Laboratory Approach to the Study of Elastic Anisotropy on Rock Samples

Transcription

1 Pure appl. geophys. 151 (1998) /98/ $ /0 Laboratory Approach to the Study of Elastic Anisotropy on Rock Samples ZDENĚK PROS, 1 TOMÁŠ LOKAJÍČEK, 1 and KAREL KLÍMA 1 Abstract The experimental approach (hardware and software) to the study of the elastic anisotropy of rocks on spherical samples under hydrostatic pressure up to 400 MPa is discussed. A substantial innovation of the existing measuring system and processing methods enabled us to make a detailed investigation and evaluation of the kinematic as well as dynamic parameters of elastic waves propagating through anisotropic media. The innovation is based on digital recording of the wave pattern with a high sampling density of both time and amplitude. Several options and results obtained with the innovated laboratory equipment are presented. Key words: Laboratory measurements, elastic anisotropy, kinematic and dynamic parameters of elastic waves, hydrostatic pressure. Introduction The elastic anisotropy of rocks is an important geophysical parameter which can be used to classify rocks in petrophysical studies, to interpret seismic field measurements, and also to study the structure of the crust and the upper mantle of the earth. The physical parameters of rocks are determined mainly by their modal composition, properties of grains, grain-size distribution, contact conditions between grains, pore space, population of microcracks and by the combined action of the above-mentioned parameters affected by external conditions, mainly by stressstate conditions. The anisotropy of P waves was studied by BIRCH (1960, 1961); BAYUK et al. (1967); GIESEL (1963); BABUŠKA (1966); PROS and BABUŠKA (1967, 1968) and others. The laboratory measuring equipment, developed in the Geophysical Institute, enables elastic anisotropy of P waves to be studied by means of ultrasonic sounding under conditions of high hydrostatic pressure reaching 400 MPa (PROS and PODROUŽKOVÁ, 1974; PROS, 1977). In this paper the actual state of the laboratory equipment and the methodical approaches used to study elastic anisotropy are presented. 1 Geophysical Institute, Academy of Sciences of the Czech Republic, Boční II/1401, Praha 4 Spořilov, Czech Republic.

2 620 Zdeněk Pros et al. Pure appl. geophys., Experimental Setup During recent years laboratory measuring equipment has been improved considerably, mainly the ultrasonic data recording and the processing equipment has been modified. Now, not only are the travel times of P-wave ultrasonic signals stored, but the whole wavelet of the recorded ultrasonic wave is stored and used for further analysis. A schematic diagram of the improved measuring system is shown in Figure 1. The entire measuring system consists of a pressure vessel, pressure generator, ultrasonic transducers and a sample-positioning control unit, a device for generating and recording of ultrasonic signals. The digitizing scope HP 54540C Figure 1 Equipment for the laboratory study of P-wave anisotropy on spherical samples under hydrostatic pressure of up to 400 MPa. T transmitter of ultrasonic signal, R receiver of ultrasonic signal. Transducers are located on opposite sides of the sample.

3 Vol. 151, 1998 Study of Elastic Anisotropy on Rock Samples 621 with a 100 MHz sampling frequency is used to record and store the form of the exciting pulse and ultrasonic signal in digital form. A PC card with an A/D converter is used to measure the rock sample position, the ultrasonic transducers positions and the acting pressure. All the measuring equipment is controlled by an IBM PC via an IEEE 488 interface. The measuring software is divided into two independent parts. First, ultrasonic data recording and storing, when the transmitted signal (40 s, time duration) is recorded with a sampling frequency of 100 MHz and stored on the computer HD. The spherical sample can be measured in 132 independent directions (with a step of 15 in both spherical coordinates) at each pressure level; or it can be measured in a particular direction under variable acting pressure reaching 400 MPa. Second, the stored data are analysed by specialized software packages; the parameters to be studied being selected semi-automatically (time of signal arrival, value of the first amplitude, etc.). The recorded data are stored in the file on the computer HD only after being corrected or confirmed by an operator. These files are used for further analysis of the P-wave anisotropy of rock samples. Experiments The following experiments only show some examples of methodical approaches to anisotropy study used in our laboratory. As an experimental approach the pulse-transmission method for the velocity measurement was used. The couple of piezoceramic transducers was used; the transmitter excited by a high voltage pulse and receiver located on the opposite side of the spherical sample. The receiver picked up the ultrasonic signal propagating through the sample. The same transducers were used for transmitting and receiving the ultrasonic signal. The transducers are equipped with an acoustic transformer with a spherical surface tapered to the sample. The active area of the piezoelectric element is 3 mm in diameter. The natural frequency of the element is 2.5 MHz. The shape of the samples in our experiments is a sphere with a diameter of 50 mm. Before the measurements the samples are vacuum dried to a constant weight at a temperature of 50 C during 12 hours. Then the samples are covered by an epoxy resin film (thickness 0.05 mm) to protect the sample against pressure media penetration. An example of the wave pattern of the ultrasonic signal picked up by a receiver at different pressure levels is depicted in Figure 2. The dashed line shows the transmitter excitation time. Significant changes can be seen in this figure, as not only the travel times of the ultrasonic signal vary, but also their amplitude and frequency contents. The measuring system enables the travel-time dependence, or of P-wave velocity to be investigated in selected directions. An example of P-wave velocity dependence on hydrostatic stress from atmospheric pressure to 400 MPa for pair of ultrasonic transducers positioned in the maximum and then in the minimum velocity direction

