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There is a complex relationship between seismic attributes, including the frequency dependence of reflections and fluid saturation in a reservoir. Observations in both laboratory and field data indicate that reflections from a fluid-saturated layer have an increased amplitude and delayed traveltime at low frequencies, when compared with reflections from a gas-saturated layer. Comparison of laboratory-modeling results with a diffusive-viscous-theory model show that low (<5) values of the quality factor Q can explain the observations of frequency dependence. At the field scale, conventional processing of time-lapse VSP data found minimal changes in seismic response of a gas-storage reservoir when the reservoir fluid changed from gas to water. Lowfrequency analysis found significant seismic-reflectionattribute variation in the range of 15-50 Hz. The field observations agree with effects seen in laboratory data and predicted by the diffusive-viscous theory. One explanation is that very low values of Q are the result of internal diffusive losses caused by fluid flow. This explanation needs further theoretical investigation. The frequencydependent amplitude and phase-reflection properties presented in this paper can be used for detecting and monitoring fluid-saturated layers.

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Summary Forced oil-water displacement and spontaneous countercurrent imbibition are the crucial mechanisms of secondary oil recovery. Classical mathematical models of both these unsteady flows are based on the fundamental assumption of local phase equilibrium. Thus, the water and oil flows are assumed to be locally distributed over their flow paths similarly to steady flows. This assumption allows one to further assume that the relative phase permeabilities and the capillary pressure are universal functions of the local water saturation, which can be obtained from steady-state flow experiments. The last assumption leads to a mathematical model consisting of a closed system of equations for fluid flow properties (velocity, pressure) and water saturation. This model is currently used as a basis for numerical predictions of water-oil displacement. However, at the water front in the water-oil displacement, as well as in capillary imbibition, the characteristic times of both processes are, in general, comparable with the times of redistribution of flow paths between oil and water. Therefore, the nonequilibrium effects should be taken into account. We present here a refined and extended mathematical model for the nonequilibrium two-phase (e.g., water-oil) flows. The basic problem formulation, as well as the more specific equations, are given, and the results of comparison with an experiment are presented and discussed. Introduction The problem of simultaneous flow of immiscible fluids in porous media, and, in particular, the problem of water-oil displacement, both forced and spontaneous, are both fundamental to the modern simulations of transport in porous media. These problems are also important in engineering applications, especially in the mathematical simulation of the development of oil deposits. The classical model of simultaneous flow of immiscible fluids in porous media was constructed in late 30s and early 40s by the distinguished American scientists and engineers M. Muskat and M.C. Leverett and their associates.1–3 Their model was based on the assumption of local equilibrium, according to which the relative phase permeabilities and the capillary pressure can be expressed through the universal functions of local saturation. The Muskat-Leverett theory was in the past of fundamental importance for the engineering practice of the development of oil deposits, and it remains so. Moreover, this theory leads to new mathematical problems involving specific instructive partial differential equations. It is interesting to note that some of these equations were independently introduced later as simplified model equations of gas dynamics. Gradually, it was recognized that the classical Muskat-Leverett model is not quite adequate, especially for many practically important flows. In particular, it seems to be inadequate for the capillary countercurrent imbibition of a porous block initially filled with oil, one of the basic processes involved in oil recovery, and for the even more important problem of flow near the water-oil displacement front. The usual argument in favor of the local equilibrium is based on the assumption that a representative sampling volume of the water-oil saturated porous medium has the size not too much exceeding the size of the porous channels. In fact, it happens that it is not always the case and that the nonequilibrium effects are of importance. A model, which took into account the nonequilibrium effects, was proposed and developed by the first author and his colleagues4–8 ; see also Ref. 9. This model was gradually corrected, modified, and confirmed by laboratory and numerical experiments. In turn, this model leads to nontraditional mathematical problems. In this paper, the physical model of the nonequilibrium effects in a simultaneous flow of two immiscible fluids in porous media is presented as we see it now. We also relate the new asymptotic time scaling of oil recovery by countercurrent imbibition in water-wet rock (Eq. 25) to experimental data. We discuss some peculiar properties of the solutions to the capillary imbibition problem clearly demonstrating nonequilibrium effects.

