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

(6.1) |

and the incompressibility conditions

(6.2) |

where

The microscopic formulation is completed by specifiying an initial
fluid distribution

(6.3) |

as well as for the fluid-fluid interface,

(6.4) |

combined with stress-balance across the fluid-fluid interface,

(6.5) |

Here

(6.6) |

where the symmetrization operator

(6.7) |

on the matrix

The pore space boundary

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

(6.8) |

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 [344].
In that case

(6.9) |

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
flow.
The pheneomenological slipping models describe the region around
the contact line microscopically. The typical size of this region,
called the “slipping length”, is around

Given a microscopic model for contact line slipping the next step is to evaluate the relative importance of the different terms in the equations of motion at the pore scale. This is done by casting them into dimensionless form using the definitions

(6.10) |

(6.11) |

(6.12) |

(6.13) |

(6.14) |

(6.15) |

where

With these definitions the dimensionless equations of motion on the pore scale can be written as

(6.16) |

(6.17) |

with dimensionless boundary conditions

(6.18) |

(6.19) |

(6.20) |

In these equations the microscopic dimensionless ratio

(6.21) |

is the Reynolds number, and

(6.22) |

is the kinematic viscosity which may be interpreted as a specific action or a specific momentum transfer. The other fluid dynamic numbers are defined as

(6.23) |

for the Froude number, and

(6.24) |

for the Weber number. The corresponding dimensionless ratios for the oil phase are related to those for the water phase as

(6.25) |

(6.26) |

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.

Typical pore sizes in an oil reservoir are of order

(6.27) |

where

(6.28) |

is the microscopic capillary number of water, and

(6.29) |

is the microscopic “gravity number” of water. The capillary number is a measure of velocity in units of

(6.30) |

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

(6.31) | ||||

(6.32) |

Many other dimensionless ratios may be defined. Of general interest are dimensionless space and time variables. Such ratios are formed as

(6.33) |

which has been called the “gravillary number” [47, 48].
The gravillary number becomes the better known bond number if the
density

(6.34) |

separates capillary waves with wavelengths below

(6.35) |

where

(6.36) |

is a characteristic time after which the influence of gravity
dominates viscous and capillary effects.
The reader is cautioned not to misinterpret the value of

Table V collects definitions and estimates for the dimensionless groups
and the numbers

Quantity | Definition | Estimate |
---|---|---|

For these estimates the values in Table IV together with the above
estimates of

(6.37) |

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 (