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 [4] as well as relating to static elasticity [5]. More importantly, core measurements can also be used to calibrate rock physics models [6] for the forward prediction of the stress dependence of seismic velocities. Reservoir rocks are seldom under dry conditions and so more realistic characterization of the non-linear rock physical properties should be examined under fluid saturated conditions. However, the number of published studies on saturated core samples are fewer than those on dry samples and this is primarily due to the difficulty of carrying out fluid saturated stress measurements of core samples (e.g., enormous equilibration times are necessary when performing saturated core measurements in comparison with dry sample measurements).

In this paper, we compare the microcrack parameters of the discrete and analytic microstructural stress-dependent model described in [3] for dry and water saturated core. The data used in this study come from the sandstone ultrasonic velocity-stress measurements of [7] and so allow a direct comparison between dry and water saturated microcrack parameters. This work follows from [6] who explore the microcrack properties of over 150 dry-core ultrasonic velocity-stress measurements. Given the influence of fluids within microcracks (and capillary forces) has important implications on the stress sensitivity of reservoir rocks [8], it is necessary to study the effect of fluid saturation on non-linear rock physics model parameters. Thus, the main objective is the paper is to study the influence of fluid saturation fluid on microcrack properties to further calibrate the microstructural non-linear rock physics models discussed in [6].

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, [11] present a 1D empirical formulation to describe vertical travel time perturbation due to changes in vertical strain and vertical velocity from 4D seismic data. Authors [12-14] use third-order elasticity theory to characterize 3D stress dependence elasticity and anisotropy. Authors [15-17] introduce nonlinear models consistent with empirically derived phenomenological equations [18]. Our interest in the discrete and analytic models described earlier (and in more detail below) is based on seeking formulations described using few and intuitive effective microstructural model parameters that can be calibrated with available data (e.g., ultrasonic core data).

Reference [7] investigated 69 sandstone core samples to examine the influence of fluid saturation on ultrasonic velocities. The measured porosity ranged between 5% and 30%, and the measured clay content ranged between 0% and 50%. The core samples ranged in length between 2 cm and 5 cm, and had diameter of 5.0 cm. It was noted that the dimensions of the core were approximately two orders of magnitude larger than the average grain size.

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. Figure 1 is a sandstone sample with high clay content (46.0%) and low porosity (13.8%). The fit between observation and the discrete and analytic predictions are very good. The dry measurements have a characteristically higher stress dependence compared to the saturated measurements. The ratio for the dry measurements show a stress dependence, decreasing from 1.5 to 0.5 with increasing stress, whereas the ratio for saturated measurements is approximately constant at 0.5. Estimates of initial crack density e⁰ is smaller whereas the initial aspect ratio a⁰ is greater for the saturated measurement compared to the dry measurement. Figure 2 is a sandstone sample with low clay content 7.0% and high porosity 27.05%. This sample shows similar stress dependent velocity characteristics to the sample shown in Figure 1. However, the ratio for the dry measurements is approximately constant around 0.75 and there is minimal change in the predicted initial crack density and initial aspect ratio between the dry and saturated measurements. Figure 3 is a sandstone sample with no clay (0.0%) and moderate porosity (19.8%). This sample displays the same velocity stress dependence (i.e., higher stress dependence for the dry measurement). The ratio is also stress dependent, but displaying a concave upward trend with increasing stress compared to the concave downward trend in Figure 1. Although the initial crack density is lower for the saturated sample, there is no change in the initial aspect ratio. [Note, the model parameters are assumed to be isotropic only because the data contain only one Pand one S-wave measurement for each dry and saturated sample. However, the formulation does consider anisotropy in the model parameters if there is sufficient ultrasonic data.]