Velocities and quality factors of sedimentary rocks at low and high effective pressures

Transcription

1 Geophys. J. Int. (1995) 123, Velocities and quality factors of sedimentary rocks at low and high effective pressures Simon M. Jones Postgraduate Research Institute for Sedimentology, University ofreading, Whiteknights, Reading, Berkshire RG6 6B, UK ccepted 1995 June 13. Received 1995 June 13; in original form 1995 February 20 1 INTRODUCTION Laboratory measurements of the velocity and attenuation of seismic waves in rocks have continually improved over the years (most notably in the latter), which has resulted in a number of recent advances in rock physics. Theoretical models of the pressure dependences of these acoustic properties in saturated rocks have been described in a number of papers (e.g. Toksoz, Cheng & Timur 1976; OConnell & Budiansky 1977; Cheng & Toksoz 1979), and have been used to model experimental data (e.g. Tao, King & Nabi-Bidhendi 1995). These theoretical approaches demonstrate that the closure of microcracks is the dominant mechanism in the variation of acoustic properties with pressure, but the models generally require reliable values of external parameters in order to carry out the forward problem. For example, the theoretical approach of Cheng & Toksoz (1979) models velocities on the pore aspect ratio spectrum. Tao et ae. (1995) used this method to establish pore spectra from the inversion SUMMRY Techniques for measuring acoustic velocities and quality factors (Q) of sedimentary rocks in the laboratory are sufficiently accurate to establish closely fitting regression curves which represent the variation of these acoustic properties with pressure. Ultrasonic P- and S-wave measurements of velocity (V,, V,) and Q (Q,, Q,) were made on 16 different water-saturated sandstone samples at various effective pressures using an ultrasonic reflection technique, and the data were fitted using a simple regression equation containing a constant, plus linear and exponential pressure-dependent terms. The redundancy of the linear term in the regression of Q data suggests that microcracks alone govern the pressure variation of Q, whereas velocities are additionally dependent on another mechanism, consistent with the changes in elastic moduli caused by intergranular compression (causing small reductions in porosity). The implied rates of microcrack closure with increasing effective pressure in all samples were found to be comparable. The regression equations describe the data closely enough to enable the extrapolation of acoustic properties to both low and high effective pressures. This is of practical use, since laboratory techniques can produce uncertain results at low pressures, due to coupling problems and low signal strengths, whilst high pressures require large confining vessels. Low-pressure and high-pressure data are applicable to shallow seismic surveys and hydrocarbon exploration, respectively. Key words: effective pressure, elastic moduli, microcracks, quality factors Q, and Q,, regression curve, ultrasonic velocities V, and V,. of their measured ultrasonic velocities, and modelled the attenuation in the saturated samples from fluid flow theory using the calculated pore spectra and effective moduli. They showed that, although the resolution of the inversion scheme was generally good, there were departures between the model curve and the experimental results, particularly at lower pressures (< 20 MPa). Curves that closely fit the experimental data are required if the pressure dependences of acoustic properties are to be generalized, particularly if predictions are needed for pressures that are outside the experimental range. For predictive purposes, we do not require a rigorous theoretical framework (this will be covered in a future paper by the author), and therefore it is preferable to use an empirical approach which provides a high fitting accuracy. Increased experimental accuracy now means that the variation of acoustic properties with pressure can be accurately fitted by least-squares regression curves. Eberhart-Phillips, Han & Zoback ( 1989) showed that the regression of the velocity-pressure data made RS

