Important features of multiphase flow in porous media that distinguish it from single-phase flow are the presence of interfaces between the fluid phases and of common lines where three phases come in contact. Despite this fact, mathematical descriptions of these flows have been lacking in rigor, consisting primarily of heuristic extensions of Darcy's law that include a hysteretic relation between capillary pressure and saturation and a relative permeability coefficient. As a result, the standard capillary pressure concept appears to have physically unrealistic properties. The present paper employs microscopic mass and momentum balance equations for phases and interfaces to develop an understanding of capillary pressure at the microscale. Next, the standard theories and approaches that define capillary pressure at the macroscale are described and their shortcomings are discussed. Finally, an approach is presented whereby capillary pressure is shown to be an intrinsic property of the system under study. In particular, the presence of interfaces and their distribution within a multiphase system are shown to be essential to describing the state of the system. A thermodynamic approach to the definition of capillary pressure provides a theoretically sound alternative to the definition of capillary pressure as a simple hysteretic function of saturation.

http://www.soilmech.polito.it/content/download/169/887/version/1/file/Hass_Gray_WRR93_Basis+of+Pc.pdf

Thermodynamic basis of capillary pressure in porous media

Review of gas diffusion cathodes for alkaline fuel cells

Activity of comets: Gas transport in the near-surface porous layers of a cometary nucleus

A two-phase relative permeability correlation for mixed-wet rock is presented and validated. It includes provisions for bounding drainage and imbibition processes and scanning hysteresis loops, and is inferred from a capillary pressure correlation. The well-known Corey-Burdine relative permeabilities were developed for water-wet rock from a Brooks-Corey power-law capillary pressure correlation and a bundle-of-tubes network model. We have extended this correlation to mixed-wet rock and now propose the ensuing relative permeability correlation for mixed-wet reservoirs. The functional form is symmetric with respect to fluid-dependent properties, because neither fluid has precedence in a mixed-wet environment. It reverts to the standard Corey-Burdine correlation for the completely wateror oil-wet cases, and exhibits the following characteristics in agreement with reported experiments: first, water-wet behavior at low water saturations and oil-wet behavior at low oil saturations; second, an inverted S-shape oil relative permeability curve with an inflection point; and, third, closed hysteresis scanning loops.

Relative Permeability Correlation for Mixed-Wet Reservoirs

A pore network model is used to describe constitutive relationships between saturation, capillary pressure, relative permeabilities, and interfacial areas over the full range of saturations. We focus on residual nonwetting-phase saturations, that is, the saturation range between main drainage and primary drainage. Interphase mass transfer and dynamic miscible transport are modeled to generate the range of saturations over which the extended constitutive relationships are calculated. To accommodate all saturations in a consistent way, we define capillary pressure as the areal average of local capillary pressures associated with each fluid-fluid interface. The extended constitutive relationships provide input for continuum-scale models covering the entire saturation range and indicate that special care is needed in the general formulation of continuum-scale equations.

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2000WR900234

Pore‐scale modeling extension of constitutive relationships in the range of residual saturations

1.

A necessary and sufficient condition for piercing of a porous medium was obtained.




2.

The degree of filling of the porous medium at very great distances with piercing was found.




3.

The form of the function of filling of the medium with gas at short distances was found.

Capillary equilibrium in porous media communication 2. Solution of equations

The role of a change in the radius of the pores along the length of the pores in capillary equilibrium it a porous body has been investigated. In a model cn non-intersecting pores consisting of successively arranged units with random values of the length and radius, the distribution of the length of the gas part of the pores has, at least at high pressures, an exponential character.

The properties of the interfacial boundary in one model of a porous body

1.

In the case of a uniform distribution of pores in all directions, the tortuosity is equal to two.




2.

Formulas were derived for the parameters of branching and sealing of the pores, relating them to the porosity, tortuosity, and distribution of pores according to radii.




3.

A numerical calculation was made of the parameters of branching and sealing for four concrete porous media.




4.

The depth at which uniform filling of the porous medium is established was found. This depth proves to be a maximum when the probability of branching of the pores is equal to the probability of sealing.




5.

A method was proposed for correcting the data of mercury porometry.

Capillary equilibrium in porous media Communication 3. Characteristics of porous media

1.

A model of a porous medium was formulated.




2.

The degree of filling of a porous medium by gas at various distances in the presence of equilibrium between the gas and wetting liquid was found.




3.

A method has been indicated for calculating the parameters of porous media according to the known characteristics.




4.

A method has been indicated for correcting the experimental capillary pressure curves according to Ritter and Drake.

Capillary equilibrium in the model of branching pores of variable cross section

1.

The problem of equilibrium between liquid and gas filling a porous medium was formulated.




2.

A model of a porous medium, considering both the change in the cross section of the pores along their length and their intersection, with the intersection of pores taking the form of three-way junctions, was formulated.




3.

A method of cycles representing a method of calculating the probability of filling according to parts was proposed to calculate the conditional probability of filling of a pore with a gas under the condition that this pore is supercritical.




4.

Equations were derived for all cycles, and a function relating the final summary probabilities of filling of the pores and junctions was obtained.

V. S. Markin

Papers

Capillary equilibrium in porous media communication 2. Solution of equations

The properties of the interfacial boundary in one model of a porous body

Capillary equilibrium in porous media Communication 3. Characteristics of porous media

Capillary equilibrium in the model of branching pores of variable cross section

Capillary equilibrium in porous media. Communication 1. Formulation of the problem and derivation of equations