We present a new robust approach to study the morphology (shapes and connectivity) of the pore space of a sedimentary rock. Our approach is based on the longestablished, fundamental concepts of mathematical morphology. In particular, we propose an efficient and stable algorithm which distinguishes between the "pore bodies" and "pore throats," and establishes their respective volumes and connectivity. Our algorithm is extensively tested on the 3D digital images of computer-generated and natural sandstones. The algorithm tests on a pack of equal spheres, for which exact results can be verified visually, confirm its stability. Computer-generated pore space images are used to investigate the impact of image resolution on the algorithm output. Presently, the proposed algorithm produces a stick-andball diagram of the rock pore space. One of distinctive features of our approach is that no image thinning is applied. Instead, the information about the skeleton is stored through the maximal balls associated with each voxel. These maximal balls retain information about the entire pore space. Comparison with the results obtained by a thinning procedure preserving some topological properties of the pore space shows that our method produces more realistic estimates of the number and shapes of pore bodies and pore throats, and the pore coordination numbers. Based on the information about the maximal ball distribution, we simulate mercury injection and compute a dimensionless drainage capillary pressure curve. We demonstrate that the calculated capillary pressure curve is a robust descriptor of the pore space geometry and, in particular, can be used to determine the quality of computer-based rock reconstruction.

Synchrotron-based X-ray microtomography (micro CT) at the Advanced Light Source (ALS) line 8.3.2 at the Lawrence Berkeley National Laboratory produces threedimensional micron-scale-resolution digital images of the pore space of the reservoir rock along with the spacial distribution of the fluids. Pore-scale visualization of carbon dioxide flooding experiments performed at a reservoir pressure demonstrates that the injected gas fills some pores and pore clusters, and entirely bypasses the others. Using 3D digital images of the pore space as input data, the method of maximal inscribed spheres (MIS) predicts two-phase fluid distribution in capillary equilibrium. Verification against the tomography images shows a good agreement between the computed fluid distribution in the pores and the experimental data. The model-predicted capillary pressure curves and tomography-based porosimetry distributions compared favorably with the mercury injection data. Thus, micro CT in combination with modeling based on the MIS is a viable approach to study the porescale mechanisms of CO 2 injection into an aquifer, as well as more general multi-phase flows.

Our studies of the underlying fundamental gas-recovery mechanisms from shale gas are motivated by expectations of the increasing role of shale gas in national energy portfolios worldwide. We use pore-scale analysis of reservoir shale samples to identify critical parameters to be employed in a gas-flow model used to evaluate wellproduction data. We exploit a number of 3D-imaging technologies to study the complexity of shale pore structure: from lowresolution X-ray computed tomography (CT) to focused ion beam and scanning electron microscopy (FIB/SEM). We observe that heterogeneity is present at all scales. The CT data show fractures, thin layers, and density heterogeneity. The nanometer-scale-resolution FIB/SEM images show that various mineral inclusions, clays, and organic matter are dispersed within a volume of few-hundred lm 3 . Samples from different regions differ sharply in the shape, size, and distribution of pores, solid grains, and the presence of organic matter. Although the samples have clearly distinguishable signatures related to the regions of origin, extremely low permeability is a common feature. This and other pore-scale observations suggest a bounded-stimulated-domain model of a horizontal well within fractured shale that accounts for both compression and adsorption gas storage. Using the method of integral relations, we obtain an analytical formula approximating the solution to the nonlinear pressure diffusion equation. This formula makes fast and simple evaluation of well production possible without resorting to complex computations. It defines a decline curve, which predicts two stages of production. During the early stage, the production rate declines with the reciprocal of the square root of time, whereas later, the rate declines exponentially. The model has been verified by successfully matching monthly production data from a number of shale-gas wells collected over several years of operation. Under appropriate conditions, scaling can collapse the data from multiple wells on a single type curve. Pore-scale image analysis and the mesoscale model suggest a dimensionless adsorption-storage factor (ASF) to characterize the relative contributions of compression and adsorption gas storage.

