The influence of fluid mobility on seismic velocity dispersion is directly observed in laboratory measurements from seismic to ultrasonic frequencies. A forceddeformation system is used in conjunction with pulse transmission to obtain elastic properties at seismic strain amplitude (10 −7 ) from 5 Hz to 800 kHz. Varying fluid types and saturations document the influence of pore-fluids. The ratio of rock permeability to fluid viscosity defines mobility, which largely controls pore-fluid motion and pore pressure in a porous medium. High fluid mobility permits pore-pressure equilibrium either between pores or between heterogeneous regions, resulting in a low-frequency domain where Gassmann's equations are valid. In contrast, low fluid mobility can produce strong dispersion, even within the seismic band. Here, the low-frequency assumption fails. Since most rocks in the general sedimentary section have very low permeability and fluid mobility (shales, siltstones, tight limestones, etc.), most rocks are not in the lowfrequency domain, even at seismic frequencies. Only those rocks with high permeability (porous sands and carbonates) will remain in the low-frequency domain in the seismic or sonic band.
The ultrasonic compressional [Formula: see text] and shear [Formula: see text] velocities and first‐arrival peak amplitude [Formula: see text] were measured as functions of differential pressure to 50 MPa and to a state of saturation on 75 different sandstone samples, with porosities ϕ ranging from 2 to 30 percent and volume clay content C ranging from 0 to 50 percent, respectively. Both [Formula: see text] and [Formula: see text] were found to correlate linearly with porosity and clay content in shaly sandstones. At confining pressure of 40 MPa and pore pressure of 1.0 MPa, the best least‐squares fits to the velocity data are [Formula: see text] and [Formula: see text]. Deviations from these equations are less than 3 percent and 5 percent for [Formula: see text] and [Formula: see text], respectively. The velocities of clean sandstones are significantly higher than those predicted by the above linear fits (about 7 percent for [Formula: see text] and 11 percent for [Formula: see text]), which indicates that a very small amount of clay (1 or a few percent of volume fraction) significantly reduces the elastic moduli of sandstones. For shaly sandstones we conclude that, to first order, more sensitive to the porosity and clay content than is [Formula: see text]. Consequently, velocity ratios [Formula: see text] and their differences between fully saturated (s) and dry (d) samples also show clear correlation with the clay content and porosity. For shaly sandstones we conclude that, to first order, clay content is the next most important parameter to porosity in reducing velocities, with an effect which is about 0.31 for [Formula: see text] to 0.38 for [Formula: see text] that of the effect of porosity.
Laboratory measurements of porosity and compressional velocity were conducted on unconsolidated brine saturated clean Ottawa sand, pure kaolinite, and their mixtures at various confining pressures. A peak in P velocity versus clay content in unconsolidated sand‐clay mixtures at 40 percent clay by weight was found. The peak in velocity is 20–30 percent higher than for either pure clay or clean sand. A minimum in porosity versus clay content at 20–40 percent clay by weight is also observed. Such behavior is explained using a micro‐geometrical model for mixtures of sand and clay in which two classes of sediments are considered: (1) sands and shaley sands, in which clay is dispersed in the pore space of load bearing sand and thus reduces porosity and increases the elastic moduli of the pore‐filling material and (2) shales and sandy shales, in which sand grains are dispersed in a clay matrix. For these sediments, the model reproduces the extrema in velocity and porosity and accounts for much of the scatter in the velocity‐porosity relationship.
We use a multivariate analysis to investigate the influence of effective pressure [Formula: see text], porosity ϕ, and clay content C on the compressional velocity [Formula: see text] and shear velocity [Formula: see text] of sandstones. Laboratory measurements on water‐saturated samples of 64 different sandstones provide a large data set that was analyzed statistically. For each sample, relationships between effective pressure and [Formula: see text] and [Formula: see text] have been determined. All samples were well fit by relationships that have an exponential increase in velocity at low [Formula: see text], tapering to a linear increase with [Formula: see text] for [Formula: see text] greater than 0.2 kbar. There are differences in the pressure dependences of velocity for different rocks, particularly at very low pressures; however, the differences cannot be attributed to ϕ or C. For the combined set of measurements from all samples, the best fitting formulations are [Formula: see text] and [Formula: see text]. While this is admittedly a very simplified parameterization, it is remarkable how well the velocity of the rocks considered here can be predicted based on only the three parameters, ϕ, C, and [Formula: see text]. The model accounts for 95 percent of the variance and has rms error of 0.1 km/s. An increased value of [Formula: see text] can indicate a decrease in [Formula: see text], a decrease in porosity, or an increase in clay content or some combination thereof.
