Dispersive surface acoustic waves
in poroelastic media
Dispersive surface acoustic waves in poroelastic media
PROEFSCHRIFT
ter verkrijging van de graad van doctor aan de Technische Universiteit Delft,
op gezag van de Rector Magnificus, prof.dr.ir J.T. Fokkema, voorzitter van het College voor Promoties,
in het openbaar te verdedigen op dinsdag 10 mei 2005 om 15:30 uur
door
Gabriel Eugenio CHAO
Licenciado en Ciencias Fisicas, Universidad de Buenos Aires geboren te Buenos Aires, Argentini¨e
Prof.dr.ir. M.E.H. van Dongen Samenstelling promotiecommissie: Rector Magnificus, voorzitter
Prof.dr.ir. J.T. Fokkema, Technische Universiteit Delft, promotor
Prof.dr.ir. M.E.H. van Dongen, Technische Universiteit Eindhoven, promotor Dr.ir. D.M.J. Smeulders, Technische Universiteit Delft, toegevoegd promotor Prof.dr.ir. C.P.A. Wapenaar , Technische Universiteit Delft
Prof.dr.ir. G. Ooms, Technische Universiteit Delft Prof.ir. M. Peeters, Colorado School of Mines, USA Prof.dr. R.J. Arts , Technische Universiteit Delft
This research was financially supported by the ISES program. Copyright c 2005 by G.E. Chao
ISBN 90-9019411-8
Printed by Universiteitsdrukkerij TU Eindhoven, Eindhoven, The Netherlands Cover design by Paul Verspaget
CONTENTS
Summary . . . xi
1. Introduction . . . 1
1.1 Introductory Remarks . . . 1
1.2 Literature Review . . . 3
1.2.1 Wave Propagation in Porous Media . . . 3
1.2.2 Attenuation Mechanisms . . . 3
1.2.3 Surface Waves in Porous Media . . . 4
1.2.4 Acoustic Borehole Modes in Porous Reservoirs . . . 5
1.2.5 Experimental Studies on Bulk and Surface Waves in Poroe-lastic Media . . . 6
1.3 Objectives of this Thesis . . . 7
1.4 Thesis Outline . . . 7
2. Wave Propagation in Fully Saturated Porous Media: Bulk Modes and Inter-face Waves . . . 9
2.1 Dynamics of a Continuum Porous Medium: The Biot theory . . . . 9
2.1.1 Stress-Strain Relations . . . 9
2.1.2 The Lagrangian Approach to the Biot Equations . . . 10
2.1.3 Displacement Potential Formulation. Body Waves . . . 12
2.1.4 The Slow Compressional Wave . . . 14
2.2 Poroelastic Surface Waves along Plane Interfaces . . . 16
2.2.1 Dispersive Surface Modes . . . 17
2.2.2 Synthetic Seismograms: The Influence of the Leaky Modes 21 2.3 Surface Modes in Borehole Configurations: Approximate Models . 25 2.3.1 Rigid Borehole Wall . . . 26
2.3.2 Quasi-1D Model for a Permeable Borehole Wall . . . 28
2.4 Appendix 2A: Matrix Coefficients for the Fluid/Porous Medium Plane Interface . . . 34
3. Surface Waves along Partially Saturated Porous Media . . . 35
3.2 Acoustic Properties of a Partially Saturated Porous Medium: The
Oscillating Gas Bubble Model . . . 36
3.3 Saturation Effects on the Velocities and Attenuation of the Surface Waves . . . 44
3.4 Conclusions and Discussion . . . 50
3.5 Appendix 3A: Gas Pocket Model Revisited . . . 51
3.6 Appendix 3B: Comparison Between the Two Models . . . 55
3.7 Appendix 3C: Matrix elements for the gas pocket model . . . 57
4. Shock-Induced Borehole Waves in Porous Formations: Theory and Experi-ments . . . 61
4.1 Introduction . . . 61
4.2 Theory Formulation . . . 63
4.3 Permeability Effects on the Pseudo-Stoneley Wave . . . 66
4.4 Experimental Setup and Results . . . 72
4.5 Conclusions and Discussion . . . 76
4.6 Appendix 4A: Potential Formulation of Biot’s Theory in Cylindrical Coordinates . . . 77
4.7 Appendix 4B: Matrix coefficients for the confined reservoir . . . 79
4.8 Appendix 4C: Matrix coefficients for the radially infinite porous for-mation . . . 82
5. Surface and Guided Waves along a Cylindrical Interface between a Liquid and a Liquid-Saturated Porous Medium . . . 83
5.1 Introduction . . . 83
5.2 Borehole modes in Poroelastic Formations . . . 84
5.2.1 Pseudo-Stoneley and Pseudo-Rayleigh Waves . . . 84
5.2.2 Radius Effects . . . 87
5.2.3 Dispersion Relation in the Bounded Reservoir . . . 89
5.3 Modification of the Shock Tube . . . 92
5.4 Experimental Results . . . 94
5.5 Conclusions . . . 100
6. Seismic Signatures of Partial Saturation on Acoustic Borehole Modes . . 103
6.1 Introduction . . . 103
6.2 Borehole Waves in Partially Saturated Porous Media . . . 104
6.3 Conclusions . . . 111
7. Conclusions and Recommendations . . . 113
Contents ix
Bibliography . . . 120 Acknowledgements . . . 129 Curriculum Vitae . . . 131
SUMMARY
In this thesis, the dispersive properties of the surface acoustic waves propagating along poroelastic materials are theoretically and experimentally studied. The influ-ence of the pore space saturation on the surface modes is numerically investigated in the cases of plane and cylindrical interfaces. The theoretical formulation is based on the two-phase Biot model which accounts for the frequency-dependent mechanical interaction between the solid and the fluid constituents. Shock-induced wave exper-iments on water-saturated porous samples were performed in order to validate the theoretical and numerical findings.
The characteristics of the pseudo-Stoneley wave along boreholes in porous for-mations are studied in a broad band of frequencies (100 Hz-200 kHz) using a dis-placement potential formulation. The influence of the permeability on the phase velocity, attenuation, radial displacement, and pore pressure is investigated and the contribution of the different bulk modes to the average radial displacement is ana-lyzed in the frequency domain. The numerical results indicate that the permeability dependence at low frequencies is caused by the Biot slow wave, due to its relatively high contribution to the displacements and pressures induced by the pseudo-Stoneley wave.
Acoustic experiments were performed using a shock tube technique to excite the pseudo-Stoneley wave in a water-filled borehole in a water-saturated Berea sand-stone cylinder. Frequency-dependent phase velocities and damping coefficients were measured using this technique. Quantitative agreement between the experimental re-sults and the theoretical predictions is found for the phase velocity in the frequency range from 10 to 50 kHz. The attenuation data are found to deviate from theory in an oscillatory pattern, and only qualitative agreement is reached. In order to im-prove the quality of the attenuation data, the shock tube was modified. A funnel-like structure was mounted inside the tube just above the sample in order to channel the acoustic energy into the bore and enhance the excitation of the surface modes. More-over, a new FFT-Prony-Spectral Ratio method is implemented to transform the data from the space-time domain to the frequency-wavenumber domain. For the Berea sandstone, a significant improvement in the accuracy of the attenuation data and an excellent agreement with theoretical predictions at low frequencies is found. The comparison of the experimental results and the numerical calculations shows that
the oscillating fluid flow at the borehole wall is the dominant loss mechanism for the pseudo-Stoneley wave and that it is properly described by the Biot theory at fre-quencies below 40 kHz. A higher frefre-quencies, a systematic underestimation of the numerical predictions is found, which can be attributed to the existence of other loss mechanisms neglected in the Biot model.
Higher-order guided modes associated with the compressional wave in the porous formation and the cylindrical geometry of the shock tube were excited in the exper-iments, and detailed information was obtained on the frequency-dependent phase velocity and attenuation in highly porous and permeable materials. The measured attenuation of the guided wave associated with the compressional wave, reveals the presence of regular oscillatory patterns that can be attributed to radial resonances. This oscillatory pattern is also numerically predicted, although the measured attenu-ation values are one order of magnitude higher than the theoretical ones. The phase velocities of these higher-order modes are generally well predicted by theory.
