Reservoir Engineering –Steady, Diffusive and Propagation Models - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 363

Since Equation 9.20 states that S w is still constant along a trajectory, the term
df w (S w ,

w / nw )/dS w is likewise constant. Thus, the integral of Equation 9.30

is simply
x - -1 df w (S w , w / nw )/dS w q(t) dt = constant (9.31)
where q(t) dt denotes the indefinite integral (e.g., q o dt = q o t is obtained for
our constant rate problem). Following a line of reasoning similar to that leading
to Equation 9.24, since S w is constant whenever the left side of Equation 9.31 is
constant, we have the equivalent functional statement
S w (x,t) = G(x - -1 df w (S w , w / nw )/dS w q(t) dt) (9.32)
Equation 9.32 is the general saturation solution for time-dependent q(t). If the
integrated function q(t) dt vanishes for t = 0, this solution satisfies the initial
condition specified by Equation 9.25. If the function does not vanish, some
minor algebraic manipulation is required to obtain the correct format.

9.1.5 Mudcake-dominated invasion.
So far, we have not stated how the velocity q(t), possibly transient, is
determined. If we assume that the flow at the inlet to our lineal core is
controlled by mudcake, as is often the case, the fluid dynamics within the core
will be unimportant in determining q(t). (This assumption is removed in our last
example.) Then, the general mudcake model in Chapter 17 of Chin (2002) for
single-phase filtrate flows provides the required q(t). In fact,
x f (t) = eff -1 {2k 1 (1- c )(1-f s )(p m -p r )t/( f f s )} (9.33)
when the effect of spurt and the presence of the formation are neglected. The
fluid influx rate q(t) through the mudcake is therefore given by
q(t) =

eff dx f (t)/dt = ½ t -½

{2k 1 (1-

c )(1-f s )(p m -p r )/( f f s )}

(9.34)

which can be substituted in the nonlinear saturation equation
S w / t + {q(t)/ }df w (S w , w / nw )/dS w S w / x = 0 (9.35)
This can be integrated straightforwardly using the method of characteristics. So
long as singularities and saturation fronts do not form, saturations obtained as a
function of space and time will be smooth and shocks will not appear.
9.1.6 Shock velocity.
We will consider the problem that arises when saturation shocks do form.
(Problems with smooth but rapidly varying properties are addressed in our
capillary pressure analysis.) In order to discuss saturation discontinuities and
steep gradients, we must complete the formulation by specifying initial and
boundary conditions. We assume that at t = 0, our core is held at the constant
water saturation S w i throughout, where the italicized i denotes initial