Page 47 - Reservoir Formation Damage

Mineralogy and Mineral Sensitivity of Petroleum-Bearing Formations 31

so that Eq. 2-4 becomes:

c = c,, z = 0, t > 0 (2-8)
and, therefore, an analytical solution of Eqs. 2-2, 3, 8, and 5 according
to Crank (1956) yields the expression for the cumulative and rate of water
absorption, respectively, as:

(2-9)

D
(2-10)

The rate of formation damage by clay swelling also depends on the
variation of the water concentration in the aqueous solution flowing
through porous rock. Whereas, the analytical expressions given above
assume constant water concentrations in the aqueous pore fluid. However,
they can be corrected for variable water concentrations by an applica-
tion of Duhamel's theorem. For example, if the time-dependent water
concentration at the pore surface is given by:

(2-11)

where F(t) is a prescribed time-dependent function, the analytic solution
can be obtained as illustrated, by Carslaw and Jaeger (1959). Then, us-
ing Eq. 2-10, the rate of water absorption can be expressed by:

(2-12)

However, in the applications presented here the water concentrations in-
volved in the laboratory experiments are essentially constant.
The preceding derivations assume a plane surface as supposed to a
curved pore surface. From the practical point of view, it appears reason-
able because of the very short depth of penetration of the water from the
solid-fluid contact surface.

Clay Swelling Coefficient

The rate of clayey formation swelling is derived from the definition
of the isothermal swelling coefficient given by (Collins, 1961):