A Theoretical Model of Hysteresis and Dynamic Effects in the Capillary Relation for Two-phase Flow in Porous Media

It is well known that the relationship between capillary pressure and saturation, in two-phase flow problems demonstrates memory effects and, in particular, hysteresis. Explicit representation of full hysteresis with a myriad of scanning curves in models of multiphase flow has been a difficult problem. A second complication relates to the fact that Pc–S relationships, determined under static conditions, are not necessarily valid in dynamics. There exist Pc–S relationships which take into account dynamic effects. But the combination of hysteretic and dynamic effects in the capillary relationship has not been considered yet. In this paper, we have developed new models of capillary hysteresis which also include dynamic effects. In doing so, thermodynamic considerations are employed to ensure the admissibility of the new relationships. The simplest model is constructed around main imbibition and drainage curves and assumes that all scanning curves are vertical lines. The dynamic effect is taken into account by introducing a damping coefficient in Pc–S equation. A second-order model of hysteresis with inclined scanning curves is also developed. The simplest version of proposed models is applied to two-phase incompressible flow and an example problem is solved.

[1]  A. Warrick.  Physical Principles of Water Percolation and Seepage , 1968 .

[2]  R. Hill.  The mathematical theory of plasticity , 1950 .

[3]  M. Krasnosel’skiǐ,et al.  Systems with Hysteresis , 1989 .

[4]  J. Philip.  Horizontal redistribution with capillary hysteresis , 1991 .

[5]  William G. Gray,et al.  Thermodynamic basis of capillary pressure in porous media , 1993 .

[6]  Norman R. Morrow,et al.  Capillary Equilibrium in Porous Materials , 1965 .

[7]  J. Meixner,et al.  S. Flügge, Herausgeber: Handbuch der Physik, Band III/3: Die nicht‐linearen Feldtheorien der Mechanik. Von C. Truesdell und W. Noll. Springer‐Verlag, Berlin/Heidelberg/New York 1965. VIII/602 Seiten. Preis: 198,‐ DM , 1967, Berichte der Bunsengesellschaft für physikalische Chemie.

[8]  Georges Vachaud,et al.  A Test of the Uniqueness of the Soil Moisture Characteristic During Transient, Nonhysteretic Flow of Water in a Rigid Soil , 1971 .

[9]  Y. Mualem,et al.  A conceptual model of hysteresis , 1974 .

[10]  Y. Mualem,et al.  Modified approach to capillary hysteresis based on a similarity hypothesis , 1973 .

[11]  Walter Noll,et al.  The thermodynamics of elastic materials with heat conduction and viscosity , 1963 .

[12]  Georges Vachaud,et al.  A Study of the Uniqueness of the Soil Moisture Characteristic During Desorption by Vertical Drainage , 1972 .

[13]  A. Klute,et al.  Comparison of Water Content‐Pressure Head Data Obtained by Equilibrium, Steady‐State, and Unsteady‐State Methods , 1967 .

[14]  Two-Phase Flow in Double Porosity Media , 2000 .

[15]  William G. Gray,et al.  Paradoxes and Realities in Unsaturated Flow Theory , 1991 .

[16]  A. Visintin.  Differential models of hysteresis , 1994 .

[17]  William G. Gray,et al.  Macroscale continuum mechanics for multiphase porous-media flow including phases, interfaces, common lines and common points , 1998 .

[18]  Walter Noll,et al.  The thermodynamics of elastic materials with heat conduction and viscosity , 1963 .

[19]  R. E. Collins.  Flow of fluids through porous materials , 1961 .

[20]  William G. Gray,et al.  General conservation equations for multi-phase systems: 3. Constitutive theory for porous media flow. , 1980 .

[21]  William G. Gray,et al.  Mechanics and thermodynamics of multiphase flow in porous media including interphase boundaries , 1990 .

[22]  Michael A. Celia,et al.  A Functional Relationship Between Capillary Pressure, Saturation, and Interfacial Area as Revealed by a Pore‐Scale Network Model , 1996 .