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Abstract Existing pressure test solutions for naturally fractured reservoirs neglect the macroscopic pore system permeability, the wellbore storage and skin factors of the observation wells, and are limited either to single-wells in infinite domains or to concentric wells in circular reservoirs. In this paper, two solutions for interpreting interference pressure tests in naturally fractured reservoirs are presented. pressure tests in naturally fractured reservoirs are presented. The model on which these solutions are based incorporates both macroscopic fracture and pore system permeabilities, an improved unsteady interporosity flow model, and allows for wellbore storage and skin factors of the wells. In the first solution, the pressure response of a shut-in observation well to arbitrarily pressure response of a shut-in observation well to arbitrarily varying unsteady pressure fields is considered under the assumption that the observation well affects the flow field only locally. In the second solution, a general technique for exactly solving the governing equations for fluid flow in naturally fractured reservoirs with multiple non-concentric circular boundaries is first developed. The procedure is then specialized to two circular boundaries to determine the pressure changes induced in a circular reservoir by an eccentric flowing well with wellbore storage and skin. These two solutions, used together, will enable the development of pressure tests for determining the location of the center of a circular reservoir In addition to its radius and its hydraulic properties. Introduction Transient pressure tests provide the most reliable information that can be acquired on the in-situ hydraulic properties of petroleum reservoirs. However, it is also well recognized that interpretation of pressure test data can be non-unique and that different kinds of reservoir heterogeneities can result in similar behaviour. For this reason, all available data of different types are used to guide test design and selection of an interpretive model that best fits the situation. Propelled by the belief that if all known features are faithfully represented the error In interpretation would be diminished, the trend in pressure test analysis has been to develop improved physical models to describe the impact of intrinsic reservoir heterogeneities such as naturally occuring fracture networks and to improve the capability to represent and detect flow boundaries within the reservoir. This paper is a step in this direction and represents an attempt to paper is a step in this direction and represents an attempt to combine an improved model for the intrinsic response of fractured reservoirs with additional flexibility in locating the external reservoir boundary. The foundation for the study of fluid flow in naturally fractured porous media was laid by Barenblatt et. al. They formulated two coupled linear diffusion equations governing the evolution of the average fracture and pore system pressures. In their general theory, they proposed the pseudo-steady interporosity flow approximation and allowed for both the fracture and pore system macroscopic permeabilities. Kazemi et. al., extending Warren's and Root's earlier work, applied Barenblatt's theory to the limiting case of isolated matrix blocks, isotropic fracture permeability, and pseudo-steady interporosity flow, and developed a line source solution for interpreting buildup tests without skin and wellbore storage. They presented exact solutions in the Laplace transform and time domains for this case. P. 311
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Productivity estimates in horizontal wells are subject to more uncertainty than comparable estimates in vertical wells. Further, it is much more difficult to interpret well test data because of 3D flow geometry. The radial symmetry usually present in a vertical well does not exist. Several flow regimes can potentially occur and need to be considered in analyzing test data from horizontal wells. Wellbore storage effects can be much more significant and partial penetration and end effects commonly complicate interpretation. In vertical wells, variables such as average permeability, net vertical thickness, and skin are used. Horizontal wells need more detail. Not only is vertical thickness important, but the horizontal dimensions of the reservoir, relative to the horizontal wellbore, need to be known. Evaluation of data from a vertical wellbore will generally center on a single flow regime, such as infinite-acting radial flow, known as the MTR. However, a pressure-transient test in a horizontal well can involve as many as five major and distinct regimes that need to be identified. These regimes may or may not occur in a given test and may or may not be obscured by wellbore storage effects.