Gassmann's (1951) equations commonly are used to predict velocity changes resulting from different porefluid saturations. However, the input parameters are often crudely estimated, and the resulting estimates of fluid effects can be unrealistic. In rocks, parameters such as porosity, density, and velocity are not independent, and values must be kept consistent and constrained. Otherwise, estimating fluid substitution can result in substantial errors. We recast the Gassmann's relations in terms of a porosity-dependent normalized modulus K n and the fluid sensitivity in terms of a simplified gain function G. General Voigt-Reuss bounds and critical porosity limits constrain the equations and provide upper and lower bounds of the fluid-saturation effect on bulk modulus. The "D" functions are simplified modulus-porosity relations that are based on empirical porosity-velocity trends. These functions are applicable to fluid-substitution calculations and add important constraints on the results. More importantly, the simplified Gassmann's relations provide better physical insight into the significance of each parameter. The estimated moduli remain physical, the calculations are more stable, and the results are more realistic.
Elastic wave velocities in sandstones vary with stress due to the presence of discontinuities such as grain boundaries and microcracks within the rock. The effect of any discontinuities on the elastic wave velocities can be written in terms of a second-rank and fourth-rank tensor that quantify the dependence of the elastic wave velocities on the orientation distribution and normal and shear compliances of the discontinuities. This allows the normal and shear compliance of these discontinuities to be obtained as a function of stress by inverting measurements of P-and S-wave velocities. Inversion of ultrasonic velocity measurements on dry and fluid saturated sandstones shows that the ratio of the normal to shear compliance of the discontinuities is reduced in the presence of fluid in the grain boundaries and microcracks. This is consistent with the expected reduction in the normal compliance of the discontinuities in the presence of a fluid with non-zero bulk modulus.
P‐wave seismic reflection data, with variable offset and azimuth, acquired over a fractured reservoir can theoretically be inverted for the effective compliance of the fractures. The total effective compliance of a fractured rock, which is described using second‐ and fourth‐rank fracture tensors, can be represented as background compliance plus additional compliance due to fractures. Assuming monoclinic or orthotropic symmetry (which take into account layering and multiple fracture sets), the components of the effective second‐ and fourth‐rank fracture compliance tensors can be used as attributes related to the characteristics of the fractured medium. Synthetic tests indicate that using a priori knowledge of the properties of the unfractured medium, the inversion can be effective on noisy data, with S/N on the order of 2. Monte Carlo simulation was used to test the effect of uncertainties in the a priori information about elastic properties of unfractured rock. Two cases were considered with Wide Azimuth (WAZ) and Narrow Azimuth (NAZ) reflection data and assuming that the fractures have rotationally invariant shear compliance. The relative errors in determination of the components of the fourth‐rank tensor are substantially larger compared to the second‐rank tensor, under the same assumptions. Elastic properties of background media, consisting in horizontal layers without fractures, do not cause azimuthal changes in the reflection coefficient variation with offset. Thus, due to the different nature of these properties compared to fracture tensor components (which cause azimuthal anomalies), simultaneous inversion for background isotropic properties and fracture tensor components requires additional constraints. Singular value decomposition (SVD) and resolution matrix analysis can be used to predict fracture inversion efficacy before acquiring data. Therefore, they can be used to determine the optimal seismic survey design for inversion of fracture parameters. However, results of synthetic inversion in some cases are not consistent with resolution matrix results and resolution matrix results are reliable only after one can see a consistent and robust behaviour in inversion of synthetics with different noise levels.
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