[5.0.1.1] The constitutive theory proposed above, contrary to the
traditional theory, does not postulate a unique
capillary pressure as a constitutive parameter function.
[5.0.1.2] On the other hand experimental evidence suggests that
capillary pressure is a useful concept to correlate
observations.
[5.0.1.3] To make contact with the established traditional theory
it is therefore important to check
whether the traditional

[5.0.2.1] Consider first the case of hydrostatic equilibrium
where

(31a) | |||

(31b) |

where

(32) | |||

implying the existence of a unique
hydrostatic saturation profile

[5.0.3.1] The nonlinear theory proposed here can be solved in the
special case of hydrostatic equilibrium.
[5.0.3.2] Mass balance (1) now implies

(33a) | |||

(33b) | |||

(33c) | |||

(33d) | |||

If one identifies

[5.1.1.1]
While it is not possible to identify a unique

(34a) | |||

(34b) | |||

(34c) | |||

(34d) |

Momentum balance becomes in the RDA

(35a) | |||

(35b) | |||

(35c) | |||

(35d) |

where the abbreviations

(36a) | |||

(36b) |

were used.
[5.1.1.5] Equations (34) and (35)
together with eq.(16b) provide
17 equations for 12 variables (

[5.1.2.1] Equations (34) and (35)
can now be compared to the traditional equations
(11)–(13) with the aim
of identifying capillary pressure and relative
permeability.
[5.1.2.2] Consider first the momentum balance eqs. (35).
[5.1.2.3] As in the traditional theory [24] viscous decoupling
is assumed to hold, i.e.

(37a) | |||

(37b) | |||

(37c) | |||

(37d) |

where barycentric velocities

(38a) | |||

(38b) |

have been introduced. [6.0.0.1] Subtracting eq. (37a) from eq. (37c), as well as eq. (37d) from eq. (37b), and equating the result gives

(39) |

where eq. (22) has also been employed. [6.0.0.2] This result can be compared to the traditional theory where one finds from eqs. (12) and (13)

(40) |

Again this seems to imply

[6.0.1.1] First, observe that adding (34a) to (34b) resp. (34c) to (34d) with the help of eq. (38a) yields the traditional mass balance eqs. (11). [6.0.1.2] Next, verify by insertion that eqs. (34b) and (34d) admit the solutions

(41a) | |||

(41b) | |||

where the displacement process is assumed to start from the initial conditions

(42a) | |||

(42b) | |||

(42c) |

at some initial instant

(43a) | |||

(43b) | |||

(43c) | |||

(43d) |

for imbibition processes (i.e.

(44a) | |||

(44b) | |||

(44c) | |||

(44d) |

for drainage processes (i.e.

[6.1.1.1] With these solutions in hand the capillary pressure can be identified up to a constant as

(45) | |||

where

(46a) | |||

(46b) |

where

[6.1.2.1] Figure 1 visualizes the results obtained by fitting
eq. (45) to experiment.
[6.1.2.2] The experimental results are depicted as
triangles (primary drainage) and squares (imbibition).
[6.1.2.3] The experiments were performed in a medium grained
unconsolidated water wet sand of porosity

[7.0.1.1] If it is further assumed that the medium is isotropic and
that the matrices

(47a) | |||

(47b) |

then the relative permeability functions are
obtained from eqs. (46).
[7.0.1.2] The result for the special case

[7.0.2.1] Note that Figures 1 through 4 show
a total of 30 different scanning curves, 5 drainage
and 5 imbibition scanning curves each for