Mud Acoustics in Modern Drilling - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 277

5.4.2 Designing a swab-surge simulator.
The finite difference equations, and their implementation and inversion
using TRIDI, were discussed in Chapter 4. Earlier in this chapter, we reviewed
several applicable uphole and downhole boundary conditions for swab-surge
applications. Here, we summarize the steps necessary in designing a dynamic
simulator.
At the hole bottom x = 0, the fluid displacement u(0,t) = 0, if the formation
is assumed to be rigid or if the impedance contrast is infinite.
At the A 1 -A 2 and A 2 -A 3 area change interfaces, area matching boundary
conditions apply.
Although the bit is moving relative to the borehole, its speed v is small
compared to the sound speed c. Computationally, it may be “frozen” during
a simulation; its position is updated by moving to the next closest grid block
only as needed.
At the surface, various models are possible. A capped model takes u = 0,
while an open reservoir model requires u x = 0. If the drillpipe is extremely
long, or if damping is sufficiently high that surface reflection is negligible,
or both, the radiation condition u f(x-ct) can be used. Since u t +c u x = -cf '
+ cf ' = 0, we enforce an outgoing wave using the statement (u imax,n -
u imax,n-1 )/ t + c (u imax,n - u imax-1,n )/ x = 0.
At the source , or bit , a “delta-p” (i.e., a jump or discontinuity in pressure)
exists. Since p = - Bu x in general, specifying a jump in p, following
Equation 4.2.95, amounts to prescribing the quantity (u x + - u x - ), that is,
(U ibit+1,n - U ibit+,n )/ x - (U ibit-,n - U ibit-1,n )/ x. Of course, U ibit-,n and
U ibit+,n are not equal by virtue of analogous consequences to our Equation
5.55; a mean dependent variable must be used along the lines indicated in
Equations 5.56 to 5.60. This logic ensures that the bit, with its bypass area,
is transparent to external propagating waves. The strength of the source can
be modeled simply. We showed in Example 5.2 that p = vc for a one-
dimensional piston acting on a column of fluid. This result applies to
source modeling, where in the present context, v represents the pipe speed.
The theoretical maximum for delta-p is just the difference at upper and
lower sides of the bit, that it, vc - (- vc) or 2 vc. Deviations from this
value are due to bypass flow around the bit and through its nozzles. Finally,
observe that the delta-p logic and area change logic are implemented near
the same spatial index; some experimentation will be required to ensure that
all of the relevant physics is captured.