Reservoir Engineering –Steady, Diffusive and Propagation Models - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 364

conditions. At the left boundary x = 0, where fluid influx occurs, we assume
that the water saturation is fixed at a constant value S w l where the italicized l
denotes left. (Normally, this value is unity for water filtrates, but it may differ
for certain water-oil muds.) That is, we take
S w (x,0) = S w i (9.36)
w (0,t) = S w l (9.37)
As discussed, we can expect shockwaves and steep saturation discontinuities to
form in time, depending on the exact form and values of our fractional flow
functions and initial conditions. We will assume that the particular functions do
lead to piston-like shock formation very close to the borehole. The shock
boundary value problem just stated can be solved in closed form, and, in fact, is
the petroleum engineering analogue of the classic nonlinear signaling problem
( t + c ( ) x = 0 , = o for x > 0, t = 0, and = g(t) for t > 0, x = 0) discussed
in the wave mechanics topic of Whitham (1974).
We will not rederive the mathematics, but will draw on Whitham's “shock
fitting” results (based on global conservation laws) only. For brevity, define for
convenience the function
Q(S w ) = {q(t)/ }df w (S w , w / nw )/dS w (9.38)
where q(t) is given in Equation 9.34. It turns out that the shock propagates with
a shock speed equal to
V shock = { Q w (S w l ) - Q w (S w i )}/(S w l - S w i )

(9.39)

If the injection rate q(t), the core porosity , and the speed of the front V shock
separating saturations S w l from S w i are known, then since S w l is available at
the inlet of the core, Equations 9.38 and 9.39 yield information relating the
initial formation saturation S w i to the fractional flow derivative
df w (S w , w / nw )/dS w . Equation 9.15 shows that the fractional flow function
satisfies f w (S w , w / nw ) = 1/{1 + (k nw w /k w nw )}. Thus, if additional
lithology information is available about the form of the relative permeability
functions, the viscosity ratio w / nw can be extracted, thus yielding nw . We
emphasize that this solution for the nonlinear saturation problem does not apply
to the linear single-phase flow where “red water displaces blue water.”

9.1.7 Pressure solution.
Now we derive the solution for the corresponding transient pressure field.
Let us substitute Equations 9.1 and 9.2 (that is, Darcy' s laws q w = - (k w /

w )

P w / x and q nw = - (k nw /

nw )

P nw / x) into Equation 9.9 (or q w + q nw =