formations
PROEFSCHRIFT
ter verkrijging van de graad van doctor aan de Technische Universiteit Delft,
op gezag van de Rector Magnificus prof.ir. K.C.A.M. Luyben, voorzitter van het College voor Promoties,
in het openbaar te verdedigen op woensdag 11 juni 2014 om 12.30 uur door Huajun FAN
Master of Geological Engineering
China University of Geosciences (Beijing), China geboren te Suichang, Zhejiang, China
Prof.dr.ir. D.M.J. Smeulders
Samenstelling promotiecommissie:
Rector Magnificus, Technische Universiteit Delft, voorzitter Prof.dr.ir. D.M.J. Smeulders, Technische Universiteit Eindhoven, promotor Prof.dr. N. Li, PetroChina Research Institute, Beijing Prof.dr.-ing. H. Steeb, Ruhr-Universit¨at Bochum
Prof.dr. G. Bertotti, Technische Universiteit Delft Prof.dr. A.V. Metrikine, Technische Universiteit Delft Prof.dr. R.J. Schotting, Universiteit Utrecht
Prof.dr.ir. C.P.A. Wapenaar, Technische Universiteit Delft
This work is financially supported by China Scholarship Council (CSC) and Delft University of Technology.
ISBN 978-90-8891-910-7
c
2014 by H. FAN.
All rights reserved. No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without permission from the author.
Published by: Uitgeverij BOX Press, Oisterwijk, The Netherlands Printed by: Proefschriftmaken.nl
Summary xi Samenvatting xiii 1 Introduction 1 1.1 Ray tracing . . . 2 1.2 Stoneley wave . . . 4 1.3 Literature survey . . . 5
1.3.1 Bulk waves in porous media . . . 5
1.3.2 Waves along flat interfaces . . . 5
1.3.3 Stoneley waves in a borehole . . . 6
1.3.4 Stoneley waves in a borehole intersected by fractures . 7 1.4 Thesis outline . . . 8
2 Constitutive relations and momentum equations 9 2.1 Constitutive equation . . . 9
2.2 Momentum equations . . . 11
2.3 One-dimensional field equations . . . 12
3 Tube wave propagation in a fluid-filled borehole intersected by a single horizontal layer 17 3.1 Introduction . . . 17
3.2 Borehole waves in flexible and fractured formations . . . 19
3.2.1 Low-frequency approximation . . . 19
3.2.2 Dynamic parameters . . . 25
3.3 Exact solutions for fracture and borehole waves . . . 29
3.3.1 Rigid fracture surfaces . . . 29
3.3.2 Fracture elasticity effects . . . 32
3.3.3 Tube wave propagation in the borehole adjacent to the fracture . . . 34
3.4 Reflection and transmission . . . 37
3.4.1 Fracture bounded by rigid solid formations . . . 37
3.4.2 Fracture bounded by rigid permeable formations . . . 43
3.5 Fast Fourier transform . . . 43
4 Experimental setup 47 4.1 Introduction . . . 47
4.2 Vacuum procedure . . . 50
4.3 Sample overview and preparation . . . 52
5 Shock-induced borehole waves and fracture effects 55 5.1 Introduction . . . 55
5.2 Theoretical formulation . . . 57
5.3 Shock-tube experiment . . . 60
5.3.1 Experiment with closed fracture . . . 60
5.3.2 Experiment with open fracture . . . 62
5.4 Conclusion . . . 63
6 Shock-induced wave propagation over porous and fractured borehole zones: theory and experiments 65 6.1 Introduction . . . 65 6.2 Theoretical formulation . . . 66 6.3 Borehole impedance . . . 68 6.3.1 Formation impedance . . . 68 6.3.2 Fracture impedance . . . 71 6.4 Experimental setup . . . 73 6.5 Experiment results . . . 74 6.5.1 Borehole fractures . . . 74 6.5.2 Porous sample . . . 84 6.6 Conclusions . . . 86
6.7 Fractured porous sample . . . 87
7 Fracture effects in mandrel sample 89
8 Conclusions 97
Appendix A Derivation of the plane fracture wave equation 99
Appendix C Porosity and permeability determination 103 C.1 Porosity determination . . . 103 C.2 Permeability determination . . . 103 Bibliography 107 Acknowledgements 117 Curriculum Vitae 119
Natural or hydraulic fractures are of major importance for the productivity of hydrocarbon reservoirs. Besides fracture detection, also the aperture and extension of the fractures are essential for a correct reservoir productivity estimate. There are many ways to detect and measure fractures, such as borehole televiewers and electrical borehole scans. A practical approach to investigate fracture properties is by means of acoustic logging. In this thesis, borehole waves along fractured media are investigated theoretically and ex-perimentally. Theoretically, the effect of a fracture intersecting a vertical borehole can be described by the introduction of a frequency-dependent (dy-namic) borehole fluid compressibility which is measured in the laboratory. The dynamic fluid bulk modulus comprises the intrinsic fluid stiffness, the borehole wall distensibility, and the radial fluid seepage into the adjacent (horizontal) permeable zones. The latter two effects tend to diminish the in-trinsic fluid’s stiffness, giving rise to a lower effective bulk modulus amplitude and thus to a lower wave speed in the borehole. The radial oscillatory fluid seepage causes viscous friction in the adjacent zones and results in a phase lag between the pressure increase and the compression of the borehole fluid, leading to attenuation of the borehole waves. This seepage effect is expressed in terms of a so-called borehole dynamic wall impedance specifying the ra-dial fluid velocity at the borehole wall as a function of the borehole pressure variations. If a borehole wave travels down from an undamaged zone into a fracture zone, it will encounter an impedance contrast causing the wave to partially reflect and partially transmit, thus revealing the presence of per-meable fracture zone adjacent to the borehole.
Stoneley wave propagation in porous and fractured formations is studied experimentally by means of a vertical shock tube facility. In this set-up, shock waves in air are generated that travel downwards into a water-saturated cyl-indrical rock sample that has a borehole drilled along the center axis. In this way, high-energy borehole waves can be generated with excellent repeatab-ility. A logging probe is installed in the borehole to measure the pressure
profiles.
Reflection from the water-sample interface and from the free water inter-face can be recorded by means of a fixed pressure transducer mounted in the wall of the shock tube above the sample in the water layer. The fractures in the formation are represented by small horizontal slits in composite cyl-inders whose upper and lower parts are separated by small spacer poles. In this way, a variable horizontal fracture (slit) aperture can be obtained. Obvi-ously these fractures form an open connection between the borehole fluid and the fluid outside the cylinder. Also mandrel samples are used for horizontal slits that are not open to the fluid outside the cylinder, thus representing fractures with finite radial extension. Wave experiments show that varying fracture widths significantly alter the recorded Stoneley wave pressure signal at fixed depth. The reflection and transmission of borehole tube waves over 1 and 5 mm fractures are correctly predicted by theory. Other wave experi-ments show that attenuation in boreholes adjacent to porous zones without fractures can be predicted by theory. This technique even allows a direct measurement of the permeability, although the acoustically measured per-meability and the perper-meability measured by falling-head technique still show a significant discrepancy. This technique is directly applicable to fractured porous reservoir core samples.
Natuurlijke en hydraulische scheurvorming is van groot belang voor de pro-ductiviteit van olie- en gasreservoirs. Voor een correcte schatting van deze productiviteit zijn, naast de detectie van de scheuren, ook de bepaling van de wijdte en de lengte van de scheuren van belang. Er zijn veel manieren om deze scheuren te detecteren en te meten, zoals met boorgatcamera’s en door middel van elektrische boorgatmetingen. Scheureigenschappen kunnen echter ook op praktische wijze onderzocht worden door middel van akoestische boor-gatmetingen. In dit proefschrift wordt de voortplanting van boorgatgolven theoretisch en experimenteel onderzocht in het geval dat het boorgat een breukzone doorsnijdt.
