1. IMPERIAL COLLEGE LONDON
Department of Earth Science and Engineering
Centre for Petroleum Studies
Biot Model for Wave Propagation in
Fluid-Saturated Porous Media
by
David Bautista Gonz´alez
A report submitted in partial fulfillment of the requirements for
the MSc and/or the DIC.
September 2015
2. 1
Declaration of Own Work
I declare that this thesis
Biot Model for Wave Propagation in Fluid-Saturated Porous Media
is entirely my own work and that where any material could be construed as the work of others, it is fully cited and
referenced, and/or with appropriate acknowledgement given.
Signature:
Name of Student: David Bautista Gonz´alez
Name of Supervisor: Prof. Robert W. Zimmerman
3. 2
Abstract
We present a new closed form expression of the wave speed for the slow and fast compressional waves as well as for the
shear wave as predicted by Biot’s equation. These expressions are derived from first principles for isotropic rock systems
in the low frequency limit by means of an appropriate Lagrangian function for the rock-fluid system and a suitable
viscodynamic operator. A detailed study of the effect of the various rock and fluid parameters on the wave speed is
made and we identify their relation with Biot’s characteristic frequency. We have successfully validated this model with
data from water-saturated clay-free sandstones at a confining pressure of 40 MPa (equivalent to a depth of burial of
approximately 1.5 km) and for two samples of sandstones at differential pressures of 15 MPa and 18 MPa. Finally, we
followed the procedure from Johnson & Chandler to investigate the relationship between the quasi-static slow Biot wave
and the pressure diffusion equation used in well-test analysis. We found an inconclusive connection between Biot theory
of poroelasticity and the diffusion equation from well-test analysis, giving rise to a factor-of-two underestimate of rock
compressibility, this could be due to an effective rock-to-fluid compressibility and fluid-to-rock compressibility that are
not taken into account in traditional models, and this could also be related to geometrical effects of the pore network and
of the actual wave front of the acoustic waves. These results will be a very helpful starting point for further work on this
problem.
4. 3
Acknowledgements
I would like to thank my supervisor Professor Robert Zimmerman, for his guidance and constant support.
I would like to express my gratitude to the Mexican Petroleum Institute (IMP) for providing the sponsorship that allowed
me to take part in this MSc course.
Many thanks to the staff of the Earth Science and Engineering department, whose support made our learning easier and
more interesting.
On a personal note, I want to thank my family, especially my mother for her constant encouragement throughout the
year.
Thanks also to my fellow students for an interesting and challenging year at Imperial College.
Thank you, Barbara.
I would like to thank the Academy.
8. 7
Biot Model for Wave Propagation in Fluid-Saturated Porous Media
David Bautista-Gonz´alez
Imperial College supervisor: Robert W. Zimmerman
Copyright 2015, Imperial College London
This report was submitted in partial fulfilment of the requirements for the Master of Science and/or the Diploma of Imperial College from
Imperial College London – 28 August 2015.
Abstract
We present a new closed form expression of the wave speed for the slow and fast compressional waves as well as for the
shear wave as predicted by Biot’s equation. These expressions are derived from first principles for isotropic rock systems
in the low frequency limit by means of an appropriate Lagrangian function for the rock-fluid system and a suitable
viscodynamic operator. A detailed study of the effect of the various rock and fluid parameters on the wave speed is
made and we identify their relation with Biot’s characteristic frequency. We have successfully validated this model with
data from water-saturated clay-free sandstones at a confining pressure of 40 MPa (equivalent to a depth of burial of
approximately 1.5 km) and for two samples of sandstones at differential pressures of 15 MPa and 18 MPa. Finally, we
followed the procedure from Johnson & Chandler to investigate the relationship between the quasi-static slow Biot wave
and the pressure diffusion equation used in well-test analysis. We found an inconclusive connection between Biot theory
of poroelasticity and the diffusion equation from well-test analysis, giving rise to a factor-of-two underestimate of rock
compressibility, this could be due to an effective rock-to-fluid compressibility and fluid-to-rock compressibility that are
not taken into account in traditional models, and this could also be related to geometrical effects of the pore network and
of the actual wave front of the acoustic waves. These results will be a very helpful starting point for further work on this
problem.
Introduction
Wave physics is one of the most directly applicable branches of physics. One can encounter waves of different nature in
gravitation, quantum mechanics, electromagnetism, fluid mechanics, and elasticity theory. It is clearly one of the most
fundamental physical phenomena, since we interact with waves all the time in our day-to-day experience. And most
importantly, we rely on seismic interpretation for exploration geophysics in the Petroleum Industry.
The existence of a moving viscous fluid in a porous medium affects its mechanical response. At the same time, the
modification in the mechanical internal state of the porous skeleton influences the behaviour of the viscous fluid inside
the pores. These two coupled deformation-diffusion phenomena rest at the heart of the theory of poroelasticity.
Namely, the two fundamental phenomena can be summarized as follows:
• Fluid-to-Solid coupling: occurs when a change in the fluid pressure or fluid mass induces a deformation of the porous
skeleton.
• Solid-to-Fluid coupling: occurs when modifications in the stress of the porous skeleton induce change in fluid
pressure or fluid mass.
In accordance with these two phenomena, the fluid-filled porous medium acts in a time-dependent manner. Assume that
the porous medium is compressed – this will result in an increment of the fluid pressure inside the pores and resulting
fluid flow. The time dependence of the fluid pressure (i.e. dissipation of the fluid pressure through the diffusive fluid flux
according to the Darcy’s law for filtration velocity) will induce a time dependence of the poroelastic stresses, which in
turn will respond back to the fluid pressure field. It is clear that the dynamical model of such process is time dependent
and only if the inertial forces are neglected then it can be considered as quasi-static.
9. 8
The purpose of this work is to clarify the validity of Biot’s model which attempts to account for the effect of pore
fluid n the speeds of P and S waves in a porous rock. Even if Biot’s equations of motion have been known for more than
half a century, nobody has developed a fully explicit closed-form relation for the wavespeeds, since everyone has decided
in the past to solve this problem numerically. Also, nobody has ever solved the fundamental paradigm for the model of
diffusivity in well-test analysis which considers merely a rigid frame instead of a deformable one that contains fluid inside
its pores. These are the two actual motivations for this work, since one would like to understand the reason why core
data is never consistent with well-test data when one tries to correlate them.
In the first stage we present the formalism for deriving an analytical expression for the wave speeds and attenuations of
these elastic waves, then perform a study of the influence of their respective parameters, after that we will present an
experimental verification of the model with data from the literature of water- saturated clay-free sandstones [11] and from
two additional samples of sandstones [15] and we will prove the connexion between the slow compressional wave from
Biot’s theory in the quasi-static limit and the diffusion equation from well-test analysis via the diffusivity.
Literature Review
The study of the propagation of elastic waves in porous media is a very important research topic in the petroleum in-
dustry because of its applications in seismic prospection and geomechanics for drilling engineering. Biot was able, by
means of phenomenological approach, to lay the foundations of poroelasticity theory in a collection of 5 essential papers
[1, 2, 3, 6, 4] and we are mostly interested in the theoretical and practical influence of those related to wave propagation
in fluid saturated porous media [2, 3]; Biot’s theory describes wave propagation in a porous saturated medium, i.e., a
medium made of a solid matrix (skeleton or frame), fully saturated with a fluid. Biot ignores the microscopic level and
assumes that continuum mechanics can be applied to measurable macroscopic quantities. He postulates the Lagrangian
and uses Hamilton’s principle to derive the equations governing wave propagation. This theory predicts the existence
of two compressional waves (compressional waves of the first and second kind) and one shear wave. We will focus on
the low-frequency range of the theory regime, indeed one must take into account that the initial theory is valid up to a
limit frequency, where the assumption of Poiseuille flow breaks down. The high-frequency range theory is developed by
considering the flow of a viscous fluid under an oscillatory pressure gradient either between parallel walls or in a circular
tube. Such study yields a complex viscosity correction factor function of the frequency (i.e. a viscodynamic operator)
through the dimensionless ratio where is a characteristic frequency of the material. Both cases indicate that the effect
of pore cross-sectional shape is well represented by taking the same function of the frequency for the viscosity correction
and simply changing the frequency scale.
The main assumptions of the theory are [13]:
1. Infinitesimal transformations occur between the reference and current states of deformation. Displacements, strains
and particle velocities are small. Thus, the Eulerian and Lagrangian formulations coincide up to the first-order.
The constitutive equations, dissipation forces, and kinetic momenta are linear.
2. The principles of continuum mechanics can be applied to measurable macroscopic values. The macroscopic quantities
used in Biot’s theory are volume averages of the corresponding microscopic quantities of the constituents.
3. The wavelength is large compared with the dimensions of a macroscopic elementary volume. This volume has
well defined properties, such as porosity, permeability and elastic moduli, which are representative of the medium.
Scattering effects are thus neglected.
4. The conditions are isothermal.
5. The stress distribution in the fluid is hydrostatic.
6. The liquid phase is continuous. The matrix consists of the solid phase and disconnected pores, which do not
contribute to the porosity.
7. In most cases, the material of the frame is isotropic. Anisotropy is due to a preferential alignment of the pores (or
cracks).
Biot states that it is quite possible that the soil particles are held together by capillary forces which behave in pretty
much the same way as the springs of the model of a system made of a great number of small rigid particles held together
by tiny helical springs. That’s the main reason why he postulates a dissipation potential with the functional form of a
harmonic oscillator as an additional element in the Lagrangian of the fluid-rock system.
Later on, Plona [7] reported all three propagatory waves in the high-frequency limit in fused glass bead samples; this
critical observation had been lacking for many years. He demonstrated the existence of a slow wave very close to the one
10. 9
predicted by Biot’s theory. He showed that this wave could only exist as a propagating wave if the following conditions
were satisfied:
a) Continuity of the solid and liquid phases (i.e. possibility of differential fluid and liquid motion).
b) Sufficiently high incident wave frequency (i.e. possibility of differential fluid and liquid motion).
c) Incident wavelength sufficiently large in comparison with pore size to avoid scattering, while the pore size must be
adequate to avoid viscous effects at the wall (skin depth effect).
d) Very different fluid and solid bulk moduli in order to separate clearly the two compressional waves.
