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Acoustic Logging in Fractured and Porous Formationsby Submitted to the Department of Earth, Atmospheric, and Planetary Sciences on June 1990 in partial fulfillment of the requirements for the degree of Doctor of Philosophy ABSTRACT
This thesis is concerned with the dynamic fluid transport properties of fractures
and porous media and their application to the estimation of formation hydraulic
properties using borehole acoustic logging techniques. In the first part of the
thesis, the dynamic response of a viscous fluid in a borehole fracture to the
oscillatory pressure excitation of borehole acoustic waves is investigated, which
leads to the theory of fracture dynamic conductivity. The distinction between
this dynamic conductivity and the conventional cubic law conductivity is whether
the viscous skin depth, d= (2n/w)1/2, is large or small compared to the thickness
of the fracture. Although this characteristics of dynamic fluid flow is obtained
using the simple plane parallel fracture model, the physics involved is universally
true for dynamic fluid flow in hydraulic conduits of rocks. The theory of fracture
dynamic conductivity is compared with the theory of dynamic permeability of a
general porous medium. It is found that the latter theory, when applied to the
fracture case, is in excellent agreement with the theory of conductivity. This
points to the general behavior of frequency-dependent fluid motion through conduits
in rocks, regardless whether they are fractures or pores. Consequently, in acoustic
logging measurements performed in a typical range of [2-20] kHz, the dynamic fluid
flow theory, instead of the conventional Darcy’s law, is the appropriate
theory for the fluid flow in the formation induced by logging acoustic waves. In
the second part of the thesis, the concept of dynamic permeability is applied
to the important problem of acoustic logging in a permeable porous formation using
borehole Stoneley waves. The interaction of the Stoneley wave with the porous
formation is decomposed into two parts. The first is the interaction of the Stoneley
with an equivalent elastic formation composed of the saturated porous matrix.
The second is the interaction with pore fluid flow governed by the dynamic permeability.
In this manner, a simple dynamic model is obtained for the Stoneley propagation
in permeable boreholes. This simple model is compared with the complete model
of the Biot-Rosenbaum theory for the effects of a porous formation on the Stoneley
propagation characteristics. It is found that the results from the two models
agree very well for a hard formation, although they differ at higher frequencies
for a soft formation because of the increased formation compressibility. The simple
model is also tested with recently published laboratory experimental data of Stoneley
wave measurements. The theory and experiment are in excellent agreement. As a
result, the application of the dynamic fluid flow theory not only clearly points
to the physical process involved in wave propagation in permeable boreholes, but
also yields a much simplified Biot-Rosenbaum model that can be applied to the
problem of acoustic logging in porous formations, especially to an inverse problem
to extract formation permeability from the Stoneley wave measurements. In
the third part, the problem of acoustic logging in a fluid-filled borehole with
a vertical fracture is investigated both theoretically and experimentally. The
Stoneley wave is used to probe the borehole. The propagation of this wave excites
fluid motion in the fracture and the resulting fluid flow at the fracture opening
perturbs the fluid-solid boundary condition at the borehole wall. The dynamic
conductivity is applied to measure the fluid flow into the fracture and a boundary
condition perturbation technique is developed to study the effects of the change
in the boundary condition on the Stoneley propagation. The results indicate that
the fracture has significant effects on the Stoneley waves, especially in the
low frequency range. Significant Stoneley wave attenuation is produced and the
Stoneley phase velocity is drastically decreased with decreasing frequency. Ultrasonic
experiments are performed to measure Stoneley propagation in laboratory fracture
borehole models. Cases of both hard and soft formations are studied. For both
formations, the experimental results are found to agree well with the theoretical
predictions. The important result of this study is that, a quantitative relationship
between the Stoneley propagation and the fracture character is found. This relationship
can be used to provide a method for characterizing a vertical borehole fracture
by means of Stoneley wave measurements. In the last part, the guided wave
propagation in a fluid-filled borehole with a horizontal fracture is investigated.
For the solution of the problem, a hybrid method is used to generate wave modes
for the two regions separated by the fracture. The modes are then summed to match
the boundary conditions at the fracture surfaces. A singularity problem arises
in matching the surface conditions and is regularized by balancing borehole fluid
flow across and into the fracture. The latter flow is characterized using the
fracture dynamic conductivity. The results show that a low frequencies, the Stoneley
wave attenuation across a fracture is controlled by the fluid flow into the fracture.
As the frequency increases, mode conversion at the fracture becomes important.
Above the cut-off frequency of the first pseudo-Rayleigh mode, the Stoneley wave
is strongly coupled with pseudo-Rayleigh waves, which is demonstrated by synthetic
microseimograms. The pseudo-Rayleigh wave is strongly attenuated and reflected
by thin as well as thick fractures. These effects are more pronounced toward the
cut-off frequencies than away from the frequencies. Consequently, in acoustic
logging measurements, the lack of wave energy across a borehole fracture may be
very good indication of the existing fracture. The substantial effects of a fracture
on a pseudo-Rayleigh waves has been verified in the laboratory by experimenting
with thin and thick fracture models. The experimental results demonstrate the
guided wave characteristics across a fracture and confirm the theoretical analysis
on these effects. The wave characteristics in the vicinity of a fracture, as described
in this study, can be used to provide useful information for the detection and
characterization of borehole fractures using an acoustic logging techniques. Return to Theses Return to ERL Home Updated: June,1999
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