4 622 Zdeněk Pros et al. Pure appl. geophys., Figure 2 Example of P-wave signals (sample RP3, orthogneiss) recorded at different values of hydrostatic pressure. is depicted in Figure 3. The directions were determined at 400 MPa at first loading cycle. This particular rock sample granite from the West Bohemia region with signature ZC19 displays a very fast change of P-wave velocity at a low value of applied hydrostatic stress. This fast change is mainly due to the closing of microcracks and will be discussed in more detail later. For higher stress values the changes of P-wave velocity are less pronounced. It also can be seen that nearly the same anisotropy is preserved in this sample for the whole range of acting pressure. Slight scatter shows the limits of measurement accuracy due to the resolution of the A/D convertor. As the measuring system records the wavelet of the ultrasonic signal as a whole, it enables us to investigate the amplitude changes of the ultrasonic signal passing through the spherical rock sample with direction. A comparison of the P-wave velocity dependence and amplitude dependence A 12 (A 12 is the difference between the first local minimum and first local maximum of the recorded signal) on hydrostatic pressure up to 400 MPa is shown in Figure 4. The P-wave velocity

5 Vol. 151, 1998 Study of Elastic Anisotropy on Rock Samples 623 varies from 5250 m/sec. to 6500 m/sec., which represents a change of about 24%. On the contrary, the A 12 amplitude varies from 8 mv to 380 mv, which is nearly 50 times higher value. The variations described above indicate that additionally the amplitude dependence of ultrasonic waves on hydrostatic pressure can also be considered as a very promising tool for describing and studying the elastic anisotropy of rocks. The acting force to sliding contact between the sample and the transducer does not vary substantially during the change of hydrostatic pressure. Only the elastic properties of oil film between them change with pressure. Due to this fact, the drastic change of A 12 amplitude with an increasing load is mainly caused by the consolidation of contact conditions between mineral grains and closing microcracks. The experimental setup also enables a detailed study of the velocity-confining pressure equation, which describes the P (or S ) velocity dependence on applied hydrostatic confining pressure. WEPFER and CHRISTENSEN (1991) proposed an empirical formula (1) relating velocity to confining pressure in the form: V(P)=A(P/100 MPa) a +B(1 e bp ), (1) Figure 3 Dependence of P-wave velocity on hydrostatic pressure in the maximum and minimum velocity direction (sample ZC19, granite).

6 624 Zdeněk Pros et al. Pure appl. geophys., Figure 4 Comparison of P-wave velocity dependence and amplitude dependence on hydrostatic pressure up to 400 MPa (sample ZC19, granite). A 12 is a difference between first local minimum and first local maximum of the signal recorded by receiver R. where P is the confining pressure in MPa, V is the velocity ( P or S ) and A, a, B and b are four fitted parameters. This equation was obtained solely from the shape of measured data. However the values of fitted parameters do not reflect the rock structure and its physical properties. We assume that two different physical processes can be observed during the confined loading of heterogeneous rock samples. First, under low acting pressure the most significant phenomenon is crack closing, which can be described by an exponential equation. Second, under high pressure the prevailing phenomenon is the linear increase of elastic parameters with pressure, as can be expected in an ideal sample without cracks, see also GREENFIELD and GRAHAM (1996). Based on the above assumptions, we propose a different formula: V(P)=Vo+kP dv10 ( P/Po), (2) where Vo is the velocity under a pressure of 0.1 MPa (in an ideal sample without cracks), k increasing velocity coefficient of an ideal sample without cracks,

