4 Constitutive Assumptions

[page 145, §1]   
[145.1.1] The porous medium is assumed to be macroscopically homogeneous

although this assumption rarely holds in practice [26]. [145.1.2] Let us further assume that the fluids are incompressible so that

where the constants ϱW,ϱO are independent of x and t.

[145.2.1] Flows through porous media often have low Reynolds numbers. [145.2.2] Thus accelerations and the inertial term

can be neglected in the momentum balance equation (10).

[145.3.1] The momentum transfer into phase i from all the other phases is assumed to arise from viscous drag,

with resistance coefficients Ri⁢j quantifying the loss due to viscous friction between phase i and j. [145.3.2] The matrix is assumed to be rigid so that v5=0. [145.3.3] Hence -Ri⁢5⁢vi is the momentum transfer from the wall into phase i. [145.3.4] Then

where R12=0 and R34=0 was used because there is no common interface and hence no direct viscous interaction between these phase pairs. [145.3.5] The viscous resistance coefficients Ri⁢j may be rewritten in term of dimensionless coefficients ri⁢j as

where μW,μO are the viscosities of water and oil, k is the absolute permeability tensor of the medium, and ri⁢j are dimensionless viscous drag coefficients. [145.3.6] Each Ri⁢j is a 3×3-matrix. [145.3.7] In practice viscous coupling terms between the two fluid phases are often neglected.

[page 146, §1]    [146.1.1] The stress tensor is written as a pressure term plus a capillary correction term. [146.1.2] The reference pressure for the nonpercolating phases is the pressure of the surrounding percolating phase [24]. [146.1.3] Thus

where Σci are capillary stresses resulting from the presence of fluid-fluid and fluid-matrix interfaces.

[146.2.1] Similarly, the body forces are augmented with capillary body forces as

with i=1,2,3,4. [146.2.2] The capillary body forces Fci are responsible for keeping the trapped fluids inside the medium. [146.2.3] They are assumed to be potential forces

where Πci are the capillary potentials.

[146.3.1] One has considerable freedom to specify the capillary stresses Σci and potentials Πci. [146.3.2] General thermodynamic considerations suggest ideas to restrict this freedom. [146.3.3] Let F denote the total Helmholtz free energy of the system with and oil-water interface, and let FW and FO denote the individual Helmholtz free energies of bulk water and bulk oil. [146.3.4] Then [27, 28]

where PW,PO are the oil and water pressure, VW,VO are the volumes of oil and water, and A~W⁢O,A~W⁢M,A~O⁢M are the total interfacial areas between oil and water, water and matrix, resp. oil and matrix. [146.3.5] The oil-water surface tension σW⁢O and the fluid-matrix interfacial tensions σW⁢M,σO⁢M are related by Youngs equation

where ϑ is the contact angle of water. [146.3.6] The interfacial areas obey

where A~i⁢j is the total interfacial area between phase i and j, and the volumes are related by eqs. (5b) and (5c).

[page 147, §1]    [147.1.1] In equilibrium d⁢F=0 holds. [147.1.2] Also, sample volume and internal surface are constant because the porous medium is rigid. [147.1.3] This implies d⁢V=0 and d⁢A~P⁢M=0. [147.1.4] Using eq. (5c) one arrives at

where Youngs equation (22) was also used.

[147.2.1] These considerations suggest one particular way to specify the capillary stresses and potentials. [147.2.2] Following earlier ideas [24] the capillary stresses are specified as

where local equilibrium was assumed and intensive quantities (per unit volume of porous medium) were introduced. [147.2.3] The capillary potentials may be associated with the last term in eq. (27). [147.2.4] They are specified as

in analogy with [23, 24].

[147.3.1] The mass transfer rates are Mi=∑j=15Mi⁢j where Mi⁢j is the mass transfer rate from phase j into phase i. [147.3.2] Neglecting chemical reactions one assumes Mi⁢j=0 for all pairs i,j except the pairs 1,2,2,1,3,4,4,3. [147.3.3] These remaining transfer rates are assumed to be given as

[page 148, §0]    with

as in [22, 23, 24]. [148.0.1] The limiting saturations for S2,S4, called irreducible water resp. residual oil saturation,

are velocity dependent, because they depend on the velocity dependent “reaction rates” bi⁢v1,v3,i=1,2,3,4. [148.0.2] The relation between residual oil saturation SO⁢r and flow velocity is also known as capillary correlation or capillary desaturation curve [29, 30, 31]. [148.0.3] The factors ΘW*,ΘO* are defined as

where ϑ denotes the contact angle of water,

are the low velocity limits of SW⁢i,SO⁢r, and

denotes the Heaviside unit step function. [148.0.4] The velocity dependent “reaction rates” bi are chosen such that they vanish for vanishing velocities. [148.0.5] In this paper it will be assumed that [page 149, §0]   

consistent with eq. (34). [149.0.1] The parameters τW,τO are time scales, and η2,η4,bi⁢j are dimensionless constants.

[149.1.1] The first terms in the curly brackets of (30) model an equilibrium reaction between nonpercolating and percolating fluids. [149.1.2] The reaction, i.e. breakup and coalescence, takes only place when both percolating phases move, i.e. have nonvanishing velocity. [149.1.3] The prefactors ΘW*,ΘO* reproduce the experimental observation that nonpercolating nonwetting fluid phases show little breakup or coalescence below the low velocity limit of the residual nonwetting saturation. [149.1.4] The prefactors also ensure that sign and dimensions are correct.

[149.2.1] The specific internal surfaces AW⁢O,AW⁢M depend on saturation. [149.2.2] Here it is assumed that

[149.2.3] The dependence of AW⁢O⁢S1,S2,S3,S4 is suggested by inverting the classic hydraulic radius theory

where A1=A31+A41+A51 and A3=A31+A32+A35.

[149.3.1] Finally, the system is closed selfconsistently using the condition

written here in its most general form. [149.3.2] It is obtained by demanding that the closure condition should be consistent with the capillary pressure saturation relation obtained in the residual decoupling limit (see [22, 23, 24]).