ABSTRACT In this paper the three-dimensional transient wave propagation is investigated due to a point force applied at the interface of a fluid and a poroelastic solid. Using the total response, it is found that a zero of the dispersion (Stoneley) equation, which is related to a pseudo interface wave mode and forms a pole contribution to the Green s function, does not necessarily capture the entire physics of the corresponding wave mode. Furthermore, on the basis of numerical modeling and experimental detection of wave modes at the interface of Bentheimer Sandstone and air, it is found that combined particle-velocity and fluid-pressure measurements can be used for impedance determination of porous materials and validation of Biot s theory. 1. INTRODUCTION Surface waves are often used to investigate materials in various applications such as ultrasonic testing of structures, borehole logging in reservoir engineering, and surface seismics in geophysics. In the case of poroelastic materials, surface waves carry information on elastic soil/rock properties but also on petrophysical properties such as porosity, permeability and fluid mobility. In the context of Biot s theory for poroelastic solids, at the plane interface of a fluid and a fluid-saturated porous solid different interface wave modes can exist depending on the specific boundary conditions. Whether or not the pores are open for pore fluid to flow across the interface, makes an important difference. In general three types of surface wave can exist: the leaky pseudo-rayleigh () wave, the leaky pseudo-stoneley (pst) wave and the non-leaky true interface wave, see [1]. Pseudo interface waves are not true interface waves in the sense that part of their energy is leaking into slower bulk modes as they propagate along the interface. Experimental evidence has been found for all of the three interface wave modes by means of various techniques where either particle motion or fluid pressure was measured, see [2-4]. Some authors have established relationships between the location in the complex plane of the zeros of the dispersion equation,
and the phase speed and attenuation of corresponding surface wave modes [1,5]. As an extension to the two-dimensional model described in [4], in this paper the three-dimensional transient wave propagation is described due to a point force applied at the interface of a fluid and a poroelastic solid. The aim of this is to verify whether the zeros of the dispersion equation that form pole singularities in the Green s function, indeed yield proper predictions for the speed and attenuation of the corresponding pseudo interface wave modes. In the second part of the paper, results are presented concerning the detection of the wave modes at the interface of air and Bentheimer Sandstone using combined particle velocity and fluid pressure measurements. The aim of these combined measurements is to clearly distinguish between various wave modes and to pave the way for a better identification of the relation between petrophysical rock properties and properties of the wave modes. The numerical model addressed above is also used to evaluate combined particlevelocity and fluid-pressure data. 2. MODEL AND SOLUTION FOR THE TRANSIENT RESPONSE In this section the transient response is analysed of a fluid half-space on top of a poroelastic half-space subjected to a point load F(t) applied at the interface (Figure 1a). The behaviour of the poroelastic solid half-space is described by Biot s equations of motion [6], assuming frequency-dependent wave motion according to [7]. The motion in the fluid half-space is described by the acousticwave equation. At the interface x 3 =, the behaviour is assumed to be governed by open-pore conditions, i.e., by continuity of volume flux and fluid pressure, and vanishing intergranular vertical and shear stresses, except at the source position where the vertical stress σ 33 = F(t)δ (x 1 )δ (x 2 ). Here δ ( ) denotes the Dirac delta function. The open-pore behaviour at the interface implies that the true interface wave will be absent in the response, see [1]. The Green s function is derived for every component (particle velocity and pressure) in the frequency-horizontal slowness domain. It consists of a superposition of the body modes. The interface conditions form a set of equations LDV mirror Fluid () F t (a) Fluid transducer (b) r r Porous solid Porous solid x 3 x 3 Figure 1. Fluid half-space on top of poroelastic half-space, subject to a force applied at the interface: model (a) and experimental set up to excite waves and detect the vertical particle velocity (b).