Theoretisch kan het effect van een breukvlak op golfvoortplanting in een verticaal boorgat dat dat breukvlak doorsnijdt, beschreven worden door de frequentie-afhankelijke (dynamische) samendrukbaarheid (bulkmodulus) van de vloeistof in het boorgat. Dit kan worden gemeten in het laboratorium. De dynamische bulkmodulus van de vloeistof beschrijft de intrinsieke stijfheid van die vloeistof, de flexibiliteit van de wand van het boorgat en de zijdelingse vloeistofpenetratie in de aangrenzende (horizontale) permeabele gesteentela-gen of breukzones. Deze laatste twee verschijnselen verminderen de intrins-ieke vloeistofstijfheid, waardoor de bulkmodulus kleiner wordt. Hierdoor neemt de golfvoortplantingssnelheid in het boorgat af. De zijdelingse osciller-ende vloeistof ondervindt wrijving in de aangrenzosciller-ende poreuze gesteentelagen of breukzones. Dit resulteert in een faseverschil tussen de druktoename in het boorgat en de samendrukking van de vloeistof in het boorgat. Het gevolg is dat de golven in het boorgat demping ondervinden. De zijdelingse vloeis-tofbeweging wordt beschreven met behulp van de zogenaamde dynamische wandweerstand van het boorgat, waarbij de zijdelingse vloeistofsnelheid ter plaatse van de wand wordt geformuleerd als functie van de drukvariaties in het boorgat. Als een drukgolf in een boorgat op zijn weg naar beneden ter plaatse van een breukzone aankomt, zal hij be¨ınvloed worden door de ver-anderende bulkmodulus van de boorgatvloeistof. Dit zorgt ervoor dat de golf
deels zal reflecteren en deels zal verdergaan, waardoor de aanwezigheid van een permeabele breukzone kan worden aangetoond daar waar hij het boorgat doorsnijdt.
De voortplanting van Stoneleygolven in poreuze formaties met breukzones is experimenteel onderzocht met behulp van een verticale schokbuisopstelling. In deze opstelling worden schokgolven opgewekt in lucht. Deze schokgolven reizen omlaag naar een met water verzadigd gesteentemonster met daarin een gecentraliseerd boorgat. Op deze manier worden hoog-energetische boorgat-golven gemaakt met een uitstekende reproduceerbaarheid. In het boorgat wordt een verplaatsbare detector gemonteerd om het drukprofiel te regis-treren.
Reflecties vanaf het grensvlak tussen het water en het gesteentemonster, en van het vrije wateroppervlak kunnen vastgelegd worden door een vaste drukopnemer die in de wand van de schokbuis is gemonteerd, in de waterlaag boven het gesteentemonster. De scheuren in de formatie worden gesimuleerd door nauwe horizontale spleetvlakken tussen kunststof cilinders, die met ver-schillende kleine afstandshoudertjes van elkaar worden gehouden. Op deze manier wordt een variabele horizontale spleetopening verkregen. Deze spleten vormen dus een open verbinding tussen de vloeistof in het boorgat en de vloeistof buiten de cilinder. Ook is gebruik gemaakt van horizontale spleten die niet in contact staan met de vloeistof buiten de cilinder. Deze doornvor-mige monsters simuleren dus breuken met een eindige lengte in de radiale richting. Uit onze golfexperimenten blijkt dat een vari¨erende spleetopening een aanzienlijke verandering teweeg brengt in de gemeten druksignalen van de Stoneleygolf op constante diepte. De reflectie en transmissie van boorgat-golven worden door de theorie correct voorspeld voor spleetopeningen van 1 en 5 mm. Andere golfexperimenten laten zien dat de theorie ook geschikt is om bij afwezigheid van breukzones de verzwakking van golven in boorgaten door poreuze gesteentelagen te kunnen voorspellen. Deze techniek is zelfs geschikt voor een directe metingen van de permeabiliteit, hoewel de akoes-tisch gemeten permeabiliteit en de stationaire permeabiliteitsmetingen nog steeds een aanzienlijk verschil vertonen. De hier besproken methodieken zijn direct toepasbaar op reservoirgesteenten die zowel poreus zijn als breukzones bevatten.
Introduction
Borehole logging is the technique to obtain quantitative information on the formation adjacent to the wellbore by means of measuring tools inside the borehole. These measuring tools are typically combined in a tool string which is lowered into the wellbore by means of a long cable. The cable thus carries
Borehole Formation Head wave Borehole Formation Source Source Wave front Formaon wave front Formaon wave front
Figure 1.1: Sketches of a borehole and the adjacent formation. A source generates a fluid
wave in the borehole. At the interface between the formation and the borehole, the fluid wave partially reflects back into the borehole and partially transmits into the formation to be converted into formation body waves (left panel). A head wave in the borehole is generated (right panel).
the weight of the tool string and also transmits measurement data to the surface. Acoustic logging is one of the borehole logging techniques. In this technique, sound chirps in the kHz frequency range are generated that travel through the formation and thus carry information on its consistency. The sound waves are recorded by the tool and interpreted for reservoir properties. Here we introduce the basic concepts of monopole acoustic logging.
In Fig. 1.1, a borehole is sketched as well as the adjacent formation. In the left panel of Fig. 1.1, a source generates a fluid wave which propagates as a spherical wave. When the fluid wave reaches the interface between the form-ation and the borehole, the wave partially reflects back into the borehole and partially transmits into the formation to be converted into formation body waves. The formation body waves comprise compressional waves (also called P-waves) and shear waves (also called S-waves). The propagation speeds Vp and Vsof the compressional and shear waves depend on the formation prop-erties. In a so-called fast formation, Vs > Vf, in a slow formation, Vs< Vf,
where Vf is the borehole fluid wave speed.
According to Huygens’ principle, each point at the fluid-solid interface becomes a secondary wave source. The secondary source generates two kinds of waves: the borehole head wave (velocity Vp or Vs) and the secondary compressional and shear waves in the formation. In most situations, we ignore the secondary body waves, but focus on the head waves in the borehole. By using this method, one can measure the Vp and Vs of the formation in the
borehole. In a slow formation, there is no shear head wave present.
1.1
Ray tracing
Another method to visualize wave propagation is by ray tracing. A ray indic-ates the direction in which the wave propagindic-ates. It is always perpendicular to the wavefront. A ray path is the fastest path between two points. At the interface between two media which have different acoustic impedances, the ray path will change direction according to Snell’s law. In the left panel of Fig. 1.2, the fluid wave (with the speed Vf) in the borehole propagates to the interface between the formation and the borehole with incident angle θ1. This wave partially reflects back into the borehole with the same angle θ1, and partially transmits into the formation with angle θ2. We distinguish compressional body waves and shear body waves. According to Snell’s law, it holds that: sin θ1 Vf = sin θp Vp = sin θs Vs . (1.1)
Strictly speaking, the ray tracing method is only valid when the fluid wavelen-gth in the borehole is much smaller than the borehole diameter or when the fluid wavefront can be considered plane. Although these conditions are not always fully met, ray tracing is still a very powerful technique for problem visualization. The ray tracing method helps to better understand the travel paths of the waves and provides valuable information for logging tool design.
