[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 relation can be viewed as a derived concept within the new theory.

[5.0.2.1] Consider first the case of hydrostatic equilibrium where for all . [5.0.2.2] In hydrostatic equilibrium all fluids are at rest. [5.0.2.3] In this case the traditional theory implies and , by mass balance eq. (11). [5.0.2.4] The traditional momentum balance eqs. (12) can be integrated to give

(31a) | |||

(31b) |

where is a point in the boundary. [5.0.2.5] Combined with the assumption (13) one finds

(32) | |||

implying the existence of a unique hydrostatic saturation profile . [5.0.2.6] Here is the capillary pressure at . [5.0.2.7] Experiments show, however, that hydrostatic saturation profiles are not unique. [5.0.2.8] As a consequence the traditional theory employs multiple relations for drainage and imbibition, and this leads to difficult problems when imbibition and drainage occur simultaneously.

[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 for all . [5.0.3.3] Integrating eqs. (2) yields

(33a) | |||

(33b) | |||

(33c) | |||

(33d) | |||

If one identifies with and with then eqs. (33a) and (33b) suggest to identify as . [5.1.0.1] Then eqs. (33c) and (33d) combined with and imply . [5.1.0.2] The capillary pressure depends not only on but also on and in hydrostatic equilibrium. [5.1.0.3] In the theory proposed here it is not possible to identify a unique relation when all fluids are at rest. This agrees with experiment.

[5.1.1.1] While it is not possible to identify a unique relation in hydrostatic equilibrium such a functional relation emerges nevertheless from the present theory when the system approaches hydrostatic equilibrium in the residual decoupling approximation. [5.1.1.2] The approach to hydrostatic equilibrium in the residual decoupling approximation (RDA) can be formulated mathematically as and . [5.1.1.3] In addition it is assumed that the velocities are small but nonzero. [5.1.1.4] In the RDA mass balance becomes

(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 ( and ).

[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. and . [5.1.2.4] Next, assuming that , , and one finds [page 6, §0]

(37a) | |||

(37b) | |||

(37c) | |||

(37d) |

where barycentric velocities defined through

(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 as already found above for hydrostatic equilibrium. [6.0.0.3] However, within the RDA additional constraints follow from mass balance (34).

[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 . [6.1.0.1] The limiting saturations , , are given by eqs. (29). [6.1.0.2] They depend only on the sign of if can be assumed to hold. [6.1.0.3] One finds in this case

(43a) | |||

(43b) | |||

(43c) | |||

(43d) |

for imbibition processes (i.e. ), resp.

(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 and are given by eqs. (41). [6.1.1.2] This result holds in the RDA combined with the assumptions above. [6.1.1.3] Furthermore, equations (37a) and (37c) are recognized as generalized Darcy laws with relative permeabilities identified as

(46a) | |||

(46b) |

where and are again given by eqs. (41).

[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 . [6.1.2.4] Water was used as wetting fluid while air resp. TCE were used as the nonwetting fluid. [6.1.2.5] The experiments were carried out over a period of several weeks at the Versuchseinrichtung zur Grundwasser- und Altlastensanierung (VEGAS) [page 7, §0] at the Universität Stuttgart. [7.0.0.1] They are described in more detail in Ref. [25]. The parameters for all the curves shown in all four figures are , , , , , , , Pa, Pa, and Pa Pa.

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

(47a) | |||

(47b) |

then the relative permeability functions are obtained from eqs. (46). [7.0.1.2] The result for the special case is shown in Figures 3 and 4. [7.0.1.3] The parameters are chosen such that and , where are the fluid viscosities and is the absolute permeability of the medium. [7.0.1.4] All other parameters for the relative permeability functions shown in Figures 3 and 4 are identical to those of the capillary pressure curves in Figures 1 and 2.

[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 and . [7.1.0.1] In addition a total of 9 different bounding curves are displayed, namely the primary drainage, secondary drainage and secondary imbibition curve for and . [7.1.0.2] Three more bounding curves namely primary imbibition for and starting from are not shown because they are difficult to obtain experimentally for a water-wet sample. [7.1.0.3] Of course the number of scanning curves can be increased indefinitely. [7.1.0.4] All of these curves have the same values of the constitutive parameters. [7.1.0.5] There is less than one parameter per curve. [7.1.0.6] The curves shown in the figures exhibit the full range of hysteretic phenomena known from experiment. [7.1.0.7] Nevertheless it should be kept in mind that these curves are obtained only under special approximations, and when these are not valid such curves do not exist.