A reservoir simulation system uses an analytical model to represent flow within a grid block as it enters or leaves a well. This model is called a well model. We give a description here of the theoretical background of a well model, including how the sandface pressure and saturation boundary conditions can be calculated and how the well boundary itself can be replaced (approximately) by a source function. This paper and the following companion paper, SPE 9770, present a unified viewpoint of material, some of which may be already familiar to simulator developers. Introduction Our concern in this paper is the theory of representation of wells and the well boundary condition in a reservoir simulator.It frequently has been noted that, except in the case of a central well in a problem involving cylindrical coordinates, it is impractical to represent a well with an internal boundary. The ratio of well radius to desired grid-block length can be of order 0.001 or less. In such cases, an alternative procedure has evolved in which the well is represented by a source. The relationship between the source strength, the wellbore flow, and the flow in the surrounding grid blocks composes an essential part of the well model. Even when the grid around a well is sufficiently fine to represent the well as an internal boundary, other features such as partial perforation, partial penetration, or skin may be important to the local flow but extend over a "small" interval in relation to the appropriate grid-block dimension. Here also, a suitable source representation is advantageous. We shall develop the source representation of a well for a variety of circumstances.The well boundary condition generally involves the sandface pressure and flow rate. However, these quantities also must be consistent with the requirements of wellbore flow - i.e., reservoir and wellbore flows are coupled, and a wellbore flow model is required. We describe a means of treating a wide variety of wellbore flows without creating a numerically cumbersome simulator. We hope that this paper may provide a basis for further work and discussion of this essential topic. Review of Literature The source representation of a well can be described as a local, approximate, steady, singular solution of the flow equations. The idea of separating a singularity of this type for special treatment is an old idea in applied mathematics. In series solutions to certain elliptic and parabolic equations, it was found that the convergence of the series could be improved considerably by first extracting the singular part. In these cases the singular solution extended through the entire domain. The analogous approach using numerical methods in place of the series solution is also well known. The use of singular solutions in a purely local role in numerical solutions was introduced before the general use of digital computers. Woods' use of a local logarithmic expression in a solution of Poisson's equation by relaxation methods corresponds closely to the source representation of a well recently proposed by Peaceman. SPEJ P. 323＾

This paper describes the technical considerations that set the major design parameters for the Prudhoe Bay Miscible Gas Project (PBMGP). This project was planned as a large hydrocarbon miscible flood. The basic project concept is to manufacture a miscible injectant from the field separator off-gas. This injectant is compressed and delivered to the EOR project area for injection in a water-alternating-gas (WAG) mode. The miscible gas supply will vary, generally increasing with time. Over the first 10 years of the project, an average supply of 200 MMscflD [5.66x 10 6 m 3 /d] is anticipated. The gas processing plant and the factors affecting miscible gas supply are described. The EOR froject area was selected by a screening process. This led to a project area definition encompassing 4.9xlO RB [779 X 10 6 m 3 ] PV tontaining 2.2xI09 STB [350 X 10 6 m 3 ] original oil in place (OOIP). The planned cumulative volume of miscible gas injected will be 10% PV.Reservoir studies indicate an incremental oil recovery by miscible flooding of some 5.2% OOIP or 115x 10 6 STB [18.3X10 6 m 3 ]. Aspects of these reservoir studies are described.The Sadlerochit reservoir is both the ultimate source of the miscible solvent and the target reservoir. This introduces several reservoir/facility interaction effects. The planning of a major EOR project in the Arctic has also involved technical considerations not routinely encountered in conventional oilfield projects. Both aspects are discussed in the paper.

The tearing of a pressure-sensitive (‘tacky’) adhesive is examined. Two flexible strips bonded by a layer of adhesive are passed between adjacent cylindrical guides and peeled apart, causing the adhesive layer to separate into two about a surface tension membrane. Treating the adhesive as a Newtonian viscous fluid, the slow-flow problem is solved by an iterative numerical scheme in which the surface tension membrane boundary in the vicinity of the region of separation is approximated by a shear-free boundary given by a sixth-degree polynomial expression. The energy dissipation rate, a measure of the ‘strength’ of the adhesive, is obtained from the flow.The solution method is also used to determine the similar flow induced by two counter-rotating rollers partially immersed in a large bath of fluid. The results are in fairly good agreement with available experimental data. The symmetrical eddies observed under the lowest point of the surface tension membrane in the stable flow between the rollers are reproduced in the solution, proving that fluid inertia effects are not essential for their existence.