Transmitter Receiver array Borehole Formation 1 1 2 Incident P-wave Reflected P-wave Transmied P(S)-wave Source 1 2 1 2
Figure 1.2: A fluid wave in the borehole propagates to the interface between the formation
and the borehole with incident angleθ1. This wave partially reflects back into the borehole with the same angle θ1, and partially transmits into the formation with angleθ2. The compressional body waves and shear body waves can be distinguished by using the Snell’s law (left panel of this figure). In the right panel of this figure, different ray paths can be identified. By using a sophisticated acoustic tool with multiple receivers (receiver array), detailed information on formation 1 and 2 can be obtained, as well as the position of the interface between them.
Acoustic logging can also be used for more complicated formations, where, for example, the formation properties are altered with radial penetration depth. In the right panel of Fig. 1.2, different ray paths can be identified. By using a sophisticated acoustic tool with multiple receivers (receiver array), detailed information on formation 1 and 2 can be obtained, as well as the position of the interface between them.
1.2
Stoneley wave
After the arrival of head waves and the fluid wave, there are also arrivals associated with the surface waves propagating along the borehole wall. The most prominent one is the Stoneley wave. The speed of the Stoneley wave VSt is lower than the shear head wave Vs and the borehole fluid wave Vf,
and it is slightly dispersive. The attenuation of the Stoneley wave is also frequency-dependent. The attenuation of the Stoneley wave is caused by viscous effects where borehole fluid interacts with porous formation or frac-tures. This attenuation is explained in more detail in the forthcoming. In the high-frequency range, the amplitude of Stoneley wave decreases very fast with the distance from the borehole wall. In the low-frequency range, the de-crease of the amplitude of Stoneley wave is less. In the limiting case of long wavelengths, the Stoneley wave is called a tube wave with constant amp-litude across the borehole diameter. Stoneley waves are typically carrying information on surface impedance, such as fractures and permeable zones. The Stoneley wave propagation through permeable formations and fractures is illustrated in Fig. 1.3. We identify transmitted Stoneley waves, reflected Stoneley waves, and fracture waves. When a Stoneley wave propagates over
Transmitter Receiver
Permeable Zone Permeable Zone
Fracture Fracture
Figure 1.3: A Stoneley wave propagating along the borehole wall is affected by permeable
permeable zones, the wave partially reflects and partially transmits as in-dicated in Fig. 1.3. The presence of a permeable zone causes the Stoneley wave to become dispersive, so that velocity and attenuation are induced as a function of frequency. When the Stoneley wave crosses a fracture, similar effects occur. One can obtain information on fractures and permeable zones by analyzing the reflected and transmitted Stoneley waves.
1.3
Literature survey
1.3.1 Bulk waves in porous media
A complete theoretical description of wave propagation in fluid-saturated infinite porous media was developed by Biot (1956a,b) and De Josselin de Jong (1956), although earlier results were published by Zwikker and Kosten (1941, 1949) and Frenkel (1944). Two compressional waves and one shear wave were predicted in Biot’s theory. The first compressional wave (also called the fast P-wave) corresponds to an in-phase motion of the solid and the fluid, while the second compressional wave (also called the slow P-wave) corresponds to an out-of-phase motion of the solid and the fluid. The latter results in high attenuation of the slow P-wave. A further contribution to the Biot theory was made by Johnson et al. (1987), who introduced the concept of “dynamic permeability”. The dynamic permeability of porous media was measured by Smeulders et al. (1992c), Johnson et al. (1994), and Kelder (1998). Wave propagation in heterogeneous porous media was investigated by Schoenberg (1984), Schoenberg and Sen (1986), Berryman and Wang (1995, 2000), and Pride and Berryman (2003a,b). In this thesis, heterogeneity is limited to radially and circumferentially isotropic layered media.
An elegant technique to study waves in porous media is by means of a shock tube facility. Van der Grinten et al. (1985, 1987), Sniekers et al. (1989), Smeulders et al. (1992b), Smeulders and van Dongen (1997), and Brown et al. (2000) performed wave experiments on water-saturated, partially saturated, and dry samples using a shock-tube facility.
1.3.2 Waves along flat interfaces
On interfaces between different media, waves may propagate. The wave which propagates at the interface between vacuum and an elastic half space is called a Rayleigh wave (Rayleigh, 1885). Stoneley (1924) investigated wave propagation at the interface between two solids and Scholte (1948) studied wave propagation at the interface between solid and fluid. Apart
from the Stoneley wave, also pseudo-Rayleigh wave exists. Viktorov (1967), ¨
Uberall (1973), and Brekhovskikh (1980) investigated acoustic surface waves in general. The first studies of surface wave propagation at the interface between fluid and fluid-saturated porous media can be attributed to Der-esiewicz (1960, 1961, 1962) and DerDer-esiewicz and Skalak (1963). By invest-igating the high-frequency range of Biot’s theory where all body waves in the porous medium are non-attenuated, Feng and Johnson (1983a,b) found that there are three types of surface waves existing at the interface between fluid and fluid-saturated porous media. These are pseudo-Rayleigh waves, pseudo-Stoneley waves and true Stoneley waves. The pseudo-Stoneley wave is attenuated as it leaks energy into the slow compressional wave. The true Stoneley wave is non-attenuated and it has a velocity lower than the slow compressional wave (Chao, 2005).
Other studies were performed by Edelman and Wilmanski (2002), Albers (2006), Markov (2009) and van Dalen et al. (2010, 2011a). For the experi-mental investigations, we refer to the work of Mayes et al. (1986), Adler and Nagy (1994), Allard et al. (2004) and van Dalen (2011b).
1.3.3 Stoneley waves in a borehole
Rosenbaum (1974) was the first to apply Biot’s theory in fluid-filled boreholes surrounded by poro-elastic formations to investigate the permeability effects on Stoneley wave propagation in the high-frequency range. Paillet and White (1982) used the plane-geometry model to illustrate the effects of the borehole fluid on surface and body waves propagating along the borehole in an elastic solid. White (1983) developed a model for tube wave propagation in rigid permeable formations in the low-frequency range. Kimball and Marzetta (1984) developed a method to detect arrivals of the borehole waves in the time domain by computing the scalar semblance for the arrival times and their associated slownesses of different wave modes. Hsui and Toks¨oz (1986) developed a model for determining formation permeability by using tube wave attenuation. Lang et al. (1987) used Prony method for processing full-waveform logging data and found that the velocity dispersion of the borehole modes have a good agreement with the theoretical computations. Schmitt et al. (1988) studied the wave response generated by a point source in the fluid-filled borehole using Biot’s theory in both time and frequency domains. They presented results for both the Stoneley wave and the first pseudo-Rayleigh wave. Chang et al. (1988) and Norris (1989) used low-frequency asymptotics of the Biot-Rosenbaum theory for the Stoneley wave propagation. The results are compared with White’s model. They confirm that the White model is a
representative model for Stoneley wave propagation in a rigid formation in the low-frequency range. Winkler et al. (1989) performed wave experiments for Stoneley wave propagation and found a good agreement between experiment and theory. Tang et al. (1991a) developed a simplified model for Stoneley wave propagation in a permeable borehole and compared it with the full Biot-Rosenbaum model. He found that the simple model and the Biot-Biot-Rosenbaum model yield practically the same result in the hard-formation case. The simplified Biot-Rosenbaum model was used for the inversion of formation permeability from Stoneley wave logs by Tang and Cheng (1996).
Borehole surface waves are commonly used for fracture evaluation around the borehole. Some field measurements showed the potential for the mode converted sonic waves at a fracture to investigate the fracture extension around the borehole (Gelinski and Cheng, 1998; Yamamoto et al., 1998). Also VSP (Vertical Seismic Profiling) data are widely used for the study of the fractures. Fracture waves induced by incident seismic waves at the fracture surfaces are identified in VSP data for fracture characterization (Beydoun et al., 1985; Cicerone and Toks¨oz, 1990; Toks¨oz et al., 1992). Similarly, the diffraction behavior of elastic waves at the fracture tip and fracture surfaces can be used for this (Van der Hijden and Neerhoff, 1984; Coutant, 1989; Liu et al., 1997). Henry (2005) used the direct scattering problem for investigat-ing the wave response of a sinvestigat-ingle horizontal fracture intersectinvestigat-ing a fluid-filled borehole.