to determine the corresponding complex amplitude factors. Its determinant yields the poroelastic Stoneley dispersion equation Δ St, which appears as the denominator in the derived Green s function. Some of its zeros are related to interface waves, see e.g. [1,5], whereas others are argued to be non-physical, see [1]. The response in the space-frequency domain is calculated by applying the inverse Fourier transform over the horizontal slowness p. The integration can be performed along the real p-axis or, using Cauchy s residue theorem, along a closed contour in the complex plane according to: 2 ω ( 2 ( ) ) p= s m 4π ( ω ) n ˆ = f dp = 2πi Res f f d p, f = n H pr p, (1) m α Cα (2) where nˆ = nˆ( x, ω) and n = n ( px, 3, ω) denote the Green s functions. H (...) denotes the Hankel function of second kind and zeroth order, r 2 = x 2 2 1 + x2, and ω denotes radial frequency. Every s m denotes a simple pole of the integrand inside the integration contour and every C α denotes a branch cut that departs from one of the branch points that are related to the body waves. The branch cuts ensure that the integrand is single-valued and constitute the Riemann sheet (R-sheet) on which the integration is performed. The contour integration shows that the spectrum of the Green s function is composed of discrete and continuous contributions formed by residues at the pole locations and branch-cut integrals, respectively. The former often constitute the surface wave contribution to the spectrum while the latter constitute the body wave part (including head waves). The branch-cut integration is performed numerically. The number of poles that lie inside the closed contour on the chosen R-sheet is determined by applying the Principle of the Argument to the Stoneley equation Δ St, see [8]. 3. NUMERICAL RESULTS AND DISCUSSION For a point at the interface x 3 = at offset r = x 1 =.1 m, the vertical particle velocity v por,3 and the fluid pressure p por due to a force of Ricker signature of centre frequency 5 khz were calculated. This was done for two configurations, both having water-saturated Bentheimer Sandstone as porous material. In the first configuration the upper half-space is occupied with air, and in the second one with some light fluid having bulk modulus K w /1 and density ρ w /8, where K w and ρ w refer to the properties of water. On the basis of Eq. (1) the various contributions to the total response can be distinguished and compared. For configuration 1 (Figure 2a), the contribution of the pseudo-rayleigh () pole does not coincide with the total response in the fluid pressure. At the -arrival, its contribution is opposite, which means that the branch-cut integrals can affect the contribution of a pole. This fact is illustrated more pronounced for configuration 2 (Figure 2b), where in both components the contribution of the -pole does not coincide with the total response. In this configuration both the -pole and the pseudo-stoneley (pst) pole are present on the chosen R-sheet. The pst-pole contribution is, however, not displayed separately because the pst-wave interferes with the fluid (F) wave. For both
v por,3 1 5 [ms 1 ] 1 S 1.3.4.5.6.7 (a) v por,3 1 5 [ms 1 ] 5 5 S F+pSt.3.4.5.6.7.8 (b) p por 1 3 [Nm 2 ] 5 P 2 S+FS 5.3.4.5.6.7 p por [Nm 2 ] 1 P 2 S 1 F+pSt.3.4.5.6.7.8 Figure 2. Total responses and its pole contributions for x 3 = and r =.1 m: -contribution (--), contribution of additional pole (- ) and total response (-) for Bentheimer/air (a); -contribution (--) and total response (-) for Bentheimer/light fluid (b). configurations, a pole that is related to an interface wave does yield a proper prediction for the corresponding interface-wave velocity. In both configurations an additional pole is present on the chosen R-sheet, located to the left of the fast compressional ( ) wave slowness: Re(p) < Re(s P1 ). In Figure 2a, for configuration 1 its contribution is compared to the total response. Apparently, it affects the contribution of the branch-cut integrals to a large extent. Therefore, although this pole does not correspond to an interface wave, it does have physical significance since it contributes to the -wave and, in general, also to the head waves that are generated by the -wave front. This contradicts to what is stated in [1], that poles with Re(p) < Re(s S ), where Re(s S ) > Re(s P1 ), have lost all physical significance. Here s S denotes the S-wave slowness. In fact, any pole that is present on the chosen R-sheet should be considered as physical. From comparison of the total responses in particle velocity and fluid pressure in Figure 2, some interesting aspects arise. The interface-wave modes are present in both components and might even dominate the response, which is advantageous for the experimental detection in both components. Furthermore, it is noted that a wave front shows up in the fluid pressure at the S-wave arrival time. It is important to realize that this is not a S-wave front, it consists of radiated head waves (P 2 S denotes the slow compressional (P 2 ) wave front that is created by the S-wave front; similarly, FS denotes a F-wave front). It is also noted that the -wave is present quite strongly in v por,3 although this component is perpendicular to the direction of propagation of the compressional wave. This is due to the lateral contraction in vertical direction which can easily occur at the sandstone/air interface. 4. EXPERIMENTAL SET UP, RESULTS AND DISCUSSION In the experiment waves were excited at the interface of a water-saturated Bentheimer Sandstone sample of dimension.5 x.32 x.2 m 3 and air. A pulse of 1 khz centre frequency was fed into an ultrasonic piezoelectric immersion-type transducer that was placed on the interface (Figure 1b). To detect