Almost right after this experimental verification, Chandler and Johnson [Chandler and Johnson, 1981] showed that the
quasi-static motion of a fluid-saturated porous matrix is describable by a homogeneous diffusion equation in fluid pressure
with a single composite elastic constant, the diffusivity, is contained in Biot’s model for the slow compressional wave in
the limit of zero frequency. They also tried to show –indeed they lacked some consistency with the diffusivity for well-test
analysis limiting case- that the analyses used in the applications concerned with the low-frequency dynamics of a porous
matrix saturated with a viscous fluid, such as well-test analysis, are all limiting cases of the more general analysis based
on the mixture theory of Biot.
Finally, Klimentos and McCann [11] presented experimental results of measurements of attenuation and wave speeds
for compressional waves from 42 water saturated sandstones at a confining pressure of 40 MPa (equivalent to a depth of
burial of about 1.5 km) in a frequency range from 0.5 to 1.5 MHz. They also have reported wave speeds for shear waves
and this is the basic paper that is used in this work for the verification of Biot’s model.
Further extensions of the theory and deeper explanations can be found in the literature [10, 13, 14].
Biot’s theory can be used for the following purposes:
• Estimating saturated-rock velocities from dry-rock velocities.
• Estimating frequency dependence of velocities, and
• Estimating reservoir compaction caused by pumping using the quasi-static limit of Biot’s poroelasticity theory.
Formalism
In the following discussion, the solid matrix is indicated by the index “b”, the solid by the index “s” and the fluid phase
by the index “f”. We’ll follow the approach from Biot [2, 3], Bourbi´e et al. and Carcione [13].
It is possible to postulate a well-defined Lagrangian function for the fluid-rock system. The Lagrangian density for
a conservative system is defined as:
L = T − V (1)
where the expressions for T and V are given in Appendix C. The dynamics of a conservative system can be described by
Lagrange’s equation of motion, which is based on Hamilton’s principle of least action [9]. The method can be extended
to non- conservative systems if the dissipative forces can be derived from a potential:
ΦD =
1
2
b
3
i=1
(vb
i − vf
i )(vf
i − vb
i ) (2)
where b is the friction coefficient obtained by comparing the classical Darcy’s law with the equation of the force derived
from the dissipation potential.
b = φ2 η
¯k
(3)
with φ the porosity, η the dynamic viscosity of the fluid, and ¯k the mean absolute permeability of the system.
And we also have:
vp
i = ∂tup
i (4)
with up
i the displacement vector component of the phase p where p = b (for matrix bulk) of p = f (for fluid).
A potential formulation such as the equation (2) is only justified in the vicinity of thermodynamic equilibrium and
it also assumes that the flow is of the Poiseuille type, i.e. low Reynolds number and low frequencies.
11. 10
Lagrange’s equations with the displacements as generalized coordinates can be written as:
δt
∂L
∂vp
i
+
3
j=1
∂j
∂L
∂(∂jup
i )
−
∂L
∂up
i
+
∂ΦD
∂vp
i
= 0 (5)
where p = b for the frame and p = f for the fluid.
From the definition of the potential and kinetic energy for the system (see Appendix C) and from the definition of
the dissipation potential, it can be shown that we have for isotropic media,
3
j=1
∂jσb
ij = ρ11∂2
ttub
i + ρ12∂2
ttuf
i + b(vb
i − vf
i ) (6)
− φ∂ipf = ρ12∂2
ttub
i + ρ22∂2
ttuf
i − b(vb
i − vf
i ) (7)
where, ρ11 = (1 − φ)(ρs + rρf ), ρ12 = (−φρf )(τ − 1), ρ22 = φρf τ, are the components of the density tensor; ρ =
(1−φ)(ρs)+φρf is the density of the saturated matrix, with τ = 1+ 1
φ − 1 r,the tortuosity where we have used r = 1/2
from Berryman.
If one expresses the stress tensor of the matrix and the hydrostatic pressure of the fluid in with respect to the dis-
placement vectors and if one writes down equations (6) and (7) in vector notation, we have the Biot Equations of Motion.
−G × U + P ( · u) + Q ( · U) =
∂2
∂t2
(ρ11u + ρ12U) + b
∂
∂t
(u − U) (8)
and
Q ( · u) + R ( · U) =
∂2
∂t2
(ρ12u + ρ22U) − b
∂
∂t
(u − U) (9)
where u = (ub
1, ub
2, ub
3)T
, U = (uf
1 , uf
2 , uf
3 )T
, are the displacement vectors for the matrix and the fluid. P = λ + 2G, where
λ and G are the Lam´e coefficients, G being the shear modulus of the matrix, R is a measure of the pressure required
on the fluid to force a certain volume of it into the aggregate while the total volume remains constant, Q is a coupling
coefficient between volume change of the solid and that of the fluid.
If one uses the next expressions:
λ = λf + Mφ(φ − 2α)2
(10)
for the Lam´e first parameter where α = 1 − Kb
Ks
M =
Ks
1 − φ − Kb
Ks
+ φ Ks
Kf
(11)
Q = Mφ(α − φ) (12)
R = M(φ)2
(13)
Kf = λf +
2
3
G (14)
Kb = λb +
2
3
G (15)
As said in [10], with Kp the bulk modulus of the phase p,
P =
(1 − φ)(1 − φ − Kb
Ks
)Ks + φ(Ks
Kb
)Kb
1 − φ − Kb
Ks
+ φ Ks
Kf
+
4
3
G (16)
Q =
φ(1 − φKb
Ks
Ks)
1 − φ − Kb
Ks
+ φ Ks
Kf
(17)
R =
φ2
Ks
1 − φ − Kb
Ks
+ φ Ks
Kf
(18)
In isotropic media, the compressional waves are decoupled from the shear waves, thus the respective equations of motion
can be obtained by taking the divergence and the curl in equations (8) and (9). And the typical procedure to obtain
the analytical expression for the wave speeds is to consider planar waves and to transform the given set of differential
12. 11
equations into a secular equation problem.
For the compressional waves we have to solve:
−(ρ2
f +
1
ω
˜Y ρ)v4
c +
i
ω
˜Y KG +
3
4
G + M(2αρf − ρ) v2
c + M(Kb +
3
4
G) = 0 (19)
where, ˜Y is the Fourier transform of viscodynamic operator. For harmonic waves it is:
˜Y =
η
a2
q2
1 − (1/q)tanh(q)
= iωm(ω) +
˜η(ω)
¯k
(20)
where m(ω) is a function of frequency and ˜η(ω) = η
3
q2
1−(1/q)tanh(q) and q = a iω
ν .
The viscodynamic operator obtained for the low frequency limit from the Lagrangian approach is:
Y (t) = m∂tδ(t) +
η
¯k
δ(t) (21)
with m = ρ22/φ2
= ρf τ/φ, δ(t) being the Dirac delta function.
The Fourier transform of Y (t) is:
˜Y = iωm +
η
¯k
(22)
By expanding (14) in powers of q2
and limiting to the first term in q2
we get:
˜Y =
3η
a2
2
5
q2
+ 1 (23)
By comparing (16) and (17) we find that at low frequencies,
m = ρf (6/5) (24)
For the high frequency limit, the viscodynamic operator that takes on account the frequency dependence of the viscous
drag is given in appendix B.
With ω angular frequency and ν = η
ρf
the kinematic viscosity, a the pore radius, we have
KG = KB + α2
M =
Ks − KB + φ Ks
Kf
− 1 KB
1 − φ − KB
Ks
+ φ Ks
Kf
(25)
We want to acquire the wave speeds from Biot’s model. Following the usual procedure, we must solve the following
equation for vc
− ρ2
f +
1
ω
˜Y ρ v4
c +
i
ω
˜Y KG +
3
4
G + M (2αρf − ρ) v2
c + M Kb +
3
4
G = 0
From the fundamental theorem of algebra it follows that the equation has four complex roots. The easier way to find
them is by regarding our polynomial as quadratic in the variable v2
c , thus
(v2
c )1,2 =
−B ±
√
B2 − 4AC
2A
(26)
A = A1 + iA2 = − ρ2
f +
1
ω
˜Y ρ (27)
B = B1 + iB2 =
i
ω
˜Y KG +
3
4
G + M (2αρf − ρ) (28)
˜Y = ˜Y1 + i ˜Y2 =
η
a2
q2
1 − q tanh(q)
(29)
q = a
iω
ν
(30)
c = M Kb +
3
4
G (31)
13. 12
Remark We’re considering that everything is real except for ˜Y , A, B, q: Just to gain more insight, let’s analyse more
carefully ˜Y
˜Y1 =
ω
2ν
aηωS+
D
(32)
˜Y2 =
ηω C+ − ω
2ν aS−
D
(33)
C± ≡ cosh
2a2ω
ν
± cos
2a2ω
ν
(34)
S± ≡ sinh
2a2ω
ν
± sin
2a2ω
ν
(35)
D ≡ a2
ωC− − 2a
νω
2
S− + νC+ (36)
Now we fix our attention on a and b
A1 = − ρ2
f +
1
ω
˜Y1ρ (37)
A2 = −
˜Y2ρ
ω
(38)
B1 =
− ˜Y2
ω
KG +
3
4
G + M (2αρf − ρ) (39)
B2 =
˜Y1
ω
KG +
3
4
G (40)
The roots for the quadratic polynomial are
(v2
c )1 =
−B +
√
B2 − 4AC
2A
(41)
(v2
c )2 =
−B −
√
B2 − 4AC
2A
(42)
And then, the four roots of the original problem shall be obtained after taking another square root
(v2
c )1+ = +
−B +
√
B2 − 4AC
2A
(43)
(v2
c )1− = −(v2
c )1+ = −
−B +
√
B2 − 4AC
2A
(44)
For the remaining roots we have an analogous setting
(v2
c )2+ = +i
B +
√
B2 − 4AC
2A
(45)
(v2
c )2− = −(v2
c )2+ = −i
B +
√
B2 − 4AC
2A
(46)
Now we shall get the real and the imaginary parts of each root, the resolution lurks within eight variables, but first allow
us to make some definitions in order to avoid cumbersome expressions
Σ± ≡ 4
A2
±2B
4
(∆)
2
cos 1
2 arg (∆) + (∆)
2
+ B2
(47)
∆ ≡ B2
− 4AC (48)
The real and imaginary parts for the first and second roots ((vc)1+, (vc)1−) are presented below. However we must state
the following
14. 13
Remark A, B ∈ C, thus so does ∆ ∈ C, we must be careful when taking roots: With that in mind we proceed as
follows
[(vc)−1
1+] =
√
2 cos
arg
2
A
√
∆ − B
(Σ−) − sin
arg
2
A
√
∆ − B
(Σ−) (49)
[(vc)−1
1−] = − [(vc)−1
1+] (50)
[(vc)−1
1+] =
√
2 cos
arg
2
A
√
∆ − B
(Σ−) + sin
arg
2
A
√
∆ − B
(Σ−) (51)
[(vc)−1
1−] = − [(vc)−1
1+] (52)
Consequently, the second and last set is given by
[(vc)−1
2+] =
√
2 cos
arg
2
A
B
(Σ+) + sin
arg
2
A
B
(Σ+) (53)
[(vc)−1
2−] = − [(vc)−1
2+] (54)
[(vc)−1
2+] =
√
2 cos
arg
2
A
√
∆ + B
(Σ+) + sin
arg
2
A
√
∆ + B
(Σ+) (55)
[(vc)−1
2−] = − [(vc)−1
2+] (56)
We note that we only have four independent expressions, now we introduce new variables
vp+ =
1
[(vc)−1
1+]
vp− =
1
[(vc)−1
2+]
αp+ = ω [(vc)−1
1+] αp− = ω [(vc)−1
2+] (57)
And the problem is solved, please note that the line of thinking here is: obviously these calculations must be dealt with
by software methods. Considering that, first we must enter the values of α, ω, ν, η; that determines ˜Y . Having ˜Y is quite
straightforward to obtain a, b and c is trivial. Finally with a, b, c we can calculate Σ±, ∆ and therefore the roots.