2 on 64 water-saturated sandstones by Han, Nur & Morgan (1986) could be used to derive empirical relationships between velocity and porosity, clay content and effective pressure. Regression curves can be extrapolated to pressures that may be outside the limits imposed by the experimental sysrem, provided that enough data are available to constrain them accurately. Measurements of acoustic properties at low pressures (< 5 MPa) are difficult to measure reliably in the laboratory, due to low signal strengths and poor coupling, whilst higher pressures require large confining vessels. Therefore, the use of accurate regression curves avoids the need foi direct measurement at these pressures. High-pressure acoustic data (>60 MPa, >3.5 km) are applicable to deepsca!e geophysics, as used in hydrocarbon,and gas exploration, and low-pressure data (< 5 MPa, < 500 m) are important in civil engineering geophysics. The latter may be used to supply information about the near-surface distribution of microcracks in shallow seismic field data, provided that the frequency difference is accounted for. coustic properties in the near surface will be dependent on weathering and the presence of both microfractures and larger fissures in the rocks. Compressional (P) wave and shear (S) wave velocities (V,, V,) and quality factors (Qp, Qs) were calculated from experimental measurements over a range of pressures on 16 different water-saturated sandstone samples, using an ultrasonic reflection technique based on the method described by Winkler & Plona (1982). The high accuracy of the results enabled the calculation of close-fitting regression curves for all samples, and it is shown that the differences between the experimental and predicted values of velocity and Q are usually smaller than, or close to, the inherent errors in the measurement of each quantity. This suggests that, particularly for velocity, extrapolation of the regression curves can accurately predict values at low pressures, and an example from the data set confirms this. Table 1. Physical and petrophysical properties of the samples. Sample lvsf(225) ZMYK(77) El(410) W8) ldyk( 101) 879 SHBP(21) 8VBP(20) FD(444) HW(448) RS(443) ISU(453) 4SU(447) 5SU(452) El(412) TS(451) Porosity (96) Grain ) o EXPERIMENTL Velocity and Q of sedimentary rocks Samples suite of 16 different sandstones was selected, consisting of 13 from quarried blocks, one from a 6 inch (15.24 cm) drill core and two from a 6 inch northern North Sea well core. The rocks were selected for their homogeneity and their wide range of physical properties, such as porosity, clay content and grain density, in order to represent a reasonable range of wellcemented sandstones. Each sample was cored to a diameter of 5 cm, then cut to a thickness of approximately 2 cm, with the faces ground flat and parallel to 1 pm. Porosities and grain densities (Table 1) were measured on the vacuum-dried samples using a helium porosimeter with an adapted core holder. The relative proportions of each clay mineral component were measured using the < 5 pm fraction from clay XRD. bsolute clay contents (Table 1) were then calculated using the clay XRD results in conjunction with the new technique of thermogravimetry/evolved water analysis (TG/EW), as described by Thornley & Primmer (1995). The corresponding errors in the clay content were calculated by assuming that the values of each clay fraction have associated errors of 110 per cent. ll samples were vacuum-saturated with distilled, de-aerated water at a pressure of Pa, using the method described in McCann & Sothcott (1992). 2.2 Ultrasonic measurements coustic (P and S wave) measurements were made on all samples using an ultrasonic pulse reflection technique (McCann & Sothcott 1992), modified from Winkler & Plona (1982), as shown in Fig. 1. The pulse was a toneburst from an renberg type PG650C pulse generator, and the. transducers used were Panametrics type V102 (P wave) and type V152 (S wave). The frequencies used in the experiments were 0.89 MHz for P waves and 0.59 MHz for S waves. The grain sizes of all Ka Clay content (9%) - Sm. Ch Total k f f f f f f f f f f f 3.4