7 Vol. 151, 1998 Study of Elastic Anisotropy on Rock Samples 625 P confining pressure in MPa, dv crack influence at the 0.1 MPa pressure level, Po pressure at which the crack influence dv decreases to 10% (see Fig. 5). For the given example Vo=6297, k=0.2771, dv= and Po= This formula was compared with the Wepfer and Christensen empirical formula (1). The comparison of the formulas indicates that formula (2) proposed in this paper yields better results in more cases than formula (1). A superior advantage of our approach is that the parameters we obtained during the fitting process provide for a better description of the physical properties and fabric of rock sample under study. The Po parameter denotes the pressure level at which most of low aspect ratio microcracks appear to be closed. The digital recording of the entire wavelet of the ultrasonic signal enables us to investigate its changes not only in the time domain, but also in the frequency domain. For this study only, the very beginning (6 s) of the ultrasonic signal in time domain was used. It is assumed that mainly the very beginning of the recorded ultrasonic signal (direct P wave) reflects the changes in the rock fabric due to the Figure 5 Dependence of P-wave velocity on hydrostatic pressure (sample ZC19, granite) and its fitted approximation. Vo velocity of an ideal sample without defects at 0.1 MPa, k increasing velocity coefficient of an ideal sample without defects, dv crack influence at the 0.1 MPa pressure level, Po pressure at which the crack influence dv decreases to 10%.

8 626 Zdeněk Pros et al. Pure appl. geophys., Figure 6 The frequency changes of the P-wave ultrasonic signal under increasing hydrostatic pressure up to 400 MPa. increasing pressure. Also after this time reflected signals will arrive at the receiver and the rest of the wavelet is superpositioned by reflected and transformed waves (PROS et al., 1969). The signal in the time domain is windowed by a Gaussian function after which an FFT analysis is applied. The changes of the signal in the frequency domain data with increasing hydrostatic pressure are displayed in Figure 6. This figure indicates that, due to the increasing hydrostatic pressure, not only the amplitude of the signal changes significantly, but also the frequency content. Although the pressure affects the transducers, the main change in amplitude observed is caused by changes of sample properties. This phenomenon can be explained by changes in the transmitting function of the sample. The amplitude spectra show two maxima, one at 1.1 MHz which represents the wavelet of P waves. The second one at 0.3 MHz is probably caused by surface waves or radial resonance of the transducer. A good approach to analysing the recorded data is to display the P-wave velocities in the rock sample by means of isolines and to calculate the elastic constants characterising the material. The comparison of P-wave velocities calculated from the elastic constants, which are in principle smooth values, with the measured velocities, provides information mainly about the samples homogeneity and also about the quality of the velocity measurement.

9 Vol. 151, 1998 Study of Elastic Anisotropy on Rock Samples 627 The calculation of elastic constants based on P-wave velocities is described in more detail in KLÍMA (1973). It consists in the solution of cubic equations with 21 unknown elastic parameters. Only 15 independent equations are available and six conditions must be added to make the problem solvable. These conditions can vary within relatively wide limits, but during reverse calculations of P-wave velocities they are not affected by them. On the contrary, the S-wave velocities, which we are not dealing with, are greatly affected by the selected limits. Nevertheless, the P-wave velocities calculated from these elastic constants are fully sufficient to describe the quality of the measurement and samples heterogeneity. In depicting the P-wave velocity distribution in terms of isolines, it is necessary to solve three separate problems in turn. Velocity approximation by a continuous function, calculation of the isolines on a spherical surface and, finally, projection of the spherical surface onto a plane. To approximate the velocity of a spline function, the 2-D harmonic analysis of velocities in a regular grid of points was used, due to the measuring step of 15 over the entire sphere. This 2-D analysis enables a simple calculation of functional values, as well as partial derivatives at every point. It has been proved that the calculations can be limited to 5 7 harmonics. The isoline shape is calculated in two steps. In the first one the approximate position of the next isoline point is determined, based on the knowledge of partial derivatives at a selected distance. The second step consists in the correction of the isoline point position in the perpendicular direction on the basis of the difference between the calculated function value at a given point and a given isoline value. Equal-area projection of calculated isolines to the lower hemisphere is used. The algorithm used to display isolines enables us to apply arbitrary transformation before the display. An example of the isolines of measured P-wave velocities is pictured in Figure 7. The isolines are displayed for all measured loading and unloading pressure levels of granite sample from the West Bohemia region (sample G6). Conclusions The experimental tests have proved that the improved measuring equipment can be used to study the elastic anisotropy of P waves under high hydrostatic pressure; mainly to investigate the effect of fabric change on the directional dependence of elastic wave velocities, its amplitude changes due to the transmitting function of the sample. It was demonstrated that increasing hydrostatic pressure changes not only the ultrasonic wave velocity, but also significantly changes the amplitude and frequency contents of the transmitted signal due to the closing microcracks and consolidation of contact conditions between mineral grains.

