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^

In this paper we present a method to integrate well test, production, shut-in pressure, log, core, and geological data to obtain a reservoir description for the Pagerungan field, offshore Indonesia. The method computes spatial distributions of permeability and porosity and generates a pressure response for comparison to field data. This technique produced a good match with well-test data from three wells and seven shut-in pressures. The permeability and porosity distributions also provide a reasonable explanation of the observed effects of a nearby aquifer on individual wells. As a final step, the method is compared to an alternate technique ͑object modeling͒ that models the reservoir as a two-dimensional channel. *Now with Chevron Petroleum Technology Co. **Now with DeGolyer and MacNaughton.

pressure transient data to obtain more realistic well and reservoir parameters.

Advanced drilling technology has been widely and successfully applied to construct multilateral wells in reservoirs. This paper presents several field applications of the modeling of complex well architectures. A generalized semi-analytical segmented model, accounting for multilateral well systems in commingled layered reservoirs was used in several field applications. Cases include evaluating the flow efficiency of different configurations of wells with multiple laterals, analyzing an interference test between three horizontal wells in Al Rayyan oil field offshore Qatar, and analyzing a well test in a commingled multilateral well in a multilayer reservoir in the Dos Cuadras field offshore California. The model can predict the production performance under either constant-rate or constant-pressure conditions of a well system with any number of arbitrarily oriented laterals of any length and nonuniform formation damage. The reservoir layers, with different porosities, anisotropic permeabilities and drainage areas, are non-communicating except through the wellbore. The solution is valid for large reservoirs and when no-flow or constant-pressure boundaries affect the pressure behavior. Results of applying this method in the field cases showed that the model enabled us to predict multilateral well performance, to obtain information about reservoir connectivity, and to estimate well and reservoir properties in a multilayer system. Uncertainty, due to the large number of unknown parameters in such a complex system, represents the main challenge in using this method. It is recommended to use other means (e.g. production logging and PNC log) together with pressure transient data to reduce the uncertainty. Whenever possible, each lateral should be tested individually to provide more reliable estimates of reservoir and well properties. The presented model and the lessons learned from the field applications provide engineers a tool to use transient data collected from multilateral wells in multilayer systems for reservoir characterization and performance forecast. Background Information Methods are becoming commonplace for drilling horizontal wells, slanted wells and one or more curved wells from a common central hole in producing formations. These wells will have productivities exceeding that of a single vertical well or a single horizontal well. The performance of these wells depends greatly on appropriate reservoir selection, substantial predrilling formation evaluation and optimized completion and stimulation practices. Several studies addressed pressure transient behavior of slanted, horizontal and multilateral wells. Examples include work on spatial transformation for deviated well in anisotropic reservoir (Besson 1990), multiple horizontal wells (Economides et al. 1996, Retnanto and Economides 1996, and Yildiz 2000), and segmentation technique (Gommanard and Horne 1996). There are also studies of transient solution in layered reservoirs (Kuchuk and Wilkinson 1988, Spath, et al. 1990, and Kuchuk 1991, etc.), and field applications of multilateral well in multilayer formation (Vo and Madden 1995 and Youcef, et al. 2008, etc.) A comprehensive single- and multilateral wellbore pressure response prediction model allows arbitrary positioning of the well laterals in layered anisotropic formations. Such a flexible and generalized model can be used to study several plausible scenarios, especially the economic advantages of drilling multiple laterals from the same wellbore in a layered reservoir.

This study is concerned with the steady flow of a fluid of constant density and viscosity between two eccentric cylinders of fixed parallel axes. The inner cylinder has a constant angular velocity and the outer one is stationary. Approximate solutions of the Navier-Stokes equations were obtained for the two-dimensional flow. These solutions together with an experiment were utilized to study the appearance of an eddy with separation and reattachment points occurring at the outer cylinder.