3 776 S. M. Jones I 5 cm 1- Pore fluid inlet Figure 1. The ultrasonic reflection technique. the sandstones were sufficiently small that no Rayleigh scattering was observed at these frequencies. Reflections from the top and bottom rockbuffer-rod interfaces were stacked 1000 times and displayed on a digital oscilloscope. The traveltimes of the pulse through the samples were measured by selecting one particular peak from the top reflection and its corresponding trough on the bottom reflection (to account for the phase reversal of the latter). The attenuation coefficients, a(f), were calculated from eq. (l), which accounts for the geometrical spreading and the acoustic impedance contrast between the rock and the Perspex buffer-rod (Papadakis, Fowler & Lynworth 1973): where x is the thickness of the sample (in cm),fis the frequency (inhz), and,(f) and,(f) are the amplitudes of the top and bottom reflections, respectively. R(f) is the reflection coefficient of the buffer-rod/rock interface, and was calculated from where pr and pb are the densities of the rock and Perspex buffer-rods (in kg m-3), respectively, and G(f) and Vb(f) are the velocities (in m s-l) of the same. Since both Pb and &(f) are pressure-dependent, R(f) will also vary with effective pressure. However, corrections for this pressure dependence, using the data from Hughes, Blankenship & Mims (1950), showed that their effects on the velocity and Q of each rock sample were negligible. The traveltimes and attenuation coefficients were corrected for the effects of wave diffraction (beam spreading) in the Fresnel zone, caused by the finite aperture of the transducers. These were taken from the tabulated values for acoustic waves propagating through a fluid (Benson & Kiyohara 1974), based on the theory of Papadakis et al. (1973), which can be shown to give acceptable corrections using this apparatus for P and S waves propagating through a solid (Green & Wang 1991). Interference from sidewall reflections due to the finite size of the samples was calculated to be negligible above 0.5 MHz (Kinder & Frey 1962, p. 506). The absence of layering in the samples eliminates the possibility of wave attenuation by interlayer reflections. The differences between the traveltimes of the reflection pairs were converted to velocities by dividing them by twice the sample thickness. The quality factors, Q(f), were determined from The corresponding errors were calculated from where E~ and E, are the standard errors in Q(f) and cr(f), respectively. Using this technique, measurements of velocity have an accuracy better than k0.3 per cent, and the attenuation coefficient is accurate to fo.l db cm-' at 0.85 MHz (McCann & Sothcott 1992). The overall effective pressure on the rock sample was taken to be the difference between the confining pressure (axial + radial) and the pore fluid pressure; the confining pressure was varied between 10 and 65 MPa, with the pore fluid pressure held constant at 5 MPa. The reflection method for measuring the acoustic properties of rocks requires that the buffer-rods are in 'welded contact with the rock sample if Q is to be correctly calculated from the amplitude ratio using eq. (3). Thus, measurements of Q cannot be calculated reliably below about 5 MPa using this technique (the lower limit will vary slightly, depending on the degree of coupling between the rock sample and buffer-rods). It is therefore necessary to use an alternative method in order to infer the low-pressure acoustic properties. 3 QUNTIFYING THE PRESSURE DEPENDENCES OF VELOCITY ND Q In order to describe empirically the velocity-pressure or Q-pressure relationships for any particular sample, it is necessary to perform a regression analysis on the data. By examining the velocity data, it can be seen that the relationship is neither linear nor entirely exponential, but is a combination of the two (Eberhart-Phillips et al. 1989). The regression equation is therefore v, = + KP, - (5) where V, is the regression value of velocity, P, is the effective pressure, and, K, B and D are the coefficients to be evaluated

4 ~ for the optimum fit. The statistical procedure used to evaluate the regression coefficients was based on a least-squares approach, with the goodness-of-fit being assessed from the standard error of the estimate. computer program was written in order to implement the data regressions rapidly by scanning through a range of likely D values and calculating, K and B for each one. The set of values that gave rise to the lowest sum of squares produced the final regression equation in each case. 4 RESULTS ND DISCUSSION Both velocity-pressure and Q-pressure data are well-fitted by eq. (9, but in the latter case, values of K were found to lie close to K=O (mean values over the 16, samples were K,= -0.15f0.35 and K,= -0.65k0.31). Negative K implies that the regression curve contains a maximum, beyond which Q decreases with increasing effective pressure. Since the error in Q increases with QZ according to eq. (4), it appears that negative K is simply a result of the greater uncertainty in high values of Q (the lower energies of S-wave data result in greater experimental uncertainty than for P-wave data). With K set to zero and the regression parameters recalculated, it was found that the curves still lay within the error bars of the highpressure Q. Fig. 2 shows the velocity and Q (K = 0) regression curves for sample El(412). The redundancy of the linear term in the Q regression equation has implications for the nature of the pressure h v1.? , Effective pressure (MPa) I//,,'. l T I,..