Page 97 - Percolation Models for Transport in Porous Media With

90 CHAPTER 5. NON-STEADY STATE TWO-PHASE FLOW

Physical premises. In the network model of heterogeneous media [25], capil-
laries form an IC, whose conductivity is determined by the capillary chains oriented
in the direction of the flow and composing the skeleton of the IC. These chains
communicate with each other through analogous capillary chains, providing flow
in the transverse direction. Some of these chains create a network of an irregular
form. As it is shown in [25), there exists an hierarchy of chains according to their
average conductivity; therefore, the phase flow velocities, including the case of dis-
placement of one phase by another, are different. At two-phase flow the injected
phase enters the chains {later called "tree trunks") oriented in the direction of the
applied pressure gradient, and through them enters the "branches of trees" -the
capillary chains which provide flow in the transverse direction. As a result, growth
of the tree formed by a trunk and branches takes place during the injection of the
displacing phase.
In their turn, the branches provide inflow of the displacing phase into the
"leaves," the capillary chains oriented parallel to the trunk. Leaves may have a
complicated arborescent form, too. In Fig.28, two interrelated trees are presented
schematically. The number 1 indicates trunks of growing trees, the number 2,
branches, and the number 3, leaves.
Thus the same chains of capillaries oriented in the direction of flow may take
part in the formation of both trunks and leaves, depending on how the displacing
phase enters them. We shall consider leaves belonging to a given tree if the dis-
placing phase enters them through branches of this tree. During the flow in the
medium, trees grow at different rates. As a result, rapidly growing trees outrun in
growth the slower-growing ones and block their further growth; this results in the
decrease in the concentration of the latter. A similar situation is observed during
the growth of leaves, which grow until the capillary chains forming them intersect
with the next tier of branches. As a result, the displacing phase is trapped in
these chains. This effect is caused by the dynamic nature of the displacement, and
the fraction of the trapped phase is determined by the ratio between the rates of
growth of the trunk and leaves.
Residual saturation of the displaced phase, trapped at the dynamic stage, may
relax to the equilibrium value. This can happen if the IC of the displaced phase
exists in the macro-volume, and the capillary forces prevent the invasion of the
displacing phase into the IC. If the pressure P of the displacing phase is greater
than the threshold pressure Pt at which the capillaries filled with the displaced
phase form an IC, then the maximum possible fraction of the displaced phase
is dynamically trapped. The above-mentioned mechanism allows to explain the
increase of the fraction of the trapped phase with the increase of the flow velocity.
Below, we consider the approach which allows to get a quantitative descrip-
tion of two-phase flow in porous media using the forest growth model, where the
"forest" is understood as the sum of trees (or one "banyan tree") formed by the