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Pressure diffusion waves in porous media
Dmitry Silin * and Valeri Korneev, Lawrence Berkeley National Laboratory,
Gennady Goloshubin, University of HoustonSummary
Pressure diffusion wave in porous rocks are under
consideration. The pressure diffusion mechanism can
provide an explanation of the high attenuation of low-
frequency signals in fluid-saturated rocks. Both single and
dual porosity models are considered. In either case, the
attenuation coefficient is a function of the frequency.
Introduction
Theories describing wave propagation in fluid-bearing
porous media are usually derived from Biot's theory of
poroelasticity (Biot 1956ab, 1962). However, the observed
high attenuation of low-frequency waves (Goloshubin and
Korneev, 2000) is not well predicted by this theory.
One of possible reasons for difficulties in detecting Biot
waves in real rocks is in the limitations imposed by the
assumptions underlying Biot's equations. Biot (1956ab,
1962) derived his main equations characterizing the
mechanical motion of elastic porous fluid-saturated rock
from the Hamiltonian Principle of Least Action. However,
using the Hamiltonian Principle for describing fluid flow in
porous media imposes certain restrictions. These
restrictions are related to the nature of Darcy's law. Darcy
fluid velocity or, equivalently, superficial velocity is
defined as the fluid flux through an elementary surface in
the bulk volume. In real rocks, the fluid flows through an
extremely complex system of numerous geometrically
irregular pore channels with different orientations and
cross-sectional areas, see Figure 1. In addition, the pore
walls are rough surfaces. Due to this complex geometry of
the pore space, individual fluid particles accelerate and
slow down all the time and the dispersion of the Lagrangian
velocities between can be very large. King Hubbert (1956)
stated: "... deductions concerning Darcy-type flow made
from the simpler Poiseuille flow are likely to be seriously
misleading". At the same time, the equations describing
elastic displacement of the solid skeleton are formulated at
a microscopic scale, where characteristic length is
comparable with the size of an individual grain and,
therefore, that of an individual pore. In case of a parallel
bundle of capillary tubes or other simplified model of
porous medium, the velocity dispersion is smaller than in
natural rocks. Therefore this scale mismatch may be not as
well pronounced.
To narrow the above-mentioned gap between the length
scales of Darcy's law and elastic equations, we propose to
use the model of slightly compressible fluid flow in anelastic porous medium. Such a model results in a parabolic
pressure diffusion equation. Its validity has been
confirmed and "canonized", for instance, in transient
pressure well test analysis, where it is used as the main tool
since 1930th, see e.g. Earlougher (1977) and Barenblatt et.
al., (1990). The basic assumptions of this model make it
applicable specifically in the low-frequency range of
pressure fluctuations.
o
0 s
22
Figure 1: A 5 m resolution image of the pore space of
2.5x2.5x2.5 mm3 piece of Fontaineblaux sandstone and
schematic representation of the connections between the pores
(courtesy of Statoil).
In the next sections, we discuss the harmonic wave
solutions to pressure diffusion equation in the contexts of
single and dual porosity models. Since diffusion and heat
conductance processes are usually described by parabolic
equations, such waves are called diffusion or thermal
waves (Tikhonov and Samarskii, 1963; Mandelis, 2001).
As early as in 1851, Stokes (1851) used diffusion waves
mechanism for measuring fluid shear viscosity.
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Silin, Dmitry; Korneev, Valeri & Goloshubin, Gennady. Pressure diffusion waves in porous media, article, April 8, 2003; Berkeley, California. (https://digital.library.unt.edu/ark:/67531/metadc735497/m1/1/: accessed April 18, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.