1.3.4 Stoneley waves in a borehole intersected by fractures
Stoneley waves for borehole fracture evaluation were investigated by Paillet and White (1982), Cheng et al. (1987) and Hsu et al. (1987). Hornby et al. (1989) used the reflected Stoneley wave to estimate the effective fracture aper-tures. Tang and Cheng (1993) considered a zone containing large amounts of microfractures to be a highly permeable zone for Stoneley wave propaga-tion. They found that the effective permeability of the fracture zone is more important than the sum of all fracture apertures. Kostek et al. (1998b) gave complete numerical models of Stoneley wave propagation including borehole washout effects and fracture effects. Formation elasticity effects were studied by Tang (1990) and Kostek et al. (1998b). Stoneley wave propagation in a borehole with vertical fractures was investigated by Holzhausen and Gooch (1985), Tang and Cheng (1989), Tang et al. (1991b), Paige et al. (1992, 1995) and Groenenboom (1998). Saito et al. (2004) used frequency-domain Stone-ley wave field data to detect permeable fractures. Field data for fracture detection by means of Stoneley wave propagation was discussed by Cheng
and Toks¨oz (1981), Hardin et al. (1987), Hornby et al. (1987), Hsu et al. (1987), Hsu and Esmersoy (1992), and Brie et al. (2000).
Hardin et al. (1987) reported an approach for borehole Stoneley wave at-tenuation based on the steady fluid flow into the fracture. Tang and Cheng (1989) developed a wave propagation theory in a single fracture bounded by two rigid half spaces based on the Navier-Stokes equations. The viscous ef-fects of the fluid were taken into account. Ferrazzini and Aki (1987) presented a theory of wave propagation in a fracture bounded in two elastic half spaces where viscous effects were neglected. Kostek et al. (1998b) and Korneev (2008, 2010) discussed wave propagation in a fracture which includes both the fluid viscous effects and the fracture surfaces elasticity effects.
1.4
Thesis outline
Natural or hydraulic fractures are of paramount importance for the pro-ductivity of hydrocarbon reservoirs. Besides fracture detection, also the aperture and extension are essential to estimate the reservoir productivity. A practical approach to investigate fracture properties is by means of acoustic monitoring. In this thesis experimental shock-induced acoustic wave data are compared with theoretical predictions. Chapter 2 gives a comprehensive introduction to wave propagation in porous media (Biot theory). In chapter 3, an overview of existing techniques to describe Stoneley wave attenuation along porous and fractured formation is given. Secondly, a full frequency-dependent theory for Stoneley waves along fractured formations is derived, and compared with existing theory. The experimental facility is described in chapter 4, and original measurements in fractured (mandrel) samples are discussed in chapter 5. New types of samples and wave experiments therein are detailed in chapter 6, where also an acoustic funnel is introduced for wave enhancement. Finally, additional mandrel sample experiments using the fun-nel for wave enhancement, are treated in chapter 7. Conclusions are given in chapter 8.
Constitutive relations and
momentum equations
2.1
Constitutive equation
We consider a fluid-filled elastic frame which has a statistical distribution of interconnected pores. We have the following assumptions for this system:
• The wavelength is larger than the pore size;
• The displacements of both the fluid and solid phases are assumed to be sufficiently small, so that the equations can be linearized;
• The fluid does not react to a shear force in the solid. The fluid cannot sustain shear forces;
• The matrix is elastic and isotropic and there is no dissipation effect in the matrix. Only the dissipation due to the interaction between the solid and the fluid is taken into account;
• There are no thermo-elastic or chemical reaction effects in the system. We consider a cube of unit size of bulk material. The total normal tension force per unit bulk area applied to the fluid faces of the unit cube τ can be described by:
τ =−pAf
Ab =−φp, (2.1)
where φ is porosity of the system, p is the pressure of the fluid in the pores, Af and Ab are total surfaces of the interconnected pores and the bulk area
respectively. The total normal tension force per unit bulk area applied to that portion of the cube faces occupied by the solid τij can be described by:
τij =−σij − (1 − φ)pδij, (2.2)
where the intergranular stress σij refers to the intergranular forces percent
bulk area. The Kronecker symbol δij is introduced because the pore fluid cannot exert nor sustain any shear forces on the macroscopic scale.
The stress-strain relations can be written as follows (Biot, 1955):
τij = 2μeij+ Aekkδij+ Qkkδij, (2.3) τ = Qekk+ Rkk, (2.4) σij =−2μeij− A−1− φ φ Q ekkδij− Q−1− φ φ R kkδij, (2.5)
where eij = 1/2(∂usi/∂xj+ ∂usj/∂xi), ij = 1/2(∂uf i/∂xj+ ∂uf j/∂xi), and summation over repeated indices is assumed. A, Q and R are generalized elastic parameters which can be related via Gedanken experiments to poros-ity, bulk modulus of the solid Ks, bulk modulus of the fluid Kf, bulk modulus of the porous drained matrix Kb, and shear modulus μ of both the drained matrix and of the composite (Biot and Willis, 1957):
A = Kb− 2μ/3 + Kf(1− φ − Kb/Ks)2/φeff, (2.6)
Q = φKf(1− φ − Kb/Ks)/φeff, (2.7)
R = φ2Kf/φeff, (2.8)
where the effective porosity φeff = φ + Kf/Ks(1− φ − Kb/Ks). If we assume
that the frame and pore fluid are much more compressible than the solids themselves (Kb/Ks 1 and Kf/Ks 1), we can write
A = (1− φ) 2 φ Kf+ Kb− 2 3μ, (2.9) Q = (1− φ)Kf, (2.10) R = φKf. (2.11)
For this case, the stress-strain relations are simplified. Substituting (2.9), (2.10) and (2.11) into (2.4) and (2.5), we obtain:
σij =−(Kb− 2 3μ)ekkδij − 2μeij, (2.12) and p =−1− φ φ Kfekk− Kfkk. (2.13)
2.2
Momentum equations
The equations of motion for a fluid-saturated porous medium were formu-lated by Biot (1956a,b). Zwikker and Kosten (1941, 1949) and Frenkel (1944) published earlier results, but some aspects were ignored in their papers. It is assumed that for long-wavelength disturbances, we can define average val-ues of the local displacements us(r, t) in the solid and uf(r, t) in the fluid. Denoting the solid and fluid densities ρs and ρf, respectively, the linearized momentum equations are written as:
