A schematic of the formation of a filter cake over the sand face during over-balanced mud circulation in a wellbore is shown in this article. The radii of the mud slurry side cake surface, the sand face over which the cake is built up, and the external surface considered for the region of influence are denoted by rc, rw and re, respectively. The formation thickness is h. The particle mass balance equation is given by (Civan, 1994, 1998a)

The slurry shear-stress at the cake surface is given by the Rabinowitsch-Mooney equation (Metzner and Reed, 1955)

The radial volumetric flux of the carrier fluid is given by

Integration of Eq. 12-54 for conditions prevailing prior to and during filter cake formation leads to the following expressions, respectively (Civan, 1999a)

Thus, eliminating (Ρc - Ρe) between Eqs. 12-55 and 56, substituting Eq.12-46, and then solving for q, yields for Darcy flow ( βf =βC = O :

Substituting Eq. 12-57 and considering the initial condition given by Eq. 12-51, Eq. 12-49 can be solved using a numerical scheme, such as the Runge-Kutta-Fehlberg four (five) method (Fehlberg, 1969). The cumulative filtrate volume is given by Eq. 12-27. The pressure difference (Pc-Pe}, or the slurry injection pressure pc when the back pressure pe is prescribed, can be calculated by Eq. 12-56. When the inertial flow terms are negligible, equating Eqs. 12-55 and 12-56 and rearranging leads to (Civan, 1998a):

where q0 is the injection rate given by Eq. 12-55 for βf = 0 before the filter cake buildup and

Thus, substituting Eqs. 12-46 and 12-62 into Eq. 12-49 and rearranging yield the filtration flow rate equation as (Civan, 1998a):

The wall shear-stress is calculated by Eq. 12-47 for the varying cake radius, rc=rc(t}. The filter cake thickness is calculated by means of Eqs. 12-46 and 62. Equations 12-65 and 66 can be solved numerically using an appropriate method such as the Runge-Kutta method. However, for thin cakes, it is reasonable to assume that the wall-shear stress is approximately constant, because rc=rw. Then, Eq. 12-65 can be integrated as (Civan, 1998a):

For constant rate filtration, Eq. 12-49 subject to Eq. 12-51 can be integrated numerically for varying shear-stress is. When the filter cake is thin, the variation of the shear-stress is by the cake radius rc can be neglected and an analytical solution can be obtained as for dynamic filtration conditions (B ≠ 0) (Civan, 1999a):

The solution for static filtration conditions (B = 0) is obtained as (Civan, 1999a):

Eq. 12-68 and 12-69 apply irrespective of whether the flow is Darcy or non-Darcy.

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