VI.B.1 Microscopic Equations of Motion
Microscopic equations of motion for two-phase flow in porous media are
commonly given as Stokes (or Navier-Stokes) equations for two
incompressible Newtonian fluids with no-slip and stress-balance
boundary conditions at the interfaces [342, 270, 322].
In the following the wetting fluid
(water) will be denoted by a subscript while the nonwetting
fluid (oil) is indexed with . The solid rock matrix, indexed
as , is assumed to be porous and rigid. It fills a closed subset
of three dimensional space. The pore space is
filled with the two fluid phases described by the two closed subsets
which are in general time
dependent, and related to each other through the condition
Note that is independent of time because is rigid
while and are not.
The rigid rock surface will be denoted as , and the
mobile oil-water interface as .
A standard formulation of pore scale equations of motion for
two incompressible and immiscible fluids flowing through a porous
medium are the Navier-Stokes equations
and the incompressibility conditions
where are the velocity fields
for water and oil, are the
pressure fields in the two phases,
the densities, the dynamic viscosities, and
the gravitational constant. The vector denotes
the coordinate vector, is the time,
the gradient operator, the Laplacian and the superscript
denotes transposition. The gravitational force is directed
along the -axis and it represents an external body force.
Although gravity effects are often small for pore scale processes
(see eq. (6.37) below), there has recently been
a growing interest in modeling gravity effects also at the pore
scale [343, 245, 246, 42].
VI.B.2 The Contact Line Problem
The pore scale equations of motion given in the preceding
section contain a self contradiction. The problem
arises from the system of contact lines defined as
on the inner surface of the porous medium. The contact lines
must in general slip across the surface of the rock in direct
contradiction to the no-slip boundary condition Eq. (6.3).
This selfcontradiction is not specific for flow in porous media
but exists also for immiscible two phase flow in a tube or in
other containers [344, 345, 346].
There exist several ways out of this classical dilemma depending
on the wetting properties of the fluids. For complete and uniform
wetting a microscopic precursor film of water wets the entire
rock surface .
In that case and thus
the problem does not appear.
For other wetting properties a phenomenological slipping model
for the manner in which the slipping occurs at the contact line
is needed to complete the pore scale description of two phase
The pheneomenological slipping models describe the region around
the contact line microscopically. The typical size of this region,
called the “slipping length”, is around m.
Therefore the problem of contact lines is particularly acute
for immiscible displacement in microporous media, and the
Navier-Stokes description of the previous section
does not apply for such media.
VI.B.3 Microscopic Dimensional Analysis
With these definitions the dimensionless equations of motion on the
pore scale can be written as
with dimensionless boundary conditions
In these equations the microscopic dimensionless ratio
is the Reynolds number, and
is the kinematic viscosity which may be interpreted as a specific action
or a specific momentum transfer. The other fluid dynamic numbers are
for the Froude number, and
for the Weber number. The corresponding dimensionless ratios
for the oil phase are related to those for the water phase as
by viscosity and density ratios.
Table IV gives approximate values for densities, viscosities and
surface tensions under reservoir conditions [47, 48].
In the following these
values will be used to make order of magnitude estimates.
Table IV: Order of magnitude estimates for densities,
viscosities and surface tension of oil and water under
Typical pore sizes in an oil reservoir are of order
and microscopic fluid velocities
for reservoir floods range around
. Combining these
estimates with those of Table IV shows that the dimensionless ratios
obey . Therefore, the
pore scale equations (6.16) reduce to the simpler Stokes form
is the microscopic capillary number of water, and
is the microscopic “gravity number” of water. The capillary number
is a measure of velocity in units of
a characteristic velocity at which the coherence of the oil-water
interface is destroyed by viscous forces. The capillary and gravity
numbers for the oil phase can again be expressed through density
and viscosity ratios as
Table V collects definitions and estimates for the dimensionless groups
and the numbers and characterizing the oil-water system.
For these estimates the values in Table IV together with the above
estimates of and have been used. Table V shows that
and hence capillary forces dominate on the pore scale
[333, 2, 47, 48].
From the Stokes equation (6.27) it follows immediately that for low
capillary number floods () the viscous term
as well as the shear term in the boundary condition (6.20) become
negligible. Therefore the velocity field drops out, and the problem
reduces to finding the equilibrium capillary pressure field.
The equilibrium configuration of the oil-water interface then defines
timeindependent pathways for the flow of oil and water.
Hence, for flows with microscopic capillary numbers
an improved methodology for a quantitative description of
immiscible displacement from pore scale physics requires
improved calculations of capillary pressures from the pore
scale, and much research is devoted to this topic
[347, 348, 246a].