[184.108.40.206] The constitutive theory proposed above, contrary to the traditional theory, does not postulate a unique capillary pressure as a constitutive parameter function. [220.127.116.11] On the other hand experimental evidence suggests that capillary pressure is a useful concept to correlate observations. [18.104.22.168] 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.
[22.214.171.124] Consider first the case of hydrostatic equilibrium where for all . [126.96.36.199] In hydrostatic equilibrium all fluids are at rest. [188.8.131.52] In this case the traditional theory implies and , by mass balance eq. (11). [184.108.40.206] The traditional momentum balance eqs. (12) can be integrated to give
where is a point in the boundary. [220.127.116.11] Combined with the assumption (13) one finds
implying the existence of a unique hydrostatic saturation profile . [18.104.22.168] Here is the capillary pressure at . [22.214.171.124] Experiments show, however, that hydrostatic saturation profiles are not unique. [126.96.36.199] 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.
If one identifies with and with then eqs. (33a) and (33b) suggest to identify as . [188.8.131.52] Then eqs. (33c) and (33d) combined with and imply . [184.108.40.206] The capillary pressure depends not only on but also on and in hydrostatic equilibrium. [220.127.116.11] In the theory proposed here it is not possible to identify a unique relation when all fluids are at rest. This agrees with experiment.
[18.104.22.168] 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. [22.214.171.124] The approach to hydrostatic equilibrium in the residual decoupling approximation (RDA) can be formulated mathematically as and . [126.96.36.199] In addition it is assumed that the velocities are small but nonzero. [188.8.131.52] In the RDA mass balance becomes
Momentum balance becomes in the RDA
where the abbreviations
[184.108.40.206] Equations (34) and (35) can now be compared to the traditional equations (11)–(13) with the aim of identifying capillary pressure and relative permeability. [220.127.116.11] Consider first the momentum balance eqs. (35). [18.104.22.168] As in the traditional theory  viscous decoupling is assumed to hold, i.e. and . [22.214.171.124] Next, assuming that , , and one finds [page 6, §0]
where barycentric velocities defined through
Again this seems to imply as already found above for hydrostatic equilibrium. [126.96.36.199] However, within the RDA additional constraints follow from mass balance (34).
[188.8.131.52] First, observe that adding (34a) to (34b) resp. (34c) to (34d) with the help of eq. (38a) yields the traditional mass balance eqs. (11). [184.108.40.206] Next, verify by insertion that eqs. (34b) and (34d) admit the solutions
where the displacement process is assumed to start from the initial conditions
at some initial instant . [220.127.116.11] The limiting saturations , , are given by eqs. (29). [18.104.22.168] They depend only on the sign of if can be assumed to hold. [22.214.171.124] One finds in this case
for imbibition processes (i.e. ), resp.
for drainage processes (i.e. ).
[126.96.36.199] With these solutions in hand the capillary pressure can be identified up to a constant as
where and are given by eqs. (41). [188.8.131.52] This result holds in the RDA combined with the assumptions above. [184.108.40.206] Furthermore, equations (37a) and (37c) are recognized as generalized Darcy laws with relative permeabilities identified as
where and are again given by eqs. (41).
[220.127.116.11] Figure 1 visualizes the results obtained by fitting eq. (45) to experiment. [18.104.22.168] The experimental results are depicted as triangles (primary drainage) and squares (imbibition). [22.214.171.124] The experiments were performed in a medium grained unconsolidated water wet sand of porosity . [126.96.36.199] Water was used as wetting fluid while air resp. TCE were used as the nonwetting fluid. [188.8.131.52] 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. [184.108.40.206] They are described in more detail in Ref. . The parameters for all the curves shown in all four figures are , , , , , , , Pa, Pa, and Pa Pa.
[220.127.116.11] If it is further assumed that the medium is isotropic and that the matrices have the form
then the relative permeability functions are obtained from eqs. (46). [18.104.22.168] The result for the special case is shown in Figures 3 and 4. [22.214.171.124] The parameters are chosen such that and , where are the fluid viscosities and is the absolute permeability of the medium. [126.96.36.199] 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.
[188.8.131.52] Note that Figures 1 through 4 show a total of 30 different scanning curves, 5 drainage and 5 imbibition scanning curves each for and . [184.108.40.206] In addition a total of 9 different bounding curves are displayed, namely the primary drainage, secondary drainage and secondary imbibition curve for and . [220.127.116.11] 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. [18.104.22.168] Of course the number of scanning curves can be increased indefinitely. [22.214.171.124] All of these curves have the same values of the constitutive parameters. [126.96.36.199] There is less than one parameter per curve. [188.8.131.52] The curves shown in the figures exhibit the full range of hysteretic phenomena known from experiment. [184.108.40.206] 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.