Reservoir Engineering –Steady, Diffusive and Propagation Models - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 358

9
Reservoir Engineering -
Steady, Diffusive and Propagation Models
Most petroleum engineering students are familiar with reservoir flow from
the context of pressure equations resembling, say “k x 2 P/ x 2 + k y 2 P/ y 2 = 0 ”
and “k x 2 P/ x 2 + k y 2 P/ y 2 = c P/ t,” which actually apply to steady and
transient compressible liquid fluids in the single-phase flow limit. The author' s
topic Quantitative Methods in Reservoir Engineering (Chin, 2002) provides a
rigorous and comprehensive exposition of mathematical models used in
different aspects of reservoir flow production, e.g., constant rate flow, well
transient analysis, multiphase effects, multilateral wells in layered media, and so
on. Much less obvious, at least to newer engineers, are wave-like properties that
are possible when multiple fluid phases are present in immiscible applications.
For detailed discussions, the reader should consult Quantitative Methods - in
this short chapter, we wish only to introduce basic formulation ideas and
demonstrate how first-order partial differential equations and solution
singularities can arise, in much the same way as they did in our treatment of
“kinematic wave theory” covered in Chapter 2. The material presented in
Chapters 5-10 illustrate the commonality behind many seemingly different areas
of petroleum engineering, and it has been the author' s experience that a
familiarity of the underlying mathematics with one sub-discipline will enhance
physical understanding not just with a single discipline, but with many.
9.1 Buckley-Leverett Multiphase Flow
For our purposes, we will study the immiscible, constant density flow
through a homogeneous lineal core where the effects of capillary pressure are
insignificant - that is, we will consider “immiscible Buckley-Leverett linear
flows without capillary pressure.” In particular, we will derive exact, analytical,
closed form solutions for the forward modeling problem for a single core. These
solutions include those for saturation, pressure and shock front velocity, for
arbitrary relative permeability and fractional flow functions. We will determine
what formations properties can be inferred, assuming the existence of a
propagating front, when the front velocity is known. The Darcy velocities are
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