IMPERIAL COLLEGE LONDON

Department of Earth Science and EngineeringCentre for Petroleum Studies

Biot Model for Wave Propagation inFluid-Saturated Porous Media

by

David Bautista Gonzalez

A report submitted in partial fulfillment of the requirements forthe MSc and/or the DIC.

September 2015

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 andreferenced, and/or with appropriate acknowledgement given.

Signature:

Name of Student: David Bautista Gonzalez

Name of Supervisor: Prof. Robert W. Zimmerman

2

Abstract

We present a new closed form expression of the wave speed for the slow and fast compressional waves as well as for theshear wave as predicted by Biot’s equation. These expressions are derived from first principles for isotropic rock systemsin the low frequency limit by means of an appropriate Lagrangian function for the rock-fluid system and a suitableviscodynamic operator. A detailed study of the effect of the various rock and fluid parameters on the wave speed ismade and we identify their relation with Biot’s characteristic frequency. We have successfully validated this model withdata from water-saturated clay-free sandstones at a confining pressure of 40 MPa (equivalent to a depth of burial ofapproximately 1.5 km) and for two samples of sandstones at differential pressures of 15 MPa and 18 MPa. Finally, wefollowed the procedure from Johnson & Chandler to investigate the relationship between the quasi-static slow Biot waveand the pressure diffusion equation used in well-test analysis. We found an inconclusive connection between Biot theoryof poroelasticity and the diffusion equation from well-test analysis, giving rise to a factor-of-two underestimate of rockcompressibility, this could be due to an effective rock-to-fluid compressibility and fluid-to-rock compressibility that arenot taken into account in traditional models, and this could also be related to geometrical effects of the pore network andof the actual wave front of the acoustic waves. These results will be a very helpful starting point for further work on thisproblem.

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 allowedme 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 andmore interesting.

On a personal note, I want to thank my family, especially my mother for her constant encouragement throughout theyear.

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.

Contents

Declaration of Own Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Experimental validation of the Biot model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17Derivation of the diffusion equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18Analysis of Diffusivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25Appendix A: Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26Appendix B: Viscodynamic operator for the high-frequency limit . . . . . . . . . . . . . . . . . . . . . . . 40Appendix C: Potential and Kinetic Energy of the Rock-Fluid System . . . . . . . . . . . . . . . . . . . . . 41Appendix D: Numerical evidence for validity of uniaxial pore compressibility . . . . . . . . . . . . . . . . 42

4

List of Figures

1 Porosity dependence Fast Compressional Wave Speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 Porosity dependence Fast Compressional Attenuation Factor . . . . . . . . . . . . . . . . . . . . . . . . . 163 Porosity dependence Slow Compressional Wave Speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 Porosity dependence Slow Compressional Attenuation Factor . . . . . . . . . . . . . . . . . . . . . . . . . 165 Porosity dependence Shear Wave Speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 Porosity dependence Shear Attenuation Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 Viscosity and mobility dependence for Fast Compressional Wave Speed. . . . . . . . . . . . . . . . . . . . 178 Biot model prediction for wave speeds (Average Group 2H2M1 & 2V1M1) . . . . . . . . . . . . . . . . . . 179 Biot model prediction for fast wave attenuation (Average Group 2H2M1 & 2V1M1) . . . . . . . . . . . . 1710 Biot model validation of attenuation factor for Sample A. Pd = 15 MPa. . . . . . . . . . . . . . . . . . . . 1811 Biot model validation of attenuation factor for Sample B. Pd = 18 MPa. . . . . . . . . . . . . . . . . . . . 1812 Numerical evidence of Biot effect of more than 10%. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

5

List of Tables

1 Sandstone sample parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 Input data for model for clay-free sandstones. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 Literature Milestones Biot Model for Wave Propagation in Fluid-Saturated Porous Media . . . . . . . . . 264 Data for numerical comparison of pore compressibilities. . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

6

7

Biot Model for Wave Propagation in Fluid-Saturated Porous Media

David Bautista-Gonzalez

Imperial College supervisor: Robert W. Zimmerman

Copyright 2015, Imperial College LondonThis report was submitted in partial fulfilment of the requirements for the Master of Science and/or the Diploma of Imperial College fromImperial 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 theshear wave as predicted by Biot’s equation. These expressions are derived from first principles for isotropic rock systemsin the low frequency limit by means of an appropriate Lagrangian function for the rock-fluid system and a suitableviscodynamic operator. A detailed study of the effect of the various rock and fluid parameters on the wave speed ismade and we identify their relation with Biot’s characteristic frequency. We have successfully validated this model withdata from water-saturated clay-free sandstones at a confining pressure of 40 MPa (equivalent to a depth of burial ofapproximately 1.5 km) and for two samples of sandstones at differential pressures of 15 MPa and 18 MPa. Finally, wefollowed the procedure from Johnson & Chandler to investigate the relationship between the quasi-static slow Biot waveand the pressure diffusion equation used in well-test analysis. We found an inconclusive connection between Biot theoryof poroelasticity and the diffusion equation from well-test analysis, giving rise to a factor-of-two underestimate of rockcompressibility, this could be due to an effective rock-to-fluid compressibility and fluid-to-rock compressibility that arenot taken into account in traditional models, and this could also be related to geometrical effects of the pore network andof the actual wave front of the acoustic waves. These results will be a very helpful starting point for further work on thisproblem.

