Summary An improved technique based on the concept of hydraulic flow units or hydraulic units (HU's) is presented to calculate permeability distribution in uncored wells. Graphical probability methods, nonlinear regression, and the Ward's analytical algorithm are presented to perform cluster analysis on core data and identify prevailing HU's in a formation. A Bayesian-based probabilistic approach is discussed next to estimate HU's and permeability distributions in logged wells. This is an inverse problem that requires constructing an a priori training database to capture implicit relationships between core-derived HU's and various well log measurements. Information from the database is drawn statistically to infer HU distributions at target logged wells. The method is applied to two heterogeneous reservoirs, a carbonate formation, and a laminated sandstone formation. Comparisons of permeabilities calculated by the HU approach and other conventional techniques are provided that demonstrate the usefulness of the HU method. Introduction Estimation of permeability in uncored but logged wells is a generic problem common to all reservoirs. Any field-scale reservoir characterization study inevitably requires knowledge of petrophysical properties at drilled wells for its starting point. Therefore, scientifically sound and geologically compatible procedures must be sought to allow for reliable calculation of permeability distributions in wells. Traditional approaches for estimation of permeability are based either on simple linear regressions or empirical inferences that use correlations among various well log responses. Usually, these empirically inferred models apply locally because there may be large differences in depositional characteristics at other locations. The regression method assumes that a linear relationship exists between core porosity and the logarithm of core permeability. Another linear regression establishes dependency between measured core porosity and well log-derived porosity. These two regression models, when combined, allow calculation of permeability at logged wells. The method explicitly ignores the scatter of data about the regression lines and implicitly attributes any scatter to measurement errors or second-order fluctuations in reservoir characteristics. A partial improvement to this method is achieved by first identifying lithological categories of the formation1–3 and then calculating linear regression lines for petrophysical core measurements that belong to each lithology class. Because the regression methods smooth data, the predicted permeability values from these regression models lack the variability seen on actual core data. For this reason, probability field simulations may be applied to add stochastic fine-scale variations to the predicted permeability values.4,5 A more reasonable approach for prediction of permeability is to address the development of permeability in reservoir rocks from fundamentals of geology and physics of flow at pore network scale. This is best achieved by attributing the nature of interdependency between permeability and porosity to geological variations in reservoir rock and by seeking functional relationships for permeability that capture geological controls on flow properties. This will require establishing causal relationships between core-derived pore-throat parameters and log-derived macroscopic petrophysical attributes.6 Such relationships are achieved best if rocks of similar fluid conductivity are identified and grouped together. Each grouping is referred to as an HU. The HU's do not necessarily correspond to lithofacies, which have been referred to as geological flow units.4 Petrophysical properties are controlled by both depositional characteristics, such as grain-size and sorting, and by diagenetic features, such as the amount and type of cement or clay minerals. Thus, an HU involves more than a genetic facies of a depositional system. Generally, the variability of petrophysical properties is large among the HU's and low within them. Permeability calculation by HU's offers an improved estimation over traditional regression-based averaged relationships by incorporating geology and fluid-flow principles. The extent of improvement depends mainly on the geological characteristics of the formation, where larger enhancements are usually achieved in more heterogeneous deposits. In any case, the method can describe permeability variations in a reservoir more comprehensively and realistically because it integrates geology, petrophysics, and engineering. This paper presents an improved technique for permeability estimation using HU's. The basic concept of HU's is first reviewed, and the theoretical basis for their classification in cored wells is provided. Next, a statistically based inference method is presented to identify HU's in uncored but logged wells and thereafter to calculate permeability distributions. Lastly, the applicability of the improved permeability estimation technique to a carbonate and a sandstone reservoirs is illustrated. Method Concept of HU's. An HU is defined as the representative volume of total reservoir rock within which geological properties that control fluid flow are internally consistent and predictably different from properties of other rocks. Thus, a flow unit is a reservoir zone that is continuous laterally and vertically and has similar flow and bedding characteristics.7 HU's are related to geological facies distributions but do not necessarily coincide with facies boundaries. The parameters that influence fluid flow are mainly pore-throat geometrical attributes. The pore geometry is in turn controlled by mineralogy (type, abundance, location) and texture (grain size, grain shape, sorting, packing). Various combinations of these geological properties can lead to distinct rock flow units that