Capillary pressure model for frequency dependent velocity-saturation relations [electronic resource]
Keywords: Geophysics.

Relating the remotely-sensed elastic properties of rock to fluid saturation has been a debated (if not unanswered) question of rock physics. This question often arises during a time-lapse seismic interpretation where the goal is to understand where and how much of injected or original pore fluid is located (Landro 2001, Lumley 2001). The Gassmann (1951) fluid substitution equation remains the cornerstone of fluid substitution. The problem of applying this equation at partial saturation is that one needs to provide the effective bulk modulus of the mixture of the fluid phases. One way of arriving at this effective bulk modulus is to assume that the fluid phases coexist at the pore scale and each pore contains parts of water and 1- parts of gas, where is the water saturation. In this uniform-saturation case, the harmonic average of the bulk moduli of the fluid phases can be used as the effective bulk modulus of the mixture. Because harmonic averaging produces the lower bound for the bulk modulus of the mixture, the uniform saturation assumption produces the lower bound for the bulk modulus of rock at partial saturation (Mavko et al., 2009). However, this assumption is not always valid. For example, Domenico (1976) demonstrated in the laboratory that the elastic properties of partially saturated rock deviate from those computed from this theory. Later, this finding was confirmed by, e.g., Cadoret (1993) and Brie (1995). In the former work, the CT scans of the partially saturated limestone samples revealed that the fluid phases were distributed in patches whose size was much larger than the individual pore size. Coincidentally, the measured velocity in these samples exceeded that predicted by the uniform saturation assumption. In the latter work, a similar situation was found in well log data, especially well pronounced in gas sands. In contrast to uniform saturation, the situation revealed by Cadoret (1993) is called patchy saturation. Several empirical (e.g., Domenico, 1976; and Brie et al., 1995) and theoretical (e.g., Dvorkin and Nur, 1998; and Sengupta, 2000) equations help relate to the elastic properties of a rock volume with patchy saturation. These equations typically produce the upper bound for the anticipated bulk modulus of partially saturated rock. The difference between these bounds (as computed, respectively, from the uniform and patchy saturation assumptions) can be large, especially in soft rock and at high . To assess this uncertainty, Knight et al. (1998) offer a process-driven theory based on the premise of capillary pressure equilibrium in wet rock subject to gas injection. This theory is physics based and does not require an a-priori assumption about fluid distribution in the pore space. In fact, it produces the fluid phase distribution in the rock as a function of water saturation. It also does not require any adjustable parameters and can directly use measurable rock properties, such as porosity, permeability, density, and the elastic P- and S-wave velocities. This theory is based on experiment-supported understanding of how fluid invades porous rock. Sen and Dvorkin (2011) have compared and contrasted the existing popular methods of fluid substitution with the theory developed in Knight et al. (1998), showing that it may be particularly useful for decreasing the uncertainty associated with large elastic bounds that may make saturation analysis difficult. In addition to understanding the fluid distribution, it is crucial to understand the effect of the fluid distribution on elastic properties and its behavior with frequency. Data for energy exploration is collected from the ultrasonic frequency (laboratory measurements) all the way up to near-zero frequency (seismic data). Porous fluids give rise to frequency dependence of velocity and amplitude attenuation due to dispersion. The measured data cannot be compared directly due to this frequency dependence. In order to correctly analyze seismic data, especially when using laboratory or well log data as a constraint, it is critical to consider how the elastic wave velocity behaves with frequency. The question still remains of how to accurately relate seismic properties to fluid saturations, particularly in reservoirs where there are fluids with significantly contrasting elastic properties (ex. Gas reservoirs, steam flood injections, CO2 injections). In addition, with current workflows, it is necessary to pre-assign an assumed fluid distribution (uniform or patchy). This can lead to significantly different saturation analysis results, and it can be difficult to determine which distribution type is most appropriate for the reservoir of interest. Here, I summarize a methodology to improve the process of fluid substitution and saturation analysis of seismic data. Preliminary work has been done by Sen and Dvorkin (2011) to show the benefits of applying the methodology established in Knight et al.(1998) to understand the elastic behavior of fluid distributions, and model behavior which falls between the uniform and patchy saturation bounds based on the capillary pressure equilibrium concept. The thesis is three fold, consisting of the following: (1) development of a rock physics model that can describe the fluid distribution within porous media and relate the behavior of elastic wave velocity with frequency, (2) use of the model to match existing laboratory data measured by Cadoret (1992) and (3) application of the workflow to a real seismic data set (BHP Macedon Reservoir) to derive a probabilistic gas saturation map. In part (1) we develop a method to incorporate frequency dependence into the original methodology from Knight et al. (1998) by introducing the concepts of measurement scale and model scale. As the procedure calls for subdividing the model reservoir, we reference the measurement scale to be the scale at which measured heterogeneity data is available (typically core or well log). The scale we wish to model then is a larger one, typically the seismic. The frequency decreases as we go from core to seismic, and can be characterized by a diffusion length, which has an inverse relationship with frequency. We employ the diffusion length to constrain which parts of the subdivided reservoir will be observed as uniform or patchy. Thus for a single reservoir, we can determine the expected relationships between velocity and saturation at any given frequency. In this section we also briefly investigate the sensitivity of the CPET workflow to spatial information, and provide a means for incorporating a variogram into the workflow. In part (2) we apply the above outlined workflow from part (1) to several laboratory measurements from Cadoret (1992). Velocity versus saturation was available for each core at 3 different ultrasonic frequencies, along with appropriate heterogeneity data. We were able to reproduce much of the observed laboratory behavior using the workflow outlined in part (1), and came away with a few particularly significant observations. First and foremost, the porosity distribution plays a major role in the curvature of the velocity versus saturation profiles. A bimodal distribution creates very different behavior as compared to a uniform distribution, all other variables held equal. Realizing this, we can glean important information about the porosity distribution simply from the shape of the curves. Additionally we find that the permeability plays a role in the transitional shape of these curves, as well as the expected irreducible water saturation. We have developed a system based on the lessons learned from matching the Cadoret (1992) data that can help a person quickly glean several insights about the reservoir in question simply from observing a single velocity-saturation curve. We also note that our workflow does appear to have a limitation, highlighted by this particular application, in that it does not appropriately account for the possibility of squirt flow. Squirt flow can occur when taking ultrasonic measurements, and for one of the Cadoret (1992) samples we were unable to obtain a match at the highest recorded frequency. While this mis-match could be the result of some other experimental procedures or a specific heterogeneity pattern at a very small scale, we suspect that the CPET workflow is unable to capture squirt flow behavior, as we were able to obtain a match for the two lower frequencies. Thus, we exercise caution in using CPET as the only model when working with very high frequency data, and recommend the possibility of coupling it with a squirt-flow model. Lastly, in part (3) we apply the CPET workflow to the BHP Macedon reservoir data set. The data consists of four angle stacks, which we used to perform both pre-stack and post-stack inversions. The CPET workflow was then used to create an impedance-saturation model at the appropriate frequency for the Macedon reservoir. This CPET model can then be used as a "key" to obtain a saturation map from velocity (and other parameters). This process created a good saturation match at the two provided well locations, with more details in variation observed in the pre-stack data analysis, likely due to the large range in angle. The Macedon data set was ideal for this kind of analysis as there were not significant changes in lithology across the reservoir, which can heavily influence observed velocities. We outline the steps and data required to create these saturation maps, and indicate the importance and effect of any given assumptions in the workflow. In addition, an error analysis is conducted to determine the sensitivity of impedance to saturation for the specific case of the Macedon.