Introduction

Wave physics is one of the most directly applicable branches of physics. One can encounter waves of different nature ingravitation, quantum mechanics, electromagnetism, fluid mechanics, and elasticity theory. It is clearly one of the mostfundamental physical phenomena, since we interact with waves all the time in our day-to-day experience. And mostimportantly, 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, themodification in the mechanical internal state of the porous skeleton influences the behaviour of the viscous fluid insidethe 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 porousskeleton.

• Solid-to-Fluid coupling: occurs when modifications in the stress of the porous skeleton induce change in fluidpressure or fluid mass.

In accordance with these two phenomena, the fluid-filled porous medium acts in a time-dependent manner. Assume thatthe porous medium is compressed – this will result in an increment of the fluid pressure inside the pores and resultingfluid flow. The time dependence of the fluid pressure (i.e. dissipation of the fluid pressure through the diffusive fluid fluxaccording to the Darcy’s law for filtration velocity) will induce a time dependence of the poroelastic stresses, which inturn will respond back to the fluid pressure field. It is clear that the dynamical model of such process is time dependentand only if the inertial forces are neglected then it can be considered as quasi-static.

8

The purpose of this work is to clarify the validity of Biot’s model which attempts to account for the effect of porefluid n the speeds of P and S waves in a porous rock. Even if Biot’s equations of motion have been known for more thanhalf a century, nobody has developed a fully explicit closed-form relation for the wavespeeds, since everyone has decidedin the past to solve this problem numerically. Also, nobody has ever solved the fundamental paradigm for the model ofdiffusivity in well-test analysis which considers merely a rigid frame instead of a deformable one that contains fluid insideits pores. These are the two actual motivations for this work, since one would like to understand the reason why coredata 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 ofthese elastic waves, then perform a study of the influence of their respective parameters, after that we will present anexperimental verification of the model with data from the literature of water- saturated clay-free sandstones [11] and fromtwo additional samples of sandstones [15] and we will prove the connexion between the slow compressional wave fromBiot’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, bymeans 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 propagationin fluid saturated porous media [2, 3]; Biot’s theory describes wave propagation in a porous saturated medium, i.e., amedium made of a solid matrix (skeleton or frame), fully saturated with a fluid. Biot ignores the microscopic level andassumes that continuum mechanics can be applied to measurable macroscopic quantities. He postulates the Lagrangianand uses Hamilton’s principle to derive the equations governing wave propagation. This theory predicts the existenceof two compressional waves (compressional waves of the first and second kind) and one shear wave. We will focus onthe low-frequency range of the theory regime, indeed one must take into account that the initial theory is valid up to alimit frequency, where the assumption of Poiseuille flow breaks down. The high-frequency range theory is developed byconsidering the flow of a viscous fluid under an oscillatory pressure gradient either between parallel walls or in a circulartube. 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 effectof pore cross-sectional shape is well represented by taking the same function of the frequency for the viscosity correctionand 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, strainsand 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 quantitiesused 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 haswell 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 notcontribute to the porosity.

7. In most cases, the material of the frame is isotropic. Anisotropy is due to a preferential alignment of the pores (orcracks).

Biot states that it is quite possible that the soil particles are held together by capillary forces which behave in prettymuch the same way as the springs of the model of a system made of a great number of small rigid particles held togetherby tiny helical springs. That’s the main reason why he postulates a dissipation potential with the functional form of aharmonic 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; thiscritical observation had been lacking for many years. He demonstrated the existence of a slow wave very close to the one

9

predicted by Biot’s theory. He showed that this wave could only exist as a propagating wave if the following conditionswere 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 beadequate 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 thequasi-static motion of a fluid-saturated porous matrix is describable by a homogeneous diffusion equation in fluid pressurewith a single composite elastic constant, the diffusivity, is contained in Biot’s model for the slow compressional wave inthe limit of zero frequency. They also tried to show –indeed they lacked some consistency with the diffusivity for well-testanalysis limiting case- that the analyses used in the applications concerned with the low-frequency dynamics of a porousmatrix saturated with a viscous fluid, such as well-test analysis, are all limiting cases of the more general analysis basedon the mixture theory of Biot.

Finally, Klimentos and McCann [11] presented experimental results of measurements of attenuation and wave speedsfor compressional waves from 42 water saturated sandstones at a confining pressure of 40 MPa (equivalent to a depth ofburial of about 1.5 km) in a frequency range from 0.5 to 1.5 MHz. They also have reported wave speeds for shear wavesand 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 phaseby the index “f”. We’ll follow the approach from Biot [2, 3], Bourbie et al. and Carcione [13].

It is possible to postulate a well-defined Lagrangian function for the fluid-rock system. The Lagrangian density fora 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 byLagrange’s equation of motion, which is based on Hamilton’s principle of least action [9]. The method can be extendedto non- conservative systems if the dissipative forces can be derived from a potential:

ΦD =1

2b

3∑i=1

(vbi − vfi )(vfi − v

bi ) (2)

where b is the friction coefficient obtained by comparing the classical Darcy’s law with the equation of the force derivedfrom the dissipation potential.

b = φ2η

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:vpi = ∂tu

pi (4)

with upi 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 andit also assumes that the flow is of the Poiseuille type, i.e. low Reynolds number and low frequencies.