It is useful to know some terms explicitly, mainly:
θ ≡
2CA2 + B1B2
B2
1 − B2
2 − 4CA1
(58)
ψ ≡ −4CA1 + B2
1 − B2
2
2
+ (2B1B2 − 4CA2)2
(59)
√
∆ = 4
ψ cos (θ) (60)
√
∆ = 4
ψ sin (θ) (61)
arg
A
√
∆ ± B
= arg(A) − arg(
√
∆ ± B) = arctan
A2
A1
− arg(
√
∆ − B) (62)
= arctan
A2
A1
− arctan
4
√
ψ sin (θ) ± B2
4
√
ψ cos (θ) ± B1
(63)
15. 14
And we end this glancing in depth (Σ±), (Σ±)
arg ∆ = 2θ (64)
Γ ≡ 16 A2
1 + A2
2 C2
− 8 A1B2
1 + 2A2B2B1 − A1B2
2 C + |B|2 2
(65)
σ1 = − 2 cos(θ) B2A2
1 − 2A2B1A1 − A2
2B2 cos
2θ
√
1 + 64θ2
+ B1A2
1 + 2A2B2A1 − A2
2B1 sin
2θ
√
1 + 64θ2
4
√
Γ
+ 2A1A2 cos
4θ
√
1 + 64θ2
√
Γ + (A2
2 − A2
1) sin
4θ
√
1 + 64θ2
√
Γ
+ 2A1A2B2
1 − 2A1A2B2
2 − 2A2
1B1B2 + 2A2
2B1B2 (66)
σ2 =≡ 2 cos(θ) B1A2
1 + 2A2B2A1 − A2
2B1 cos
2θ
√
1 + 64θ2
+ −B2A2
1 + 2A2B1A1 + A2
2B2 sin
2θ
√
1 + 64θ2
4
√
Γ
+ A2
1 − A2
2 cos
4θ
√
1 + 64θ2
√
Γ + 2A1A2 sin
4θ
√
1 + 64θ2
√
Γ
+ A2
1B2
1 − A2
2B2
1 − A2
1B2
2 + A2
2B2
2 + 4A1A2B1B2 (67)
(Σ±) = cos
1
4
σ1
σ2
|a| 2 cos(θ) ±B1
4
ψ cos
4θ
√
1 + 64θ2
+ B2
4
ψ sin
4θ
√
1 + 64θ2
+ ψ sin
2θ
√
1 + 64θ2
+ 2B1B2
2
+ cos(θ) ±2B2
4
ψ cos
2θ
√
1 + 64θ2
−2B1
4
ψ sin
2θ
√
1 + 16θ2
+ ψ cos
4θ
√
1 + 64θ2
+ B2
1 − B2
2
2 −1/8
(68)
Whereas for the imaginary part we have
(Σ±) = sin
1
4
σ1
σ2
|a| 2 cos(θ) ±B1
4
ψ cos
4θ
√
1 + 64θ2
+ B2
4
ψ sin
4θ
√
1 + 64θ2
+ ψ sin
2θ
√
1 + 64θ2
+ 2B1B2
2
+ cos(θ) ±2B2
4
ψ cos
2θ
√
1 + 64θ2
−2B1
4
ψ sin
2θ
√
1 + 16θ2
+ ψ cos
4θ
√
1 + 64θ2
+ B2
1 − B2
2
2 −1/8
(69)
For the shear wave, it is easy to show that, by following an analogue reasoning and by separating real and imaginary
parts, we get
vc = (vc) + i (vc) (70)
vc =
|z|−1/2
G1/2
ρ −
˜Y2ωρ2
f
˜Y 2
1 + ˜Y 2
2
+ |z|
ρ −
˜Y2ωρ2
f
˜Y 2
1 + ˜Y 2
2
+ |z|
2
+
˜Y1ωρ2
f
˜Y 2
1 + ˜Y 2
2
2
+ i
|z|−1/2
G1/2
˜Y1ωρ2
f
˜Y 2
1 + ˜Y 2
2
ρ −
˜Y2ωρ2
f
˜Y 2
1 + ˜Y 2
2
+ |z|
2
+
˜Y1ωρ2
f
˜Y 2
1 + ˜Y 2
2
2
(71)
|z| = ρ −
˜Y2ωρ2
f
˜Y 2
1 + ˜Y 2
2
2
+
˜Y1ωρ2
f
˜Y 2
1 + ˜Y 2
2
2
(72)
Thus
v−1
c =
(vc) − i (vc)
(vc)2 + (vc)2
(73)
The phase velocity for the shear wave is:
vs = [ (v−1
c )]−1
=
(vc)
(vc)2 + (vc)2
−1
16. 15
The attenuation factor for the shear wave is:
αS = ω[ (v−1
c )] = −ω
(vc)
(vc)2 + (vc)2
Study of wave speed and attenuation parameters
For the three wave types, the waves speed rises with increasing frequency due to the fact that inertial forces increase
simultaneously. Since there is a contrast in inertial forces for the fluid and the solid part, there is a consequent differential
movement between the fluid and the fluid/solid combination due to permeability effects which involves less fluid entraining
a decrease in mass in the overall movement as the frequency increases. The attenuation also increases with frequency for
the three cases, since the dissipation is proportional to the square of the angular frequency. The slow compressional wave
attenuation is highly attenuated in comparison with the other two.
It is evident that wave speeds increase with increasing elastic moduli. As elastic moduli depend on porosity, it is
possible that there is a relationship between porosity, permeability, and ultrasonic parameters. Velocity is expected to
decrease and attenuation to increase as porosity increases However, those trends may differ for saturated materials. In
Biot’s model, compressional wave speeds and attenuations behave as expected as well as shear attenuation, however shear
wave speed behaves inversely. This is numerically shown in the Figures 1 to 6. Note also that permeability only affects
the abscissa scale. Due to the expression of the characteristic frequency.
fc =
φ
2πρf κ
=
φη
2πρf
¯k
(74)
where
κ =
¯k
η
(75)
is the mobility.
As the mobility approaches 0 (or towards infinity), the characteristic frequency tends inversely towards infinity (or
towards 0) .The rise in the curves with increasing frequency on an absolute scale is accordingly less (or more) pronounced.
This is due to the fact that the lower (or higher) the permeability, the less (or more) are the differential movements
(fluid/matrix) privileged and the less (or more) Biot’s effects are pronounced. Biot’s theory takes into account only the
dissipation due to mean differential movements and not those due to absolute movements of the fluid, this latter becomes
more important in the high frequency regime, thus it is logical that the lower the viscosity, the more the differential
movement may be pronounced and hence the differential velocity is greater and the dissipation increases. Therefore, the
lower the viscosity and hence the higher the mobility, the greater the attenuation.
Biot [2] claimed that the low frequency theory is valid up to f < 0.15fc and in a general porous medium, we may
assume that the transition occurs when inertial and viscous forces, from the expression of the viscodynamic operator for
low frequencies we can infer that, that happens when iωm = η/¯k, this defines another criterion of validity for the theory:
fl =
η
m¯k
=
φη
2πτρf
¯k
(76)
This frequency indicates the upper limit for the validity of low-frequency Biot’s theory.
17. 16
Figure 1: Porosity dependence Fast Compres-
sional Wave Speed
Figure 2: Porosity dependence Fast Compres-
sional Attenuation Factor
Figure 3: Porosity dependence Slow Compres-
sional Wave Speed
Figure 4: Porosity dependence Slow Compres-
sional Attenuation Factor
Figure 5: Porosity dependence Shear Wave
Speed
Figure 6: Porosity dependence Shear Attenua-
tion Factor
The above plots use the parameter values for Group B from while varying porosity. The next plot shows the dependence
of the fast compressional wave speed with viscosity, and thus with mobility, once again the remaining parameter values
correspond to those from Group B from [11]. The behaviour of the remaining wave speeds and attenuation is analogous
and this example thus representative of the effect on the abscissa scale due to mobility.
18. 17
Figure 7: Viscosity and mobility dependence for Fast Compressional Wave Speed.
One can appreciate in general a velocity contrast of 1% to 2% in consolidated sandstones at 40 MPa confining pressure.
Since the elastic moduli are increasing functions of confining pressure, in order to observe a larger spread of the values
for velocity (from 5% to 10% difference), one needs to have a low confining pressure or smaller values for elastic moduli
as is the case of sediments, of the order to 107
Pa to 108
Pa for the frame elastic moduli.