Summary For many rocks of high economic interest such as chalk, diatomite, shale, tight gas sands, or coal, a submicron-scale resolution is needed to resolve the 3D pore structure, which controls the flow and trapping of fluids in the rocks. Such a resolution cannot be achieved with existing tomographic technologies. A new 3D imaging method based on serial sectioning, which uses the focused-ion-beam (FIB) technology, has been developed. FIB technology allows for the milling of layers as thin as 10 nm by using accelerated gallium (Ga+) ions to sputter atoms from the sample surface. After each milling step, as a new surface is exposed, a 2D image of this surface is generated, and the 2D images are stacked to reconstruct the 3D pore structure. Next, the maximum-inscribed-spheres (MIS) image-processing method computes the petrophysical properties by direct morphological analysis of the pore space. The computed capillary pressure curves agree well with laboratory data. Applied to the FIB data, this method generates the fluid distribution in the chalk pore space at various saturations. Introduction Field-scale oil-recovery processes are the result of countless events happening in individual pores. To model multiphase flow in porous media at pore scale, the resolution of the 3D images must be adequate for the rock of interest. Chalk formations in the oil fields of Texas, the Middle East, the North Sea, and other areas hold significant oil reserves. The extremely small typical pore sizes in chalk impose very high requirements on imaging resolution. In the last decade, X-ray microtomography has been used extensively for direct visualization of the pore system and the fluids within sandstone (Jasti et al. 1993; Coles et al. 1998; Wildenschild et al. 2003; Seright et al. 2003). While this approach is fast and nondestructive, its applicability is limited mostly to micron resolutions, although recent developments are bringing the resolution to submicron range (Stampanoni et al. 2002). For chalk pore systems, which are characterized by submicron- to nanometer-length scales, 3D stochastic methods based on 2D scanning-electron-microscope (SEM) images of thin sections have been used to reconstruct the pore system (Talukdar et al. 2001). The advent of FIB technology has it made possible to reconstruct submicron 3D pore systems for diatomite and chalk (Tomutsa and Radmilovic 2003) (Fig. 1). FIB technology is used in microelectronics to access individual components with nanoscale accuracy for design verification, failure analysis, and circuit modification (Orloff et al. 2002). FIB has been used in material sciences for sectional sample preparation for SEM and for 3D imaging of alloy components (Kubis et al. 2004). In earth sciences, the FIB also has been used for sample preparation for SEM and to access inner regions for performing microanalysis (Heaney et al. 2001).To access the pore structure at submicron scale, the FIB mills successive layers of the rock material as thin as 10 nm. As successive 2D surfaces are exposed, they are imaged with either the electron or the ion beam. After processing, the images are stacked to reconstruct the 3D pore structure. The geometry of the pore space of the obtained structure can be analyzed further to estimate petrophysical rock properties through computer simulations. To analyze the 3D chalk images obtained by the FIB method, we applied the MIS technique (Hazlett 1995; Silin et al. 2003, 2004; Silin and Patzek 2006). The MIS method analyzes the 3D pore-space image directly, without construction of pore networks. It bypasses the nontrivial task of extracting a simple but representative network of pore throats linking pore bodies from the 3D data (Lindquist 2002). Moreover, the pore-network extraction methods, which are based on relatively simple grain and pore shapes in sandstones (Øren and Bakke 2002), may not always be feasible for the complex pore structures of carbonates. Although a pore-network-based flow-modeling approach enjoyed a significant interest from the researchers and resulted in theoretically and practically sound conclusions (Øren et al. 1998; Xu et al. 1999; Patzek 2001; Blunt 2001), we believe that direct pore-space analysis deserves more attention. In addition, direct analysis of the pore space provides an opportunity to study alteration of the rock flow properties (e.g., those resulting from mechanical transformations or mineralization) (Jin et al. 2003).