10

Lagrange’s equations with the displacements as generalized coordinates can be written as:

δt

(∂L

∂vpi

)+

3∑j=1

∂j

[∂L

∂(∂jupi )

]− ∂L

∂upi+∂ΦD∂vpi

= 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 ofthe dissipation potential, it can be shown that we have for isotropic media,

3∑j=1

∂jσbij = ρ11∂

2ttu

bi + ρ12∂

2ttu

fi + b(vbi − v

fi ) (6)

− φ∂ipf = ρ12∂2ttu

bi + ρ22∂

2ttu

fi − b(v

bi − v

fi ) (7)

where, ρ11 = (1 − φ)(ρs + rρf ), ρ12 = (−φρf )(τ − 1), ρ22 = φρfτ , are the components of the density tensor; ρ =

(1−φ)(ρs)+φρf is the density of the saturated matrix, with τ = 1+(

1φ − 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(ρ11~u+ ρ12~U) + b

∂

∂t(~u− ~U) (8)

and

Q∇(∇ · ~u) +R∇(∇ · ~U) =∂2

∂t2(ρ12~u+ ρ22~U)− b ∂

∂t(~u− ~U) (9)

where ~u = (ub1, ub2, u

b3)T , ~U = (uf1 , u

f2 , u

f3 )T , are the displacement vectors for the matrix and the fluid. P = λ+ 2G, where

λ and G are the Lame coefficients, G being the shear modulus of the matrix, R is a measure of the pressure requiredon the fluid to force a certain volume of it into the aggregate while the total volume remains constant, Q is a couplingcoefficient 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 Lame first parameter where α = 1− Kb

Ks

M =Ks

1− φ− Kb

Ks+ φKs

Kf

(11)

Q = Mφ(α− φ) (12)

R = M(φ)2 (13)

Kf = λf +2

3G (14)

Kb = λb +2

3G (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+ φKs

Kf

+4

3G (16)

Q =φ(1− φKb

KsKs)

1− φ− Kb

Ks+ φKs

Kf

(17)

R =φ2Ks

1− φ− Kb

Ks+ φKs

Kf

(18)

In isotropic media, the compressional waves are decoupled from the shear waves, thus the respective equations of motioncan be obtained by taking the divergence and the curl in equations (8) and (9). And the typical procedure to obtainthe analytical expression for the wave speeds is to consider planar waves and to transform the given set of differential

11

equations into a secular equation problem.

For the compressional waves we have to solve:

−(ρ2f +1

ωY ρ)v4c +

[i

ωY

(KG +

3

4G

)+M(2αρf − ρ)

]v2c +M(Kb +

3

4G) = 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) = η3

[q2

1−(1/q)tanh(q)

]and q = a

√iων .

The viscodynamic operator obtained for the low frequency limit from the Lagrangian approach is:

Y (t) = m∂tδ(t) +η

kδ(t) (21)

with m = ρ22/φ2 = ρfτ/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

5q2 + 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 viscousdrag is given in appendix B.

With ω angular frequency and ν = ηρf

the kinematic viscosity, a the pore radius, we have

KG = KB + α2M =Ks −KB + φ

(Ks

Kf− 1)KB

1− φ− KB

Ks+ φKs

Kf

(25)

We want to acquire the wave speeds from Biot’s model. Following the usual procedure, we must solve the followingequation for vc

−(ρ2f +

1

ωY ρ

)v4c +

[i

ωY

(KG +

3

4G

)+M (2αρf − ρ)

]v2c +M

(Kb +

3

4G

)= 0

From the fundamental theorem of algebra it follows that the equation has four complex roots. The easier way to findthem is by regarding our polynomial as quadratic in the variable v2c , thus

(v2c )1,2 =−B ±

√B2 − 4AC

2A(26)

A = A1 + iA2 = −(ρ2f +

1

ωY ρ

)(27)

B = B1 + iB2 =

[i

ωY

(KG +

3

4G

)+M (2αρf − ρ)

](28)

Y = Y1 + iY2 =η

a2

[q2

1− q tanh(q)

](29)

q = a

√iω

ν(30)

c = M

(Kb +

3

4G

)(31)

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Remark We’re considering that everything is real except for Y , A,B, q: Just to gain more insight, let’s analyse morecarefully Y

Y1 =

√ω

2ν

aηωS+D

(32)

Y2 =ηω(C+ −

√ω2ν aS−

)D

(33)

C± ≡ cosh

(√2a2ω

ν

)± cos

(√2a2ω

ν

)(34)

S± ≡ sinh

(√2a2ω

ν

)± sin

(√2a2ω

ν

)(35)

D ≡ a2ωC− − 2a

√νω

2S− + νC+ (36)

Now we fix our attention on a and b

A1 = −(ρ2f +

1

ωY1ρ

)(37)

A2 = − Y2ρω

(38)

B1 =

[−Y2ω

(KG +

3

4G

)+M (2αρf − ρ)

](39)

B2 =Y1ω

(KG +

3

4G

)(40)

The roots for the quadratic polynomial are

(v2c )1 =−B +

√B2 − 4AC

2A(41)

(v2c )2 =−B −

√B2 − 4AC

2A(42)

And then, the four roots of the original problem shall be obtained after taking another square root

(v2c )1+ = +

√−B +

√B2 − 4AC

2A(43)

(v2c )1− = −(v2c )1+ = −

√−B +

√B2 − 4AC

2A(44)

For the remaining roots we have an analogous setting

(v2c )2+ = +i

√B +

√B2 − 4AC

2A(45)

(v2c )2− = −(v2c )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 allowus to make some definitions in order to avoid cumbersome expressions

Σ± ≡ 4

√√√√ A2

±2B4

√(∆)

2cos(12 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 statethe following

13

Remark A,B ∈ C, thus so does ∆ ∈ C, we must be careful when taking roots: With that in mind we proceed asfollows

<[(vc)−11+] =

√2

(cos

(arg

2

(A√

∆−B

))< (Σ−)− sin

(arg

2

(A√

∆−B

))= (Σ−)