Experimental validation of the Biot model
Data for fast compressional and shear wave speeds and fast compressional attenuation factor for 42 water-saturated
sandstones divided in 3 groups (i.e., Groups A, B and the combination of 2H2M1 & 2V1M1), was obtained from a report
from Klimentos and McCann at a frequency f = 1 MHz and at a confining pressure of 40 MPa. However one of the groups
(Group A) cannot be used for experimental validation since the attenuation factor is calculated taking on account the
clay content of the sandstones, thus the remaining two groups can be considered as clay-free samples. On the other hand,
group B cannot be used since the measurements were made at a frequency that is higher than the breakdown frequency
for this model. So we can only use the combination of groups 2H2M1 & 2V1M1.
Figure 8: Biot model prediction for wave speeds
(Average Group 2H2M1 & 2V1M1)
Figure 9: Biot model prediction for fast wave
attenuation (Average Group 2H2M1 & 2V1M1)
19. 18
Data for fast compressional attenuation factor were extracted from another paper [15] and the frequency dependence
was confirmed for low frequencies. Sample A is tight sandstone with low permeability and low porosity, Sample B is a
sandstone with high permeability and high porosity.
For the remaining parameters we used are Ks = 35GPa, Kb = 1.7 GPa, µb = 1.855 GPa, Kf = 2.4 GPa, ρf = 1000kg/m3
,
Figure 10: Biot model validation of attenuation
factor for Sample A. Pd = 15 MPa.
Figure 11: Biot model validation of attenuation
factor for Sample B. Pd = 18 MPa.
and η = 1 cP (Carcione, 1998a).
Table 1: Sandstone sample parameters.
Sample Porosity (φ), Permeability (¯k), Solid Length, Diameter,
% mD density (ρs), ×10−3
m ×10−3
m
kg/m2
A 14.8 7.8 2099 70 38
B 20.6 590 2261 70 38
Table 2: Input data for model for clay-free sandstones.
Parameter Unity Average Average Average
Group B 2H2M1/2V1M1 Group A
Porosity (φ) % 15 ± 1 2.5 15 ± 1
Permeability (¯k) mD 175 ± 65 0.01 48 ± 6
Solid density (ρs) kg/m3
2628 2628 2628
Clay content % 0.68 ± 0.3 0 15
(volume)
Kb GPa 19 30 12
µb GPa 30 40 28
Vp+ (f = 1MHz) m/s - 5884 ± 50 -
Vs (f = 1MHz) m/s - 3397 ± 68 -
α (f = 1MHz) 1/s - 0.012 ± .005 -
Derivation of the diffusion equation
We will derive the diffusion equation from Biot’s equations in the quasi-static limit. The typical procedure is to consider
planar waves and finding roots of the secular equation associated to such wave. We shall use the notation of the paper
from [8], furthermore we are neglecting the inertial terms for low frequencies (i. e. ¨u = 0 =
¨
U) with G = µ ; Y = Q and
20. 19
P = λ + 2µ, we finally have
−G × U + P ( · u) + Q ( · U) = b
∂
∂t
(u − U) (77)
Q ( · u) + R ( · U) = −b
∂
∂t
(u − U) (78)
The authors propose a change of variables to normal mode coordinates
ξ = u − U ζ = u +
R + Q
P + Q
U (79)
Rendering it in matrix notations yields
ξ
ζ
=
1 −1
1 R+Q
P +Q
u
U
(80)
We invert the matrix in order to have a similar expression for the original variables
u
U
=
1
P + R + 2Q
R + Q P + Q
−(P + Q) P + Q
ξ
ζ
(81)
This expression gives origin to the next two equations
(P + R + 2Q)
∂ξ
∂t
=
− G × × (P + Q)ζ + (R + Q)ξ + (P + Q)2
( · ξ) + (PR − Q2
) ( · ξ) (82)
− G × × ζ +
R + Q
P + Q
ξ + (P + R + 2Q) ( · ζ) = 0 (83)
Where the last equation is the sum of (77) and (78), after doing the sum the change of variables is implemented. Allow
us to define F = ( · ζ) and use the Helmholtz decomposition theorem, which states that F may be decomposed into a
curl-free component and a divergence-free component, i. e.:
F = − Φ + × A (84)
Φ(r) =
1
4π V
· F (r )
|r − r |
dV −
1
4π S
ˆn ·
F (r )
|r − r |
dS (85)
A(r) =
1
4π V
× F (r )
|r − r |
dV −
1
4π S
ˆn ×
F (r )
|r − r |
dS (86)
If V = R3
which of course is unbounded, therefore F must vanish faster than 1
r as r → ∞ if we want to avoid divergences
in the boundary term. This is equivalent to ask for bulk solutions that vanish at infinity, thus giving
Φ(r) =
1
4π V
· F (r )
|r − r |
dV (87)
A(r) =
1
4π V
× F (r )
|r − r |
dV (88)
But it’s evident now that × F = 0 and · F = 0 too, so we must have F ≡ 0. Applying this and simplifying one step
further we get
× × ζ +
R + Q
P + Q
ξ = 0 (89)
(RP − Q2
) ( · ξ) = b(P + R + 2Q)
∂ξ
∂t
(90)
By taking the curl of the last equation it’s clear that × ξ is time independent; the author fixes it to zero by saying
that if × ξ = 0 at some time in the past (before the experiment starts) it shall remain zero forever; this is a convenient
initial condition for this derivation. Finally we can further simplify the equations
× ζ = 0 ( · ζ) = 0 (PR − Q2
) ( · ξ) = b(P + R + 2Q)
∂ξ
∂t
(91)
21. 20
If we use
Kf = λf +
2
3
G Kb = λb +
2
3
G (92)
α = 1 −
Kb
Ks
M =
Ks
1 − φ − Kb/Ks + φKs/Kf
(93)
λ = λf + Mφ(φ − 2α) λ = λb + M(α − φ)2
(94)
γ = Mφ(α − φ) R = Mφ2
(95)
We get
P =
(1 − φ)(1 − φ − Kb/Ks)Ks + φ(Ks/Kf )Kb
1 − φ − Kb/Ks + φKs/Kf
(96)
Q =
(1 − φ − Kb/Ks)φKs
1 − φ − Kb/Ks + φKs/Kf
(97)
R =
φ2
Ks
1 − φ − Kb/Ks + φKs/Kf
(98)
If one follows Biot’s papers we can see that σij = (3P − 4G) · u + R · U. The total dilational stress on the aggregate
is σkk = σii − 3φp with
σkk = (3P − 4G + 3Q) · u + 3(Q + R) · U (99)
So we have
p
σkk
=
−Q/φ −R/φ
(3P − 4Q + 3Q) 3(Q + R)
· u
· U
(100)
=
−Q/φ −R/φ
(3P − 4Q + 3Q) 3(Q + R)
1
P + R + 2Q
· ξ
· ζ
(101)
=
1
P +R+2Q
R(P +Q)−Q(R+Q)
φ −Q(P +Q)+R(P +Q)
φ
(3P −4G+3Q)(R+Q)−3(Q+R)(P +Q) (P +Q)(3P −4G+6Q+3R)
· ξ
· ζ
(102)
Thus, inverting we arrive to
· ξ
· ζ
=
det−1
A
P +R+2Q
(P +Q)(3P −4G+6Q+3R)
Q(P +Q)+R(P +Q)
φ
3(Q+R)(P +Q)−(3P −4G+3Q)(R+Q) −Q(R+Q)+R(P +Q)
φ
p
σkk
(103)
−1
det A =
φ(P + R + 2Q)2
Ω
(104)
Ω = ([P + Q][3P − 4G + 6Q + 3R][−Q(R + Q) − R(P + Q)]
−[Q(P + Q) + R(P + Q)][3(Q + R)(P + Q) − [3P − 4G + 3Q][R + Q]) (105)
By using this new basis and the expressions for P, Q and R it can be shown that
· ζ ∝ p +
Kb + 4
3 G
4G(1 − Kb
Ks
)
σkk (106)
· ξ ∝ p +
1 − Kb
Ks
3Kb[ 1
Kb
+ φ
Kf
− 1+φ
Ks
]
σkk (107)
Thus, by taking the divergence of (91) and by replacing these expressions we get
2
p +
Kb + 4/3G
4G(1 − Kb/Ks)
σkk = 0 (108)
CD
2
+
∂
∂t
p +
(1 − Kb/Ks)σkk
3Kb[1/Kb + φ/Kf − (1 + φ)/Ks]
= 0 (109)
With
CD =
PR − Q2
b(P + R + 2Q)
(110)
=
¯kKf
ηφ
1 +
Kf
φ(Kb + 4/3G)
1 +
1
Ks
4
3
G(1 − Kb/Ks) − Kb − φ(Kb + 4/3G)
−1
(111)
If the argument in the Laplace’s equation vanishes for some reason, then the previous diffusion equation is either in P or
in σkk at the same time. Thus in the limit of low frequencies, Biot’s equations become in fact a diffusion equation.
22. 21
Analysis of Diffusivity
We have previously derived from [8] the expression for the diffusivity from Biot’s theory in the quasi-static limit, which
can be re ordered as:
CD =
¯kKf
ηφ
1 +
Kf
φ Kb + 4
3 G
1 −
Kb
Ks
(1 + φ) +
4G
3Ks
(α − φ) (112)
From [12], we find that,
1
Kb + 4
3 G
=
1
Kb − 2
3 G + 2G
=
1
(λ + 2G)
=
(1 + ν)
3(1 − ν)Kb
(113)
With λ the Lam´e first parameter and ν the Poisson ratio of the frame, we also know that
G =
3Kb(1 − 2ν)
2(1 + ν)
(114)
Chydro
pp =
1
φ
1
Kb
−
(1 + φ)
Ks
(115)
Cm =
1
Ks
(116)
Cf =
1
Kf
(117)
By plugging (113),(114), (115), (116) and (117) into (112), we get
CD =
¯k
ηφCf
1 +
1
Cf
Chydro
pp −
2(1 − 2ν)α
3(1 − ν)
Chydro
pp + Cm −
Chydro
pp
(1 − 2ν)
+ Cm
1
α
+
1
φ
+ 1
−1
(118)
Also from[12], we know that the expression for the uniaxial pore compressibility is:
CUni
pp = Chydro
pp −
2(1 − 2ν)α
3(1 − ν)
(Chydro
pp + Cm) (119)
By inspection of relation (118), we find (119) plus a term that has to be equated to zero, and we get,
CD =
¯k
ηφCf
1 +
1
Cf
(CUni
pp )
−1
(120)
Which leads to
CD =
¯k
ηφ(Cf + CUni
pp )
=
¯k
ηφCT
(121)
With CT the total compressibility used in well test analysis .