In order to investigate the effect of partial gas saturation on the surface waves, the Biot model is extended to include additional interactions between the distinct phases, namely acoustic radiation, heat exchange and viscous dilatation. Within this theoret-ical framework, a numertheoret-ical study of the dispersive phase velocity and attenuation of the surface waves along a liquid/partially saturated poroelastic plane interface reveals an important dependence on the gas fraction. Increasing gas fraction causes increased attenuation over the entire frequency range. Maximum values in the attenuation coef-ficient of the pseudo-Stoneley wave are obtained in the 10-20 kHz frequency interval, which is relevant for borehole logging purposes. The attenuation level and the char-acteristic frequency of this maximum depend on gas fraction. In the high-frequency limit, where only compressibility effects govern the dynamics of the surface modes, a transition is found between a leaky pseudo-Stoneley wave and a true Stoneley mode. This transition occurs at a typical gas fraction where the slow compressional bulk wave and the acoustic wave in the liquid half-space have identical phase velocities.
The saturation effects are also studied for a borehole configuration. In this case a poroelastic formation saturated by a brine-air mixture is considered. A clear de-pendence of the damping of the pseudo-Stoneley wave on gas fraction is found. The damping increases with gas fraction over the complete range of frequencies studied (10 Hz-50 kHz). The interpretation of the results indicates that the compressibility of the mixture governs the dynamics of the pseudo-Stoneley wave and accounts for the saturation effects observed in the phase velocity, damping and pore pressure. A quantitative analysis demonstrates the high sensitivity of the damping coefficient to small amounts of gas in the pores. This effect not only holds for bulk modes, but for surface modes as well.
1. INTRODUCTION
1.1 Introductory Remarks
Surface elastic waves are quite common phenomena in nature. Perhaps the most illustrative example follows from seismological observations. When an earthquake occurs in the earth’s interior, part of the elastic energy radiates in the form of surface seismic waves. Their energy is confined to a region close to the surface as the waves spread out along the two-dimensional surface instead of the three-dimensional do-main. These waves propagate almost unattenuated along the earth’s surface and are responsible for most of the damage due to their large amplitude and associated ro-tational movement (Aki and Richards, 1980). In this case, the wavelengths involved are on the order of magnitude of meters to kilometers.
On the other hand, in ultrasound technological applications related to interdigital transducers (IDTs) and more recently to material characterization by means of laser-generated surface acoustic waves (SAW), the typical wavelengths are on the microm-eters scale (Maznev et al., 2003). In fact, the study of surface elastic waves covers a broad band of wavelengths and frequencies as illustrated in Figure 1.1. Allard and coauthors (2003; 2004) reported studies on laser-irradiated surface waves along porous materials. The analysis of the surface deformation caused by the surface mode provides information on the mechanical properties of the porous solid. Among other applications, the inversion of surface wave data for subsurface imaging is relevant for seismological studies in general and in shallow geophysics for local soil charac-terization. The wide spectrum of applications extends to bone biomechanics where measurements of the distribution of material elastic constants of trabecular bones and anisotropy in cortical bones have been lately carried out with promising results (Jorgensen and Kundu, 2002; Raum et al., 2004).
Particularly in the frequency band relevant to seismic and borehole exploration (10 Hz- 30 kHz) it is challenging to exploit at maximum the information provided by the surface waves in order to obtain an accurate reservoir characterization. In this respect there has been an increasing interest in understanding the fluid effects on wave propagation over the last years. Most of the work is focused on the flow-induced loss mechanisms in the compressional P wave. Scarce studies on surface waves have shown the great sensitivity of these kind of waves to relevant parameters
W a v e l e n g t h F r e q u e n c y [ H z ] k m I n f r a s o u n d 1 S e i s m i c W a v e s 1 0 2 m A u d i b l e W e l l l o g g i n g s o u n d 1 0 4 c m m m 1 0 6 U l t r a s o u n d I D T s 1 0 8 S A W p u l s e s m m S A W g r a t i n g s 1 0 1 0
Figure 1.1: Surface (elastic) acoustic waves are found over a broad frequency spectrum.
Cur-rent research extends from seismic waves in the infrasound region to interdigital transducers (IDTs) and to laser-generated SAW (surface acoustic waves) pulses and transient gratings in the ultrasound region (adapted from Hess (2002)).
of the reservoir. Theoretical and experimental investigations in boreholes surrounded by poroelastic formations provide promising evidence about the dependence of the permeability and porosity of the reservoir on the frequency-dependent velocities and attenuations of the surface modes. However, there is a lack of a full understanding of the underlying physical mechanisms which govern the propagation of the surface waves along poroelastic formations. Moreover, further experimental studies under well-defined laboratory conditions are required to validate the theoretical models.
The particle motion driven by a surface wave follows an elliptical trajectory. This rather complicated elliptical polarization can be expressed as a combination of the compressional and shear bulk modes in the medium. This is a fundamental differ-ence between the surface waves and the bulk or body waves. While the bulk waves propagate independently in an unbounded medium, the presence of the boundaries induces a coupled motion of the compressional and shear waves which characterizes the surface waves. Therefore an accurate description of the bulk waves dynamics in a porous material is an essential step in order to gain a better understanding of the surface waves propagating along a poroelastic layer.
1.2. Literature Review 3
1.2 Literature Review
In this section a literature review is presented for the research on propagation of bulk and surface waves in poroelastic materials. Special attention is devoted to the exam-ination of the existing knowledge on attenuation mechanisms and the propagation of acoustic borehole modes in porous formations. The experimental findings and the experimental techniques are reviewed in detail with emphasis on shock wave tech-niques.
1.2.1 Wave Propagation in Porous Media
From a purely historical point of view, the first investigations on acoustic wave propa-gation in porous media can be traced back to the works of Zwikker and Kosten (1941) and Frenkel (1944). Despite their significance and correct qualitative description of the wave phenomena, these models neglected significant compressibility (Zwikker and Kosten) and inertial (Frenkel) effects. Maurice A. Biot developed a theory which includes the viscous dissipation due to the relative fluid/solid motion, and describes the mechanical interaction between the fluid and solid phases (Biot, 1956a,b; Biot and Willis, 1961). An important aspect of this theory is the accurate description of a second compressional wave which propagates through a fully saturated porous medium. This so-called slow compressional wave has since then been detected exper-imentally and shown to correctly predict wave speeds (Plona, 1980; van der Grinten
et al., 1985; Kelder and Smeulders, 1997). However, Biot’s theory fails to account
for the level of attenuation of the fast compressional wave observed in laboratory and field experiments. This suggests the existence of loss mechanisms which are not con-sidered in the original theory. A critical comparison of the attenuation predicted by the Biot theory with existing experimental data for natural rocks was given by Gist (1994a).
1.2.2 Attenuation Mechanisms
The unability of the Biot theory to properly describe the attenuation values observed in liquid-saturated rocks and sediments encouraged a series of investigations on the influence of the wave induced fluid-flow loss mechanisms. At present it is assumed that the presence of cracks, gas inclusions and lithological inhomogeneities are re-sponsible for the most relevant dissipative mechanisms ignored in the Biot theory.
The influence of cracks at the grain scale on the wave propagation through a porous material has been discussed by Mavko and Nur (1979), Dvorkin and Nur (1993) and Dvorkin et al. (1995). At this microscopic scale the dilatation of the rock due to the passage of an elastic wave causes the fluid to flow from the crack to the pore space. This dissipative mechanism is denominated “squirt flow” and
can be described in terms of a characteristic squirt-flow length. It has been shown that the incorporation of this local dissipative mechanism significantly increases the attenuation values, but still is unable to fully explain the observed attenuation in seismic surveys.