)(49)

<[(vc)−11−] = −<[(vc)

−11+] (50)

=[(vc)−11+] =

√2

(cos

(arg

2

(A√

∆−B

))= (Σ−) + sin

(arg

2

(A√

∆−B

))< (Σ−)

)(51)

=[(vc)−11−] = −=[(vc)

−11+] (52)

Consequently, the second and last set is given by

<[(vc)−12+] =

√2

(cos

(arg

2

(A

B

))= (Σ+) + sin

(arg

2

(A

B

))< (Σ+)

)(53)

<[(vc)−12−] = −<[(vc)

−12+] (54)

=[(vc)−12+] =

√2

(cos

(arg

2

(A√

∆ +B

))= (Σ+) + sin

(arg

2

(A√

∆ +B

))< (Σ+)

)(55)

=[(vc)−12−] = −=[(vc)

−12+] (56)

We note that we only have four independent expressions, now we introduce new variables

vp+ =1

<[(vc)−11+]

vp− =1

<[(vc)−12+]

αp+ = ω=[(vc)−11+] αp− = ω=[(vc)

−12+] (57)

And the problem is solved, please note that the line of thinking here is: obviously these calculations must be dealt withby software methods. Considering that, first we must enter the values of α, ω, ν, η; that determines Y . Having Y is quitestraightforward 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

B21 −B2

2 − 4CA1(58)

ψ ≡(−4CA1 +B2

1 −B22

)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)

14

And we end this glancing in depth <(Σ±),=(Σ±)

arg ∆ = 2θ (64)

Γ ≡ 16(A2

1 +A22

)C2 − 8

(A1B

21 + 2A2B2B1 −A1B

22

)C +

(|B|2

)2(65)

σ1 =

(− 2 cos(θ)

[(B2A

21 − 2A2B1A1 −A2

2B2

)cos

(2θ√

1 + 64θ2

)+(B1A

21 + 2A2B2A1 −A2

2B1

)sin

(2θ√

1 + 64θ2

)]4√

Γ

+ 2A1A2 cos

(4θ√

1 + 64θ2

)√Γ + (A2

2 −A21) sin

(4θ√

1 + 64θ2

)√Γ

+ 2A1A2B21 − 2A1A2B

22 − 2A2

1B1B2 + 2A22B1B2

)(66)

σ2 =≡(

2 cos(θ)

[(B1A

21 + 2A2B2A1 −A2

2B1

)cos

(2θ√

1 + 64θ2

)+(−B2A

21 + 2A2B1A1 +A2

2B2

)sin

(2θ√

1 + 64θ2

)]4√

Γ

+(A2

1 −A22

)cos

(4θ√

1 + 64θ2

)√Γ + 2A1A2 sin

(4θ√

1 + 64θ2

)√Γ

+A21B

21 −A2

2B21 −A2

1B22 +A2

2B22 + 4A1A2B1B2

)(67)

<(Σ±) = cos

(1

4

σ1σ2

)√|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

)

−2B14√ψ sin

(2θ√

1 + 16θ2

))+√ψ cos

(4θ√

1 + 64θ2

)+B2

1 −B22

)2)−1/8

(68)

Whereas for the imaginary part we have

=(Σ±) = sin

(1

4

σ1σ2

)√|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

)

−2B14√ψ sin

(2θ√

1 + 16θ2

))+√ψ cos

(4θ√

1 + 64θ2

)+B2

1 −B22

)2)−1/8

(69)

For the shear wave, it is easy to show that, by following an analogue reasoning and by separating real and imaginaryparts, we get

vc = <(vc) + i=(vc) (70)

vc =

|z|−1/2G1/2

(ρ− Y2ωρ

2f

Y 21 +Y 2

2

+ |z|)

√(ρ− Y2ωρ2f

Y 21 +Y 2

2

+ |z|)2

+

(Y1ωρ2fY 21 +Y 2

2

)2+ i

|z|−1/2G1/2

(Y1ωρ

2f

Y 21 +Y 2

2

)√(

ρ− Y2ωρ2fY 21 +Y 2

2

+ |z|)2

+

(Y1ωρ2fY 21 +Y 2

2

)2(71)

|z| =

(ρ−

Y2ωρ2f

Y 21 + Y 2

2

)2

+

(Y1ωρ

2f

Y 21 + Y 2

2

)2

(72)

Thus

v−1c =<(vc)− i=(vc)

<(vc)2 + =(vc)2(73)

The phase velocity for the shear wave is:

vs = [<(v−1c )]−1 =

[<(vc)

<(vc)2 + =(vc)2

]−1

15

The attenuation factor for the shear wave is:

αS = ω[=(v−1c )] = −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 increasesimultaneously. Since there is a contrast in inertial forces for the fluid and the solid part, there is a consequent differentialmovement between the fluid and the fluid/solid combination due to permeability effects which involves less fluid entraininga decrease in mass in the overall movement as the frequency increases. The attenuation also increases with frequency forthe three cases, since the dissipation is proportional to the square of the angular frequency. The slow compressional waveattenuation 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 ispossible that there is a relationship between porosity, permeability, and ultrasonic parameters. Velocity is expected todecrease and attenuation to increase as porosity increases However, those trends may differ for saturated materials. InBiot’s model, compressional wave speeds and attenuations behave as expected as well as shear attenuation, however shearwave speed behaves inversely. This is numerically shown in the Figures 1 to 6. Note also that permeability only affectsthe abscissa scale. Due to the expression of the characteristic frequency.

fc =φ

2πρfκ=

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 (ortowards 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 thedissipation due to mean differential movements and not those due to absolute movements of the fluid, this latter becomesmore important in the high frequency regime, thus it is logical that the lower the viscosity, the more the differentialmovement may be pronounced and hence the differential velocity is greater and the dissipation increases. Therefore, thelower 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 mayassume that the transition occurs when inertial and viscous forces, from the expression of the viscodynamic operator forlow frequencies we can infer that, that happens when iωm = η/k, this defines another criterion of validity for the theory:

fl =η

mk=

φη

2πτρf k(76)

This frequency indicates the upper limit for the validity of low-frequency Biot’s theory.