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Abstract The existing methods for evaluating the well productivity index are based on solution of transient problems. One approach is to consider the single well problem in infinite domain and subsequently apply the method of images. This puts restrictions on the geometry of the well and of the drainage volume. Another approach is to solve the transient problem in the bounded domain for late times. While in this case restrictions on the well geometry are less severe, the shape of the drainage volume is still limited to the simplest ones. In addition, for the constant rate case highly accurate wellbore pressures, for the constant pressure case highly accurate wellbore rates are required and that puts an extreme computational burden on the semi-analytical or numerical methods involved. Even with the most powerful methods and hardware available, the calculation of the productivity index of directionally drilled and partially penetrating wells, especially in more complex drainage volumes is a formidable task. An additional problem is that in general, the productivity index for a well produced under constant pressure condition is different, although very close, from the productivity index of a well produced with constant rate. In this work we present a new technique to evaluate single well productivity indices both for constant pressure and for constant rate conditions. The approach is based on the solution of two steady state boundary value problems with constant pressure prescribed on the wellbore. The two productivity indices, (for constant rate and constant wellbore pressure, respectively) are then computed as integral characteristics of the solutions of the corresponding time dependent boundary value problem. The two productivity indices are computed independently. The method can be applied to any geometry of the reservoir (both 2-D and 3-D, regular or irregular), and any direction and length of penetration of the well. Designer wells (with a freely prescribed path) can be also considered. Introduction We consider a bounded reservoir with no flow outer conditions. The fluid is single phase, slightly compressible. A well producing with either constant pressure or constant rate is characterized by the productivity index defined as [19]:Equation 1 where q(t) is the production rate, pw(t) is the flowing bottomhole pressure and pa(t) is the average reservoir pressure. We are particularly interested in the stabilized (late time) value of the PI. For the constant production rate stabilization means that the difference of average and wellbore pressure (the denominator) becomes time invariant. This flow regime is called pseudo-steady state (PSS). In the case of constant wellbore pressure, both the numerator and denominator keep changing with time, but their ratio stabilizes, leading to the flow regime called boundary-dominated (BD).
Abstract This paper presents new analytical solutions for two-well systems with storage and skin in both wells. The solutions are generated by the application of the addition theorem and the Laplace transformation. Previously presented research in this area is limited to one-dimensional Previously presented research in this area is limited to one-dimensional radial flow. In contrast, the new solutions presented in this paper are true two-dimensional, and do not make use of the superposition principIe for solving the basic cases. Hence, limitations stemming from the principIe for solving the basic cases. Hence, limitations stemming from the incorrect application of the superposition principle in the presence of finite sources are overcome. The paper describes the mathematical forulation for sixteen different configurations and some calculations showing the pressure transient response of both the active well and the interference well. Thc results of this study show that under certain conitions, storage in the observation well may have a significant effect on the analysis procedure. The presented solutions show that an impermeable linear boundary has a significant effect on the pressure response of a slug test. In addition, the traditional constant rate test with storage and skin can also be used for linear boundary detection. However, current techniques are limited in that the boundary effects must occur after the storage effects have ended. The new solutions presented here show the combined effects of an impermeable boundary and wellbore storage and skin. Introduction Interference testing has been a subject of great interest in the fields of groundwater hydrology and petroleum engineering for many years. As early as 1935, Theis matched interference pressure responses in an aquifer using the constant rate line source solution. There are few papers published about the effects of storage and skin in the observation well on transient pressure testing. Two papers on constant rate tests with storage and skin in the observation well were pubished by Tongpenyai and Raghavan and by Ogbe and Brigham. pubished by Tongpenyai and Raghavan and by Ogbe and Brigham. Yet, other kinds of tests such as interference slug tests or interference constant pressure tests have never been considered with storage and skin in the observation well. Linear boundary detection, be it a sealing or a constant pressure boundary, has also been a subject of great importance and discussion in the literature. There is no attempt to present a complete discussion of all the literature about linear boundary detection, only a few papers are mentioned. As early as 1952, Stallman published a set of type curves for a constant rate line source well near a linear boundary. Other classical papers concerning linear boundary detection were authored by Ferris