[1.1.1.1] A predictive macroscopic theory of two phase
flow inside a rigid porous medium is of
fundamental importance for many
applied sciences such as hydrology, catalysis,
petrophysics or filtration technology
[22, 24, 5, 1, 18, 8, 13, 12].
[1.1.1.2] Despite being used
in innumerable physics and engineering applications
the accepted theory (see below) based on
capillary pressure

[1.1.2.1] Modern investigations often abandon
the traditional theory and resort to microscopic models
(e.g. network models) as an alternative to predict macroscopic
immiscible displacement in porous media
[7, 9, 6, 3, 2, 10].
[1.1.2.2] An important motivation for these alternative
investigations are the
unresolved problems with the traditional
macroscopic theory based on capillary pressure and
relative permeabilities and the necessity to
relate these functions to pore scale parameters.
[1.1.2.3] It is therefore adequate to remind the reader of
the traditional theory, introduced more than 60 years
ago [28, 23, 19], and its problems.
[1.1.2.4] One formulation of the traditional theory
starts from the
fundamental balance laws of continuum mechanics for two
fluids (called water

(1) |

where

(2) |

where

[1.2.1.1] Defining the saturations

(3) |

[1.2.1.3] In order to get the traditional theory these balance laws
for mass, momentum and volume have to be combined with
specific constitutive assumptions for

[1.2.2.1] Great simplification is afforded by assuming that the porous medium is macroscopically homogeneous

(4) |

although this assumption can be relaxed, and is rarely valid in practice [17]. [1.2.2.2] Let us further assume that the fluids are incompressible so that

(5a) | |||

(5b) |

where the constants

(6a) | |||

(6b) |

where

(7) |

[page 2, §0] can be neglected in the momentum balance equation (2). [2.0.0.1] It is further assumed that the body forces

(8a) | |||

(8b) |

are given by gravity. [2.0.0.2] As long as there are no chemical reactions between the fluids the mass transfer rates vanish, so that

(9) |

holds. [2.0.0.3] Momentum transfer between the fluids and the rigid matrix is governed by viscous drag in the form

(10a) | |||

(10b) |

where

[2.0.1.1] Inserting the constitutive assumptions (4)–(10) into the mass balance eqs. (1) yields

(11a) | |||

(11b) |

while the momentum balance eqs. (2)

(12a) | |||

(12b) |

give the generalized Darcy laws
for the Darcy velocities

[2.0.2.1] Equations (11) and (12) together
with eq. (3) provide 9 equations
for 10 unknowns

(13) |

where

[2.1.1.1] This concludes my presentation of the traditional theory. [2.1.1.2] Equations (3),(11),(12) and (13) appropriately supplemented with initial and boundary conditions constitute the traditional theory of macroscopic capillarity in porous media. [2.1.1.3] In numerous engineering applications eqs.(3),(11),(12) and (13) are solved by computer programs [12].

[2.1.2.1] Serious problems limit the predictive power of
equations (3),(11),
(12) and (13).
[2.1.2.2] The biggest problem arises from eq. (13),
because it is not unique and does not account for residual saturations.
[2.1.2.3] It is tacitly assumed that fluids
trapped in pendular rings, ganglia or blobs
behave in the same way as fluids that percolate
to the sample surface.
[2.1.2.4] Moreover, combining eq. (13) with
eq. (12) implies that
in static equilibrium, when

[2.1.3.1] Other problems with

[2.1.4.1] Most practitioners ignore these problems and continue
to use the traditional set of equations.
[2.1.4.2] Many physicists on the other hand try to overcome these
problems by resorting to microscopic model calculations
in an attempt to predict macroscopic behaviour starting
from the pore scale or below [7, 20].
[2.1.4.3] My objective in this paper is to present a purely
macroscopic approach in the same spirit as the
traditional theory, but without requiring
capillary pressure

[2.1.5.1] Before defining percolating and nonpercolating fluid phases in the next section, it is appropriate to comment on the relation to other approaches. [2.1.5.2] Firstly, several authors (including the present one) have emphasized the importance of introducing the fluid-fluid surface area as a state variable (see [14, 16, 15] and references therein). [2.1.5.3] The present approach differs from such theories. [2.1.5.4] The present theory is based only on volume fractions. [2.1.5.5] It avoids surface area and its concomitant proliferation of unknowns and constitutive relations. [2.1.5.6] Secondly, some models generalize eq. (13) into a so called dynamic capillary pressure by including a dependence on rates of saturation change (see e.g. [11] and references therein). [2.1.5.7] The present approach includes dynamic (or viscous) effects on capillarity [page 3, §0] in a more fundamental way by avoiding the concept of capillary pressure. [3.0.0.1] Thirdly, there exist several ad-hoc models for hysteresis loops based on rescaling the main loop formed by the bounding drainage and imbibition curves (see e.g. [27] and references therein). [3.0.0.2] Again, such approaches differ fundamentally from the present one, because they are based on the traditional concepts of capillary pressure and relative peremeability, while the present approach challenges the basis of these concepts. [3.0.0.3] It will be seen below that the present theory requires fewer parameters than previous theories.