- l - ~ ~, ~~~r~~.. ~ Effective pressure (MPa) Figure 2. Experimental data and regression curve for sample E1(412), showing the variation with effective pressure of (a) P-wave velocity (V,) and (b) P-wave Q (Q,) RS, GJI 123, Velocity and Q of sedimentary rocks 777 dependence of Q in comparison with that of velocity. It is generally agreed that at lower effective pressures, microcrack closure with increasing hydrostatic stress is responsible for the rapid increases in velocity (e.g. Toksoz et al. 1976; Nur & Murphy 1981) and the decreases in attenuation (e.g. Winkler & Nur 1979; Johnston & Toksoz 1980), which are commonly observed in most rocks. The non-zero values of K in the velocity regressions indicate that, when microcracks are closed at higher pressures, the velocity continues to increase linearly with pressure. This has been demonstrated by other work on dry and saturated sandstones, for which the confining pressures were increased to over 100MPa (Johnston & Toksoz 1980; Nur & Murphy 1981; Freund 1992). The more compressible shales and siltstones, which contain clays at grain contacts, produce greater linear regions with larger slopes, due to their reduction in stiffness (Johnston& Toksoz 1980; Jones & Wang 1981; Tosaya & Nur 1982). Conversely, sandstones with little grain contact clay and high aspect ratio pores possess little or no linear region, e.g. St. Peter Sandstone (Tosaya & Nur 1982). The linear increase of velocity with pressure is consistent with an increase in the elastic moduli due to intergranular compression, causing a small decrease in porosity. This reverses the effect of the slight increase in density, which acts to decrease the velocity. The compressional values of K are greater than the shear values in all samples, since both the bulk and shear moduli are affected by the intergranular compression at higher pressures. Q, however, becomes independent of pressure when the microcracks are closed, and therefore it is these cracks alone that govern the Q variation with pressure. The reciprocal of the asymptotic value of Q is therefore a measure of the inherent attenuation of a rock, and is independent of microcracks. Despite some significant scatter in the results of previous work on the variation of Q with effective pressure, it is evident that Q is generally pressure-invariant in saturated sandstones and shales at higher pressures (Johnston & Toksoz 1980; Tittmann et al. 1981). The regression parameters for V,, V, (K 20) and Q,, Q, (K = 0) are given in Tables 2 and 3, along with the values of the standard errors of the estimate (3 for each curve. The standard errors are generally low for both velocity and Q data, and although s, values are mostly smaller for Q, the fractional errors will be larger due to the smaller magnitude of Q. The parameters K, B and D do not correlate with the petrophysical properties, such as porosity or clay content, but the pressureindependent parameter shows a linear decrease with porosity. There is little variation in D across the samples for either velocity or Q, and the values are similar in both P- and S- wave data; mean values for velocity and Q are 0.115k0.016 and , respectively. The systematic differences between values of D for velocity and Q are an indication of their divergent underlying dependences, which become more evident at higher pressures. The fact that D is relatively constant indicates that, although the crack densities at atmospheric pressure may vary widely, their rates of closure with increasing effective pressure do not. This was also noted by Eberhart-Phillips et al. (1989), who found that D was similar for 128 P- and S-wave velocity measurements from the data of Han et al. (1986); they calculated a mean value of D= 0.167k0.053 (for P, in MPa). It is clear, therefore, that the linear pressure dependence of velocity at high effective pressures (> 60 MPa), and the corres-

