physica A 359, 119 (2006)
https://doi.org/10.1016/j.physa.2005.05.086

submitted on
Friday, May 20, 2005

A macroscopic theory for capillarity in porous media is presented. The capillary pressure function in this theory is not an input parameter but an outcome. The theory is based on introducing the trapped or residual saturations as state variables. It allows to predict spatiotemporal changes in residual saturation. The theory yields process dependence and hysteresis in capillary pressure as its main result.

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Heterogeneous Materials Porous Media Two-Phase Flow

Dimensional analysis and upscaling of two-phase flow in porous media with piecewise constant heterogeneities

Post author By R Hilfer
Post date March 15, 2004

R. Hilfer, R. Helmig

Advances in Water Resources 27, 1033 (2004)
https://doi.org/10.1016/j.advwatres.2004.07.003

submitted on
Monday, March 15, 2004

Dimensional analysis of the traditional equations of motion for two-phase flow in porous media allows to quantify the influence of heterogeneities. The heterogeneities are represented by position dependent capillary entry pressures and position dependent permeabilities. Dimensionless groups quantifying the influence of random heterogeneities are identified. For the case of heterogeneities with piecewise constant constitutive parameters (e.g. permeabilities, capillary pressures) we find that the upscaling ratio defined as the ratio of system size and the scale at which the constitutive parameters are known has to be smaller than the fluctuation strength of the heterogeneities defined e.g. as the ratio of the standard deviation to the mean value of a fluctuating quantity.

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Porous Media Two-Phase Flow

Macroscopic Two Phase Flow in Porous Media

Post author By R Hilfer
Post date July 6, 1999

R. Hilfer, H. Besserer

Physica B 279, 125 (2000)
https://doi.org/10.1016/S0921-4526(99)00694-8

submitted on
Tuesday, July 6, 1999

A system of macroscopic equations for two-phase immiscible displacement in porous media is presented. The equations are based on continuum mixture theory. The pairwise character of interfacial energies is explicitly taken into account. The equations incorporate the spatiotemporal variation of interfacial energies and residual saturations. The connection between these equations and relative permeabilities is established, and found to be in qualitative agreement with experiment.

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Porous Media Two-Phase Flow

Trapping and Mobilization of Residual Fluid During Capillary Desaturation in Porous Media

Post author By R Hilfer
Post date April 21, 1998

L. Anton, R. Hilfer

Physical Review E 59, 6819 (1999)
https://doi.org/10.1103/PhysRevE.59.6819

submitted on
Tuesday, April 21, 1998

We discuss the problem of trapping and mobilization of nonwetting fluids during immiscible two-phase displacement processes in porous media. Capillary desaturation curves give residual saturations as a function of capillary number. Interpreting capillary numbers as the ratio of viscous to capillary forces the breakpoint in experimental curves contradicts the theoretically predicted force balance. We show that replotting the data against a novel macroscopic capillary number resolves the problem for discontinuous mode displacement.

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Porous Media Two-Phase Flow

Macroscopic Equations of Motion for Two Phase Flow in Porous Media

Post author By R Hilfer
Post date January 20, 1998

R. Hilfer

Physical Review E 58, 2090 (1998)
https://doi.org/10.1103/PhysRevE.58.2090

submitted on
Tuesday, January 20, 1998

The usual macroscopic equations of motion for two-phase immiscible displacement in porous media are known to be physically incomplete because they do not contain the surface tension and surface areas governing capillary phenomena. Therefore, a more general system of macroscopic equations is derived here that incorporates the spatiotemporal variation of interfacial energies. These equations are based on the theory of mixtures in macroscopic continuum mechanics. They include wetting phenomena through surface tensions instead of the traditional use of capillary pressure functions. Relative permeabilities can be identified in this approach that exhibit a complex dependence on the state variables. A capillary pressure function can be identified in equilibrium that shows the qualitative saturation dependence known from experiment. In addition, the proposed equations include a description of the spatiotemporal changes of residual saturations during immiscible displacement.

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Tags capillarity

fluid flow Porous Media Two-Phase Flow

Old Problems and New Solutions for Multiphase Flow in Porous Media

Post author By R Hilfer
Post date November 20, 1997

R. Hilfer, H. Besserer

in: Porous Media: Physics, Mo\-dels, Simulation
edited by: A. Dmitrievsky and M. Panfilov
World Scientific Publ. Co., Singapore, 133-144 (2000)
https://doi.org/10.1142/9789812817617_0008
ISBN: 978-981-02-4126-1

submitted on
Thursday, November 20, 1997

The existing macroscopic equations of motion for multiphase flow in porous media are unsatisfactory in two general respects. On the one hand characteristic experimental features, such as relationships between capillary pressure and saturations, cannot be predicted. On the other hand the theoretical derivation of the equations from the well-known laws of hydrodynamics has not yet been accomplished. In this paper we discuss these deficiencies and present an alternative description which is based on energy balances. Our description includes surface tensions as parameters and interface areas as a new macroscopic state variable. The equations are obtained from general multiphase mixture theory by explicitly accounting for the pairwise character of interfacial energies. For the special case of two immiscible fluids in a porous medium the most important ingredient is the distinction between a connected and a disconnected subphase of each fluid phase. In this way it becomes possible to handle also the spatiotemporal variation of residual saturations. The connection between the new approach and the established formulation is given by identifying a generalized Darcy Law with generalized relative permeabilities. The new equations reproduce qualitatively the saturation dependent behaviour of capillary pressure in gravitational equilibrium.

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Tags capillarity, capillary pressure