(1− φ)ρs ∂2 ∂t2usi =− ∂σji ∂xj − (1 − φ) ∂p ∂xi + fi, (2.14) φρf ∂2 ∂t2uf i=−φ ∂p ∂xi − fi . (2.15)
The interaction term is specified in its linear form: fi = b0∂ ∂t+ φρf(α∞− 1) ∂2 ∂t2 (uf i− usi). (2.16)
The viscous damping factor b0= ηφ2/k0, with η the dynamic fluid viscosity and k0 the permeability. The tortuosity is denoted α∞. The equations of motion resulting from momentum conservation and the stress-strain relations can now be written as:
μ∇2u s+ (A + μ)∇∇ · us+ Q∇∇ · uf = ∂2 ∂t2(ρ11us+ ρ12uf) +b0∂ ∂t(us− uf), (2.17) Q∇∇ · us+ R∇∇ · uf = ∂2 ∂t2(ρ12us+ ρ22uf)− b0 ∂ ∂t(us− uf), (2.18) where the density terms are given by
ρ12=−(α∞− 1)φρf, (2.19)
ρ11= (1− φ)ρs− ρ12, (2.20)
2.3
One-dimensional field equations
Essential features of wave propagation in porous media are brought out by considering the one-dimensional situation. Introducing P = A + 2μ, (2.17) and (2.18) can be rewritten as:
ρ11 ∂ 2 ∂t2 + b0 ∂ ∂t − P ∂2 ∂x2 usx = Q ∂ 2 ∂x2 + b0 ∂ ∂t − ρ12 ∂2 ∂t2 uf x, (2.22) ρ22 ∂2 ∂t2 + b0 ∂ ∂t − R ∂2 ∂x2 uf x= Q ∂2 ∂x2 + b0 ∂ ∂t− ρ12 ∂2 ∂t2 usx. (2.23)
Combination of both equations yields that ρ22 ∂ 2 ∂t2 + b0 ∂ ∂t − R ∂2 ∂x2 ρ11∂ 2 ∂t2 + b0 ∂ ∂t − P ∂2 ∂x2 usx = Q ∂ 2 ∂x2 + b0 ∂ ∂t − ρ12 ∂2 ∂t2 2 usx. (2.24)
Developing this expression, we find that (ρ11ρ22− ρ212)∂ 4 ∂t4 − Γ ∂4 ∂t2∂x2 + (P R− Q 2) ∂4 ∂x4 +b0ρ∂ ∂t ∂2 ∂t2 − H ρ ∂2 ∂x2 usx= 0, (2.25) where Γ = P ρ22+ Rρ11− 2Qρ12, ρ = ρ11+ ρ22+ 2ρ12, and H = P + R + 2Q. Dividing by ρ11ρ22− ρ212, we find that ∂4 ∂t4 − (V 2 ++ V−2) ∂ 4 ∂t2∂x2 + V 2 +V−2 ∂ 4 ∂x4 + θ −1 ∂ ∂t ∂2 ∂t2 − V 2 0 ∂ 2 ∂x2 usx = 0, (2.26) with V+,−2 = Γ± Γ2− 4(ρ11ρ22− ρ212)(P R− Q2) 2(ρ11ρ22− ρ212) , (2.27) V02 = H ρ. (2.28)
We notice that V+,− are real-valued high-frequency phase velocities, and V0 is a low-frequency phase velocity. Moreover,
θ−1= b0ρ
is the frequency which characterizes the transition from low-frequency vis-cosity dominated flow to high-frequency inertia dominated flow. Equation (2.26) can be rewritten as ∂2 ∂t2 − V 2 + ∂ 2 ∂x2 ∂2 ∂t2 − V 2 − ∂ 2 ∂x2 + θ−1 ∂ ∂t ∂2 ∂t2 − V 2 0 ∂ 2 ∂x2 usx = 0.(2.30) We now consider harmonic waves of the form exp i(ωt− κx), where κ is the wavenumber. Substitution into (2.30) and introduction of the phase speed c = ω/κ, yields that c4 1−iθ−1 ω − c2V2 ++ V−2− iθ−1 ω V 2 0 + V+2V−2 = 0. (2.31)
From (2.31), the squared phase speed is expressed as c2 = V 2 ++ V−2− i(ωθ)−1V02± √ D 2[1− i(ωθ)−1] , (2.32) with D = (V+2− V−2)2+ 2i(ωθ)−1[2V+2V−2− V02(V+2+ V−2)]− (ωθ)−2V04. (2.33) The result of the dispersion relation (2.32) is that there are two distinct longitudinal modes which we call mode 1 (first wave) and mode 2 (second wave). The solution (2.32) is somewhat modified by introducing a frequency-dependent friction factor. It corrects for the fact that above the characteristic frequency θ−1, deviations from low-frequency Stokes’ flow become important owing to inertia effects. In the limit of high frequencies, the viscous skin depth δ = 2ν/ω (ν denotes the kinematic viscosity η/ρf) eventually be-comes much smaller than a characteristic viscous length scale Λ (Smeulders, 1992a). The steady-state friction factor b0 in (2.29) is then replaced by a more realistic frequency-dependent friction factor b(ω) = b0F (ω), where the viscous correction factor F (ω) is given by:
F (ω) = b(ω) b0 = 1 + 1 2iM ω ωc , (2.34) where ωc = ηφ α∞k0ρf (2.35)
10−2 10−1 100 101 102 0 500 1000 1500 2000 2500 3000 3500 Phase Velocity (m/s) ω/ω c Slow Shear Fast 10−2 10−1 100 101 102 10−6 10−4 10−2 100 102 Attenuation (m − 1) ω/ω c Slow Shear Fast
Figure 2.1: Phase velocity and attenuation factors of the fast, slow and shear wave for a
water-saturated porous medium. The parameters of the porous medium are listed in Table 6.1.
is the frequency for which inertia and viscous forces are of equal importance in rigid porous solids, and M is the viscous shape factor defined as
M = 8α∞k0
It was shown that for many porous materials the shape factor M = 1 in good approximation, so that the characteristic viscous length scale is directly related to other material properties by Λ = 8k0α∞/φ = 8ν/ωc. We
remark that there is direct relation between the characteristic frequencies ωc and θ−1: ωc = θ−1(1− φ)ρs+ (1− α −1)φρ f (1− φ)ρs+ φρf . (2.37)
In the limiting case of large tortuosity, ωc = θ−1. From (2.17) and (2.18) we have for the shear wave
μ ∂ 2 ∂x2usy = ∂2 ∂t2(ρ11usy+ ρ12uf y) + b0 ∂ ∂t(usy− uf y), (2.38) 0 = ∂ 2 ∂t2(ρ12usy+ ρ22uf y)− b0 ∂ ∂t(usy− uf y). (2.39)
The above equations can also be written as ρ11∂ 2 ∂t2 + b0 ∂ ∂t − μ ∂2 ∂x2 usy = b0 ∂ ∂t − ρ12 ∂2 ∂t2 uf y, (2.40) b0 ∂ ∂t − ρ12 ∂2 ∂t2 usy = ρ22∂ 2 ∂t2 + b0 ∂ ∂t uf y. (2.41)
Combination of both equations yields that ρ11∂ 2 ∂t2 + b0 ∂ ∂t − μ ∂2 ∂x2 ρ22∂ 2 ∂t2 + b0 ∂ ∂t usy = b0 ∂ ∂t − ρ12 ∂2 ∂t2 2 usy. (2.42)
Developing this expression, we find that (ρ11ρ22− ρ212)∂ 4 ∂t4 − μρ22 ∂4 ∂t2∂x2 + b0ρ ∂ ∂t ∂2 ∂t2 − μ ρ ∂2 ∂x2 usy = 0, (2.43) where ρ = ρ11+ ρ22+ 2ρ12. Dividing by ρ11ρ22− ρ212, we find that
∂4 ∂t4 − V 2 s ∂4 ∂t2∂x2 + θ −1∂ ∂t ∂2 ∂t2 − V 2 0 ∂ 2 ∂x2 usy = 0, (2.44)
with
Vs2 = μρ22
ρ11ρ22− ρ212, (2.45)
V02 = μ
ρ. (2.46)
We note that Vs is real-valued high-frequency phase velocity of the shear wave, and V0 is a low-frequency phase velocity of the shear wave. As already described in (2.29), θ−1 is the transition frequency. Repeating the same procedure as for the compressional waves, we obtain
c2 = V 2
s − i(ωθ)−1V02
1− i(ωθ)−1 . (2.47)
As an example, we plot the phase velocities and the attenuation factors of the fast, slow and shear waves in the water-saturated case (see Fig. 2.1). The parameters of the porous medium are listed in Table 6.1. The value of the tortuosity is estimated from porosity by Berryman (1980) and can be expressed as
α∞= 1 + φ
2φ . (2.48)
It is clear that the phase velocity of the slow wave is frequency-dependent while the velocities of the other two waves are almost frequency-independent. The velocity of the fast wave is around 3000 m/s, the velocity of the shear wave is around 1300 m/s, the velocity of the slow wave is below 1000 m/s for all frequencies. The attenuation factor of the fast wave has the smallest value, the attenuation factor of the slow wave has the largest value because of the out-of-phase character of the slow wave which is described before, the attenuation factor of the shear wave has the value between the fast wave and the slow wave.