If and only if we fulfil the condition,
Chydro
pp = Cm(1 − 2ν)
1
α
+
1
φ
+ 1 (122)
From [12] we know that,
Chydro
pp = Cbc
α
φ
− (1 − α) (123)
Cbc = Cbp + Cm (124)
By inspecting (122) and (123), we conclude that,
Cbp = Cm
(1 − 2ν)
α
α
φ − (3 − α)
α
φ − (1 − α)
− 1
< 0 (125)
However, equation (125) can never be satisfied; if we plug typical values from [12], the expression between parentheses
is always negative, so the only reason why equation (123) works instead of the correct equation (118) is due to the fact
that for typical reservoirs CUni
pp and the Biot’s equivalent compressibility differ by a factor of two, as can be shown for
some cases in appendix D, and since we have so much uncertainty on measurements this factor of 2 becomes negligible
so that equation (123) holds.
23. 22
If one investigates equation (118), when we ask the next condition:
−
Chydro
pp
(1 − 2ν)
+ Cm
1
α
+
1
φ
+ 1 = 0 (126)
We can interpret this result as asking the effective fluid-to-rock compressibility and the effective rock-to-fluid compress-
ibility to cancel up as Newtonian addition of forces in the pore network, but this condition cannot be satisfied which
possibly means that locally, the rock-to-fluid and the fluid-to-rock interactions are not symmetric and we must be locally
out of dynamic equilibrium. This model is limited since we consider planar waves and it’s more reasonable to consider
either spheroidal or spherical waves, even Bessel waves would be a better geometrical ansatz for seismic prospection, so
we lose some insight due to this geometrical simplification. Another, limitation is the pore network geometry which must
be integrated in some manner as well.
Discussion
Biot’s poroelasticity theory predicts the existence of two compressional waves, due to two possible uniaxial vibrational
modes (one longitudinal wave in phase and one longitudinal wave in antiphase), and of one shear wave.
The analytical expressions for the wave speeds and attenuation are derived by means of elementary complex analy-
sis; since finding the real and imaginary parts for some expressions requires, to some extent, being familiar with the
behaviour of some complex functions, however the procedure for calculating them is quite straight forward after some
mechanical algebraic manipulations. It is true that the expressions are quite complex and one is forced to use a simple
sequential algorithm to plot the curves, for this purpose a simple excel spreadsheet is sufficient. In order to avoid possible
oscillations of the model, one must remain below the breakdown frequency, above which the model is no longer valid.
The study of the behaviour of the wave speed and attenuation parameters is done using as base case the data given
in table 2 for the average group B. The understanding of the parameter influence on the model is mainly based on
rock/fluid coexistence mechanisms. Even if most of the parameters are correlated, it is still interesting to appreciate the
model’s different features based on these simple numerical experiments. One explanation for the inverse behaviour for the
shear wave speed with respect to porosity is that the shear modulus of the frame dominates over the bulk moduli, as the
shear wave only travels through the rock part of the frame and not through the fluid itself (which has zero shear modulus).
One of the most outstanding features is the mobility dependence that expands or contracts the abscissa scale depending on
the characteristic frequency of the system. For the studied case, we appreciate a 1.6 % variation in wave speed, however, It
is found numerically that, when the frame bulk and shear moduli are of the order of 109
Pa at the same time, the variation
in wave speed is of the order of 11% for very tight sandstones. This is shown in the next plot using the remaining data
from Carcione for the remaining elastic moduli (Carcione, 1998a) and data from 2H2M1 & 2V1M1 for the fluid and rock
parameters, this means that the sandstone must have low permeability and low porosity as well. The experimental vali-
Figure 12: Numerical evidence of Biot effect of more than 10%.
dation of the model is clear for the attenuation at low frequencies (Samples A and B) , at high frequencies (but less than
the limit frequency) it is possible to predict a single value (average group 2H2M1 & 2V1M1) due to lack of more available
data. The former is a less strong verification, since we are not able to verify the frequency dependence for the wave speeds.
24. 23
However, we are able to predict the correct values within the range given in table 2. This is thus, to a certain ex-
tent, strong evidence of the applicability of Biot’s model to reservoir rocks such as consolidated and tight sandstones.
Finally, we have been able to clarify Johnson and Chandler’s approach for deriving the diffusion equation from Biot’s
equation in the quasi-static limit by ignoring the inertial forces, and doing some algebraic and vector manipulations. We
derived the diffusion equation from the quasi-static limit of the theory, in the diffusivity we recognized the expression
for the uniaxial strain pore compressibility [12], and we tried to make a connection with the diffusivity from well test
analysis which led to inconsistent results, however the former model works only thanks to the usual values for the elastic
parameters found in typical reservoirs, where CUni
pp and the Biot’s equivalent compressibility differ by a factor of two, as
can be shown for some cases in appendix D.
We found an inconclusive connection between Biot theory of poroelasticity and the diffusion equation from well-test
analysis, giving rise to a factor-of-two underestimate of rock compressibility, this could be due to an effective rock-to-fluid
compressibility and fluid-to-rock compressibility that are not taken into account in traditional models, and this could also
be related to geometrical effects of the pore network and of the actual wave front of the acoustic waves. These results
will be a very helpful starting point for further work on this problem.
From the above series of arguments, it is now clear that the initial objectives of this work have been satisfied, how-
ever, due to insufficient data we haven’t been able to verify the frequency dependence of the wave speeds, and we suggest
to perform experimental measurements in a geomechanics laboratory, instead of looking for data on the literature in order
to have more control of the experimental settings. It has been very difficult to find data in the literature, and it is also
experimentally difficult to observe the slow Biot wave at low frequencies according to Plona [7]. However, some important
work still has to be done in the future, mainly:
• To verify the high- frequency Biot’s model, the expressions for the real and imaginary parts of the viscodynamic
operator in the high frequency limit is given on appendix B.
• To verify the anisotropic version of Biot’s model.
• To implement a more robust method of experimental verification such as a full wave inversion algorithm for the
Biot Model.
Conclusion
The main conclusions of this work are the following:
• Analytical closed form relations for the fast and slow compressional and shear wave speeds and attenuations were
derived from first principles of Biot’s poroelasticity theory by means of elementary complex analysis.
• A study of their behaviour was carried out by identifying the dominant parameters of the model, mainly porosity
and mobility. From numerical experiments, we concluded that increasing porosity will decrease wave speed and will
increase attenuation, except for the case of the shear wave speed since the dominant parameter, in the former case,
is the shear modulus of the frame which can be correlated with porosity in a complex fashion, we also observed that
the effect of mobility is the modulation of the amplitude of the abscissa scale via de characteristic frequency.
• By using typical values for the model parameters for water saturated sandstones, we appreciate a very mild Biot
effect in wave speed varying from 1% to 2 % [10], it is also found numerically that we can get greater Biot effects
of the order of 10% for softer rocks with frame shear and bulk moduli of approximately.
• The frequency dependence of attenuation model has been successfully validated with water saturated clay free
sandstones for a frequency range from 0 Hz to 100 Hz (samples A and B) and the wave speed model is locally
verified at a frequency of 1 MHz (average group 2H2M1 & 2V1M1).
• We derived the diffusion equation from the quasi-static limit of the theory, in the diffusivity we recognized the
expression for the uniaxial strain pore compressibility [12], and we tried to make a connection with the diffusivity
from well test analysis which led to inconsistent results, however the former model works only thanks to the usual
values for the elastic parameters found in typical reservoirs.
• We found an inconclusive connection between Biot theory of poroelasticity and the diffusion equation from well-
test analysis, giving rise to a factor-of-two underestimate of rock compressibility, this could be due to an effective
rock-to-fluid compressibility and fluid-to-rock compressibility that are not taken into account in traditional models,
and this could also be related to geometrical effects of the pore network and of the actual wave front of the acoustic
waves. These results will be a very helpful starting point for further work on this problem.
25. 24
Nomenclature
α = Biot’s coefficient [−] (Dimensionless)
αp+ = Attenuation of the fast compressional wave [1/m]
αp− = Attenuation of the slow compressional wave [1/m]
αs = Attenuation of the shear wave [1/m]
CD = Diffusivity [m2
/s]
Cbp = Pore compressibility of the bulk volume [1/Pa]
Cf = Compressibility of the pore fluid [1/Pa]
Cm = Compressibility of the solid grains [1/Pa]
Chydro
pp = Hydrostatic pore compressibility of the pore vol-
ume [1/Pa]
CUni
pp = Uniaxial strain pore compressibility of the pore vol-
ume [1/Pa]
η = Dynamic viscosity [Pa· s]
φ = Porosity [−] (Dimensionless)
f = Frequency [Hz]
fc = Characteristic frequency [Hz]
fl = Limit frequency [Hz]
G = Frame shear modulus [Pa]
(z) = Imaginary part of the complex number z
Kb = Frame bulk modulus [Pa]
Kf = Fluid bulk modulus [Pa]
KG = Gassmann bulk modulus [Pa]
KS = Solid grains bulk modulus [Pa]
¯k = Mean absolute permeability [m2
]
κ = Hydraulic permeability [m2
/Pa · s]
λ = Lam´e First parameter [Pa]
ν = Kinematic viscosity [m3
Pa · s/kg]
ω = Angular frequency [rad/s]
Pc = Confining pressure [Pa]
PD = Differential pressure [Pa]
Pe = Effective pressure [Pa]
(z) = Real part of the complex number z
σij = Component (i, j) of the stress tensor [Pa]
v = Poisson’s ratio of the frame [−] (Dimensionless)
vp+ = Wave speed of the fast compressional wave [m/s]
vp− = Wave speed of the slow compressional wave [m/s]
vs = Wave speed of the shear wave [m/s]
˜Y = Viscodynamic operator [Pa· s/m2
]
˜Y1 = Real part of the viscodynamic operator [Pa· s/m2
]
˜Y2 = Imaginary part of the viscodynamic operator [Pa·
s/m2
]
26. 25
References
[1] Maurice A. Biot, General Theory of Three-Dimensional
Consolidation, Journal of Applied Physics, Vol. 12, pp.