have similar fluid transport properties. Therefore, an HU can include several rock facies types, depending on their depositional texture and mineralogical content. The grouping of rocks based on their fundamental geological flow attributes is the basis of HU classification. If a porous medium is simulated as a bundle of straight capillary tubes, the following expression for rock permeability is obtained by combining Darcy's law for flow in porous media and Poiseuille's law for flow in tubes8:Equation 1 Concept of HU's. An HU is defined as the representative volume of total reservoir rock within which geological properties that control fluid flow are internally consistent and predictably different from properties of other rocks. Thus, a flow unit is a reservoir zone that is continuous laterally and vertically and has similar flow and bedding characteristics.7 HU's are related to geological facies distributions but do not necessarily coincide with facies boundaries. The parameters that influence fluid flow are mainly pore-throat geometrical attributes. The pore geometry is in turn controlled by mineralogy (type, abundance, location) and texture (grain size, grain shape, sorting, packing). Various combinations of these geological properties can lead to distinct rock flow units that have similar fluid transport properties. Therefore, an HU can include several rock facies types, depending on their depositional texture and mineralogical content. The grouping of rocks based on their fundamental geological flow attributes is the basis of HU classification. If a porous medium is simulated as a bundle of straight capillary tubes, the following expression for rock permeability is obtained by combining Darcy's law for flow in porous media and Poiseuille's law for flow in tubes8:Equation 1

Summary This paper presents an interpretation method for injectivity and falloff testing in a single-layer oil reservoir that is under waterflooding and develops analytical solutions for pressure and saturation distributions. The effects of relative permeability, wellbore storage, and skin are considered in these solutions. New field-dependent type curves for falloff tests, which exhibit features that do not appear in the currently available single-phase-flow type curves, are also presented. Matching of field data on these curves yields fluid mobilities in various banks, skin, formation permeability, and flood-front location. Field data interpretation with the new method shows that falloff tests can be used to monitor the progress of waterfloods. Introduction Numerous waterflooding projects are now under way throughout the world to increase oil recovery. In large oil fields, particularly in offshore reservoirs, water injection is initiated during the early stages of reservoir development. Because of differences in oil and water properties, a saturation gradient is established in the reservoir soon after injection begins, forming a region of high water saturation around the wellbore. Outside this region, water saturation decreases as we move away from the wellbore until the flood front is reached. The oil bank with initial water saturation is located ahead of the injection front, The fluid mobility in each bank differs from those in the surrounding banks. The knowledge of variation of mobilities and saturations in the reservoir is needed to model the reservoir effectively and to conduct waterflooding operations properly. Pressure-transient testing, often in the form of falloff tests, can provide valuable information about the parameters of an injection scheme. These tests are usually run to detect near-wellbore damage, to provide interwell average reservoir pressure, and to determine formation permeability. Proper analysis of falloff tests can lead to determining saturation distribution around the injection wells, to monitoring movement of fluid banks, and to evaluating the well injectivity and average reservoir pressure as they change with time. Several models have been proposed for the analysis of falloff tests. Almost all are for two-bank systems that assume that the injected fluid displaces the formation fluid in a piston-like manner. Therefore, saturation gradients within each bank are not considered, and the mobility and compressibility in each bank are assumed constant. Abrupt changes in properties occur at the interface of the banks. Such models are often inadequate for the interpretation of falloff tests because they ignore fluid mobility and diffusivity variations in the reservoir. Weinstein examined pressure-falloff data with a numerical model including the relative permeability and dependence of viscosity on temperature. He investigated only cases with very favorable mobility ratios, representing essentially piston-like displacements. Sosa et al. considered the effect of saturation distribution in the flooded region on water-injection falloff tests. They used a radial numerical simulator to account for the relative permeability characteristics of the porous medium. Their study showed that the existence of the transition bank between the oil and single-phase water banks had noticeable effects on the falloff data. The study provided some qualitative information about the water-flooding system, but did not provide the analysis procedure for the interpretation of falloff data. In this paper, we first examine the two-bank system with a step change in saturation and illustrate its features. Next, we extend the study to the case with a region of variable saturation around the injector. The paper presents an interpretation method for falloff tests that allows a reservoir engineer to calculate the parameters of an injection system. Finally, we present field data to demonstrate the application of the proposed interpretation method. Two-Bank Falloff Solution Fig. 1 is a schematic of the two-bank system in