A reservoir simulation system uses an analytical model to represent flow within a grid block as it enters or leaves a well, This model is called a well model. This paper presents a succinct but comprehensive description of the installation of a well model in a simulator, including problems which may be encountered and possible remedies. This and the preceding paper, SPE 7697, present possible remedies. This and the preceding paper, SPE 7697, present a unified viewpoint of material, some of which may be already familiar to simulator developers. Introduction Our concern in this paper is the inclusion of a well model and well boundary conditions in a reservoir simulator. The source representation and the wellbore flow model are the basic components of the well model. The usefulness of the working version finally installed in a reservoir simulator depends greatly on the numerical implementation. We accordingly discuss numerical aspects of the well model for black-oil, compositional, and thermal well models.We have omitted a discussion of the incorporation into well models of surface gathering facilities and what could be called "well group constraints" such as lease, platform or pipeline constraints. These subjects easily could be the topics of several other papers.A satisfactory well model is frequently a key to successful simulation. Many of the details of well model development have not appeared in the petroleum literature. It is our hope that this paper may provide a basis for further work and discussion of this paper may provide a basis for further work and discussion of this essential topic. Implementation We shall discuss the implementation of the following equation (developed in Part 1) for the flow of each phase per completion interval. (1) Here, p is the phase (either oil, water, or gas). We note here certain aspects of this well model.1. The rates are in standard units.2. The relative permeability is calculated using the grid-block (average) fluid saturation from a well (i.e., not necessarily the grid-block) relative permeability table. It is at this point that the saturation boundary condition is imposed.3. The oil pressure is used to calculate the potential for all phases. Thus, capillarity, is not treated (i.e., no capillary end effect or water block). Also, the difference in phase pressures within a grid block due to gravity segregation is ignored.4. Zk is the vertical distance from the center of the kth completion interval to the center of the (k + 1)th completion interval (positive downward).5. The viscosity, formation volume factor, solution GOR, and density are calculated at gridblock pressure. Only the grid block for the completion interval is used.6. The skin and well radius are the same for every completion interval for each well.7. The external radius re of the grid block is a function of the grid-block, geometry. JPT P. 339

A gas blowout may be brought under control by injecting water into the formation through relief wells. By avoiding direct contact between relief well and blowout well, this technique reduces the inflow of gas by creating sufficient backpressure in the formation itself. It guarantees a feasible, successful relief-well injection rate, no matter how large the lifting capacity of the blowout well may be. A constraint condition on relief-well injection pressures is found that ensures killing of the pressures is found that ensures killing of the blowout. The minimum number of relief wells then follows from injection-pressure limitations. The positions of the relief wells are kept arbitrary in positions of the relief wells are kept arbitrary in the analysis, but the results indicate that their landing points should be close to the blowout well and that direct communications with the latter (e.g., by formation fracturing) should be avoided. The analysis yields no information as to shutoff times or cumulative injection requirements. These must be found from a separate study, which could be guided by the results presented in this paper. Introduction Control over a blowout may be gained by any technique that blocks the escaping reservoir fluid either in the wellbore or in the formation. The method most frequently used is wellbore blockage the recapping of a wild well, for example, or the drilling of a relief well to establish direct connection with the wild-well borehole, followed by the injection of heavy mud at a rate greater than the lifting capacity of the blowing well. There are, however, reservoirs in which blowout conditions may become too severe to allow successful surface operations and also reservoirs in which bottom-hole pressures exceed the pressure that could be pressures exceed the pressure that could be balanced by feasible mud injection rates. Complications that rule out surface operations may also arise when the uncontrolled production from one formation "blows in" at another lower-pressure formation. In such cases the only safe and effective remedy may be to inject water into the formation through relief wells deliberately aimed off the wild-well landing point. This restricts the escape of reservoir fluid by a pressure buildup resulting from the flow of water through the formation, and by the continuous narrowing of the passageways open to the escaping reservoir fluid between spreading water-saturated volumes. When all passageways are closed off by the water, the wild well is under "dynamic" control and may produce a large fraction of the water that is being continuously injected it final plugging operation is still necessary to gain permanent control over the well. The termination of permanent control over the well. The termination of relief-well water injection must then be timed carefully, particularly when dealing with an overpressured gas reservoir.We are concerned here with only the reservoir engineering aspects of bringing a well under "dynamic control by continuously injecting water through relief wells. In considering such an operation, the most important matters to be decided are the following:1. The number of relief wells and their location with respect to the blowout well,2. The water injection rate, and3. The total quantity of water injected at shut-off.In the following we present a simple formula for estimating the minimum successful water-injection rate. The minimum number of relief wells required is then obtain from injection pressure limitations. Using this result, it is possible to determine the optimal strategy for locating relief wells. No information is obtained on cumulative injection requirements or shut-off times. This lies beyond the scope of simple analysis; but such a study -which would probably be undertaken on a computer could clearly be shortened by using the results of this paper as a screening tool. ANALYSIS OF A TWO-DIMENSIONAL PROBLEM In formulating the interrelation of the most important parameters governing the conditions for shut-off, we are forced to idealize.We assume that the fluid flow is two-dimensional in a plane homogeneous reservoir of uniformly thick layers. P. 321