Gas inclusions in the pore space of a liquid-saturated porous rock are also re-sponsible for losses since wave-induced oscillations of the trapped gas bubbles cause local dissipation. In this case, two immiscible fluid phases coexist in the void space. White (1975) assumed that the gas phase distribution can be represented by spherical gas pockets regularly arranged in a cubic lattice and formulated the so-called “gas pocket” model. Numerical studies based on Biot’s theory for this system have been published by Dutta and Ode (1979a,b). Their results show an important increase in the attenuation though they are restricted to frequencies where the wavelengths are considerably larger than the size of the inhomogeneities. A similar model has been proposed by Smeulders and van Dongen (1997). The main difference with respect to the work of Dutta and coauthors lies in the dissipative mechanisms induced by the bubble oscillations. In this model the thermal interaction between the bubble and the surrounding porous medium is also included. Their theoretical results were compared with experimental data measured in shock wave experiments.
One of the basic assumptions of the Biot model is related to the isotropy of the material that constitutes the rock. Heterogenous composites have wave-induced flow losses due to compressibility changes which are not considered in the original Biot formulation. Recently, Pride and Berryman (2003a), (2003b) have developed from first principles a set of governing equations for double-porosity, dual-permeability materials. These heterogeneous materials are constituted by two distinct porous ma-terials with different lithological and poromechanical properties.
1.2.3 Surface Waves in Porous Media
The first investigations on surface wave propagation in poroelastic materials are those of Deresiewicz (1960), (1961) and Deresiewicz and Skalak (1963). In this series of papers, the boundary conditions at the interface between a fluid and a fully saturated porous material were derived. Obviously these conditions govern the dynamics and the propagation of the surface modes. The most important difference with respect to the elastic case is the existence of the second compressional wave, which adds an additional degree of freedom and therefore an extra boundary condition is required to completely describe and close the problem. Deresiewicz and Skalak linked this extra boundary condition with the fluid pressure continuity at the boundary and introduced the concept of surface permeability, which controls the pressure discontinuity at the interface.
1.2. Literature Review 5
Feng and Johnson (1983a,b) applied these boundary conditions to study the ve-locity and attenuation of surface waves propagating along air/porous and liquid/po-rous plane interfaces. Their investigations are on the high-frequency limit of the Biot theory where the second compressional wave, as well as the first compressional wave and the shear wave, propagate without attenuation. They found that there are three different surface waves that may propagate depending on the stiffness of the porous medium and the surface permeability. Two of them, the pseudo-Rayleigh wave and the pseudo-Stoneley wave are the generalization of the pseudo-Rayleigh wave and the Stoneley wave which propagate along a liquid/elastic plane interface. In this case the pseudo-Rayleigh wave radiates energy into the liquid. It is important to remark that in the porous case the pseudo-Stoneley wave becomes attenuated because of ra-diation into the second compressional wave. A new surface mode is predicted which propagates slower than the second compressional wave and is denominated the true Stoneley wave.
Recently, the influence of the Biot frequency-dependent viscous dissipative mech-anisms on the propagation of surface waves for a very high-permeable porous medium have been studied by Gubaidullin et al. (2004). Their numerical results for the frequency-dependent phase velocity and attenuation of the surface waves indicate that the true Stoneley wave becomes attenuated due to viscous friction. This loss mechanism also governs the attenuation of the Stoneley wave and the pseudo-Rayleigh wave, particularly at low frequencies. Studies of surface wave propagation along porous surfaces have also been carried out based on other models than the Biot theory (Edelman and Wilmanski, 2002; Edelman, 2004).
1.2.4 Acoustic Borehole Modes in Porous Reservoirs
The propagation of surface and head waves along cylindrical interfaces between a liquid and a liquid-saturated porous reservoir has drawn attention due to the practi-cal implications in the oil industry related to reservoir characterization. Rosenbaum (1974) applied Biot’s theory to solve the high-frequency modes of a fluid-filled bore-hole surrounded by a poroelastic formation in order to correlate the permeability of the formation with the properties of the second compressional wave. The low-frequency limit, where the (pseudo) Stoneley wave is referred as the tube wave, was derived by Chang et al. (1988). Norris (1989) included the compressibility of the solid matrix. Schmitt et al. (1988) studied the wave response to a point source in both the time and frequency domain using Biot’s theory modified according to a ho-mogenization theory. They presented results for the (pseudo) Stoneley mode and the first pseudo-Rayleigh mode. Cheng et al. (1987) applied Biot’s theory to interpret the permeability dependence of the properties of the (pseudo) Stoneley wave observed in field data. Liu and Johnson (1997) simulated the effect of a mudcake layer that
may exist on the wall of the borehole. The influence of a thin impairment layer on the wall of the borehole on the dispersion relation of the (pseudo) Stoneley wave was analyzed by Tichelaar et al. (1999).
1.2.5 Experimental Studies on Bulk and Surface Waves in Poroelastic Media The first experiments which unambiguously linked a slower arrival with the second compressional wave were reported by Plona (1980). The experiments were per-formed in water-saturated sintered glass porous materials at ultrasonic frequencies using a mode conversion technique. Using a shock tube technique, van der Grin-ten et al. (1985), (1987) performed experiments in air- and water-saturated porous columns consisting of agglutinated sand particles. They were able to detect the sec-ond compressional wave. Using this method also the slow compressional wave in water-saturated natural rocks was detected (Smeulders, 1992; Brown et al., 2000). Nagy et al. (1990) successfully measured the second compressional wave in air-saturated natural rocks. Using ultrasound, experimental evidence of the slow wave in water-saturated rocks was also provided by Kelder and Smeulders (1997) and Kelder (1998).
An interpretation of laboratory velocity measurements in a wide range of partially gas-saturated rocks is given by Gist (1994b). Cadoret et al. (1995), (1998) present experimental results using a resonant-bar technique to determine the velocity and attenuation of acoustic waves in partially saturated limestones at a sonic frequency of 1 kHz . Similar experiments were previously performed by Lucet et al. (1991). Wave propagation experiments in porous media saturated by a mixture of water and air were performed by Smeulders and van Dongen (1997).
Experimental data regarding surface waves along plane interfaces were reported by Mayes et al. (1986) and Nagy (1992). The aim of their investigations was to detect the new surface mode predicted in the earlier work of Feng and Johnson (1983a). Recently, Allard et al. (2002), (2003), (2004) reported experimental data on surface wave propagation along poroelastic materials using a laser-based technique. Wisse
et al. (2002) reported data for shock-induced guided waves along the outer surface of
water-saturated porous cylinders.
Laboratory data for surface waves along boreholes in porous formations are scarce. Winkler et al. (1989) measured phase velocities and damping coefficients of (pseudo) Stoneley waves in porous samples saturated with silicone oil. Hsu et al. (1997) studied the influence of a cylindrical permeable mandrel on the tube wave in an elastic formation. They performed measurements using stainless steel and polyethylene as formations.
1.3. Objectives of this Thesis 7
1.3 Objectives of this Thesis
From the literature review presented in the previous section it is obvious that there are several open questions regarding the propagation of surface waves along poroelastic surfaces. In the borehole case, the cause for the dependence of the pseudo-Stoneley wave on the permeability of the formation is not fully understood. Moreover there is an increasing demand for accurate independent experimental data to validate the the-oretical models, since the only set of experimental data available is mostly restricted to frequencies above 10 kHz (Winkler et al., 1989). In this thesis a shock wave tech-nique is applied to study the propagation of borehole modes in poroelastic formations. One of the advantages of this method is the broad band of frequencies involved in the experiments (1 kHz- 150 kHz). We aim to obtain accurate frequency-dependent data of the phase velocities and attenuation coefficients of the surface waves. These results are compared with numerical calculations based on Biot’s theory in order to assert the validity of the Biot loss mechanisms to describe the attenuation of the surface waves. Also a theoretical analysis is performed to quantitatively explain the dependence of the frequency-dependent pseudo-Stoneley wave on the permeability.
The study of the dissipative mechanisms induced by the presence of gas in the pore space is relevant to many applications. It has been proven that even small frac-tions of gas significantly affect the acoustic properties of a porous medium. The main efforts so far have been conducted towards a description of the bulk acoustic proper-ties of partially saturated porous materials. In this thesis the analysis is extended to consider the influence of distributed bubbles on the properties of the surface waves. Plane and cylindrical interfaces between a liquid and a partially saturated porous medium are considered. Results on the flat interface problem are relevant to explo-ration seismology in layered one-dimensional structures. Applications concerning the cylindrical interface configuration are obviously related to borehole geophysics and acoustic borehole logging techniques.
