The effect of a sand-filled vertical fracture of limited radial extent and finite capacity (fracture capacity is the product of the permeability and width of the fracture) on the flow behavior of a cylindrical reservoir producing an incompressible fluid through a centrally located well has been investigated mathematically. The shape of the lines of equal pressure near the fracture is essentially independent of the size of the reservoir, provided that the field radius is of the order of the fracture length or larger. For reasonable values of production rates and of fluid, reservoir and fracture properties, the total pressure drop between the end of the fracture and the well is generally negligible compared with the pressure drop in the reservoir. With regard to production response, the effect of vertical fractures can be represented by the production response of an equivalent or effective well radius. For a high-capacity fracture, the effective well radius is a quarter of the total fracture length, decreasing with the fracture capacity. When invasion effects are simulated by decreasing the width of the damaged zone with distance from the well, the effect of formation damage around a fracture on the production response is not so serious as indicated by the literature. This suggests that frac fluids with a conventional filter-loss response are better than high-spurt-loss frac fluids, provided the effective permeability of the damaged zone is the same. Introduction This paper considers the effect of the fracture capacity, as well as the formation damage which can result from fracture treatments, on the productive capacity of vertically fractured wells. Other publications, notably those of van Poollen, consider these same effects. In addition to providing more general results for vertical fractures than are available from the literature, the present paper gives the equivalent well radius of fractures having different lengths and Capacities and, also, includes pressure distributions in and around the fractures. The effect of a damaged zone around a fracture on the production response was not found to be so great as that reported by van Poollen. This difference probably stems from the fact that we consider a damaged zone which is widest (but is still small) near the well and thins out toward the extremities of the fracture, whereas van Poollen considers a damaged zone having a uniform width for the entire fracture length. Simple, but adequate, equations which describe the effect of these variables on production response are presented (in Appendixes A and B). Thus, results can easily be extended to values of the variables not specifically considered here.
The critical Rayleigh number characterizing the onset of instability in porous mediums due to a positive gradient of temperature with depth is independent of the magnitude of a horizontal fluid current. However, the presence of a horizontal fluid current will cause temperature and velocity fluctuations with time. Recognition of these oscillations is necessary for the proper interpretation of phenomena occurring in industrial operations and various natural processes.
The pressure and production behavior of a homogeneous cylindrical reservoir producing a single fluid through a centrally located vertical fracture of limited lateral extent was determined by using mathematical methods to solve the appropriate differential equation. It is assumed that there is no pressure drop within the fracture - that is, that the fracture capacity is infinite. It was found that the production-rate decline of such a reservoir is constant (except for very early times) when the flowing bottom-hole pressure remains constant. The production-rate decline increases as the fracture length increases. Thus, the lateral extent of fractures can be determined from the production-rate declines before and after fracturing or from the decline rate after fracturing when the properties of the formation and fluids are known. The production behavior over most of the productive life of such a fractured reservoir can be represented by an equivalent radial-flow reservoir of equal volume. The effective well radius of this equivalent reservoir is equal to one-fourth the total fracture length (within 7 per cent); the outer radius of this equivalent reservoir is very nearly equal (within 3.5 per cent) to that of the drainage radius of the fractured well. The effective well radius of a reservoir producing at semisteady state is also very nearly equal to one-fourth the total fracture length. It thus appears that the behavior of vertically fractured reservoirs can be interpreted in terms of simple radial-flow reservoirs of large wellbore. Introduction An earlier report has considered the effect of a vertical fracture on a reservoir producing an incompressible fluid. That investigation of the fractured reservoir producing an incompressible fluid was started because of its simplicity. Thus, pertinent behavior of fractured reservoirs was obtained at an early date, while experience was being gained of value in the solution of more complicated fracture problems. One of these more complicated problems, and the one discussed in this report, considers the effect of a compressible fluid (instead of incompressible fluids) on the production behavior of a fractured reservoir. In the incompressible-fluid work mentioned, it was shown that the production rate after fracturing could be described exactly by an effective well radius equal to one-fourth the fracture length whenever the pressure drop in the fracture was negligible. Because of the simplification in interpretation, it is a matter of much interest to determine whether the production behavior of reservoirs producing a compressible liquid could be described in terms of an effective well radius which remains essentially constant over the producing life of the field. The details of the mathematical investigation are given in the Appendixes. IDEALIZATION AND DESCRIPTION OF THE