the fluid pressure, the same type of transducer was placed on the interface. Some light pressure was applied on top of the transducers for better contact with the sample. For the detection of the vertical particle velocity, a Laser Doppler Vibrometer (LDV) was used with a 633 nm laser beam (Figure 1b). To ensure signal strength of the reflected laser beam, a small piece of reflective tape was sticked on the sample surface. The recorded data were averaged 2 times to improve signal-to-noise ratio. A band-pass filter between 1-15 khz was applied to the particle-velocity data. The measurements were performed for different source positions. The result is displayed in Figures 3a and 3c. In the particle-velocity data, the -wave of approximate velocity 29 ms -1, the -wave of 14 ms -1 and the F-wave of 34 ms -1 can be distinguished, while in the fluid-pressure data only the first two waves are visible. An incoherency in the decreasing amplitude trend can be observed in some of the traces of both particle-velocity and fluid-pressure data. This is due to varying coupling between source transducer and sample. Analysis shows that the frequency content of the particle-velocity data is higher than that of the fluidpressure data, which might be caused by the coupling of measuring transducer to the sample. The computed responses are displayed in Figures 3b and 3d, where the total response according to Eq. (1) is used, rather than only separate contributions. Given the lower frequency content of the measured data, in the model the Ricker source is chosen to have 6 khz centre frequency. The predicted -wave velocity corresponds with the measured value. The velocity of the indicated -event, which is in fact an interference of the -wave and a relatively weak S-wave, is approximately 16 ms -1. In both particle velocity and fluid pressure the -wave.5.4.6.8.1.12.14.16 (a).5.4.6.8.1.12.14.16 (b).1.15.1.15.2.25.3 F.2.25.3.35.35.5.1.15.4.6.8.1.12.14.16 (c).5.1.15.4.6.8.1.12.14.16 (d).2.25.2.25 F.3.3.35.35 Figure 3. Vertical particle velocity, modelled (a) and detected (b), and fluid pressure, modeled (c) and detected (d), at the Bentheimer/air interface. The traces were scaled by exp(ar), where a =.15.
is much stronger than the -wave, whereas in the measured data the amplitudes are in the same order of magnitude. The reason for this deviation might be the presence of strong capillary forces in the pore fluid at the pore water/air interface. The capillary forces can create a surface tension by which the boundary conditions are modified such that the behaviour is more governed by closed-pore rather than by open-pore conditions, see also [3]. Mainly because of the large differences in amplitude between the -wave and the -wave, proper trace by trace comparison between predicted and experimental results cannot be made yet. However, the fact that different waves can be detected in both components is promising. Combined measurements seem to be suitable for validation of Biot s theory and investigation of materials since the ratio of fluid pressure and particle velocity yields an impedance of the porous medium. 5. CONCLUSIONS It is concluded that the poroelastic Stoneley dispersion equation has zeros that are not related to interface wave modes, but that do have physical significance due to their contribution to the total response. Furthermore, it can be stated that a zero that is related to a pseudo interface wave mode, in general does not necessarily capture the entire physics of that wave mode. Also branch-cut integrals might contain part of the physics. Therefore, using only the location of the zeros in the complex plane to predict the entire kinematics of a pseudo interface wave might yield erroneous results. The fast compressional wave and the pseudo-rayleigh wave have been successfully detected in both vertical particle-velocity and fluid-pressure component. Combined measurements seem to be suitable for validation of Biot s theory and investigation of materials since the ratio of fluid pressure and particle velocity can be used for impedance determination of the porous medium. 6. REFERENCES 1. Feng, S. and D. L. Johnson. 1983. High-frequency acoustic properties of a fluid/porous solid interface. I. New surface mode, J. Acoust. Soc. Am., 74 (3): 96-914. 2. Mayes, M. J., P. B. Nagy, L. Adler, B. P. Bonner, and R. Streit. 1986. Excitation of surface waves of different modes at fluid-porous solid interface, J. Acoust. Soc. Am., 79 (2): 249-252. 3. Adler, L. and P. B. Nagy. 1994. Measurements of acoustic surface waves on fluid-filled porous rocks, J. Geoph. Res., 99 (B9): 17863-17869. 4. Allard, J. F., M. Henry, C. Glorieux, W. Lauriks, and S. Petillon. 24. Laser induced surface modes at water-elastic and poroelastic interfaces, J. Appl. Phys., 95 (2): 528-535. 5. Gubaidullin, A. A., O. Y. Kuchugurina, D. M. J. Smeulders, and C. J. Wisse. 24. Frequencydependent acoustic properties of a fluid/porous solid interface, J. Acoust. Soc. Am., 116 (3): 1474-148. 6. Biot, M. A. 1956. Theory of propagation of elastic waves in a fluid-saturated porous solid. I. Low-frequency range, J. Acoust. Soc. Am., 28: 168-178. 7. Johnson, D. L., J. Koplik, and R. Dashen. 1987. Theory of dynamic permeability and tortuosity in fluid-saturated porous-media, J. Fluid Mech., 176: 379-42. 8. Fuchs, B. A., B. V. Shabat, and J. Berry. 1964. Functions of a Complex Variable and some of their Applications. Pergamon Press, Oxford.