We also aim to study the surface waves in the time domain. An algorithm is developed to calculate the reflected pressure field for a spherical wavefront imping-ing upon a liquid/poroelastic flat interface. In this way we can compute synthetic seismograms and analyze the influence of the surface waves in an actual waveform.
1.4 Thesis Outline
A review of the Biot theory is given in Chapter 2 where the main features of the slow compressional wave are highlighted. Surface waves along liquid/porous plane inter-faces are also discussed in this chapter in terms of the frequency-dependent phase velocities and attenuations. The pressure response in the time domain for the re-flected pressure field due to an incident spherical waveform is numerically solved
following a Sommerfeld formulation. At the end of this chapter approximate models for surface wave propagation along boreholes in porous formations are described and developed. Chapter 3 is focused on the influence of the gas saturation on the speed and attenuation of the surface modes propagating along a liquid/poroelastic plane interface.
The shock-induced experiments performed in a Berea sandstone are reported and compared with numerical calculations based on Biot’s theory in Chapter 4. An anal-ysis of the permeability dependence of the properties of the pseudo-Stoneley wave is also given in Chapter 4, followed by a quantitative explanation based on the relative contribution of the bulk waves to the radial displacement induced by this mode. A series of experiments in a modified shock tube configuration, aimed to improve the enhancement of the surface waves, is analyzed in Chapter 5. Also in this chapter acoustic borehole modes in poroelastic formations are studied in detail. The purpose of Chapter 6 is to investigate the saturation effects on the frequency-dependent phase velocity and attenuation of the pseudo-Stoneley wave that propagates along a liq-uid/poroelastic cylindrical interface. The main results of this thesis are summarized in Chapter 7.
2. WAVE PROPAGATION IN FULLY SATURATED POROUS MEDIA: BULK MODES AND INTERFACE WAVES
2.1 Dynamics of a Continuum Porous Medium: The Biot theory
2.1.1 Stress-Strain Relations
The formulation of a compact and consistent theory for the mechanical interaction between the fluid and solid phase in a porous material can be attributed to M.A. Biot, who developed a theory which also includes the viscous dissipation due to the relative fluid/solid motion (Biot, 1956a,b; Biot and Willis, 1961). In this theory the stress in the solid and the pressure in the fluid are coupled and the stress-strain relations are written as:
τij = (P − 2N) ukkδij + 2N uij+ QUkkδij (2.1)
and
p = −1
φ (Qukk+ RUkk) , (2.2)
where summation over repeated indexes is assumed, τij is the stress in the solid, p
the pore pressure, u denotes the displacement of the solid matrix, U denotes the fluid displacement, uij = 12(∂ui/∂xj + ∂uj/∂xi), and δij is the Kronecker delta. The
porosity of the material φ is the ratio between the volume of the void space and the total volume in the representative element of volume. The representative element of volume REV, is a volume for which the average of all physical quantities over that volume is independent of the volume size. Therefore the REV is a volume cell containing all the information regarding the geometry and the physical properties of the material. N is the shear modulus of the composite material and P , R and Q are the so-called generalized elastic coefficients. They are related to the porosity, the solid frame bulk modulus Kb, the solid grain bulk modulus Ks, the pore fluid
modulus Kf, and N according to the following expressions:
P = (1 − φ) 1 − φ − KKb s Ks+ φ Ks Kf Kb 1 − φ −KKb s + φKs Kf +4 3N, (2.3)
Q = 1 − φ −KKb s φKs 1 − φ − KKb s + φKs Kf , (2.4) R = φ 2K s 1 − φ − KKb s + φKs Kf . (2.5)
The above expressions for the generalized elastic coefficients are obtained through ‘Gedanken’ experiments (see e.g. Allard (1993)).
2.1.2 The Lagrangian Approach to the Biot Equations
In this subsection the Biot equations are derived following a Lagrangian formulation (see e.g. Johnson (1986)). For a rigorous treatment of the Lagrangian formulation for continuous systems the reader is referred to Goldstein (1980). Consider the two-phase system consisting of the solid matrix and the fluid saturating the pore space. The kinetic energy T can be written as follows:
T = 1 2
ρ11| ˙u|2+ 2ρ12u · ˙U + ρ˙ 22| ˙U|2
, (2.6)
where ρ11, ρ12and ρ22are the density terms that are related to the density of the fluid
phase ρf and the solid phase ρsby:
ρ11= (1 − φ) ρs+ (α∞− 1) φρf, (2.7)
ρ12= (1 − α∞) φρf, (2.8)
ρ22= α∞φρf. (2.9)
The parameter α∞ is the tortuosity. It takes into account the microgeometry
of the channels in the pore space. Its value is always larger than 1. The potential energy V associated with the deformation of the porous material can be expressed as (Johnson, 1986):
V = 1 2
h
P (∇ · u)2+ 2Q (∇ · u) (∇ · U) + R (∇ · U)2+ N |∇ × u|2i. (2.10) Equations 2.6 and 2.10 define the volumetric Lagrangian density L = T − V for this continuum system. In order to describe the viscous dissipation mechanism due to the relative fluid-solid motion, the dissipation pseudo-potential D is introduced:
D = 1 2 ηφ2 κ0 ˙ u − ˙U2, (2.11)
2.1. Dynamics of a Continuum Porous Medium: The Biot theory 11
where η is the viscosity of the pore fluid and k0 is the steady-state permeability. The
coupled equations of motion are derived from the Hamilton principle which implies that the Lagrangian should satisfy the following equation:
∂ ∂t ∂L ∂ ˙qi + ∂ ∂xj ∂L ∂∂qi ∂xj −∂q∂L i + ∂D ∂ ˙qi = 0. (2.12)
In this case the generalized coordinates qiareu and U. The equations obtained are:
ρ11 ∂2u ∂t2 + ρ12 ∂2U ∂t2 = P ∇ (∇ · u) + Q∇ (∇ · U) − N∇ × ∇ × u + +ηφ 2 κ0 ∂U ∂t − ∂u ∂t (2.13) and ρ12 ∂2u ∂t2 + ρ22 ∂2U ∂t2 = R∇ (∇ · U) + Q∇ (∇ · u) − ηφ2 κ0 ∂U ∂t − ∂u ∂t . (2.14) Due to the intrinsic frequency-dependent nature of the viscous interaction, it is useful to consider the Biot equations in the frequency domain. Assuming a temporal depen-dence e−iωtit is possible to transform the Biot equations to the frequency domain by
the Fourier Transform to obtain: −ω2ρe11eu + eρ12Ue = P ∇ (∇ · eu) + Q∇(∇ · eU) − N∇ × ∇ × eu, (2.15) −ω2ρe12eu + eρ22Ue = R∇∇ · eU+ Q∇ (∇ · eu) , (2.16)
where the density terms are now written in their generalized frequency-dependent form: e ρ11= (eα − 1) φρf + (1 − φ) ρs, (2.17) e ρ12= (1 − eα) φρf, (2.18) and e ρ22= eαφρf. (2.19)
Here we have introduced a frequency-dependent tortuosity: e
α(ω) = α∞+
iηφF (ω) k0ωρf
where F (ω) is the viscodynamic operator describing the interaction between the pore fluid and the solid matrix. Johnson et al. (1987) argued that the simplest possible expression for F (ω) is:
F (ω) = s
1 −iωρ2ηφfα∞κ0
. (2.21)
For the sake of completeness we also give the expression for the frequency-dependent permeability κ(ω), which reads,
κ(ω) = iηφ
α(ω)ωρf
. (2.22)
2.1.3 Displacement Potential Formulation. Body Waves
In this subsection the main properties of the body waves’s solutions of the Biot equa-tions will be discussed. The most remarkable result is the prediction of a highly dispersive compressional wave, usually called slow P wave. In agreement with the classical results of linear elasticity, this two-phase theory predicts the presence of P and S waves. However, the existence of viscous interaction due to the relative motion of the fluid with respect to the solid matrix causes the classical compressional and shear waves to become dispersive and attenuated. It is first shown how the Biot equa-tions can be solved in the frequency domain by means of a displacement potential formulation. Then the expressions for the velocity of the different bulk modes are de-rived. In order to simplify the notation, in the remaining of the chapter the tilde above the functions and quantities in the frequency domain is omitted. The tilde above the density terms and the tortuosity is used to denote the frequency-dependent nature of these parameters.