FRACTURED SYSTEM It is assumed that a horizontal oil-producing layer of constant thickness and of uniform porosity and permeability is bounded above and below by impermeable strata. The reservoir has an impermeable circular cylindrical outer boundary of radius r e. The fracture system is represented by a single, plane, vertical fracture of limited radial extent, bounded by the impermeable matrix above and below the producing layer (reservoir). It is assumed that there is no pressure drop in the fracture due to fluid flow. Fig. 1 indicates the general three-dimensional geometry of the fractured reservoir just described. When gravity effects are neglected, the flow behavior in the reservoir is independent of the vertical position in the oil sand. Thus, the flow behavior in the fractured reservoir is described by the two-dimensional flow behavior in a horizontal cross-section of the reservoir, such as the one shown in Fig. 2. SPEJ P. 87^
This paper identifies and assesses the importance of phenomena common to thermal recovery processes, discusses the extent of current operations, and anticipates the potential of these processes. The status of in-situ combustion, cyclic steam injection, and steam drive also is reported. Introduction Thermal recovery processes have been used extensively since the early 1950's. Emphasis during that first decade was on the in-situ combustion process, with the use of steam (in both cyclic injection and drives) coming of age in the following decade. These processes are by far the most commonly used thermal recovery methods, and this paper briefly discusses their status. paper briefly discusses their status. I assume that the reader is at least somewhat familiar with these thermal recovery processes, and if not that he can learn about them from the published literature. Selected references on the development, applications, appraisals, and reviews of each process are given in the references. It is not my intention process are given in the references. It is not my intention to discuss what is well established. Here, the emphasis is onidentifying and assessing the importance of phenomena common to thermal recovery processes, phenomena common to thermal recovery processes,discussing the extent of current thermal recovery operations, both in an absolute sense as well as relative to enhanced recovery processes other than waterflooding, andanticipating the potential of thermal recovery processes. processes. Brief Description of the Thermal Recovery Processes Processes In-Situ Combustion In-situ combustion (ISC), also known as underground combustion (UC), underground combustion drive (UCD), and fireflooding, is the name applied to a broad class of recovery processes in which part of the crude oil is burned in the reservoir. Air almost invariably is injected to support combustion processes. But these terms also are used to denote die dry forward combustion process, in which only air is injected and in which the process, in which only air is injected and in which the combustion front moves in the same direction (cocurrent) as the injected air.Variations of the dry forward combustion process include reverse combustion (also dry, with the combustion front moving countercurrent to the direction of the injected air) and wet combustion (a cocurrent process in which water is injected with the air). A number of terms are used to suggest the ratio of water-to-air injection rates in the wet-combustion process. Optimal wet, partially quenched, and superquenched are variations of the wet-combustion process, each succeeding term implying increased use of water. The wet-combustion process also is generally known as COFCAW (combination of forward combustion and waterflooding). Recycle-combustion is yet another process and is characterized by diluting the injected air with other gases, generally gases produced in the process itself, but the process seldom is used produced in the process itself, but the process seldom is used in oilfield operations.These variations of the combustion process arose from specific needs. JPT P. 1129
Original analytical results provide estimates of the heat remaining in a homogeneous and uniform reservoir during injection of steam or hot water, for the practical cases where the temperature is not vertically uniform within the reservoir. Temperatures quadratic in the vertical position within the reservoir are used to accommodate the boundary conditions for three cases: steam extends to the boundary of both, only one, or none of the formations above and below the reservoir. The first case corresponds to steam extending over the full thickness of the reservoir, already published. The second case approximates steam flooding with gravity override, and the third case approximates hot water flooding. Two graphs, sufficient to estimate the time dependence of the heat remaining in the reservoir, are provided. Results depend on the thermal properties of the reservoir and its adjacent formations, which may differ. Results also depend on the square of the reservoir thickness and the rate of heat injection. Heat efficiency is defined as the fraction of the injected heat remaining in the reservoir. Comparison of our hot water heat efficiency with Rubinshtein's formal analytical results1 (for equal thermal properties) shows excellent agreement. The heat in the reservoir is not a function of the injectant temperature, a result which had been shown previously only for the case that steam extends over the full reservoir thickness. Nor is the heat efficiency dependent on the rate of heat injection when the latter is constant. Because of low temperatures at the boundaries with the adjacent formations, the heat efficiencies of hot water floods and steam floods with override are higher than those for steam extending over the full reservoir thickness. For hot water floods, the increase is as much as 16%. For steam floods with override the increase is more modest.
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