Tube wave propagation in a
fluid-filled borehole intersected by
a single horizontal layer
3.1
Introduction
We consider a single horizontal layer, having a height h, intersecting the borehole (see Fig. 3.1). The layer can be a fracture with aperture h, or a permeable zone. The layer is of infinite extent in the radial (r) direction. The borehole has radius b. The plane z = 0 is in the center of the layer. The z-coordinate is pointing downward. We assume that the borehole fluid pressure is uniform across the borehole. The 1-D borehole wave equation is
d2ψ dz2 + κ
2ψ = 0, (3.1)
where ψ is the displacement potential, and κ is the wavenumber. The fluid pressure p and the axial fluid displacement U in the borehole are given by
p = ρfω2ψ, (3.2)
U = dψ
dz, (3.3)
where ρf is the fluid density, and ω is the angular frequency. Borehole wave propagation in the z-direction is described by
ψ = A+e−iκ1z+ A−eiκ1z for z <−h/2, (3.4)
ψ = B+e−iκ2z+ B−eiκ2z for − h/2 < z < h/2, (3.5) 17
Borehole Formation Viscous fluid Formation Formation Formation 2b r z h Δz Δuz ub
Figure 3.1: Borehole intersected by a single horizontal fracture. The distortion of an
elementary volume of borehole fluid by fluid compression and wall expansion is indicated by the dashed box. The axial compression of the volume element is denoted by Δuz. The radial wall displacement is denotedub.
ψ = C+e−iκ1z for z > h/2. (3.6)
In the region z < −h/2, A+e−iκ1z and A−eiκ1z represent the incident wave
propagating in the positive z direction and the reflected wave propagating in the negative z direction, respectively. Note that κ1is the fluid wavenumber in the undisturbed borehole, and κ2is the fluid wavenumber where the borehole is intersected by the layer. A+ is the incident amplitude, and A− is the reflected amplitude (see Fig. 3.2). In the region −h/2 < z < h/2, B+ and B−are the amplitudes of the waves propagating in the positive and negative z directions, respectively. C+ is the amplitude of the transmitted wave in the region z > h/2. The fluid displacement and the pressure should be continuous at z = h/2 and z =−h/2. The coefficients A−, B+, B−and C+ can now be calculated as a function of the incident amplitude coefficient term A+:
A−/A+=−2i(κ22− κ21) sin(κ2h)/D, (3.7)
A- A+ B+ C+ B- r z h 2b
Figure 3.2: Stoneley wave propagation in the borehole intersected by a single horizontal
fracture. A+ and A− represent incident and reflected Stoneley wave amplitude in z <
−h/2 respectively; B+ and B− represent incident and reflected Stoneley wave amplitude
in −h/2 < z < h/2 respectively; C+ represents transmitted Stoneley wave amplitude in
z > h/2. B−/A+ = 2κ1(κ2− κ1)e−iκ2h/D, (3.9) C+/A+= 4κ1κ2eiκ1h/D, (3.10) where D is given by D = (κ1+ κ2)2eiκ2h− (κ 1− κ2)2e−iκ2h. (3.11)
The above equations were also found by Tang and Cheng (1993). We will now discuss tube wave propagation over different types of layers.
3.2
Borehole waves in flexible and fractured
form-ations
3.2.1 Low-frequency approximation
The description of wave propagation in the fluid in a borehole is greatly simplified if the wavelengths involved are long compared with the borehole
diameter (low frequencies). In that case we can assume piston-like motion in the axial (z) direction. Due to a pressure increase in the water, (i) the water is compressed, (ii) the walls of the borehole are displaced outward over a distance ub, and (iii) water is squeezed into the surrounding permeable formation. For the first contribution (i), we have that the volume change of the water ΔV1 = πb2(∂uz/∂z)Δz (see Fig. 3.1), so that the relative volume
change of the water in this case ΔV1/V0= ∂uz/∂z. For the second contribu-tion (ii), the volume change of the water ΔV2 = 2πbΔzub, so that the relative volume change becomes ΔV2/V0= 2ub/b. The third contribution (iii) will be discussed later. We know that the relation between excess pressure change p (p = pe+ p, where peis the equilibrium pressure) and the volume change is given by p Kbf =− ΔV V0 =− ΔV1+ ΔV2 V0 , (3.12)
where Kbf is the incompressibility of the bore fluid. Substitution of both
contributions yields that − p Kbf = ∂uz ∂z + 2ub b . (3.13)
This equation must be further developed into an expression involving only pressure and axial displacement. The relation between radial displacement and pressure change can be expressed in terms of a distensibility D:
D = b 2
p
ub. (3.14)
Substitution of this relations yields that the mass balance equation becomes ∂uz ∂z =−p 1 Kbf + 1 D . (3.15)
The momentum balance in the z direction yields that ρbf
∂2uz
∂t2 =− ∂p
∂z, (3.16)
where ρbf is the density of the borehole fluid. Combination (3.16) with (3.15) yields that ρbf∂ 2u z ∂t2 = ∂2uz ∂z2 1 Kbf + 1 D −1 . (3.17)
This equation has the well-known solution: uz = u+(z− cTt) + u−(z + cTt), where u+ and u− are the downward and upward waves respectively, and cT is the tube wave velocity. It shows that a fluid column in a flexible tube is capable of supporting pulses of any waveform, travelling in either direction, without dispersion, where the speed of the tube waves can be expressed as
1 c2T = ρbf 1 Kbf + 1 D . (3.18)
For the third contribution (iii), we assume an outward fluid velocity wb at the wall. Assuming harmonic variations wb = ˆwbexp(iωt), the outward fluid displacement is simply ˆwb/(iω). The third contribution thus becomes ΔV3/V0 = 2 ˆwb/(iωb), and the wave velocity is subsequently modified to be-come 1 c2T = ρbf 1 Kbf + 1 D + 2 iωbZb , (3.19)
where the relation between radial velocity and pressure change is expressed in terms of a wall impedance Zb:
Zb = p
wb. (3.20)
For further evaluation, we need to specify D and Zb. For a thick-walled tube of inner radius b, and outer radius a, Lamb (1928) derived the classical relation
D = 1 2
E(a2− b2)
(1 + ν)(a2+ b2)− 2νb2, (3.21)
where E is Young’s modulus and ν is Poisson’s ratio. For a borehole in an infinite solid, a is very large compared with b, and D approaches E/2(1 + ν), which equals the shear rigidity G of the solid.
