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Stress dependent rock physics models are being used more routinely to link mechanical deformation and stress perturbations to changes in seismic velocities and seismic anisotropy. In this paper, we invert for the effective non-linear microstructural parameters of 69 dry and saturated sandstone core samples. We evaluate the results in terms of the model input parameters of two non-linear rock physics models: A discrete and an analytic microstructural stress-dependent formulation. The results for the analytic model suggest that the global trend of the initial crack density is lower and initial aspect ratio is larger for the saturated samples compared to the corresponding dry samples. The initial aspect ratios for both the dry and saturated samples are tightly clustered between 0.0002 and 0.001, whereas the initial crack densities show more scatter. The results for the discrete model show higher crack densities for the saturated samples when compared to the corresponding dry samples. With increasing confining stress the crack densities decreases to almost identical values for both the dry and saturated samples. A key result of this paper is that there appears to be a stress dependence of the compliance ratio
B_{N}/B_{T} within many of the samples, possibly related to changing microcrack geometry with increasing confining stress. Furthermore, although the compliance ratio
B_{N}/B_{T} for dry samples shows a diffuse distribution between 0.4 and 2.0, for saturated samples the distribution is very tightly clustered around 0.5. As confining stresses increase the compliance ratio distributions for the dry and saturated samples become more diffuse but still noticeably different. This result is significant because it reaffirms previous observations that the compliance ratio can be used as an indicator of fluid content within cracks and fractures. From a practical perspective, an overarching purpose of this paper is to investigate the range of input parameters of the microstructural models under both dry and saturated conditions to improve prediction of stress dependent seismic velocity and anisotropy observed in time-lapse seismic data due to hydro-mechanical effects related to fluid production and injection.

Non-linear or stress dependent rock physics models are being applied increasingly to model the influence of stress perturbations due to reservoir production and injection activities on seismic velocities. Laboratory measurement of non-linear rock physical properties of dry core samples can provide valuable information on the stress dependent elastic properties of reservoir rocks [1- 3] and have the potential for up scaling to seismic frequencies [

In this paper, we compare the microcrack parameters of the discrete and analytic microstructural stress-dependent model described in [

We examine the microcrack properties of two non-linear rock physics models: A discrete microcrack model defined by a secondand a fourth-rank crack density tensor [1-3] and an analytic microcrack model defined by an initial crack density and initial aspect ratio [3,9]. Although the discrete model describes the non-linear dependence of velocity with stress, the input parameters are two tensor quantities that are not necessarily intuitive. The analytic formulation provides a model based on physically intuitive input parameters to forward model the non-linear stress dependence of velocity, yet requires assuming that the microcracks are penny-shaped (i.e., the scalar crack approximation). Although penny-shaped cracks offer intuitive parameterization of the pore space and a reduction in the model complexity, the scalar crack approximation is not totally consistent with ultrasonic core data [6,10]. However, it should be noted that the analytic formulation still captures some of the essential stress dependent behaviour of sedimentary rocks and has utility for forward modelling applications.

It should be noted that the non-linear formulation we examine in this paper is one of many approaches to model the influence of stress on seismic velocity. For instance, [

Reference [

[19,20] where δ_{ij} is the Kronecker delta and summation convention is being used. The secondand fourth-rank crack density tensors α_{ij} and β_{ijkl} are defined

and

where V is volume and n is the unit normal to the displacement discontinuity set m (i.e., microcrack or grain boundary). and are the normal and tangential compliances across the microcrack set m having surface area S^{m}. The effective compliance S of a rock can be expressed

where S^{0} is the background (or intact) rock compliance estimated from either mineral composition [

The key assumptions for this model are that the microcracks are rotationally invariant and thin. Since ultrasonic measurements are only performed in one direction, we assume the samples are isotropic [_{ijkl} is isotropic)

and

where

and (7)

[

To enable forward modelling of 4D seismic effects related to perturbations in stresses [23-25], [

where

and are the effective initial number crack density and effective initial aspect ratio, λ_{i} and μ_{i} are the Lame constants, and is the principal effective stress in the i-th direction. The second-rank microcrack density term is

where (10 )

is a normalization factor [

Reference [_{ijkl} is often neglected to enable characterization of the nonlinear stress dependent elasticity based solely from contribution of the second-rank term α_{ij} [2,3,27]. Deviations from the scalar crack assumption (i.e., where β_{ijkl} is small such that = 1) potentially result from several factors, namely presence of fluids with nonzero bulk modulus, clay within cracks, cementation, and complex crack geometries. In this paper, we know the saturating fluid as well as an estimate of the clay content of the rock specimens (see discussion below), but lack quantitative measures of cementation or microcrack geometry.

In [3,6], the analytic model parameter inversion involved implementing a simple grid search over model parameters to minimize the misfit between model predictions and observed ultrasonic data. However, such a simple grid search may not be an efficient method for determining the best-fitting model parameters. For the discrete model parameter inversion in [3,6], a Newton-Ralphson approach was used to minimize the misfit between the model predictions and observations based on derivatives of the elasticity tensor components with respect to model parameters (see

find the true global minimum and so solutions may be biased towards local minima if the initial starting model is not chosen carefully. In this paper, we use the neighbourhood algorithm of [

Reference [

The Pand S-wave velocities were measured with the pulse transmission technique based on picking the first arrival peak amplitude. The confining and pore pressure were controlled separately, where the differential pressure limited to 50 MPa and the pore pressure to 1 MPa. The ultrasonic frequency of the Pand S-wave transducers were 1.0 MHz and 0.6 MHz, respectively. Based on the measured velocities, the average dominant wavelengths of the P and S waves were at least five times the mean grain size of the samples. Under dry conditions, the P-wave arrival times were picked to within 0.003 μsec (which equates to ≤1% error in velocity). The S-wave velocity errors were estimated to be less than 2%, except for the poorly consolidated samples at low confining stresses where the velocity errors were up to 3%. The samples were preloaded to 50 MPa and the velocities were measured on the unloading path to reduce the effects of hysteresis. Although hysteresis was observed the magnitude was small (i.e., ≤1%). For water-saturated conditions, the samples were fully saturated with water. However, for the samples having high clay content, the saturating fluid used was brine to minimize chemical alteration effects. [No velocity differences were observed between the water and brine saturation samples.] Velocities were measure during loading and unloading with only minor hysteresis being observed (≤1% for well consolidated samples and ≤2% for poorly consolidated samples).

Figures 1-3 show the results of the inversion for three of the sandstone samples. ^{0} is smaller whereas the initial aspect ratio a^{0} is greater for the saturated measurement compared to the dry measurement.

^{0} versus a^{0} estimates for the analytic model for both the dry and saturated measurements. Also shown (inset) are histograms for e^{0} and a^{0}. For the dry and saturated measurements, the initial aspect ratios show similar clustering centred around 0.0005 and is consistent with that observed by [^{0} show more scatter, with values between 0.0 and 0.4 for the dry measurements and 0.0 and 0.25 for the saturated measurements. In