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 WaveSpeed

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 dependenceof the fast compressional wave speed with viscosity, and thus with mobility, once again the remaining parameter valuescorrespond to those from Group B from [11]. The behaviour of the remaining wave speeds and attenuation is analogousand this example thus representative of the effect on the abscissa scale due to mobility.

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 valuesfor velocity (from 5% to 10% difference), one needs to have a low confining pressure or smaller values for elastic modulias 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-saturatedsandstones divided in 3 groups (i.e., Groups A, B and the combination of 2H2M1 & 2V1M1), was obtained from a reportfrom 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 theclay 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 frequencyfor 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 waveattenuation (Average Group 2H2M1 & 2V1M1)

18

Data for fast compressional attenuation factor were extracted from another paper [15] and the frequency dependencewas confirmed for low frequencies. Sample A is tight sandstone with low permeability and low porosity, Sample B is asandstone 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 attenuationfactor for Sample A. Pd = 15 MPa.

Figure 11: Biot model validation of attenuationfactor 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), ×10−3m ×10−3m

kg/m2

A 14.8 7.8 2099 70 38B 20.6 590 2261 70 38

Table 2: Input data for model for clay-free sandstones.

Parameter Unity Average Average AverageGroup B 2H2M1/2V1M1 Group A

Porosity (φ) % 15± 1 2.5 15± 1Permeability (k) mD 175± 65 0.01 48± 6Solid density (ρs) kg/m3 2628 2628 2628

Clay content % 0.68± 0.3 0 15(volume)

Kb GPa 19 30 12µb 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 considerplanar 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

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+QP+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 acurl-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

~n′ ·~F (~r′)

|~r − ~r′|dS′ (85)

~A(~r) =1

4π

∫V

∇′ × ~F (~r′)

|~r − ~r′|dV ′ − 1

4π

∮S

~n′ ×~F (~r′)

|~r − ~r′|dS′ (86)

If V = R3 which of course is unbounded, therefore ~F must vanish faster than 1r 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 stepfurther 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 sayingthat if ∇× ~ξ = 0 at some time in the past (before the experiment starts) it shall remain zero forever; this is a convenientinitial 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)

20

If we use

Kf = λf +2

3G Kb = λb +

2

3G (92)

α = 1− Kb

KsM =

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 =φ2Ks

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 aggregateis σkk = σii − 3φp with

σkk = (3P − 4G+ 3Q)∇ · ~u+ 3(Q+R)∇ · ~U (99)

So we have (pσkk

)=

(−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−1A

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

)(pσkk

)(103)

−1detA =

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 + 43G

4G(1− Kb

Ks)σkk

)(106)

∇ · ~ξ ∝

(p+

1− Kb

Ks

3Kb[1Kb

+ φKf− 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)σkk3Kb[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

3G(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 orin σkk at the same time. Thus in the limit of low frequencies, Biot’s equations become in fact a diffusion equation.

21

Analysis of Diffusivity

We have previously derived from [8] the expression for the diffusivity from Biot’s theory in the quasi-static limit, whichcan be re ordered as:

CD =kKf

ηφ

[1 +

Kf

φ(Kb + 4

3G) ((1− Kb

Ks(1 + φ)

)+

4G

3Ks(α− φ)

)](112)

From [12], we find that,1(

Kb + 43G) =

1(Kb − 2

3G+ 2G) =

1

(λ+ 2G)=

(1 + ν)

3(1− ν)Kb(113)

With λ the Lame first parameter and ν the Poisson ratio of the frame, we also know that

G =3Kb(1− 2ν)

2(1 + ν)(114)

Chydropp =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

(Chydropp − 2(1− 2ν)α

3(1− ν)

(Chydropp + Cm −

Chydropp

(1− 2ν)+ Cm

(1

α+

1

φ+ 1

)))]−1(118)

Also from[12], we know that the expression for the uniaxial pore compressibility is:

CUnipp = Chydropp − 2(1− 2ν)α

3(1− ν)(Chydropp + 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(CUnipp )

]−1(120)

Which leads to

CD =k

ηφ(Cf + CUnipp )=

k

ηφCT(121)

With CT the total compressibility used in well test analysis .

If and only if we fulfil the condition,

Chydropp = Cm(1− 2ν)

(1

α+

1

φ+ 1

)(122)

From [12] we know that,

Chydropp = Cbc

(α

φ− (1− α)

)(123)

Cbc = Cbp + Cm (124)

By inspecting (122) and (123), we conclude that,

Cbp = Cm

( (1− 2ν)

α

)(αφ − (3− α)

)(αφ − (1− α)

)− 1

< 0 (125)

However, equation (125) can never be satisfied; if we plug typical values from [12], the expression between parenthesesis always negative, so the only reason why equation (123) works instead of the correct equation (118) is due to the factthat for typical reservoirs CUnipp and the Biot’s equivalent compressibility differ by a factor of two, as can be shown forsome cases in appendix D, and since we have so much uncertainty on measurements this factor of 2 becomes negligibleso that equation (123) holds.