et al (1962), Davis and Hawkins (1963), Vela (1977), and Tiab and Crichlow (1979). Most of the papers dealing with linear boundary detection, be it for active well or interference well analysis, apply the method of images, or superposition of line sources in space and time to generate the boundary effects on the pressure responses. This method is mathematically correct only if the wells are constant rate line source wells. In other words, the wells cannot have wellbore storage associated with them, and may not be finite in radius. Hence, the lack of any literature on even the simplests realistic case of a constant rate finite radius well with wellbore storage near a linear boundary. This case can only be handled today, from a practical point of view, if wellbore storage effects are small, and if they do not distort the boundary effects. In other words, wellbore storage effects must be negligible at the time that the linear boundary effects become prominent. There are also several other simple well-reservoir prominent. There are also several other simple well-reservoir configurations that have not been considered, such as a constant pressure well near a linear boundary, or a slug test near a linear pressure well near a linear boundary, or a slug test near a linear boundary. In all the three linear boundary cases described above, there is no literature about interference wells. Common to all these cases is that the mathematical descriptions of them cannot be assembled using the simple superposition technique, and true two-dimensional solutions are required. The purpose of this paper is to present a new solution technique for several well-reservoir configurations and show the significance of these new solutions by presenting some characteristic pressure responses of these configurations. The solutions of the following cases are presented:Line source rate well - no-flow linear boundaryLine source rate well - constant pressure linear boundaryLine source rate well - observation well with storageLine source rate well - observation well with storage and skinFinite radius rate well - no-flow linear boundaryFinite radius rate well - constant pressure linear boundaryFinite radius rate well - observation well with storageFinite radius rate well - observation well with storage and skinStorage and skin rate well - no-flow linear boundaryStorage and skin rate well - constant pressure linear boundary Storage and skin rate well - observation well with storage Storage and skin rate well - storage and skin observation well P. 221
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In the analysis of field data it is important to bear in mind the proper relationship between the radial flow period and the linear proper relationship between the radial flow period and the linear flow period. The transition between these periods is quite long and can be misinterpreted as being the linear flow period. Another factor that can complicate analysis is turbulence. Introduction The trend in gas well testing has been to rely more on be early-time flow data of drawdown and buildup tests than on stabilized flow tests. The stabilized testing methods often are not adequate for complicated modem gas reservoirs because many of these reservoirs have extremely low permeabilities and the transient flow period of a well test can last for days or weeks. It has become common to augment the flow capacity of gas wells by hydraulically fracturing their producing formations. In deep reservoirs the induced producing formations. In deep reservoirs the induced fractures are generally vertical and tend to follow a single plane of weakness. The presence of a vertical fracture at the wellbore complicates the transient flow behavior of a low permeability gas well. The flow is further complicated when turbulence occurs near the wellbore. Russell and Truitt published transient drawdown solutions for vertically-fractured liquid wells. They developed methods of drawdown and buildup testing utilizing these solutions, which were based on numerical simulation. Clark applied the basic Russell-Truitt solutions to analyze fractured water injection wells by falloff tests. Field examples were given to substantiate the method of analysis. The Russell-Truitt solutions and well testing methods can be extended to gas well flow. Millheim and Cichowicz presented the drawdown equations for ideal gas flow, including the effects of formation damage and turbulence. Two actual field cases were presented, both of which exhibited extensive fractures presented, both of which exhibited extensive fractures and turbulent flow. For the pressure range of these well tests, the use of ideal gas equations was acceptable. These field cases were significant in being the first published data showing the occurrence of turbulence in fractured gas wells. We, also, observed such cases in practice. Our purpose here is to extend the theory of fractured gas well testing to the flow of real gases. To determine the effects of wellbore storage and turbulence on well test interpretation, we developed a finite-difference model to simulate well test conditions. Since gas wells usually are widely spaced and have high compressibility, the emphasis has been put on the early transient behavior before the effects of the outer boundary are noticeable at the wellbore. The Mathematical Model The geometry of our mathematical model is similar to that used by Prats. The well is centered in a circular uniform reservoir. A vertical fracture of infinite flow capacity penetrates the formation and passes through die wellbore. The wellbore itself is passes through die wellbore. The wellbore itself is not important. Fig. 1 shows a sketch of the problem. Fig. 2 shows the idealized model. JPT P. 625