155- 164, 1941.
[2] Maurice A. Biot, Theory of Propagation of Elastic
Waves in a Fluid-Saturated Porous Solid. I. Low-
Frequency Range, The Journal of The Acoustical Society
of America, Vol. 28, No. 2, pp. 168-178, 1956.
[3] Maurice A. Biot, Theory of Propagation of Elastic
Waves in a Fluid-Saturated Porous Solid. II. Higher Fre-
quency Range, The Journal of The Acoustical Society of
America, Vol. 28, No. 2, pp. 179-191, 1956.
[4] M. A. Biot and D. G. Willis, The Elastic Coefficients
of the Theory of Consolidation, Journal of Applied Me-
chanics, Vol. 24, pp. 594- 601, 1957.
[5] L.D. Landau and E.M. Lifshitz, Theory of Elasticity, In-
stitute of Physical Problems, U.S.S.R. Academy of Sci-
ences, Vol. 7 of Course of Theoretical Physics. pp. 98-
115, 1959.
[6] Maurice A. Biot, Mechanics of Deformation and Acous-
tic Propagation in Porous Media, Journal of Applied
Physics, Vol. 33,No. 4, pp. 1482- 1498, 1962.
[7] Thomas J. Plona, Observation of a second bulk com-
pressional wave in a porous medium at ultrasonic fre-
quencies, Applied Physics Letters, Vol. 52, No. 5, pp.
3391-3395, 1980.
[8] Richard N. Chandler and David L. Johnson, The equiva-
lence of quasi-static flow in fluid-saturated porous media
and Biot’s slow wave in the limit of zero frequency, Jour-
nal of Applied Physics, Vol. 12, pp. 155- 164, 1981.
[9] J. Achenbach, Wave propagation in elastic solids ,
Achenbach, J. (1973). (North-Holland series in applied
mathematics and mechanics ; v.16). Amsterdam ;Lon-
don: North-Holland. pp. 61.
[10] T. Bourbi´e, O. Coussy and B. Zinszne, Acoustics of
Porous Media, Institut Fran¸cais du P´etrole Publications
Geophysics Petroleum Engineering, pp. 63- 95, 1987.
[11] T. Klimentos and C. McCann, Relationships among
compressional wave attenuation, porosity, clay content,
and permeability in sandstones, Society of Exploration
Geophysicists Geophysics, Vol. 55, No.8, pp. 998-1014,
1990.
[12] R.W. Zimmerman, Implications of Static Poroelasticity
for Reservoir Compaction, Proc. 4th North Amer. Rock
Mech. Symp., A.A. Balkema, Rotterdam, pp. 169-172,
2000.
[13] Jos´e M. Carcione, Wave Fields in Real Media: Wave
Propagation in Anisotropic, Anelastic and Porous Me-
dia, Seismic Exploration Volume 31 Elsevier pp. 219-
261, 2001.
[14] G. Mavko, T. Mukerji and J. Dvorkin, The Rock
Physics Handbook: Tools for Seismic Analysis of Porous
Media, Cambridge University Press pp.266-272, 2009.
[15] V. Mikhaltsevitch, M. Lebedev and B. Gurevich, An
Experimental Study of Low-Frequency Wave Disper-
sion and Attenuation in Water Saturated Sandstones,
Poromechanics V ASCE 2013 “Proceedings of the Fifth
Biot Conference on Poromechanics”, 2013.
27. 26
Appendix A: Literature Review
Table 3: Literature Milestones Biot Model for Wave Propagation in Fluid-Saturated Porous Media
Paper or Book Year Title Authors Contribution
Journal of Applied “General Theory of Three- Maurice First to formulate a mathematical treatment for
Physics, Vol. 12, 1941 Dimensional Consolidation” A. Biot consolidation by means of operational calculus.
pp. 155- 164.
The Journal of The “Theory of Propagation of Elastic Maurice First to formulate a theory for propagation of
Acoustical Society of 1956 Waves in a Fluid-Saturated Porous A. Biot waves in fluid-saturated porous media by means
America, Vol. 28, Solid. of Lagrangian mechanics for Poiseuille flow
No. 2, pp. 168-178. I. Low-Frequency Range” valid up to a critical frequency.
The Journal of the 1956 “Theory of Propagation of Elastic Maurice First to formulate a theory for propagation of
Acoustical Society Waves in a Fluid-Saturated Porous A. Biot waves in fluid-saturated porous media by means
of America, Vol. 28, Solid. of Lagrangian mechanics and viscodynamic
No. 2, pp. 179-191. II. Higher Frequency Range” operators for the breakdown of Poiseuille flow
beyond the critical frequency.
Journal of Applied 1957 “The Elastic Coefficients of the M.A. Biot First to describe methods of measurements of the
Mechanics, Vol. 24, Theory of Consolidation.” D.G. Willis elastic coefficients from Biot’s theory.
pp. 594- 601
Institute of Physical “Theory of Elasticity” L.D. Landau First to formalize the theory of elasticity for
Problems, U.S.S.R. 1959 solids by means of a rigorous mathematical
Academy of Sciences, description.
Vol. 7 of Course of E.M. Lifshitz
Theoretical Physics.
Journal of Applied 1962 “Mechanics of Deformation and Maurice A. A unified treatment of the mechanics of
Physics, Vol. 33, Acoustic Propagation in Porous Biot deformation and acoustic propagation in porous
No. 4, pp. 1482- 1498. media is presented, and some new results and
Media.” generalizations are derived.
Applied Physics “Observation of a second bulk Thomas J. First to observe experimentally the slow
Letters, Vol. 52, No. 5, 1980 compressional wave in a porous Plona compressional wave predicted by Biot’s theory.
P pp. 3391-3395. medium at ultrasonic
frequencies”
Journal of Applied “The equivalence of quasi-static flow Richard N. First to prove the equivalence between Biot’s
Physics, Vol. 12, 1981 in fluid-saturated porous media and Chandler slow wave equation and a Diffusion Equation for
pp. 155- 164. Biot’s slow David L. Pressure.
wave in the limit of zero frequency” Johnson
Institut Fran¸cais du 1987 “Acoustics of Porous Media” T. Bourbi´e First to explain in a clear fashion Biot’s theory
P´etrole Publications O. Coussy and other relevant elements of applied elasticity
Geophysics theory to rock mechanics.
Petroleum Engineering B. Zinszner
Society of Exploration 1990 “Relationships among compressional T. Klimentos Experimental measures of wave speeds and
Geophysicists wave attenuation, attenuation for the fast compressional wave and
Geophysics, Vol. 55, porosity, clay content, and C. McCann the shear wave in 42 water saturated sandstones
No.8, pp. 998-1014. permeability in sandstones” under confining pressure.
Proc. 4th North Amer. 2000 Implications of Static Poroelasticity R.W. First to derive an analytical expression for the
Rock Mech. Symp., A.A. for Reservoir Compaction Zimmerman uniaxial pore compressibility.
Balkema, Rotterdam,
pp. 169-172,
Seismic Exploration “Wave Fields in Real Media: Wave Jos´e M. Rigorous and very mathematical approach to
Volume 31 2001 Propagation in Anisotropic, Anelastic Carcione Biot’s theory with its extension to anisotropic
Elsevier and Porous Media.” systems.
Cambridge University “The Rock Physics Handbook: Tools G. Mavko Summary of Biot’s theory results and its
Press 2009 for Seismic Analysis of Porous T. Mukerji applications and limitations.
Media.” J. Dvorkin
Poromechanics V 2013 “An Experimental Study of Low- V. Data for frequency dependence validation of
ASCE 2013 Frequency Wave Dispersion and Mikhaltsevitch attenuation factor for fast compressional wave.
“Proceedings of the Attenuation in
Fifth Biot Conference Water Saturated Sandstones” M. Lebedev
on Poromechanics” B. Gurevich
28. 27
Journal of Applied Physics, Vol. 12, pp. 155- 164 (1941)
General Theory of Three-Dimensional Consolidation
Authors: Maurice A. Biot
Contribution to the understanding of Biot Model for Wave Propagation in Fluid-Saturated Porous Media:
The phenomenon of consolidation is explained by using the model of a fluid being squeezed out of a porous medium. It
gives the simplified theory for the case most important in practice of a soil completely saturated with water.
Objective of the Paper:
To describe the basic concepts and equations governing consolidation of rocks.
Methodology used:
Introduces the mathematical formulation of the physical properties of the soil and the number of constants necessary to
describe these properties. Gives a discussion of the physical interpretation of these various constants. Establishes the
fundamental equations for the consolidation and an application is made to the one-dimensional problem corresponding
to a standard soil test. Gives the simplified theory for the case most important in practice of a soil completely saturated
with water.
Conclusion reached:
1. The number of these constants including Darcy’s permeability coefficient is found equal to five in the most general
case.
2. It is quite possible that the soil particles are held together by capillary forces which behave in pretty much the same
way as the springs of the model of a system made of a great number of small rigid particles held together by tiny
helical springs.
Comments:
It is shown how the mathematical tool known as the operational calculus can be applied most conveniently for the calcu-
lation of the settlement without having to calculate any stress or water pressure distribution inside the soil. This method
of attack constitutes a major simplification and proves to be of high value in the solution of the more complex two- and
three-dimensional problems.
29. 28
The Journal of The Acoustical Society of America, Vol. 28, No. 2, pp. 168-178. (1956)
Theory of Propagation of Elastic Waves in Fluid-Saturated Porous Solid.
I. Low-Frequency Range
Authors: Maurice A. Biot
Contribution to the understanding of Biot Model for Wave Propagation in Fluid-Saturated Porous Media:
A theory is developed for the propagation of stress waves in a porous elastic solid containing a compressible viscous fluid.
The emphasis of the present treatment is on materials where fluid and solid are of comparable densities as for instance
in the case of water-saturated rock. The paper denoted here as Part I is restricted to the lower frequency range where
the assumption of Poiseuille flow is valid.
Objective of the Paper:
To establish a theory of propagation of elastic waves in a system composed of a porous elastic solid saturated by a viscous
fluid.
Methodology used:
Introduces the concept of dissipation potential into the Lagrangian of the rock-fluid system and by solving Euler-Lagrange
equations for both rotational and dilatational waves, gets the Biot’s equation of motion. Uses planar wave analysis to
obtain phase velocities and attenuation.