For compressional P waves the displacements of the solid and fluid phase can be described by the following scalar potentials:
u = ∇Φs (2.23)
and
U = ∇Φf. (2.24)
The above potentials describe irrotational waves associated with the solid, Φs,
and the fluid, Φf. In terms of the potentials, the Biot equations in the frequency
domain can be written in a matrix form as follows:
2.1. Dynamics of a Continuum Porous Medium: The Biot theory 13
where ρ is the density matrix given by: ρ = e ρ11 ρe12 e ρ12 ρe22 , (2.26)
Φis a vector defined as [ΦsΦf]T and M is a matrix containing the generalized elastic coefficients: M= P Q Q R . (2.27)
Assuming that the solutions of equations 2.25 can be expressed as Φc1 = Ac1ekc1·rand Φ2 = Ac2ek2·r, the linear system is reduced to an eigenvalue
problem. The eigenvalues are the squared complex-valued wavenumbers of the com-pressional waves that propagate in the porous material. The wavenumbers kc1 and
kc2are given by:
k2c1(ω) = ω 2 2 (P R − Q2) h P eρ22+ Reρ11− 2Qeρ12− √ ∆i (2.28) and k2c2(ω) = ω 2 2 (P R − Q2) h P eρ22+ Reρ11− 2Qeρ12+ √ ∆i, (2.29) where ∆ is: ∆ = (P eρ22+ Reρ11− 2Qeρ12)2− 4 P R − Q2 e ρ11ρe22− eρ212 . (2.30)
The velocities of the compressional waves follow from: cj(ω) =
ω Re[kcj(ω)]
j = 1, 2. (2.31)
The attenuation of the compressional modes can be expressed in terms of the damping coefficient, Dj(ω) = |Im[kcj(ω)]| or the inverse quality factor,
Q−1j (ω) = 2Im[kcj(ω)] Re[kcj(ω)] j = 1, 2. (2.32)
The potentials Φsand Φf are related to the eigenvectors Φc1and Φc2, associated
with the fast and slow P wave respectively, by the following relations:
and
Φf = Gc1Φc1+ Gc2Φc2, (2.34)
where Gc1and Gc2are given by:
Gc1= P − v 2 c1ρe11 v2 c1ρe12− Q (2.35) and Gc2= P − v 2 c2ρe11 vc22 ρe12− Q . (2.36)
Shear waves are described by a vector potential Ψsh. The displacements induced
in the solid and fluid phase are respectively:
u = ∇ × Ψs, (2.37) U = Gsh∇×Ψs, (2.38) where Gsh= α − 1e e α . (2.39)
The velocity of the shear wave becomes: csh = N e ρ11− eρ212/eρ22 1 2 . (2.40)
2.1.4 The Slow Compressional Wave
The accurate prediction of the frequency-dependent velocity and attenuation of the slow compressional wave is the most important result of the Biot theory. This wave was detected experimentally by Plona (1980) at ultrasonic frequencies (500 kHz-2.25 MHz). The experiments were performed in water-saturated porous materials made out of glass beads. Some years later, van der Grinten et al. observed the slow compressional wave in shock wave experiments on porous columns consisting of agglutinated sand particles (van der Grinten et al., 1985, 1987). Nagy et al. (1990) observed the slow wave in air-saturated natural rocks. Kelder and Smeulders (1997) provided experimental evidence for the slow compressional wave in water-saturated rocks. They observed the effects of the slow wave in transmission experiments in a Nivelsteiner sandstone. A detailed review of the different attempts to measure the slow wave using different experimental techniques is given by Smeulders (2005).
2.1. Dynamics of a Continuum Porous Medium: The Biot theory 15
Figure 2.1: Velocity and specific attenuation of the slow compressional wave for a
water-saturated Berea sandstone. Biot’s theory predicts an important dependence of the properties of this wave on the frequency and the permeability. The slow wave is propagative at high frequencies and diffusive at low frequencies. Out-of-phase movement between the fluid and the solid phase characterizes the slow wave, therefore an increase in permeability results in lower values for the specific attenuation and higher velocities.
Berea sandstone Synthetic glass
Solid density ρs(kg/m3) 2644 2539
Porosity φ 0.20 0.465
Permeability k0(D) 0.36 10
Tortuosity α∞ 2.4 1.93
Frame bulk modulus Kb(GPa) 10.37 2.53
Shear modulus N (GPa) 7.02 2.9
Grain bulk modulus Ks(GPa) 36.5 36.5
Fluid density ρf (kg/m3) 1000 1000
Fluid viscosity η (mPa s) 1.0 1.0
Fluid bulk modulus Kf (GPa) 2.25 2.25
Table 2.1: Physical properties of the porous formations.
The slow wave is highly dispersive. It is diffusive in the low-frequency limit where the viscous dissipation dominates and propagative at high frequencies where the inertial effects prevail. The transition between these two different regimes is marked by the critical frequency:
fc =
ηφ 2πκ0α∞ρf
. (2.41)
The slow compressional wave is characterized by an out-of-phase displacement of the solid matrix with respect to the pore fluid. Therefore, a large influence of the per-meability and the viscosity on the properties of this wave is to be expected. This is analyzed in Figure 2.1 where the permeability and frequency effects on the velocity and attenuation of the slow compressional wave are depicted for a Berea sandstone saturated with water. The material properties of the rock and the saturating fluid are given in Table 2.1. The velocity increases with increasing permeability and fre-quency. In the zero-frequency limit the velocity tends towards zero. The specific attenuation increases with decreasing permeability due to increasing friction. The diffusive character of the slow wave at low frequencies results in a flattening of the attenuation coefficient. At higher permeability values, the relative flow of the fluid with respect to the solid matrix facilitates and therefore the attenuation decreases.
2.2 Poroelastic Surface Waves along Plane Interfaces
Surface waves are localized phenomena. They occur in the vicinity of an interface between two different media. The energy of these waves is usually limited to the re-gion close to the interface as these waves fade away exponentially when moving away
2.2. Poroelastic Surface Waves along Plane Interfaces 17
from the interface. The penetration in the material is on the order of the wavelength involved. The properties of the surface modes depend on the properties of the two media and the characteristics of the interface itself. In the case that one of the media is a porous material the flow impedance of the surface plays a significant role. Feng and Johnson (1983a,b) were the first to systematically study the velocity and attenua-tion of the surface waves that propagate along a plane interface between a fluid and a fully saturated porous medium. Their results are limited to the high-frequency limit where the viscous effects are neglected and the slow compressional wave is unatten-uated. In this limit there is also no influence of the permeability of the formation on the features of the surface modes. Gubaidullin et al. (2004) included the viscous effects and extended this study to consider the frequency-dependent properties of the surface modes for the case of a plane interface between water and a water-saturated highly permeable material (308 Darcy).
In this section the general procedure to compute the dispersion of the surface modes is outlined. Then the surface waves for a water/Berea sandstone interface are studied. Synthetic seismograms are obtained using a Sommerfeld integral formula-tion for the incident and the reflected pressure fields.
2.2.1 Dispersive Surface Modes
In this subsection a displacement potential formulation is developed in order to de-scribe the surface waves that propagate along a plane interface between a fluid half-space and a liquid-saturated porous half-half-space. The configuration studied is displayed in Figure 2.2. The surface modes propagate parallel to the interface, depend expo-nentially on the distance z from the interface and can be expressed in terms of the bulk mode solutions. In the liquid (z < 0), the compressional waves are described by the following potential:
Φf = Afeγfzei(kxx−ωt). (2.42)
The potentials associated to each of the bulk modes which propagate in the porous half-space are:
Φc1= Ac1e−γc1zei(kxx−ωt), (2.43)
Φc2= Ac2e−γc2zei(kxx−ωt), (2.44)
and
Ψsh= Be−γshzei(kxx−ωt)eˆy, (2.45)
where ˆey is the cartesian basis vector in the y− direction. The above potentials
x
z
f l u i d
f u l l y s a t u r a t e d p o r o u s m e d i u m
Figure 2.2: Plane interface between a liquid and a fully saturated poroelastic medium. The
surface waves propagate along the interface in the x−direction and decay expo-nentially in the z−direction.
z−direction are related to the horizontal wavenumber kxthrough the following
rela-tions: γj = s k2 x− ω2 c2 j , j = 1, 2, sh, f, (2.46)
where cj is the velocity of the corresponding bulk mode.