5 778 S. M. Jones Table 2. P-wave regression coefficients for velocity and Q data from all samples. Sample 8VBP(2O) ldyk(io1) ZMYK(77) B79 5SU(452) FD(444) HW(448) RS(443) TS(451) lsu(453) 4SU(447) W8) 8HBP(21) lvsf(225) El(410) E l(4 12) VP r K , ion mrameters B D I Table 3. S-wave regression coefficients for velocity and Q data from all samp smple vs gvbp(20) ldyk(101) ZMYK(77) B79 5SU(452) FD(444) Hw(448) RS(443) TS(451) lsu(453) 4SU(44T) W8) 8HBP(21) 1 VSF(225) El(410) El(412) K O.Oo O.Oo O.Oo B D ponding asymptotic value of Q, can be readily predicted from regression curves. 4.1 Prediction of acoustic properties at low effective pressures In order to predict the low-pressure acoustic properties by extrapolation of the regression curves, the scatter of the data about the regression line must be minimal, with sufficient data at lower pressures to constrain the exponential term in eq. (5). Figs 3 and 4 show how the differences between the experimental and predicted values of velocity and Q compare with the inherent errors in each quantity, for all samples and pressures. The standard errors in Q were calculated from eq. (4). Points that lie beneath, or close to, the line of equal error are therefore the most preferable, and velocities are more consistent in this respect than are values of Q [most of the exceptional velocities s, S QP Qs :ression p B meters I = 0) D S mion parameters <=O) B D s are from the anomalously poor-fitting sample 4SU(447)]. Points that lie close to, or above, the line are generally associated with lower pressures, and this affects Q to a greater extent than velocity. In order to investigate the effect of the experimental data on the predicted low-pressure results, I calculated the theoretical values of velocity and Q at zero effective pressure for a typical sample [5SU(452)], using data from 5-60, and 20-60MPa effective pressures. The results are shown in Table 4. It is clear that the predicted zero-pressure values of velocity and Q will converge to the true values as more lowpressure data are added to the regression. Velocities (especially compressional) have a greater stability than Q, and can therefore give potentially accurate low-pressure estimates. Q may also be extrapolated to low effective pressure, but care must be taken that the values have little scatter about the regression line, and that measurements are made at effective pressures of

6 i ' Standard error in V (m/s) Figure 3. Comparison between the velocity regression error (I VV' I) and the standard error in velocity F 2o Ol x P-wave data + S-wave data Standard error in Q Figure 4. Comparison between the Q regression error (I Q-Q' I) and the standard error in Q. Table 4. Zero-pressure regression predictions of velocity and Q for sample 5SU(452). Zer+pressure regression estimates WMPa MPa MPa VP (mls) Vs (mk) QP Qs at least 5 MPa. In practice, the accuracy of the low-pressure predictions will increase for non-zero effective pressures, and this can be tested by comparing the 5 MPa results from the experimental data with those from the MPa regression. For sample 5SU(452), the 5 MPa experimental/predicted results were: V,=4076/4077 m s-', V,=2400/2367 m s-', Qp= 10.9/8.0 and Q,= 11.1/ CONCLUSIONS Using the ultrasonic reflection technique, velocity-pressure and Q-pressure data can yield accurate fits to a simple regression equation, containing only a constant and an exponential pressure term, plus an additional linear pressure term for velocity data. The differences between the predicted and experimental values of velocity and Q were generally found Velocity and Q of sedimentary rocks 779 to be smaller than, or similar to, the standard errors in the respective quantities. The redundant linear pressure term in the regression of Q-pressure data suggests that only microcracks govern the pressure dependence of Q, whereas its presence in the velocity regression equation implies that an additional mechanism, such as intergranular compression (causing porosity reduction), is occurring. Therefore, the asymptotic value of Q for a rock sample at high pressure is an inherent property which is independent of microcracks. Regression analyses of 16 different sandstones saturated with distilled water indicate that the rates of microcrack closure with increasing effective pressure are comparable. The results for velocities are in agreement with those of Eberhart-Phillips et ~ l (1989),. based on the data of Han et al. (1986). The regression equations