22

If one investigates equation (118), when we ask the next condition:

−Chydropp

(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 whichpossibly means that locally, the rock-to-fluid and the fluid-to-rock interactions are not symmetric and we must be locallyout of dynamic equilibrium. This model is limited since we consider planar waves and it’s more reasonable to considereither spheroidal or spherical waves, even Bessel waves would be a better geometrical ansatz for seismic prospection, sowe lose some insight due to this geometrical simplification. Another, limitation is the pore network geometry which mustbe integrated in some manner as well.

Discussion

Biot’s poroelasticity theory predicts the existence of two compressional waves, due to two possible uniaxial vibrationalmodes (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 thebehaviour of some complex functions, however the procedure for calculating them is quite straight forward after somemechanical algebraic manipulations. It is true that the expressions are quite complex and one is forced to use a simplesequential algorithm to plot the curves, for this purpose a simple excel spreadsheet is sufficient. In order to avoid possibleoscillations 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 givenin table 2 for the average group B. The understanding of the parameter influence on the model is mainly based onrock/fluid coexistence mechanisms. Even if most of the parameters are correlated, it is still interesting to appreciate themodel’s different features based on these simple numerical experiments. One explanation for the inverse behaviour for theshear wave speed with respect to porosity is that the shear modulus of the frame dominates over the bulk moduli, as theshear 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 onthe characteristic frequency of the system. For the studied case, we appreciate a 1.6 % variation in wave speed, however, Itis found numerically that, when the frame bulk and shear moduli are of the order of 109Pa at the same time, the variationin wave speed is of the order of 11% for very tight sandstones. This is shown in the next plot using the remaining datafrom Carcione for the remaining elastic moduli (Carcione, 1998a) and data from 2H2M1 & 2V1M1 for the fluid and rockparameters, 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 thanthe limit frequency) it is possible to predict a single value (average group 2H2M1 & 2V1M1) due to lack of more availabledata. The former is a less strong verification, since we are not able to verify the frequency dependence for the wave speeds.

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’sequation in the quasi-static limit by ignoring the inertial forces, and doing some algebraic and vector manipulations. Wederived the diffusion equation from the quasi-static limit of the theory, in the diffusivity we recognized the expressionfor the uniaxial strain pore compressibility [12], and we tried to make a connection with the diffusivity from well testanalysis which led to inconsistent results, however the former model works only thanks to the usual values for the elasticparameters found in typical reservoirs, where CUnipp and the Biot’s equivalent compressibility differ by a factor of two, ascan be shown for some cases in appendix D.

We found an inconclusive connection between Biot theory of poroelasticity and the diffusion equation from well-testanalysis, giving rise to a factor-of-two underestimate of rock compressibility, this could be due to an effective rock-to-fluidcompressibility and fluid-to-rock compressibility that are not taken into account in traditional models, and this could alsobe related to geometrical effects of the pore network and of the actual wave front of the acoustic waves. These resultswill 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 suggestto perform experimental measurements in a geomechanics laboratory, instead of looking for data on the literature in orderto have more control of the experimental settings. It has been very difficult to find data in the literature, and it is alsoexperimentally difficult to observe the slow Biot wave at low frequencies according to Plona [7]. However, some importantwork 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 viscodynamicoperator 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 theBiot 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 werederived 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 porosityand mobility. From numerical experiments, we concluded that increasing porosity will decrease wave speed and willincrease 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 thatthe 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 Bioteffect in wave speed varying from 1% to 2 % [10], it is also found numerically that we can get greater Biot effectsof 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 freesandstones for a frequency range from 0 Hz to 100 Hz (samples A and B) and the wave speed model is locallyverified 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 theexpression for the uniaxial strain pore compressibility [12], and we tried to make a connection with the diffusivityfrom well test analysis which led to inconsistent results, however the former model works only thanks to the usualvalues 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 effectiverock-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 acousticwaves. These results will be a very helpful starting point for further work on this problem.

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]Chydropp = Hydrostatic pore compressibility of the pore vol-ume [1/Pa]CUnipp = 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 zKb = 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 = Lame 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 zσij = 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]

25

References

[1] Maurice A. Biot, General Theory of Three-DimensionalConsolidation, Journal of Applied Physics, Vol. 12, pp.155- 164, 1941.

[2] Maurice A. Biot, Theory of Propagation of ElasticWaves in a Fluid-Saturated Porous Solid. I. Low-Frequency Range, The Journal of The Acoustical Societyof America, Vol. 28, No. 2, pp. 168-178, 1956.

[3] Maurice A. Biot, Theory of Propagation of ElasticWaves in a Fluid-Saturated Porous Solid. II. Higher Fre-quency Range, The Journal of The Acoustical Society ofAmerica, Vol. 28, No. 2, pp. 179-191, 1956.

[4] M. A. Biot and D. G. Willis, The Elastic Coefficientsof 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 AppliedPhysics, 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 mediaand 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 appliedmathematics and mechanics ; v.16). Amsterdam ;Lon-don: North-Holland. pp. 61.

[10] T. Bourbie, O. Coussy and B. Zinszne, Acoustics ofPorous Media, Institut Francais du Petrole PublicationsGeophysics Petroleum Engineering, pp. 63- 95, 1987.

[11] T. Klimentos and C. McCann, Relationships amongcompressional wave attenuation, porosity, clay content,and permeability in sandstones, Society of ExplorationGeophysicists Geophysics, Vol. 55, No.8, pp. 998-1014,1990.

