[page 145, §1]
[145.1.1] The porous medium is assumed to be macroscopically homogeneous
![]() |
(12) |
although this assumption rarely holds in practice [26]. [145.1.2] Let us further assume that the fluids are incompressible so that
![]() |
![]() |
(13a) | |
![]() |
![]() |
(13b) | |
![]() |
![]() |
(13c) | |
![]() |
![]() |
(13d) |
where the constants are independent
of
and
.
[145.2.1] Flows through porous media often have low Reynolds numbers. [145.2.2] Thus accelerations and the inertial term
![]() |
(14) |
can be neglected in the momentum balance equation (10).
[145.3.1] The momentum transfer into phase from all the other phases
is assumed to arise from viscous drag,
![]() |
(15) |
with resistance coefficients quantifying
the loss due to viscous friction between phase
and
.
[145.3.2] The matrix is assumed to be rigid so that
.
[145.3.3] Hence
is the momentum transfer from the
wall into phase
.
[145.3.4] Then
![]() |
![]() |
(16a) | |
![]() |
![]() |
(16b) | |
![]() |
![]() |
(16c) | |
![]() |
![]() |
(16d) |
where and
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
may be rewritten
in term of dimensionless coefficients
as
![]() |
![]() |
(17a) | |
![]() |
![]() |
(17b) |
where are the viscosities of water and oil,
is the absolute permeability tensor of the medium,
and
are dimensionless viscous drag coefficients.
[145.3.6] Each
is a
-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
![]() |
![]() |
(18a) | |
![]() |
![]() |
(18b) | |
![]() |
![]() |
(18c) | |
![]() |
![]() |
(18d) |
where 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
![]() |
(19) |
with .
[146.2.2] The capillary body forces
are responsible
for keeping the trapped fluids inside the medium.
[146.2.3] They are assumed to be potential forces
![]() |
(20) |
where are the capillary potentials.
[146.3.1] One has considerable freedom to specify the capillary
stresses and potentials
.
[146.3.2] General thermodynamic considerations suggest ideas
to restrict this freedom.
[146.3.3] Let
denote the total Helmholtz free energy of the
system with and oil-water interface, and let
and
denote the individual Helmholtz free energies
of bulk water and bulk oil.
[146.3.4] Then [27, 28]
![]() |
(21) |
where are the oil and water pressure,
are the volumes of oil and water, and
are the total
interfacial areas between oil and water, water and matrix, resp. oil and matrix.
[146.3.5] The oil-water surface tension
and the fluid-matrix
interfacial tensions
are related by Youngs equation
![]() |
(22) |
where is the contact angle of water.
[146.3.6] The interfacial areas obey
![]() |
![]() |
(23) | |
![]() |
![]() |
(24) | |
![]() |
![]() |
(25) | |
![]() |
![]() |
(26) |
where is the total interfacial area between phase
and
,
and the volumes are related by eqs. (5b) and
(5c).
[page 147, §1]
[147.1.1] In equilibrium holds.
[147.1.2] Also, sample volume and internal surface are constant
because the porous medium is rigid.
[147.1.3] This implies
and
.
[147.1.4] Using eq. (5c) one arrives at
![]() |
(27) |
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
![]() |
![]() |
(28a) | |
![]() |
![]() |
(28b) | |
![]() |
![]() |
(28c) | |
![]() |
![]() |
(28d) |
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
![]() |
![]() |
(29a) | |
![]() |
![]() |
(29b) | |
![]() |
![]() |
(29c) | |
![]() |
![]() |
(29d) |
[147.3.1] The mass transfer rates are
where
is the mass transfer rate from phase
into phase
.
[147.3.2] Neglecting chemical reactions one assumes
for all pairs
except the
pairs
.
[147.3.3] These remaining transfer rates are assumed to be given as
![]() |
![]() |
(30a) | |
![]() |
![]() |
(30b) |
[page 148, §0] with
![]() |
![]() |
(31a) | |
![]() |
![]() |
(31b) | |
![]() |
![]() |
(31c) | |
![]() |
![]() |
(31d) |
as in [22, 23, 24].
[148.0.1] The limiting saturations for ,
called irreducible water resp. residual oil saturation,
![]() |
![]() |
(32a) | |
![]() |
![]() |
(32b) |
are velocity dependent, because they depend on
the velocity dependent “reaction rates”
.
[148.0.2] The relation between residual oil saturation
and
flow velocity is also known as capillary correlation or
capillary desaturation curve [29, 30, 31].
[148.0.3] The factors
are defined as
![]() |
![]() |
(33a) | |
![]() |
![]() |
(33b) |
where denotes the contact angle of water,
![]() |
![]() |
(34a) | |
![]() |
![]() |
(34b) |
are the low velocity limits of , and
![]() |
(35) |
denotes the Heaviside unit step function.
[148.0.4] The velocity dependent “reaction rates” are
chosen such that they vanish for vanishing velocities.
[148.0.5] In this paper it will be assumed that
[page 149, §0]
![]() |
![]() |
(36a) | |
![]() |
![]() |
(36b) | |
![]() |
![]() |
(36c) | |
![]() |
![]() |
(36d) |
consistent with eq. (34).
[149.0.1] The parameters are time scales, and
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 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 depend on saturation.
[149.2.2] Here it is assumed that
![]() |
(37) |
[149.2.3] The dependence of
is suggested by inverting the classic hydraulic radius
theory
![]() |
(38) |
where and
.
[149.3.1] Finally, the system is closed selfconsistently using the condition
![]() |
![]() |
||
![]() |
|||
![]() |
|||
![]() |
(39) |
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]).