The wall impedance can be evaluated from reservoir considerations. We simply apply Darcy’s law as
φwr=−
k0 η
∂p
∂r, (3.22)
where η is the viscosity of the fluid inside the pore, and wris the fluid velocity in the pore. Mass balance for a cylindrical coordinate system is
∂p ∂t + ρf ∂wr ∂r + wr r = 0, (3.23)
where ρf is the density of the fluid inside the pore. Combining (3.22) and (3.23), we find that ∂p ∂t = Dh ∂2p ∂r2 + 1 r ∂p ∂r , (3.24)
where Dh= k0Kf/(ηφ) is the hydraulic diffusivity, with Kf the bulk modulus of the pore fluid. For harmonic variations p = ˆp exp(iωt), (3.24) becomes
∂2 ∂r2 + 1 r ∂ ∂r − κ 2p = 0,ˆ (3.25) with κ2= ω 2 c2f =− iω Dh, (3.26)
where cf is the speed of sound in the fluid. The solution to (3.25), which is finite at large distance is
ˆ
p = AK0(κr), (3.27)
where K0(κr) is a modified Bessel function of zeroth order and complex argument, and A is an arbitrary constant. Substitution of the boundary condition ˆp(b) = ˆp0 into (3.27) yields that
ˆ
p = ˆp0K0(κr)
K0(κb). (3.28)
Following Darcy’s law (3.22) , we find that the fluid velocity in the pores can be expressed as φ ˆwr= k0 η pˆ0κ K1(κr) K0(κb), (3.29)
with K1(κr) a modified Bessel function of first order and complex argument. At the borehole wall, we have that
φ( ˆwr)b = k0 ηbpˆ0κb
K1(κb)
K0(κb), (3.30)
so that the wall impedance can be expressed as 1 Zb = ˆ wb ˆ p0 = φ( ˆwr)b ˆ p0 = k0 ηbκb K1(κb) K0(κb). (3.31)
10−2 10−1 100 101 102 10−2 10−1 100 101 102 η b/(k 0 Z b ) b(ω/D h) 1/2 Re Im
Figure 3.3: Real and imaginary parts of the reduced wall impedance.
A plot of the reduced wall impedance is given in Fig. 3.3. For high frequencies, both the real and imaginary parts tend to infinity as √ω. From (3.19) and (3.31), the tube wave number can be obtained
κT = ω cT = ω cf 1 +2k0Kbfκ iωηb K1(κb) K0(κb) + Kbf G . (3.32)
It is now possible to compute the tube wave velocity as a function of per-meability and porosity. For a borehole radius of 10 cm, a fluid bulk modulus Kbf = 2.0 GPa, and a fluid viscosity η = 1.0 mPa·s, we first compute some values for the hydraulic diffusivity (see Table 3.1). Assuming that the
bore-Table 3.1: Parameters values.
φ (%) k0 (mD) Dh (m2/s) ωc (MHz)
16 10.0 0.125 0.107
20 100.0 1 0.667
26 1000.0 7.692 4.41
hole is filled with the same fluid as the reservoir and that ρf = ρbf = 1000 kgm−3, we now compute the wave speed and attenuation of the tube wave as a function of frequency using (3.32). For the shear modulus we take G = 3.0
100−2 100 102 104 106 200 400 600 800 1000 1200 Phase Velocity (m/s) f (Hz) a b c 10−2 100 102 104 106 10−4 10−3 10−2 10−1 100 101 Attenuation (m −1 ) f (Hz) a b c
Figure 3.4: Tube wave phase velocity and attenuation along a pore formation from (3.32).
a: k0=10 mD,φ=16%, b: k0=100 mD,φ=20%, c: k0=1000 mD,φ=26%.
GPa. Results are plotted in Fig. 3.4. The dependence on porosity and per-meability is clearly visible. The higher the perper-meability, the more attenuation is observed due to the increased infiltration into the formation.
When the permeable layer is replaced by a horizontal fracture (see Fig. 3.1), the wavenumber κ2 in the fracture zone can also be obtained from (3.19), but
with different Zb. The difference with respect to the fluid flow in the per-meable formation is that the relation between fracture velocity and fracture pressure is now given by Poiseuille’s law:
ˆ wr=− h 2 12η ∂ ˆp ∂r. (3.33)
The wall impedance Zb can then be written as 1 Zb = h 2 12ηbκb K1(κb) K0(κb), (3.34)
and the effective fluid bulk modulus is given by 1 Keff = 1 Kbf + 1 D + h2κ 6iωηb K1(κb) K0(κb). (3.35)
We thus easily obtain the wavenumber κ2 in the borehole adjacent to the fracture: κ2= ω cT = ω cf 1 + h2Kbfκ 6iωηb K1(κb) K0(κb) + Kbf D . (3.36)
Obviously it is assumed here that 1-D approach is valid which is realistic for λ/2b 1, for b the borehole radius. The vertical motion of the layer’s wall is discussed in section 3.3.2.
3.2.2 Dynamic parameters
For higher frequencies, Stokes’ flow can no longer describe the flow behavior as inertia effects become important. This can be described by a modified per-meability and wave number in the porous media to be substituted in (3.32). A model for dynamic permeability k(ω) of porous media was developed by Johnson et al. (1987):
k0 k(ω) = i
ω
ωc + F (ω), (3.37)
where the viscous correction factor F (ω) is defined in (2.34), and the rollover frequency ωc is defined in (2.35).
If the porous medium is replaced by a fracture, we can also calculate the dynamic permeability of the fracture. For a fracture with aperture h, we know that k0/φ = h2/12, so the viscous shape factor M from (2.36) has a value of 2/3 (Λ = h in this case). The critical frequency ωc is given by
ωc = 12η
100−2 100 102 104 106 200 400 600 800 1000 1200 Phase Velocity (m/s) f (Hz) a b c 10−2 100 102 104 106 10−4 10−3 10−2 10−1 100 101 Attenuation (m −1 ) f (Hz) c b a
Figure 3.5: Tube wave phase velocity and attenuation along a porous formation from
(3.32). The solid lines are for constant permeability and the dashed lines are for dy-namic permeability (3.37) and wavenumber (3.40). a: k0=10 mD,φ=16%, b: k0=100 mD,
φ=20%, c: k0=1000 mD,φ=26%.
The dynamic permeability of fracture can now be obtained from (3.37): k0 k(ω) = iωh2ρf 12η + 1 +iωh2ρf 36η . (3.39)
100−2 100 102 104 106 200 400 600 800 1000 1200 Phase Velocity (m/s) f (Hz) a b c 10−2 100 102 104 106 10−4 10−2 100 102 104 Attenuation (m −1 ) f (Hz) a b c
Figure 3.6: Tube wave phase velocity and attenuation in the borehole adjacent to the
fracture from (3.36) (solid lines) compared with where the dynamic parameters are included (dashed lines). Fracture apertures are a: 1μm, b: 10 μm, c: 100 μm.
The introduction of inertia effects will also affect the wave number κ, which was so far diffusive in nature (3.26). The modified wave number in the porous medium can be written as (Johnson et al., 1987)
˜ κ = ω
cf
10−2 100 102 104 10−2 100 102 104 Phase Velocity (m/s) f (Hz) 10−2 100 102 104 100 101 102 103 104 Attenuation (m −1 ) f (Hz)
Figure 3.7: Tube wave phase velocity and attenuation in the borehole adjacent to the
fracture from (3.36) (solid lines) compared with where the dynamic parameters are included (dashed lines). Fracture aperture is 1 cm.
We noted that the dynamic interaction between solid and fluid in porous media can be described by a dynamic permeability which extends the low-frequency viscous effects toward high-low-frequency inertia effects. Likewise, all these effects are also comprised in a dynamic tortuosity, which extends the
high-frequency inertia effects toward low-frequency viscous effects: α(ω) = α∞−iF (ω)b0
φωρf , (3.41)
where b0 = ηφ2/k0 is the viscous damping factor of the porous media. For a fracture, (3.41) becomes
α(ω) = 1− 12iη h2ωρfF (ω) = 1− 12iη h2ωρf 1 + iωh2ρf 36η . (3.42)
This means that eqs. (3.31), (3.32), (3.34) and (3.36) still hold, but we simply replace k0 by k(ω) and κ by ˜κ.
