Incorporating the effects of fine particles deposition according to Arshad (1991) and cake compaction according to Tien et al. (1997), Civan (1998a) estimates the cake-thickness-average porosity by the following constitutive equation:

Considering the fine particles deposition and cake compaction, Civan (1998a) estimates the cake-thickness-average permeability by the Tien et al. (1997) constitutive equation:

In Eqs. 12-142 and 143, ɸ)° and k°c represent the fine particles-free and non-compacted cake porosity and permeability; respectively, α, n, Ρa, β, a1, a2, and ẟ are the empirically determined parameters.

## Thickness-Averaged Fluid Pressure and Cake Porosity

The average fluid pressure in the filter cake for linear filtration can be expressed similar to Dake (1978) as:

The following expression can be derived from Eqs. 12-144 and 12-145:

Note xw is a constant, but xc = xc(t) varies by time. Similarly, the following three expressions can be written for radial flow:

Eq. 12-147 is given by Dake (1978). Note that rw is a constant, but rc = rc(t) varies by time. Eqs. 12-146 and 149 define the average fluid pressure, but they cannot be used directly because the pressure distribution over the cake thickness is not a priori known. Civan (1998b, 1999b) circumvented this problem by applying a procedure similar to Jones and Roszelle (1978) to express a local function value in terms of its average. The local cake porosity at the slurry side of the cake can be expressed in terms of the cake-thickness-average porosity. For linear filtration Civan (1998b) differentiated Eq. 12-145 to obtain:

Similar to Tiller and Crump (1985), the cake-thickness-average drag force, ps, created by the flow of the suspension of fine particles through the filter cake is determined using

in which Pc is the pressure of the slurry applying at the progressing filter cake surface and p is the cake-thickness-average pressure of the uspension of fine particles flowing through the cake. For linear filtration, Pc and p can be related by differentiating Eq. 12-146 and then substituting Eq. 12-150 to obtain (Civan, 1998b):

For computational convenience, Eq. 12-153 can be reformulated in a form of an ordinary differential equation as (Civan, 1999b):

Differentiating Eq. 12-149 and then substituting Eq. 12-151 for radial flow, Eqs. 12-153 and 12-154 are replaced, respectively, by (Civan, 1998b):

Eqs. 12-154 or 12-156 can be solved numerically subject to the initial condition

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