Mud Acoustics in Modern Drilling - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 275

Similarly, the u difference equation at i = i*-1 involves the value of u at i = i*-.
Since {A (p) /A (c) }u i*+,n = u i*-,n , we determine that u m n = ( u i*-,n + u i*+,n )/2 = 1/2
{A (p) /A (c) +1}u i*+,n . Thus,
u i*+,n = 2 u m n /{A (p) /A (c) +1} (5.60)
Therefore, the complete set of numerical unknowns, namely, u 1 , u 2 , u 3 , ..., u i*-1 ,
u m , u i*+1 , ... , u imax-1 , u imax-1 , u imax , can be determined from the solution of
computationally stable tridiagonal equations. The procedure described above is
taken, incidentally, from a standard aerospace algorithm for multivalued
velocity potentials modeling lifting wing flows. There, the elliptic governing
equation is not wave-like, but similar tridiagonal equation algorithms are used to
iteratively converge the problem in a time-like manner. For more details about
solutions to elliptic and (time-dependent) parabolic boundary value problems,
refer to the author' s topics Modern Reservoir Flow and Well Transient Analysis
(Chin, 1993) and Quantitative Methods in Reservoir Engineering (Chin, 2002)
5.4 Swab-Surge Modeling
In rotary drilling, “blow-outs” often occur when drillpipe is being
withdrawn from the hole, despite the fact that while drilling, the muds used
contain sufficient hydrostatic head for well control. Reductions in pressure
evidently can occur while making a trip, which may be greater than the amount
by which the hydrostatic pressure exceeds formation pressure. In the literature,
low tripping speeds are recommended. Burkhardt (1961), Fontenot and Clark
(1973), and Clark and Fontenot (1974) discuss operational issues and solutions
using steady flow methods. Modeling parameters include borehole geometry,
mud properties and return area about drill bits, the objective being the prediction
of safe tripping speeds. Early work has shown that traditional steady flow
models alone may not suffice. Lal (1983) reviews earlier swab-surge models
and proposes a dynamic acoustical model for the transient compressible wave
flow, which is solved using the method of characteristics. Mitchell (1988) also
uses a characteristics model, accounting for pipe elasticity, improved turbulent
flow friction factors, and variable fluid and formation properties, and gives
comparisons with field data.
5.4.1 Wave physics of swab-surge.
In order to understand the wave nature of “dynamic swab-surge,” recall
from Example 1-1 that stress (or pressure) waves reflect with the same sign at
solid boundaries such as wellbore bottoms. Let us refer to Figure 5.3, and
imagine that the drillstring is suddenly lifted upwards, assuming that the mud
moves from rest. Immediately uphole of the bit, the fluid is compressed because
the bit moves into the mud, and an over-pressure (relative to ambient conditions)
is transmitted up the annulus.