The surface modes can be written as a frequency-dependent linear combination of the potentials stated above. The different contributions of the bulk modes are determined by the boundary conditions, namely: continuity of averaged normal dis-placement, total stress and pressure. The displacements of the solid phase and the fluid phase in the porous medium can be expressed as follows:
u = ∇ (Φc1+ Φc2) + ∇ × Ψsh, (2.47)
and
U = Gc1∇Φc1+ Gc2∇Φc2+ Gsh∇ × Ψsh, (2.48)
whereu refers to the displacement of the matrix and U to the displacement of the pore
fluid. In the liquid half-space, the displacementUf is ∇Φf. Therefore the continuity
of average normal displacement at the interface,
2.2. Poroelastic Surface Waves along Plane Interfaces 19
can be expressed as: (1 − φ + φGc1) ∂Φc1 ∂z + (1 − φ + φGc2) ∂Φc2 ∂z + + (1 − φ + φGsh) ∂Ψsh ∂x = ∂Φf ∂z . (2.50)
The continuity of the normal component of the total stress implies:
τzz− φp = −pf. (2.51)
Using the Biot’s stress-strain relations (equations 2.1 and 2.2), the above equation can be written in terms of the potentials as follows:
[P − 2N + Q + Gc1(Q + R)] ∇2Φc1+ 2N ∂2Φc1 ∂z2 + [P − 2N + Q + +Gc2(Q + R)]∇2Φc2+ 2N ∂2Φc2 ∂z2 + 2N ∂2Ψsh ∂z∂x = −ω 2ρ wΦf.(2.52)
The absence of tangential stress in the liquid requires τxz = 0at the interface, and
this condition implies that: N 2 ∂2Φ c1 ∂z∂x + ∂2Φ c2 ∂z∂x +∂ 2Ψ sh ∂x2 − ∂2Ψ sh ∂z2 = 0. (2.53)
Finally, the continuity of pressure leads to
−1φ(Q + RGc1) ∇2Φc1+ (Q + RGc2) ∇2Φc2
= ρwω2Φf. (2.54)
Substituting equations (2.42-2.45) into equations (2.50,2.52-2.54) and after some algebraic manipulations a linear system for the amplitudes of the potentials is found:
N(kx, ω) · a = 0, (2.55)
where the matrix N contains information about the mechanical properties of the fully saturated porous medium and the water half-space and a is a vector containing the amplitude of the wave potentials, aT = (A
0, Ac1, Ac2, B). The elements of the
matrix N are given in Appendix 2A. The surface modes satisfy the condition that the determinant of N equals zero:
det [N (kx, ω)] = 0. (2.56)
At a fixed frequency ω, equation 2.56 is numerically solved for complex kxusing
103 104 105 1350 1400 1450 1500 1550 1600 1650 1700 1750 1800 PSfrag replacements p-Stoneley p-Rayleigh fluid shear Frequency (Hz) V elocity (m/s) (a) 103 104 105 10−4 10−3 10−2 10−1 100 101 PSfrag replacements p-Stoneley p-Rayleigh fast slow shear Frequency (Hz) Q − 1 (b)
Figure 2.3: Frequency-dependent velocities (a) and specific attenuations (b) of the surface
modes (solid lines). The velocities and specific attenuations of the bulk modes are indicated in dashed lines. The material properties used in the numerical cal-culations are given in Table 2.1
2.2. Poroelastic Surface Waves along Plane Interfaces 21
V (ω) = ωRe−1(kx), and specific attenuation coefficients are obtained. The
spe-cific attenuation coefficient, Q−1(ω) = |2Im(k
x)/Re(kx)|, represents the fraction of
energy dissipated into heat in an oscillation period.
There are three types of surface waves that may propagate on a flat interface be-tween a fully saturated porous medium and a liquid: the Stoneley wave, the pseudo-Stoneley wave and the pseudo-Rayleigh wave. The existence of these waves depends on the mechanical properties of the porous material and the liquid and the perme-ability of the interface. Feng and Johnson (1983a) have demonstrated that for the completely open interface only the pseudo modes exist for hard rocks. The Stoneley wave leaks energy into the slow compressional wave while the pseudo-Rayleigh wave radiates into both the slow compressional wave and the acoustic wave in the fluid (z < 0). Due to radiation and viscous mechanisms, these waves are sig-nificantly damped along the direction of propagation. Adler and Nagy (1994) mea-sured the velocities and attenuation coefficients of the pseudo modes in diverse fluid-saturated rocks at ultrasonic frequencies. Their results are consistent with the theo-retical predictions of Feng and Johnson. In this subsection the frequency-dependent phase velocities and specific attenuations of the pseudo modes are calculated for a water/Berea sandstone plane interface. The open-pore boundary conditions are con-sidered. The properties of the rock are listed in Table 2.1. The viscous effects are considered in the complete range of frequencies studied (400 Hz-200 kHz).
Figure 2.3 shows the dispersion relation corresponding to the surface waves. The two pseudo surface modes are found. The pseudo-Stoneley wave propagates slower than the acoustic wave in the fluid at every frequency. The attenuation is larger than that of the shear wave and has a maximum around 25 kHz. The velocity of the pseudo-Rayleigh wave is very similar to the speed of propagation of the shear wave. This pseudo wave has an attenuation which is almost frequency-independent. These results are consistent with previous calculations carried out by Gubaidullin et al. (2004). The results presented here correspond to a more realistic material in geo-physical applications.
2.2.2 Synthetic Seismograms: The Influence of the Leaky Modes
The purpose of this subsection is to study the reflected pressure field in the liquid half-space due to the incidence of a spherical waveform, in order to investigate the influence of the surface waves in the time domain. An acoustic monopole point source M is located at a height z0in the fluid as depicted in Figure 2.4. The spherical
wavefront is decomposed in cylindrical waves by means of the Sommerfeld integral. For further details about this decomposition the reader is referred to Aki and Richards
M
L i q u i d - s a t u r a t e d p o r o e l a s t i c m e d i u m
O
r
z
Figure 2.4: A spherical wave front impinges upon a liquid/poroelastic plane interface. The
reflected pressure response is calculated at the observation point O using a Som-merfeld formulation for the reflected field.
(1980). The incident pressure wave field Pin,
Pin =
A(ω)
R e
i(k0R−ωt), (2.57)
can then be expressed as (see e.g. Brekhovskikh (1980), Chapter 4): Pin = A(ω)e−iωtik0
Z π 2−i∞
0
J0(k0r sin θ)eik0cos θ|z−z0|sin θ dθ, (2.58)
where the integration is performed over the complex angle θ. The path of integration is chosen in the form of the contour Γ0, shown in Figure 2.5. In equation 2.58, A(ω)
denotes the amplitude of the incident wave, R is the distance between the source and the observation point O, k0 is the wavenumber of the acoustic waves in the liquid,
z0is the vertical position of the source with respect to the interface, r and z are the
horizontal and vertical coordinates of the observation point respectively, and J0is the
Bessel function of zero order and first kind.