The phase velocity and attenuation of the tube wave in porous forma-tions from (3.32) are compared with and without dynamic parameters (see Fig. 3.5). For the phase velocity the two situations agree very well in both low- and high- frequency ranges. For the attenuation, they agree very well in low frequencies but have some discrepancy in the high-frequency range. Above the critical frequencies ωc (see Table 3.1), inertia effects become dom-inant and the attenuation departures from the viscosity-dominated region (see Fig. 3.5).
Tube wave phase velocity and attenuation in the borehole adjacent to the fracture from (3.36) are compared with where the dynamic parameters are included (see Fig. 3.6). For 1 μm and 10 μm fractures, the two situations agree well in all frequencies with regard to both the phase velocity and atten-uation. When the fracture aperture increases to 100 μm, they begin to show different behavior around 100 Hz. When the fracture aperture increases to 1 cm (see Fig. 3.7), discrepancies become very large as inertia effects become dominant.
3.3
Exact solutions for fracture and borehole waves
3.3.1 Rigid fracture surfaces
For plane fractures, it is also possible to solve the full Navier-Stokes equation. We now consider the wave propagation inside the fracture. Tang and Cheng (1989) derived a theory of wave propagation in a fractured borehole, where also the wave effects in the fracture itself were included. The linearized Navier-Stokes equation can be written as:
iω ˆw =−∇ˆp ρf + ν∇ 2w +ˆ ν 3 ∇ (∇ · ˆw) , (3.43)
where ν is the kinematic viscosity. Using vector decomposition (Miklowitz, 1978),we obtain
ˆ
w =∇ˆΦ + ∇ × ˆΨ. (3.44)
We substitute (3.44) into (3.43) and we obtain that iω ∇ˆΦ + ∇ × ˆΨ + ∇ˆp ρf = ν ∇2 ∇ˆΦ + ∇ × ˆΨ + ν 3 ∇ ∇ · ∇ˆΦ + ∇ × ˆΨ . (3.45)
After multiplication of this equation by iω, we find that −ω2 ∇ˆΦ + ∇ × ˆΨ = iω 3 ν + c 2 f ∇ ∇ · ∇ˆΦ + ∇ × ˆΨ + iων∇2 ∇ˆΦ + ∇ × ˆΨ , (3.46)
where we have used that ∇ˆp ρf =− 1 iωc 2 f∇(∇ · ˆw). (3.47) From (3.46), we obtain ∇ × −ω2Ψˆ − iων∇2Ψˆ=∇ 4iω 3 ν + c 2 f ∇2Φ + ωˆ 2Φˆ . (3.48)
The above equation is satisfied if ∇2Φ +ˆ ω2 c2f +43iων ˆ Φ = 0, (3.49) ∇2Ψ +ˆ ω iν ˆ Ψ = 0. (3.50)
We now consider cylindrical coordinates (r, φ, z). We obtain ∂2Φˆ ∂r2 + 1 r ∂ ˆΦ ∂r + ∂2Φˆ ∂z2 + ω2 c2f +43iων ˆ Φ = 0, (3.51) ∂2ψˆ ∂r2 + 1 r ∂ ˆψ ∂r + ∂2ψˆ ∂z2 − ˆ ψ r2 + ω iν ˆ ψ = 0. (3.52)
The solution to (3.51) and (3.52) is given by ˆ
ˆ
ψ = H1(κr)[C cos(f z) + D sin(f z)], (3.54)
where the ‘effective’ wavenumbers f and ¯f are given by f2 = ω 2 c2f+ 43iων − κ 2, (3.55) ¯ f2 = ω iν − κ 2. (3.56)
Using Helmholtz decomposition (3.44), we know that wr= ∂ ˆΦ ∂r − ∂ ˆψθ ∂z , (3.57) wz = ∂ ˆΦ ∂z + ∂ ˆψ ∂r + ˆ ψ r. (3.58)
Substitution of (3.53) and (3.54) in (3.57) and (3.58) yields that wr=−H1(κr)[Aκ cos(f z) + Bκ sin(f z)
−C ¯f sin( ¯f z) + D ¯f cos( ¯f z)], (3.59) wz = H0(κr)[−fA sin(fz) + fB cos(fz)
+Cκ cos( ¯f z) + Dκ sin( ¯f z)]. (3.60)
From the boundary conditions, z = ±h/2, wr = wz = 0, we obtain the system of equations Ax = 0, where the elements of A are specified in the Appendix A. Solving this system, we obtain the characteristic equation
κ2tan h 2f¯ + f ¯f tan h 2f = 0. (3.61)
The parameters are given by
B = 0, (3.62) C = 0, (3.63) D =−κ cos(f h 2) ¯ f cos( ¯fh2)A. (3.64)
Thus the fracture wavenumber κ can be obtained by solving (3.61). Res-ults are plotted in Fig. 3.8. Phase velocity of the fracture wave increases with increasing the fracture apertures and the frequency, attenuation of the fracture wave increases with the increasing frequency but it decreases when enlarging the fracture apertures. The results are also compared with that obtained from (3.40), where α(ω) is the frequency-dependent tortuosity of the fracture from (3.42). We notice that they are identical, for all practical purposes.
100 1 102 103 104 105 500 1000 1500 Phase Velocity (m/s) f (Hz) c b a 101 102 103 104 105 10−1 100 101 102 103 104 Attenuation (m − 1) f (Hz) a b c
Figure 3.8: Phase velocity and attenuation of the wave in the fracture from the exact
solution (3.61) (solid lines) compared with the generalized solution (3.40) (dashed lines). Fracture apertures are a: 1μm, b: 10 μm, c: 100 μm.
3.3.2 Fracture elasticity effects
The dispersion relation for guided waves in a fluid-filled infinite fracture was developed by Ferrazzini and Aki (1987). In their approach the viscous effects of the fluid in the fracture is neglected. From continuity of shear and normal
100 1 102 103 104 105 500 1000 1500 Phase Velocity (m/s) f (Hz) 10 μm 10 μm 100 μm 100 μm 1 mm 1 mm
Figure 3.9: Phase velocity of the wave in the fracture bounded by two elastic half spaces
from (3.65) (dashed lines) compared with two rigid half spaces from (3.61) (solid lines). Three fracture apertures 10μm, 100 μm and 1 mm are considered.
forces at the fluid-solid interface, and continuity of normal displacement, it can be derived that:
coth κh 2 1− 2f +ρs ρf 1− 2f 4s × ⎡ ⎣(2 − 2s)2 1− 2p − 4 1− 2s ⎤ ⎦ = 0,(3.65) where f = ω/(κcf), p = ω/(κcp), and s = ω/(κcs). The parameter ρs is
the density of the elastic formation, cp and cs are the compressional wave and shear wave velocity in the elastic formation respectively. The above equation is also given by Kostek et al. (1998b). We assume ρs = 2495 kg/m3,
cp = 1687 m/s and cs = 1097 m/s (see Table 6.1). Results are plotted in Fig. 3.9. In Fig. 3.9, three fracture apertures 10 μm, 100 μm and 1 mm are considered. The phase velocities of the fracture waves increase when increasing the frequency and the fracture apertures. As a reference, we also plot the results from (3.61) where a rigid formation is considered. Apparently, the introduction of fracture elasticity lowers the propagation speed as we have seen in borehole waves as well for elastic formations. However, in order to take all effects into account, full Biot solution should be considered for fracture elasticity on porous formation which is beyond the scope of the present thesis.