can be used to interpolate accurately the acoustic properties between experimental data points, and may be extrapolated to both high and low effective pressures. The reliability of the latter was gauged by its stability to a reduction in the amount of data provided for the regression; velocities were found to be more stable than Q at low pressures. Prediction of acoustic properties at low and high effective pressures is of practical use, because direct measurements in the laboratory are difficult to achieve at these pressures. coustic data can become unreliable at low pressures (<5 MPa) due to poor coupling and low signal strengths, whereas high effective pressures (> 60 MPa) require large confining vessels. CKNOWLEDGMENTS This work forms part of the author's PhD research at the Postgraduate Research Institute for Sedimentology (University of Reading), and was funded by the Natural Environment Research Council (NERC) and British Gas Plc. The author wishes to thank Clive McCann for his advice on the implications of the results, and Jeremy Sothcott for his technical support during the experimental work. This paper is publication number 407 of the Postgraduate Research Institute for Sedimentology, University of Reading, UK. REFERENCES Benson, G.C. & Kiyohara, O., Tabulation of some integral functions describing diffraction effects in the ultrasonic field of a circular piston source, J. acoust. SOC. m., 55, Cheng, C.H. & Toksoz, M.N., Inversion of seismic velocities for the pore aspect ratio spectrum of a rock, J. geophys. Res., 84, Eberhart-Phillips, D., Han, D.-h. & Zoback, M.D., Empirical relationships among seismic velocity, effective pressure, porosity, and clay content in sandstone, Geophysics, 54, Freund, D., Ultrasonic compressional and shear velocities in dry clastic rocks as a function of porosity, clay content, and confining pressure, Geophys. J. Int., 108, Green, D.H. & Wang, H.F., Shear wave diffraction loss for circular plane-polarized source and receiver, J. acoust. SOC. m., 90, Han, D.-h., Nur,. & Morgan, D., Effects of porosity and clay content on wave velocities in sandstones, Geophysics, 51, Hughes, D.S., Blankenship, E.B. & Mims, R.L., Variation of elastic moduli and wave velocity with pressure and temperature in plastics, J. appl. Phys., 21, Johnston, D.H. & Toksoz, M.N., Ultrasonic P and S wave

7 780 S. M. Jones attenuation in dry and saturated rocks under pressure, J. geophys. Res., 85, Jones, L.E.. & Wang, H.F., Ultrasonic velocities in Cretaceous shales from the Williston basin, Geophysics, 46, Kinder, L.E. & Frey,.R., Fundamentals of acoustics, John Wiley & Sons, Inc., New York. McCann, C. & Sothcott, J., Laboratory measurements of the seismic properties of sedimentary rocks, in Geological pplications of Wireline Logs II, Geological Society Special Publication No. 65, pp , eds Hurst,., Griffiths, C.M. & Worthington, P.F., Geological Society of London, London. Nur,. & Murphy, W., Wave velocities and attenuation in porous media with fluids, in Continuum models ofdiscrete systems 4: Proc. 4th Int. Con$ on Continuum Models of Discrete Systems, pp , eds Brulin, 0. & Hsieh, R.K.T., North-Holland, msterdam. OConnell, R.J. & Budiansky, B., Viscoelask properties of fluidsaturated cracked solids, J. geophys. Res., 82, Papadakis, E.P., Fowler, K.. & Lynworth, L.C., Ultrasonic attenuation by spectrum analysis of pulses in buffer rods: Method and diffraction corrections, J. acoust. SOC. m., 53, Tao, G., King, M.S., & Nabi-Bidhendi, M., Ultrasonic wave propagation in dry and brine-saturated sandstones as a function of effective stress: laboratory measurements and modelling, Geophys. Prospect., 43, Thornley, D.M. & Primmer, T.J., Thermogravimetry/evolved water analysis (TG/EW) combined with XRD for improved quantitative whole-rock analysis of clay minerals in sandstones, Clay Minerals, 30, Tittmann, B.R., Nadler, H., Clark, V.., hlberg, L.. & Spencer, T.W., Frequency dependence of seismic dissipation in saturated rocks, Geophys. Res. Lett., 8, Toksoz, M.N., Cheng, C.H. & Timur,., Velocities of seismic waves in porous rocks, Geophysics, 41, Tosaya, C. & Nur,., Effects of diagenesis and clays on compressional velocities in rocks, Geophys. Res. Lett., 9, 5-8. Winkler, K. & Nur,., Pore fluids and seismic attenuation in rocks, Geophys. Res. Lett., 6, 1-4. Winkler, K.W. & Plona, T.J., Technique for measuring ultrasonic velocity and attenuation spectra in rocks under pressure, J. geophys. Rex, 87,