Analogously, the reflected pressure wave field Pr can be expressed in terms of
cylindrical wave fronts as follows: Pr = A(ω)e−iωtik0
Z π 2−i∞
0
R(θ)J0(k0r sin θ)eik0cos θ|z+z0|sin θ dθ. (2.59)
In the above equation R(θ) is the reflection coefficient for a plane wave which im-pinges upon a plane interface between a liquid (in this case water) and a fully satu-rated porous material. Reflection and transmission coefficients for this configuration
2.2. Poroelastic Surface Waves along Plane Interfaces 23
p / 2 R e ( q )
I m ( q )
G 0
Figure 2.5: The path of integration Γ0in the complex plane used to evaluate the Sommerfeld
integral.
were studied by Wu et al. (1990), Santos et al. (1992) and Denneman et al. (2002). The reflection coefficient follows from the boundary conditions at the water/porous solid interface for oblique incidence and is obtained numerically at each frequency of interest. Further algebraic manipulation of the integral in equation 2.59 enables us to write the reflected pressure field as:
Pr = A(ω)e−iωt[ik0
Z π 2
0
R(θ)J0(k0r sin θ)eik0cos θ|z+z0|sin θ dθ +
+k0 Z ∞ 0 R(ζ)J0(k0r p ζ2+ 1)ek0|z+z0|ζ dζ], (2.60) where ζ = −i cos(θ).
The point source located at z0 is assumed to emit a Blackman-Harris acoustic
pulse of the form: A(t) = 3 X j=1 Bj 2πj T0 2 cos 2πjt T0 , (2.61)
for t [0, T0]and 0 elsewhere. T0 is the duration of the pulse and B1 = −0.48829,
B2 = 0.14128, and B3= −0.01168. We chose these coefficients in agreement with
the source used in the high-frequency investigations of Feng and Johnson (1983b). The incident pulse is shown in Figure 2.6.
The complex amplitudes A(ω) in equation 2.60 are obtained through the Fast Fourier Transform of the incident wave Pi. The modified Sommerfeld integrals of
0 5 10 15 20 −5 −4 −3 −2 −1 0 1 2 3 PSfrag replacements Time (µs) Pressure (a.u.)
Figure 2.6: Incident Blackman-Harris pressure pulse.
equation 2.60 are then solved numerically for each frequency using a recursive adap-tive Simpson quadrature algorithm. Finally the pressure traces are obtained via the Inverse Fast Fourier Transform. This routine is implemented in a Matlab algorithm.
First the features of the reflected wave for the water-Berea sandstone interface are investigated. The acoustic source is placed at z0= -0.3 cm and the incident pulse has
a duration of 10 µs. The pressure response at z = -0.1 cm and r = 40 cm is depicted in Figure 2.7. Both the source and the observation point are situated close to the interface in order to improve the observation of the surface waves. The calculations in the frequency domain discussed in the previous subsection showed that there are two surface waves that can propagate along the interface in this case: the pseudo-Stoneley wave and the pseudo-Rayleigh wave. In our computations, it is possible to clearly recognize the pseudo-Stoneley wave. Also the head wave associated with the fast compressional wave and the reflected fluid wave are observed. The arrows indicating the arrivals in Figure 2.7 are calculated using the velocities values of the different waves at the typical frequency of 150 kHz, which corresponds to the main frequency of the incident wave. The pseudo-Rayleigh wave is not observed due to its high attenuation. From the dispersion relation, the calculated value for the imaginary part of the wavenumber predicts that the amplitude of the wave has decreased at least to a factor of e−1 of its original amplitude for distances larger than 3 cm. The same
2.3. Surface Modes in Borehole Configurations: Approximate Models 25 0.1 0.2 0.3 0.4 0.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 PSfrag replacements Time (ms) Pressure (a.u.) Fast P FluidP-Stoneley
Figure 2.7: Reflected pressure from a water/Berea sandstone plane boundary. The incident
pressure wave from the liquid is a Blackman-Harris spherical wavefront (Figure 2.6). The coordinates of the source are r0 = 0 and z0 = -0.3 cm. The
observa-tion point is placed at r = 40 cm and z = -0.1 cm. The properties of the Berea sandstone are given in Table 2.1. Shear modulus N = 4.02 GPa.
which practically disappears when the travelling distances are beyond 6 cm.
We also investigated the behavior of the amplitude of the pseudo-Stoneley wave as a function of the vertical position of the observation point z. The results for three different values of z are shown in Figure 2.8. From the computed pressure signals it can be concluded that the amplitude of the pseudo-Stoneley wave decreases when the distance with respect to the interface is increased. We also remark that the fast wave remains practically unaffected over the distances considered.
2.3 Surface Modes in Borehole Configurations: Approximate Models
So far in this chapter only plane interfaces have been considered. In this section the attention is focused on the study of the propagation of head and interface waves along cylindrical interfaces. Practical applications are obviously related to hydro-carbon prospecting. Acoustic borehole logging is a common technique where the information carried by the guided modes, head and surface waves travelling along the borehole wall is used to infer the properties of the porous formation and the satu-rating fluid.
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 −3 −2 −1 0 1 2 3 4 5 PSfrag replacements Time (ms) Pressure (a.u.) z = -0.1 cm z = -0.5 cm z = -1 cm P-Stoneley P-Stoneley P-Stoneley Fast P Fast P Fast P
Figure 2.8: Pressure response at different values of z. The position of the source and the
properties of the porous medium are the same as in Figure 2.7, r = 40 cm.
2.3.1 Rigid Borehole Wall
In this subsection a review is presented for the acoustic guided waves that can prop-agate along a fluid-filled cylindrical cavity in a rigid solid. This configuration has been studied in the past and has an analytical solution for these modes White (1983). Therefore, it constitutes a test for the accuracy of our numerical search routine.
We consider a water-filled hole with a radius R of 38.5 mm. The wave equation in cylindrical coordinates for guided acoustic waves travelling in the fluid can be solved. It follows that the potential function is given by:
ϕ = AcwJ0(krr)ei(ωt−kz) (2.62)
where Acw is the amplitude of the wave, J0 the zero-order Bessel function, kr the
radial wavenumber, ω the angular frequency and k the wavenumber of the guided waves in the z− direction. The term corresponding to the Hankel function is excluded because it diverges at r = 0.
The relation between krand k can be written as:
kr= s ω2 c2 w − k 2, (2.63)
2.3. Surface Modes in Borehole Configurations: Approximate Models 27 0 10 20 30 40 50 60 70 80 90 100 0 1000 2000 3000 4000 5000 6000 7000 8000 PSfrag replacements Frequency (kHz) Phase V elocity (m/s) 0 n=1 n=2 n=3 n=4
Figure 2.9: Analytical (solid lines) and numerical results (squares) for the phase velocities of
undamped waves in a fluid filled cylindrical cavity in a rigid solid. The fluid is water, cw=1500 m/s. The radius of the cavity is 38.5 mm. Numerically only the
non-trivial modes are computed.
where cw is the speed of sound in water, 1500 m/s. The boundary condition at the
wall implies the radial displacement urto vanish, which leads to:
ur(R) = −Acw s ω2 c2 w − k 2 J1(krr)ei(ωt−kz) = 0, (2.64)
The solution k(ω) of equation 2.64 gives the dispersion relation. Phase velocities cn(ω)can be obtained as the ratio between ω and k. Analytical solutions can easily
be found: cn(ω) = cw 1 −xRωncw2 −1/2 (2.65) where xnis the nth zero of the J1(x)Bessel function.
A graphical representation of these modes is given in Figure 2.9, where both the analytical (solid lines) and numerical results (square dots) are given. The numerical calculations are performed with the zero-search Newton-Raphson algorithm in the complex plane that was previously used to study the surface modes in the flat in-terface. There is one wave which is non-dispersive and propagates at the speed of
sound in water. The remaining waves are characterized by cut-off frequencies. The cut-off frequency increases with the order of the mode. Figure 2.9 shows an excellent agreement between numerical and analytical results.
