Page 114 - Formation Damage during Improved Oil Recovery Fundamentals and Applications

96 Thomas Russell et al.

3.3.2 Exact analytical solution for 1D problem
Substitution of the equation for straining rate (Eq. (3.43)) into the mass
balance Eq. (3.42) and accounting for the fact that the attached concen-
tration is constant after the initial detachment yields a first-order hyper-
bolic equation:

@C @C
1 α 52 ΛαC: (3.48)
@T @X
Here, the filtration coefficient is considered to be constant.
Eq. (3.48) can be solved separately for the two regions behind and
ahead of the suspended particle front using the method of characteristics.
Ahead of the particle front (X . αT), along parametric curves given by:
dX
5 α; (3.49)
dT
Eq. (3.48) reduces to the ordinary differential equation:

dC
52 ΛαC: (3.50)
dT
Integrating Eq. (3.50) subject to initial condition (Eq. (3.46)) yields
the solution for the suspended concentration ahead of the particle front:

CX; TÞ 5 e 2αΛT : (3.51)
ð
Similarly, behind the concentration front (X , αT) Eq. (3.48) reduces
to the ordinary differential equation:

dC
52 ΛC; (3.52)
dX
Along parametric curves given by:
dT 1
5 ; (3.53)
dX α

Using the boundary condition (Eq. (3.47)) corresponding to the
absence of suspended particles in the injected fluid, Eq. (3.52) can
be integrated to yield zero suspended concentration behind the particle
front, C(X,T) 5 0.
Integrating Eq. (3.43) using separation of variables accounting for the
derived solution for C(X,T) enables determination of the strained con-
centration S s (X,T).