[12] R.W. Zimmerman, Implications of Static Poroelasticityfor Reservoir Compaction, Proc. 4th North Amer. RockMech. Symp., A.A. Balkema, Rotterdam, pp. 169-172,2000.

[13] Jose M. Carcione, Wave Fields in Real Media: WavePropagation 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 RockPhysics Handbook: Tools for Seismic Analysis of PorousMedia, Cambridge University Press pp.266-272, 2009.

[15] V. Mikhaltsevitch, M. Lebedev and B. Gurevich, AnExperimental Study of Low-Frequency Wave Disper-sion and Attenuation in Water Saturated Sandstones,Poromechanics V ASCE 2013 “Proceedings of the FifthBiot Conference on Poromechanics”, 2013.

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 ContributionJournal of Applied “General Theory of Three- Maurice First to formulate a mathematical treatment forPhysics, 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 ofAcoustical Society of 1956 Waves in a Fluid-Saturated Porous A. Biot waves in fluid-saturated porous media by meansAmerica, Vol. 28, Solid. of Lagrangian mechanics for Poiseuille flowNo. 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 ofAcoustical Society Waves in a Fluid-Saturated Porous A. Biot waves in fluid-saturated porous media by meansof America, Vol. 28, Solid. of Lagrangian mechanics and viscodynamicNo. 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 theMechanics, Vol. 24, Theory of Consolidation.” D.G. Willis elastic coefficients from Biot’s theory.pp. 594- 601Institute of Physical “Theory of Elasticity” L.D. Landau First to formalize the theory of elasticity forProblems, U.S.S.R. 1959 solids by means of a rigorous mathematicalAcademy of Sciences, description.Vol. 7 of Course of E.M. LifshitzTheoretical Physics.Journal of Applied 1962 “Mechanics of Deformation and Maurice A. A unified treatment of the mechanics ofPhysics, Vol. 33, Acoustic Propagation in Porous Biot deformation and acoustic propagation in porousNo. 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 slowLetters, 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’sPhysics, Vol. 12, 1981 in fluid-saturated porous media and Chandler slow wave equation and a Diffusion Equation forpp. 155- 164. Biot’s slow David L. Pressure.

wave in the limit of zero frequency” JohnsonInstitut Francais du 1987 “Acoustics of Porous Media” T. Bourbie First to explain in a clear fashion Biot’s theoryPetrole Publications O. Coussy and other relevant elements of applied elasticityGeophysics theory to rock mechanics.Petroleum Engineering B. ZinsznerSociety of Exploration 1990 “Relationships among compressional T. Klimentos Experimental measures of wave speeds andGeophysicists wave attenuation, attenuation for the fast compressional wave andGeophysics, Vol. 55, porosity, clay content, and C. McCann the shear wave in 42 water saturated sandstonesNo.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 theRock Mech. Symp., A.A. for Reservoir Compaction Zimmerman uniaxial pore compressibility.Balkema, Rotterdam,pp. 169-172,Seismic Exploration “Wave Fields in Real Media: Wave Jose M. Rigorous and very mathematical approach toVolume 31 2001 Propagation in Anisotropic, Anelastic Carcione Biot’s theory with its extension to anisotropicElsevier and Porous Media.” systems.Cambridge University “The Rock Physics Handbook: Tools G. Mavko Summary of Biot’s theory results and itsPress 2009 for Seismic Analysis of Porous T. Mukerji applications and limitations.

Media.” J. DvorkinPoromechanics V 2013 “An Experimental Study of Low- V. Data for frequency dependence validation ofASCE 2013 Frequency Wave Dispersion and Mikhaltsevitch attenuation factor for fast compressional wave.“Proceedings of the Attenuation inFifth Biot Conference Water Saturated Sandstones” M. Lebedevon Poromechanics” B. Gurevich

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. Itgives 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 todescribe these properties. Gives a discussion of the physical interpretation of these various constants. Establishes thefundamental equations for the consolidation and an application is made to the one-dimensional problem correspondingto a standard soil test. Gives the simplified theory for the case most important in practice of a soil completely saturatedwith water.

Conclusion reached:

1. The number of these constants including Darcy’s permeability coefficient is found equal to five in the most generalcase.

2. It is quite possible that the soil particles are held together by capillary forces which behave in pretty much the sameway as the springs of the model of a system made of a great number of small rigid particles held together by tinyhelical 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 methodof attack constitutes a major simplification and proves to be of high value in the solution of the more complex two- andthree-dimensional problems.

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 instancein the case of water-saturated rock. The paper denoted here as Part I is restricted to the lower frequency range wherethe 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 viscousfluid.

Methodology used:Introduces the concept of dissipation potential into the Lagrangian of the rock-fluid system and by solving Euler-Lagrangeequations for both rotational and dilatational waves, gets the Biot’s equation of motion. Uses planar wave analysis toobtain 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 andthat 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 Poiseuilleflow.

Comments:The phenomenological parameters used are cumbersome and had to be expressed in terms of the more physical parameterslike bulk moduli and shear modulus.

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 isextended to higher frequencies. The breakdown of Poiseuille flow beyond the critical frequency is discussed for pores offlat and circular shapes. As in Part I the emphasis of the treatment is on cases where fluid and solids are of comparabledensities.

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 frequencyassumption.

Methodology used:Introduces the concept of a new viscodynamic operator for high frequencies that takes into account the breakdown ofPoiseuille 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 operatorfor Poiseuille flow that is very important for my thesis, and it is the one that I used for this work.

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 determinethe coefficients.