Conclusion reached:
1. There are 2 dilatational waves and one rotational wave, the dilatational wave of the first kind is a normal wave and
that of the second kind is highly attenuated and is of the nature of a diffusion process.
2. There is a characteristic frequency above which the theory doesn’t work any further due to breakdown of Poiseuille
flow.
Comments:
The phenomenological parameters used are cumbersome and had to be expressed in terms of the more physical parameters
like bulk moduli and shear modulus.
30. 29
The Journal of The Acoustical Society of America, Vol. 28, No. 2, pp. 179-191. (1956)
Theory of Propagation of Elastic Waves in a Fluid-Saturated Porous Solid.
II. High-Frequency Range
Authors: Maurice A. Biot
Contribution to the understanding of Biot Model for Wave Propagation in Fluid-Saturated Porous Media:
The theory of propagation of stress waves in a porous elastic solid developed in Part I for the low-frequency range is
extended to higher frequencies. The breakdown of Poiseuille flow beyond the critical frequency is discussed for pores of
flat and circular shapes. As in Part I the emphasis of the treatment is on cases where fluid and solids are of comparable
densities.
Objective of the Paper:
The purpose of this paper is to extend the theory to the full frequency range without the limitation of the cut off frequency
assumption.
Methodology used:
Introduces the concept of a new viscodynamic operator for high frequencies that takes into account the breakdown of
Poiseuille flow.
Conclusion reached:
1. The previous theory is successfully extended for the higher frequency range.
2. The only upper bound is when the wavelength becomes of the order of the pore size.
Comments:
The whole mathematical derivation of the viscodynamic operator is quite interesting, there is a viscodynamic operator
for Poiseuille flow that is very important for my thesis, and it is the one that I used for this work.
31. 30
Journal of Applied Mechanics, Vol. 24, pp. 594- 601 (1957)
The Elastic Coefficients of the Theory of Consolidation
Authors: Maurice A. Biot, D.G. Willis.
Contribution to the understanding of Biot Model for Wave Propagation in Fluid-Saturated Porous Media:
First to describe methods of measurements of the elastic coefficients from Biot’s theory.
Objective of the Paper:
To describe the methods of measurements of the elastic coefficients from Biot’s theory. To discuss the physical interpre-
tation of the elastic coefficients in various alternate forms.
Methodology used:
To use any combination of measurements which is sufficient to fix the properties of the system as a basis to determine
the coefficients.
Conclusion reached:
For an isotropic system, in which there are four coefficients, the four measurements of shear modules, jacketed and
unjacketed compressibility, and coefficient of fluid content, together with a measure of porosity appear to be the most
convenient. The porosity is not required if the variables and coefficients are expressed in the proper way. The coefficient of
fluid content is a measure of the volume of fluid entering the pores of a solid sample during an unjacketed compressibility
test. The stress-strain relations may be expressed in terms of the stresses and strains produced during the various mea-
surements, to give four expressions relating the measured coefficients to the original coefficients of the consolidation theory.
Comments:
The same method is easily extended to cases of anisotropy. The theory is directly applicable to linear systems but also
may be applied to incremental variations in nonlinear systems provided the stresses are defined properly.
32. 31
Institute of Physical Problems, U.S.S.R. Academy of Sciences, Vol. 7 of Course of Theoretical Physics.
(1959)
Theory of Elasticity
Authors: L.D. Landau, E.M. Lifshitz.
Contribution to the understanding of Biot Model for Wave Propagation in Fluid-Saturated Porous Media:
First to formalize the theory of elasticity for solids by means of a rigorous mathematical description.
Objective of the Book:
To describe the basic concepts and equations governing linear elasticity theory.
Methodology used:
Covers elasticity theory of solids, including viscous solids, vibrations and waves in crystals with dislocations, and a chapter
on the mechanics of liquid crystals.
Conclusion reached:
The linear theory of elasticity is powerful and the wave equations are a result of the elastic properties of a medium but
it’s applicable to real solids only up to a certain extent, of small displacements.
Comments:
It is my favourite book on elasticity, from one of my favourite authors. It really is the best text book to understand
elasticity theory if the reader has some passion for theoretical physics.
33. 32
Journal of Applied Physics, Vol. 33, No. 4, pp. 1482- 1498 (1962)
Mechanics of Deformation and Acoustic Propagation in Porous Media
Authors: Maurice A. Biot
Contribution to the understanding of Biot Model for Wave Propagation in Fluid-Saturated Porous Media:
A unified treatment of the mechanics of deformation and acoustic propagation in porous media is presented, and some
new results and generalizations are derived.
Objective of the Paper:
The purpose of this paper is to reformulate in a more systematic manner and in a somewhat more general context the
linear mechanics of fluid saturated porous media and also to present some new results and developments with particular
emphasis on viscoelastic properties and relaxation effects.
Methodology used:
Introduces the use of viscoelastic thermodynamic operators to the theory of consolidation. The writer’s earlier theory of
deformation of porous media derived from general principles of non-equilibrium thermodynamics is applied. The fluid-
solid medium is treated as a complex physical-chemical system with resultant relaxation and viscoelastic properties of a
very general nature. Specific relaxation models are discussed, and the general applicability of a correspondence principle
is further emphasized.
Conclusion reached:
Darcy’s law is derived from thermodynamic principles. This is a consequence of the isomorphism between thermos-
elasticity and the theory of porous media. For similar reasons, the wave propagation equations are also applicable to a
thermoviscoelastic continuum.
Comments:
The theory of acoustic propagation is extended to include anisotropic media, solid dissipation, and other relaxation effects.
Some typical examples of sources of dissipation other than fluid viscosity are considered.
34. 33
Applied Physics Letters, Vol. 52, No. 5, pp. 3391-3395. (1980)
Observation of a second bulk compressional wave in a porous medium at ultrasonic frequencies.
Authors: Thomas J. Plona.
Contribution to the understanding of Biot Model for Wave Propagation in Fluid-Saturated Porous Media:
First to observe experimentally the slow compressional wave predicted by Biot’s theory.
Objective of the Paper:
To describe the experimental conditions for observing the progressive slow wave and to exhibit its presence on a fully
controlled real laboratory artificial rock system.
Methodology used:
In order to observe the Biot’s progressive slow wave, it is necessary to fulfil the next set of experimental conditions:
1. Continuity of liquid and solid phases, open system.
2. High frequency content of the incident wave.
3. Low saturating fluid viscosity (high hydraulic permeability).
4. High saturating fluid density (less important),
5. High pore size and pore access radius, high absolute permeability.
Conclusion reached:
In conclusion, Plona demonstrated the existence of a slow wave very close to the one predicted by Biot’s theory. He
showed that this wave could only exist as a propagation wave if the following conditions were satisfied:
1. Continuity of the solid and liquid phases (i.e. possibility of differential fluid and liquid motion),
2. Sufficiently high incident wave frequency (i.e. possibility of differential fluid and liquid motion),
3. Incident wavelength sufficiently large in comparison with pore size to avoid scattering, while the pore size must be
adequate to avoid viscous effects at the wall (skin depth effect),
4. Very different fluid and solid bulk moduli in order to separate clearly the two compressional waves.
Comments:
The use of synthetic rocks is a very clever approach to create a controlled experimental system, the rocks are made of
sintered glass beads. It really helped me understand an experimental setting for analysing waves in a porous medium.
35. 34
Journal of Applied Physics, Vol. 12, pp. 155- 164. (1981)
The equivalence of quasi-static flow in fluid-saturated porous media and Biot’s slow wave in the limit of zero frequency.
Authors: Richard N. Chandler, David L. Johnson.
Contribution to the understanding of Biot Model for Wave Propagation in Fluid-Saturated Porous Media:
First to prove the equivalence between Biot’s slow wave equation and a Diffusion Equation for Pressure.
Objective of the Paper:
To show that the quasi-static motion of a fluid-saturated porous matrix is describable by a homogeneous diffusion equa-
tion in fluid pressure with a single composite elastic constant, the diffusivity, is contained in Biot’s model for the slow
compressional wave in the limit of zero frequency.
To show that the analyses used in the applications concerned with the low-frequency dynamics of a porous matrix sat-
urated with a viscous fluid, such as well-test analysis, are all limiting cases of the more general analysis based on the
mixture theory of Biot.
Methodology used:
To take the quasi-static limit of Biot’s Diffusivity equation, and to make a change to normal mode coordinates and some
vector algebra theorems (Helmholtz decomposition theorem) to get the exact expression for the diffusion equation on
pressure and stress. To inspect limiting cases in order to analyse the expression of the respective diffusivity constant.
Conclusion reached:
Biot’s slow wave equation is of a diffusive nature and the general diffusivity constant is the one that takes into account
all the poro-elastic properties of the system. In applications, simpler versions of the diffusivity are used.
Comments:
The mathematical treatment is correct, it’s just not very clear for the average since the author skips some important
steps, considering that the reader is not used to follow vector algebra manipulations.
36. 35
Institut Fran¸cais du P´etrole Publications, Geophysics, Petroleum Engineering, pp. 63- 94 (1987)
Acoustics of Porous Media
Authors: T. Bourbi´e, O. Coussy, B. Zinszner.
Contribution to the understanding of Biot Model for Wave Propagation in Fluid-Saturated Porous Media:
First to explain in a clear fashion Biot’s theory and other relevant elements of applied elasticity theory to rock mechanics.
Objective of the Book:
To describe in an informative fashion the results from Biot’s theory in the limit of low frequencies and those from Plona’s
experimental verification.
Methodology used:
It follows the exact same logical path as Biot’s original paper but always trying to reproduce Biot’s results and trying to
explain them in a clear fashion.
Conclusion reached:
1. Biot’s equations are invariant with respect to the author.
2. In order to get the desired expressions for wave speeds and attenuation, one must perform a planar wave analysis.
3. The parameter dependence of the wave speed model is carefully explained.
4. The experimental verification of Biot’s model is analysed and accepted.
Comments:
These book only shows the logical path to get the analytic expressions for wave speeds and attenuation but there is no
evident expression of it. There are some useful graphs of their dependence with frequency, but nothing else.