2.3.2 Quasi-1D Model for a Permeable Borehole Wall
In this subsection we illustrate the effects of the permeability on the borehole waves using an approximate quasi-1D model. The cavity is now within a rigid perme-able solid. Different authors have addressed this problem using slightly different approaches. The model presented here resembles in some aspects the work done by White (1983), Norris (1989) and Tang et al. (1991). Conservation laws are applied to the bore-fluid and it is assumed that the exchange of volume fluid with the sur-rounding formation occurs only in the radial direction and therefore is governed by the radial component of the velocity. Figure 2.10 depicts the control volume CV for a cylindrical fluid cavity. We assume that the borehole wall is permeable and the solid
d S d S l i q u i d P o r o u s r e s e r v o i r C V z r
Figure 2.10: Borehole configuration: a liquid cylindrical cavity surrounded by a porous
reser-voir. The mass and momentum balance equation are applied in the control vol-ume CV in order to obtain the dispersion relation of the surface wave. Radial fluid volume exchange with the porous reservoir is considered.
matrix is rigid. The linearized mass balance equation for the fluid ∂ρ
2.3. Surface Modes in Borehole Configurations: Approximate Models 29
applied to the control volume reads: Z CV ∂ρ ∂t + ρ0 ∂vz ∂z + 1 r ∂vr ∂r dV = 0. (2.67)
Applying Gauss theorem in the second term and performing an integration over the cross section yields:
∂ρ ∂t + ρ0 ∂vz ∂z + 2 Rρ0vr(R) = 0. (2.68)
The linearized momentum balance equation in the z−direction, after neglecting vis-cous terms, is:
ρ0
∂vz
∂t + ρ0g + ∂p
∂z = 0. (2.69)
Equations 2.68 and 2.69 govern the dynamics of the surface wave. In order to find the dispersion relation a travelling harmonic dependence is assumed for the velocity
vz = v0ei(ωt−kzz), and the pressure p = p0ei(ωt−kzz). The coupling to the properties
of the porous formation is expressed in this model by the radial component of the velocity at the borehole wall vr(R)which satisfies the following boundary condition:
vr(R) = φvr,I(R). (2.70)
In the above equation vr,I is the radial velocity of the pore fluid in the rigid porous
formation. The subscript I is used to distinguish the dynamic quantities in the porous medium from the ones in the borehole liquid.
In order to find an analytical expression for vr,I, we combine the mass balance
equation for the pore fluid 1
ρI
∂ρI
∂t + ∇ · vI = 0, (2.71)
and the constitutive equation: 1 ρI ∂ρI ∂t = 1 Kf ∂pI ∂t , (2.72)
where Kf is the bulk modulus of the fluid, to obtain:
1 Kf
∂pI
The relation between the pressure and the velocity of the fluid is given by the Darcy law. A dynamic permeability is considered here in the way stated by Johnson et al. (1987). The Darcy law then can be written:
vI = −κ(ω)
ηφ ∇pI. (2.74)
Combining Darcy’s law and equation 2.73 we can write in cylindrical coordi-nates: ∂pI ∂t = D ∂2p I ∂r2 + 1 r ∂pI ∂r , (2.75) where D = κ(ω)Kf ηφ . (2.76)
In order to solve equation 2.75 a solution of the type pI(r, z, t) = A(r)ei(ωt−kzz) is
assumed. Substitution of the solution in equation 2.75 leads to: iωA(r) = D d2A(r) dr2 + 1 r dA(r) dr , (2.77)
which can be written as: d2A(r) dr2 + 1 r dA(r) dr − λ 2A(r) = 0, (2.78)
where λ2 = iω/D. This is the modified Bessel equation of zeroth order. Assuming
that the pressure is bounded when r increases the general solution is:
A(r) = bK0(λr), (2.79)
where K0 is the zeroth-order Kelvin function. It is assumed that there is no flow
restriction at the boundary between the pore fluid and the bore fluid which implies the continuity of pressure. Therefore the solution at the borehole wall, r = R, should match the solution for the pressure induced by the surface mode inside the borehole p = p0ei(ωt−kzz). With these considerations it follows that the pore pressure in the
reservoir can be written as: pI(r, z, t) = p0
K0(λr)
K0(λR)
2.3. Surface Modes in Borehole Configurations: Approximate Models 31
From Darcy’s law (equation 2.74) we finally obtain for the radial velocity in the porous medium: vr,I = − κ(ω) ηφ ∂pI(r, z, t) ∂r . (2.81)
Substitution of the harmonic solutions for vzand pztogether with vr,I into
equa-tions 2.68 and 2.69 yields the following system of equaequa-tions: v0 iω c2 0 + 2ρ0κ(ω)λ Rη K1(λR) K0(λR) + p0(−iρ0kz) = 0 (2.82) v0 g c2 0 − ik z + p0(iρ0ω) = 0, (2.83)
where c0 = (p0/ρ0)1/2is the acoustic velocity in the liquid.
The surface mode is the nontrivial solution and its dispersion relation ω(kz) is
found by imposing that the determinant of the associated matrix equals zero. The following characteristic equation is obtained:
ρ0kz2+
iρ0g
c2 0
kz+ C(ω) = 0, (2.84)
where C(ω) is given by: C(ω) = iρ0ω iω c2 0 +2ρ0κ(ω) Rη λ K1(λR) K0(λR) . (2.85)
The quadratic equation for kzcan be solved directly:
kz= − 1 2 ig c2 0 − s − g c2 0 2 −4C(ω)ρ 0 . (2.86)
The frequency-dependent results for the phase velocity and the damping coeffi-cient are shown in Figure 2.11. The material properties correspond to the synthetic (glass) sample in Table 2.1. Two different values of the steady-state permeabil-ity are compared and the effects of the frequency-dependent dynamic permeabilpermeabil-ity are analyzed with respect to the quasi-static model, which considers a frequency-independent steady-state value for the permeability. This steady-state permeability is the low-frequency limit of the dynamic permeability. Obviously at low frequencies both models agree. Deviations are observed in the phase velocities for frequencies higher than the critical frequency, 38.3 kHz for the 1 Darcy formation and 3.83 kHz
102 103 104 105 106 0 250 500 750 1000 1250 1500 PSfrag replacements Frequency (Hz) Phase V elocity (m/s) κ0=10D κ0=1D (a) 102 103 104 105 106 10−1 100 101 102 103 PSfrag replacements Frequency (Hz) Damping coef ficient (1/m) κ0=10D κ0=1D (b)
Figure 2.11: Phase velocity (a) and damping coefficient (b) for the surface wave predicted by
the quasi 1D model (solid lines). The results are compared with the output of a steady state model where the permeability is taken equal to its low-frequency value (dotted lines).
2.3. Surface Modes in Borehole Configurations: Approximate Models 33
for the 10 Darcy formation. The discrepancy is considerably larger when the per-meability is increased. It is also interesting to note that while the quasi-static model predicts significant differences in the entire range of frequencies (100 Hz-1 MHz), the dependence on permeability in the full dynamic model is limited to relative low fre-quencies and both curves converge for frefre-quencies higher than 100 kHz. Interesting features are observed in the damping coefficient. The quasi static model predicts a monotonous increase of the damping coefficient with frequency. The dynamic model predicts the existence of a maximum for the damping coefficient. The position of the maximum in the frequency axis depends on the steady state permeability and it decreases with increasing values of steady state permeability. The existence of a maximum in the attenuation coefficient Q−1has also been noticed by Norris (1989)
and Schmitt et al. (1988). It is generally accepted that the cause of this maximum is a transition from a bore-dominated dispersion to a high frequency regime where the surface wave behaves as in a plane interface since its wavelength is much smaller than the bore radius. Note that the quasi-static model does not take into account the inertial effects that exist at high frequencies. These effects are incorporated in the dynamic formulation. The quasi static model predicts an increase on the damping coefficient with increasing permeability due to pore fluid diffusion to the surrounding medium, this diffusion eases at higher values of permeability.
Appendixes to Chapter 2
2.4 Appendix 2A: Matrix Coefficients for the Fluid/Porous Medium Plane
Interface
In this appendix the elements nij of the matrix N of eq. 2.55 are explicitly given:
n11= γf, n12= γc1(1 − φ + φGc1), n13= γc2(1 − φ + φGc2), n14= −ikx(1 − φ + φGsh), n21= 0, n22= 2N γc1ikx, n23= 2N γc2ikx, n24= (γsh2 + kx2)N, n31= ω2ρf, n32= −[(P − 2N) + Q + Gc1(Q + R)]( ωc1)2+ 2N γc12 , n33= −[(P − 2N) + Q + Gc2(Q + R)]( ωc2)2+ 2N γc22 , n34= −2Nikxγsh, n41= ω2ρf, n42= −( ωc1)2φ1(Q + RGc1), n43= −( ωc2)2φ1(Q + RGc2), n44= 0.