Conclusion reached:For an isotropic system, in which there are four coefficients, the four measurements of shear modules, jacketed andunjacketed compressibility, and coefficient of fluid content, together with a measure of porosity appear to be the mostconvenient. The porosity is not required if the variables and coefficients are expressed in the proper way. The coefficient offluid content is a measure of the volume of fluid entering the pores of a solid sample during an unjacketed compressibilitytest. 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 alsomay be applied to incremental variations in nonlinear systems provided the stresses are defined properly.

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 chapteron 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 butit’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 understandelasticity theory if the reader has some passion for theoretical physics.

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 somenew 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 thelinear mechanics of fluid saturated porous media and also to present some new results and developments with particularemphasis 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 ofdeformation 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 avery general nature. Specific relaxation models are discussed, and the general applicability of a correspondence principleis 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 athermoviscoelastic 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.

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 fullycontrolled 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. Heshowed 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 beadequate 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 ofsintered glass beads. It really helped me understand an experimental setting for analysing waves in a porous medium.

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 slowcompressional 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 themixture 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 somevector algebra theorems (Helmholtz decomposition theorem) to get the exact expression for the diffusion equation onpressure 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 accountall 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 importantsteps, considering that the reader is not used to follow vector algebra manipulations.

35

Institut Francais du Petrole Publications, Geophysics, Petroleum Engineering, pp. 63- 94 (1987)Acoustics of Porous Media

Authors: T. Bourbie, 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’sexperimental 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 toexplain 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 noevident expression of it. There are some useful graphs of their dependence with frequency, but nothing else.

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 watersaturated sandstones under confining pressure.

Objective of the Paper:To present experimental results of measurements of attenuation and wave speeds for compressional waves from 42 watersaturated sandstones at a confining pressure of 40MPa (equivalent to a depth of burial of about 1.5km) in a frequencyrange from 0.5to1.5MHz.

Methodology used:The compressional wave measurements were made using a pulse–echo method in which the sample (5 cm diameter, 1.8cmto 3.5cm long) was sandwiched between perspex (lucite) buffer rods inside the high pressure rig. The attenuation of thesample was estimated from the logarithmic spectral ratio of the signals. Data are presented to demonstrate that intrapore clays in sandstones are important in causing the attenuation of compressional waves and in controlling permeabilityof the sandstones. The data are important because this mechanism of attenuation has not been recognized before, andbecause the results bring closer the possibility of using accurate measurements of the attenuation of compressional wavesto estimate the permeability of rocks in situ in boreholes, or in the laboratory. Conclusion reached: The results showthat 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 relationshipbetween attenuation and permeability is less well defined; those samples with permeabilities less than 50 md have highattenuation coefficients (generally greater than 1 dB/cm) while those with permeabilities greater than 50 md have lowattenuation 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/Uricktheory of compressional wave attenuation in porous, fluid-saturated media.

37

Proc. 4th North Amer. Rock Mech. Symp., A.A. Balkema, Rotterdam, pp. 169-172, 2000Implications 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, anexpression is derived for the pore compressibility under uniaxial strain conditions. Finally, the influence of pore pressureon 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 strainconditions.

Comments:This paper helped to get a new expression for the pore compressibility of the bulk volume.

38

Seismic Exploration, Volume 31, Elsevier Chapter 7. (2001)Wave Fields in Real Media: Wave Propagation in Anisotropic, Anelastic and Porous Media.

Authors: Jose 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 frequencylimit 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 thingI needed to understand Biot’s theory and to create an analytic expression for the wave speed and attenuation, by simplyfollowing its guidance.

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 wavespeeds and attenuations, however they don’t take into account very carefully the low frequency viscodynamic operatoras Carcione did, even though both have the same limiting case.

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 highfrequency 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

Z22R + Z2

2I

(130)

TR =−Z1RZ2I + Z1IZ2R

Z22R + 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 =

√ωa2ρfη

(136)

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 = Aϑ2bd2b +G+ Cϑbϑf +Dϑ2f (137)

With ϑp = ep11 + ep22 + ep33 (p = b or f) and epij the component (i, j) of the strain tensor of the phase p.

d2b = dbijdbji (138)

dbij = epij −1

3ϑpδij (139)

A =1

2

((1− φ)(1− φ− Kb

Ks)Ks + Ks

KfKb

1− φ− Kb

Ks+ φKs

Kf

)(140)

C =(φ)(1− φ− Kb

Ks)Ks

1− φ− Kb

Ks+ φKs

Kf

(141)

D =1

2

φ2Ks

1− φ− Kb

Ks+ φKs

Kf

(142)

Kinetic Energy

We postulate a quadratic form with a coupling term, namely,

T =1

2Ωb(ρ11v

bi vbi + 2ρ12v

bi + ρ22v

fi v

fi ) (143)

With Ωb being the volume of the elementary macroscopic and representative region of porous material.

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 hydrostaticpore compressibility as:

Cunipp

CHydropp Zimm

=

[1− 2(1− 2ν)α

3(1− ν)

](144)

However, we get from our own calculation and by taking the same limit,

CUnipp

CHydropp 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 νCuni

pp

CHydropp Zimm

CUnipp

CHydropp Biot

Cunipp

CHydropp Zimm

/CUni

pp

CHydropp Biot

Ruhr Sandstone 0.65 0.02 0.12 0.63 1.12 1.78Berea Sandstone 0.79 0.19 0.2 0.61 1.27 2.08Ohio Sandstone 0.74 0.19 0.18 0.61 1.21 1.99Pecos Sandstone 0.83 0.2 0.16 0.55 1.21 2.20Boise 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.