Reservoir Engineering –Steady, Diffusive and Propagation Models - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 365

q(t)). Also, from Equation 9.3, we find that P nw = P w . Thus, we obtain the
governing pressure equation

{(k w (S w )/

w ) + (k nw (S w )/ nw )} P w / x = - q(t)

(9.40)

so that the pressure gradient satisfies

P w / x = - q(t)/{(k w (S w )/

w ) + (k nw (S w )/ nw )}

(9.41)

Since the saturation function S w (x,t), following Whitham's solution to the
signaling problem is a simple step function in the x direction whose hump
moves at the shock velocity, we conclude that the pressure gradient in Equation
9.41 takes on either of two constant values, depending on whether S w equals S w i
or S w l locally. Thus, on either side of the shock front, we have different but
linear pressure variations with space, when time is held fixed. This situation is
shown in Figure 9.1. At the shock front itself, the requirement that pressure be
continuous and single-valued, a consequence of our zero capillary pressure
assumption, is itself sufficient to uniquely define the time-varying pressure
distribution across the entire core.
Now we outline the computational procedure. At the left of the core, the
saturation specification S w l completely determines the value of the linear
variation P w (S w l )/ x, following the arguments of the preceding paragraph.
Since the exact value of pressure P l is assumed to be known at x = 0 (that is, the
interface between the rock core and the mudcake), knowledge of the constant
rate of change of pressure throughout completely defines the pressure variation
starting at x = 0. Unlike reservoir engineering problems, we are not posing a
pressure problem for the core in order to calculate flow rate; our flow rate is
completely prescribed by the mudcake. In this problem, saturation constraints
fix both pressure gradients, which in turn fix the right-side pressure. The radial
flow extension of this procedure leads to an estimate for reservoir pore pressure.

P

Shock front

x

Figure 9.1. Pressure in lineal core.

In finite length core flows without cake, it is appropriate to specify both the left
and right pressures P l and P r , and determine the corresponding q(t). Since q(t)