37. 36
Society of Exploration Geophysicists, Geophysics, Vol. 55, No.8, pp. 998-1014, (1990)
Relationships among compressional wave attenuation, porosity, clay content, and permeability in sandstones.
Authors: T. Klimentos, C. McCann.
Contribution to the understanding of Biot Model for Wave Propagation in Fluid-Saturated Porous Media:
Experimental measures of wave speeds and attenuation for the fast compressional wave and the shear wave in 42 water
saturated sandstones under confining pressure.
Objective of the Paper:
To present experimental results of measurements of attenuation and wave speeds for compressional waves from 42 water
saturated sandstones at a confining pressure of 40MPa (equivalent to a depth of burial of about 1.5km) in a frequency
range from 0.5to1.5MHz.
Methodology used:
The compressional wave measurements were made using a pulse–echo method in which the sample (5 cm diameter, 1.8cm
to 3.5cm long) was sandwiched between perspex (lucite) buffer rods inside the high pressure rig. The attenuation of the
sample was estimated from the logarithmic spectral ratio of the signals. Data are presented to demonstrate that intra
pore clays in sandstones are important in causing the attenuation of compressional waves and in controlling permeability
of the sandstones. The data are important because this mechanism of attenuation has not been recognized before, and
because the results bring closer the possibility of using accurate measurements of the attenuation of compressional waves
to estimate the permeability of rocks in situ in boreholes, or in the laboratory. Conclusion reached: The results show
that for these samples, compressional wave attenuation (a, dB/cm) at 1MHz and 40MPa is related to clay content (C,
percent) and porosity (φ, percent) by a = 0.0315φ+0.241C −0.132 with a correlation coefficient of 0.88. The relationship
between attenuation and permeability is less well defined; those samples with permeabilities less than 50 md have high
attenuation coefficients (generally greater than 1 dB/cm) while those with permeabilities greater than 50 md have low
attenuation coefficients (generally less than 1 dB/cm) at 1 MHz at 40 MPa.
Comments:
These experimental data can be accounted for by modifications of the Biot theory and by consideration of the Sewell/Urick
theory of compressional wave attenuation in porous, fluid-saturated media.
38. 37
Proc. 4th North Amer. Rock Mech. Symp., A.A. Balkema, Rotterdam, pp. 169-172, 2000
Implications of Static Poroelasticity for Reservoir Compaction
Authors: Robert W. Zimmerman
Contribution to the understanding of Biot Model for Wave Propagation in Fluid-Saturated Porous Media:
It has several useful expressions for compressibilities, the most important being the uniaxial strain pore compressibility,
used for the last part of the project, the theoretical connexion between Biot’s and Gringarten diffusivity.
Objective of the Paper:
Some implications of the static theory of linear poroelasticity for reservoir compaction are discussed.
Methodology used:
First, the relationship between the bulk compressibility and the uniaxial compaction coefficient is reviewed. Then, an
expression is derived for the pore compressibility under uniaxial strain conditions. Finally, the influence of pore pressure
on lateral stresses, under uniaxial strain conditions, is discussed.
Conclusion reached:
From Biot’s theory of poroelasticity one can get useful expressions for the pore compressibility under uniaxial strain
conditions.
Comments:
This paper helped to get a new expression for the pore compressibility of the bulk volume.
39. 38
Seismic Exploration, Volume 31, Elsevier Chapter 7. (2001)
Wave Fields in Real Media: Wave Propagation in Anisotropic, Anelastic and Porous Media.
Authors: Jos´e M. Carcione.
Contribution to the understanding of Biot Model for Wave Propagation in Fluid-Saturated Porous Media:
Rigorous and very mathematical approach to Biot’s theory with its extension to anisotropic systems. My model is cali-
brated with his own model and I used his expressions for shear wave speed, viscodynamic operators in the low frequency
limit and the same mathematical treatment for calculating the compressional wave speeds.
Objective of the Book:
To describe the basic concepts and equations for, Wave Propagation in Anisotropic, Anelastic and Porous Media.
Methodology used:
Covers the topics of:
1. Anisotropic elastic media.
2. Viscoelasticity and wave propagation.
3. Isotropic anelastic media.
4. Anisotropic anelastic media.
5. The reciprocity principle.
6. Reflection and transmission of plane waves.
7. Biot’s theory for porous media.
8. Numerical Methods.
Conclusion reached:
1. Biot’s equations are invariant with respect to author.
2. The formalism explained here is the more detailed, and I decided to follow it to calibrate my own model.
Comments: Even though, the notation can be quite cumbersome, this book is phenomenal, it describes every single thing
I needed to understand Biot’s theory and to create an analytic expression for the wave speed and attenuation, by simply
following its guidance.
40. 39
Cambridge University Press, Chapter 6. pp. 266 (2009)
The Rock Physics Handbook: Tools for Seismic Analysis of Porous Media.
Authors: G. Mavko, T. Mukerji, J. Dvorkin.
Contribution to the understanding of Biot Model for Wave Propagation in Fluid-Saturated Porous Media:
Summary of Biot’s theory results and its applications and limitations.
Objective of the Book:
To describe the basic concepts and equations fluid effects on wave propagation.
Methodology used:
Covers Biot’s velocity relations, its uses, assumptions, limitations and extensions.
Conclusion reached:
Biot’s theory can be used for the following purposes:
• Estimating saturated-rock velocities from dry-rock velocities.
• Estimating frequency dependence of velocities, and
• Estimating reservoir compaction caused by pumping using the quasi-static limit of Biot’s poroelasticity theory.
Assumptions and limitations:
The uses of Biot’s equations requires the following considerations:
• The rock is isotropic.
• All minerals making up the rock have the same bulk and shear moduli,
• The fluid-bearing rock is completely saturated.
• The pore fluid is Newtonian.
• The wavelength, even in the high-frequency limit, is much larger than the gran or pore scale.
Comments:
The authors recommend and approach totally analogous to that of Carcione for obtaining the expression for the wave
speeds and attenuations, however they don’t take into account very carefully the low frequency viscodynamic operator
as Carcione did, even though both have the same limiting case.
41. 40
Appendix B: Viscodynamic operator for the high-frequency limit
Using the notation from the formalism section we have the next expression for the viscodynamic operator in the high
frequency limit that could be used as an extension of this work
˜YHF = ˜Y1FR + i( ˜Y2 + FI) (127)
With
FR =
ζ
4
TR(1 − 2TI − ζ) + 2TRTI
(1 − 2TI − ζ)2 + (2TR)2
(128)
FI =
ζ
4
TI(1 − 2TI − ζ) − 2TRTI
(1 − 2TI − ζ)2 + (2TR)2
(129)
Where
TR =
Z1RZ2R + Z1IZ2I
Z2
2R + Z2
2I
(130)
TR =
−Z1RZ2I + Z1IZ2R
Z2
2R + Z2
2I
(131)
And
Z2R =
exp( ζ√
2
)
√
4πζ
1 +
1
√
2
cos
ζ
√
2
+ sin
ζ
√
2
−
1
√
2
sin
ζ
√
2
− cos
ζ
√
2
(132)
Z2I =
exp( ζ√
2
)
√
4πζ
1
√
2
cos
ζ
√
2
+ sin
ζ
√
2
+ 1 +
1
√
2
sin
ζ
√
2
− cos
ζ
√
2
(133)
Z1R =
exp( ζ√
2
)
√
2πζ
1 +
1
√
2
cos
ζ
√
2
−
1
√
2
sin
ζ
√
2
(134)
Z1I =
exp( ζ√
2
)
√
2πζ
1
√
2
cos
ζ
√
2
+ 1 +
1
√
2
sin
ζ
√
2
(135)
ζ =
ωa2ρf
η
(136)
42. 41
Appendix C: Potential and Kinetic Energy of the Rock-Fluid System
Potential Energy
In the notation from the formalism, we postulate a quadratic form with a coupling term, namely,
V = Aϑ2
bd2
b + G + Cϑbϑf + Dϑ2
f (137)
With ϑp = ep
11 + ep
22 + ep
33 (p = b or f) and ep
ij the component (i, j) of the strain tensor of the phase p.
d2
b = db
ijdb
ji (138)
db
ij = ep
ij −
1
3
ϑpδij (139)
A =
1
2
(1 − φ)(1 − φ − Kb
Ks
)Ks + Ks
Kf
Kb
1 − φ − Kb
Ks
+ φ Ks
Kf
(140)
C =
(φ)(1 − φ − Kb
Ks
)Ks
1 − φ − Kb
Ks
+ φ Ks
Kf
(141)
D =
1
2
φ2
Ks
1 − φ − Kb
Ks
+ φ Ks
Kf
(142)
Kinetic Energy
We postulate a quadratic form with a coupling term, namely,
T =
1
2
Ωb(ρ11vb
i vb
i + 2ρ12vb
i + ρ22vf
i vf
i ) (143)
With Ωb being the volume of the elementary macroscopic and representative region of porous material.
43. 42
Appendix D: Numerical evidence for validity of uniaxial pore compress-
ibility
From [12] if we can take the limit, and we’ll have the ratio between the uniaxial pore compressibility and the hydrostatic
pore compressibility as:
Cuni
pp
CHydro
pp Zimm
= 1 −
2(1 − 2ν)α
3(1 − ν)
(144)
However, we get from our own calculation and by taking the same limit,
CUni
pp
CHydro
pp Biot
= 1 −
2(1 − ν)α
3(1 − ν)
+
2α
3(1 − ν)
= 1 +
4να
3(1 − ν)
(145)
We can see that the ratio from the last column of this table varies from 1.8 to 2.3, so the use of the uniaxial pore
Rock α φ ν
Cuni
pp
CHydro
pp Zimm
CUni
pp
CHydro
pp Biot
Cuni
pp
CHydro
pp Zimm
CUni
pp
CHydro
pp Biot
Ruhr Sandstone 0.65 0.02 0.12 0.63 1.12 1.78
Berea Sandstone 0.79 0.19 0.2 0.61 1.27 2.08
Ohio Sandstone 0.74 0.19 0.18 0.61 1.21 1.99
Pecos Sandstone 0.83 0.2 0.16 0.55 1.21 2.20
Boise Sandstone 0.85 0.26 0.15 0.53 1.20 2.26
Table 4: Data for numerical comparison of pore compressibilities.
compressibility is incorrect by a factor of two in the model from well test analysis.