10 628 Zdeněk Pros et al. Pure appl. geophys., Figure 7 Isolines of P-wave velocities calculated from elastic constants (sample G6, granite). The isolines are displayed for the complete loading and unloading cycle. Upper line denotes measured pressure values p (in MPa). Numbers under the individual graphs denote appropriate velocity limits of isolines (in km/s). The velocity-confining pressure equation, based on the superposition of the linear and exponential dependence, fits the experimental data very well. The calculated coefficients can be used to describe rock behaviour under hydrostatic pressure. The number of pressure values (at least 6) for which the P-wave velocity is determined, does not change the fitted coefficients significantly. The comparison of Vp and Ap dependencies indicated that the amplitude changes of the ultrasonic signal should be used to study rock fabric due to their high sensitivity to the hydrostatic pressure applied. For hydrostatic pressures extending 400 MPa, velocities were observed to change by as much as 24%. On the contrary, amplitude changes were nearly 50 times. We believe that quantified elastic parameters obtained by the above-described measuring approach can be used as a powerful tool for realistic seismic modelling of anisotropic crustal environments. Acknowledgements We thank Prof. H. Kern and an anonymous reviewer for their constructive comments which helped us improve the manuscript. Part of this project was supported by the Grant Agency of the Academy of Sciences of the Czech Republic, Grant No. A and also by the Grant Agency of the Czech Republic, Grant Nos. 205/95/0263 and 205/97/0905.

11 Vol. 151, 1998 Study of Elastic Anisotropy on Rock Samples 629 REFERENCES BABUŠKA, V. (1966), Velocity of Compressional Wa es and Anisotropy of Some Igneous and Metamorphic Rocks, Travaux Inst. Geophys. Acad. Tchecosl. Sci. 223, BAYUK, E. I., VOLAROVICH, M. P., KLÍMA, K., PROS, Z., and VANĚK, J. (1967), Velocity of Longitudinal Wa es in Eclogite and Ultrabasic Rocks under Pressures to 4 Kilobars, Studia Geoph. et Geod. 11, BIRCH, F. (1960), The Velocity of Compressional Wa es in Rocks to 10 Kilobars, Part I, J. Geophys. Res. 65, BIRCH, F. (1961), The Velocity of Compressional Wa es in Rocks to 10 Kilobars, Part II, J. Geophys. Res. 66, GIESEL, W. (1963), Elastische Anisotropie in tektonisch erformten Sedimentgesteinen, Geophys. Prosp. 11, GREENFIELD, R. J., and GRAHAM, E. K. (1996), Application of a Simple Relation for Describing Wa e Velocity as a Function of Pressure in Rocks Containing Microcracks, J. Geophys. Res. 101, KLÍMA, K. (1973), The Computation of the Elastic Constants of an Anisotropic Medium from the Velocities of Body Wa es, Studia Geoph. et Geod. 17, PROS, Z. (1977), In estigation of anisotropy of elastic properties of rocks on spherical samples at high hydrostatic pressure. InHigh Pressure and Temperature Studies of Physical Properties of Rocks and Minerals (eds. Volarovich, M. P. and Stiller, H.) (Naukova Dumka, Kijev 1977) pp (in Russian). PROS, Z., and BABUŠKA, V. (1967), A Method for In estigating the Elastic Anisotropy on Spherical Rock Samples, Zeitschrift für Geophysik 33, PROS, Z., and BABUŠKA, V. (1968), An Apparatus for In estigating the Elastic Anisotropy on Spherical Samples, Studia Geoph. et Geod. 12, PROS, Z., and PODROUŽKOVÁ, Z. (1974), Apparatus for In estigating the Elastic Anisotropy on Spherical Rock Samples at High Pressure, Veröff. Zentralinst. Physik d. Erde 22, PROS, Z., VANĚK, J., KLÍMA, K., and BABUŠKA, V. (1969), Experimentelle Untersuchung des Wellenbildes bei der Ultraschall-Durchstrahlung einer Kugel, Zeitschrift für Geophysik 35, WEPFER, W. W., and CHRISTENSEN, N. I. (1991), A Seismic Velocity-Confining Pressure Relation, With Applications, Int. J. Rock. Mech. Min. Sci. and Geomech. Abstr. 28 (5), (Received December 6, 1996, revised June 15, 1997, accepted August 10, 1997).