an infinite reservoir. The fluid properties are constant within each bank, but change sharply at the bank interface. The following assumptions are made in the modeling of pressure transients:the reservoir is homogeneous and isotropic,the formation consists of a single layer with constant thickness,fluids are slightly compressible,flow is isothermal, andgravitational effects are negligible. Therefore, the diffusivity equation in terms of pressure describes the flow within each bank. Exact Falloff Solution. Two solutions are presented for the falloff period in Appendix A; one assumes that the interface remains stationary upon shut-in and the other allows for its movement. The assumption of stationary interface is generally acceptable becausefluid compressibilities are small-hence, volumetric expansion or compression of fluids is negligible;the first bank is often large at the time of shut-in-therefore, any volume change expressed in terms of radial distance produces a negligible change in the location of the interface; andthe duration of a falloff test is often short relative to the injection time-hence, any movement of the interface during the test is small. Comparisons between the stationary- and moving-interface solutions, which are presented in Results and Discussion, show that the two solutions produce virtually the same results. The stationary-interface solution implies that the falloff period corresponds to pressure decay in the radially composite reservoir that is formed at the end of the injection period. By definition, a composite reservoir refers to a system that consists of two stationary regions with differing properties. The pressure distribution at the beginning of the falloff in this composite system is nonuniform and is given by the injection solution at the time of shut-in. Verigin presented exact solutions for the pressure distribution in a two-bank system during injection. His solutions are given by Eqs. A-9 through A-13 and are used as the initial condition in the derivation of the falloff solution. The falloff solution (Eqs. A-23 and A-24) is converted to real-time space by the Stehfest algorithm. Combining the stationary-interface solution (Eq. A-23) with the velocity relationship at the interface (Eq. A-25) produces the moving-interface solution (Eq. A-26). Approximate Falloff Solution by Superposition. The moving-boundary condition during the injection phase introduces nonlinearity into the problem. Therefore, the principle of superposition generally may not be used to generate the falloff solution from that of the injection period. A superposition based on the single-phase injectivity solution of the composite-reservoir model, however, may be used because the two-bank system resembles a radially composite reservoir during falloff. This approach results in an approximate falloff solution: (1) The pcD terms on the right side of Eq. 1 represent single-phase injectivity solutions of the radially composite reservoir. Eq. 1 satisfies the governing partial-differential equations and boundary conditions of the falloff problem. The initial condition is satisfied only if the single-phase composite-reservoir injectivity solution and the two-bank injection solution 1 are identical at the time of shut-in. This amounts to approximating the solution of a moving-boundary problem with a stationary model. SPERE P. 115^

We have developed a phenomenological model for critical condensate saturation. This model reveals that critical condensate saturation is a function of surface tension and contact angle hysteresis. On the other hand, residual oil saturation does not have such a dependency. Consequently, the selection of fluids in laboratory measurements for gas condensate systems should be made with care. Introduction Gas condensate reservoirs are becoming increasingly important; many new hydrocarbon reservoirs found in recent years are gas condensate reservoirs. These reservoirs, from a recovery and deliverability standpoint, may have significant differences from oil reservoirs. Prior to reaching the dew point pressure, the flow of gas in porous media is similar to that of undersaturated oil. When the pressure, either in the wellbore, or in the reservoir, drops below the dew point pressure, a new liquid condensate phase appears. The saturation of the new phase has to reach a threshold value in order to become mobile, provided the surface tension is higher than a critical value. This threshold value is called critical condensate saturation.

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This paper presents fine-scale numerical simulations and mathematical analysis of the empirical foam model for representing foam-surfactant flow in a vertical column of laboratory sand-pack based on two sets of experimental data conducted at variable total velocities and variable foam qualities.The empirical foam model of CMG-STASRS is used for parametric matching of laboratory data, and relevant foam parameters are calibrated.The paper discusses experimental setup, procedure and measurements to provide apparent foam viscosity data needed for foam modeling. In the first set of lab tests, foam quality is constant and the total fluid superficial velocity varies for foam shear thinning effect; while in the second tests, foam quality is varied at a fixed total superficial velocity to capture different flow regimes and foam dry-out characteristics.Employing an analytical method and 1-D numerical simulations of the foam flow in the sand-pack, the empirical foam model is tuned to the first data set of variable velocity and used to predict the second data set of variable quality as a consistency check.The model predictions for the second data set as well as the associated sensitivity analysis prove that the foam modeling procedure of this paper is unique and applicable for large-scale predictions.