This paper proposed the use of two-point upstream weighting of fluid mobility as an alternative to the generally employed single-point approximation Use of the two-point formula results in the reduction of both numerical dispersion of flood fronts and the sensitivity of predicted areal displacement performance to grid orientation. Stability analysis performance to grid orientation. Stability analysis provides the time-step limitation for control of provides the time-step limitation for control of solution oscillations. This together with limitations for control of overshoot and truncation error provides a practical basis for the automatic selection of time steps. Introduction As an indication of the growing concern for controlling the total cost of large-scale reservoir simulations, the emphasis of a number of recent publications has been directed toward increasing publications has been directed toward increasing computing efficiency. In this paper, two methods to increase the computing efficiency of reservoir simulators are described. The use of two-point upstream weighting of fluid mobility is described and compared with the commonly used single-point upstream approximation. The two-point approximation generally requires fewer grid blocks to obtain a given accuracy than does the single-point approximation. In addition, the calculated performance of areal models is less sensitive to grid performance of areal models is less sensitive to grid orientation when using the two-point approximation. Computing efficiency is also increased with the use of an automatic time-step selector. Time-step limitations are described in this paper for controlling stability, overshoot (negative saturations), and truncation error. In general, these limitations change each time step as conditions change. If any of the limitations is exceeded, the results of the simulation may be meaningless. An automatic time-step selector detects and avoids running difficulties by using the proper time-step size. Using these methods, simulation proper time-step size. Using these methods, simulation results are obtained with less expenditure of engineering and computer time. TWO-POINT APPROXIMATIONS FOR FLUID MOBILITY The majority of general-purpose reservoir simulators reportedly in use today are based on the solution of finite-difference analogs to the conservation equations describing multiphase flow in porous media. Thus, the continuous domain of a reservoir is divided into a number of discrete blocks, and solutions for pressure and saturations are obtained at the grid block centers (or grid points). Central-difference approximations are normally used for the spatial derivatives in the discrete formulation of the conservation equations. As described below, this scheme necessitates the evaluation of flow coefficients (kk,/muB) at the planes separating adjacent grid blocks. As fluid and planes separating adjacent grid blocks. As fluid and reservoir properties are only defined at grid points, some method must be devised for approximating interblock flow coefficients based on values at the grid points. Of the terms that make up the flow coefficients, only the saturation-dependent relative permeability changes rapidly enough from grid block to grid block to cause significant difficulty. Although several weighting schemes have been employed in the past for evaluating the relative permeability at a block face, only single-point upstream weighting appears to be in general use. Unfortunately, use of this weighting scheme is well known to cause excessive numerical dispersion of flood fronts. In addition, areal displacement performance is found to be quite sensitive to the grid orientation for grid meshes of practical extent for large-scale reservoir simulations. This has been demonstrated qualitatively by Garrett and will be described both qualitatively and quantitatively later in this paper. As an alternative to single-point weighting of relative permeability, a two-point weighting of relative permeability, a two-point scheme is now described which results in both reduced numerical dispersion of flood fronts and decreased sensitivity of predicted areal performance to grid orientation. SPEJ P. 515
For most reservoirs the reservoir thickness and dip vary with position. For such reservoirs, the use of a Cartesian coordinate system is awkward as the coordinate surfaces are planes and the finite-difference grid elements are rectangular parallepipeds. However, these reservoirs may be efficiently parallepipeds. However, these reservoirs may be efficiently modeled with a curvilinear coordinate system that has coordinate surfaces that coincide with the reservoir surfaces. A procedure is presented that may be used to determine a curvilinear coordinate system that will conform with the geometry of the reservoir. The reservoir geometry is described by the depth of the top of the reservoir and the thickness. The mass conservation equations are presented in curvilinear coordinates. The finite-difference equations differ from the usual Cartesian coordinate formulation by a factor multiplying the pore volume and transmissibilities. A numerical example is presented to illustrate the magnitude of the error that may occur in the computed oil recovery if the Cartesian coordinate system is simply modified to yield the correct depth and pore volumes. Introduction Many reservoirs have a shape that is inconvenient and possibly inaccurate to model with Cartesian coordinates. The use of a curvilinear coordinate system that follows the shape of the reservoir can be advantageous for such reservoirs. The formulation discussed here will have the greatest advantage in modeling thin reservoirs but will have little advantage in modeling a reservoir whose thickness is greater than its radius of curvature, such as a pinnacle reef. pinnacle reef. In this paper the reader is introduced to various grid systems used to model reservoirs. A brief discussion of some concepts of differential geometry contrasts differences between Cartesian coordinates and curvilinear coordinates. A curvilinear coordinate system for modeling reservoir geometry is presented. Formulation of the conservation equations in curvilinear coordinates and the necessary modifications to pore volume and transmissibility are discussed. A numerical example illustrates the magnitude of the error that may result from some coordinate systems. COORDINATE SYSTEMS AND RESERVOIR GRID NETWORKS A reservoir is usually described with the depth, thickness, boundaries, etc., shown on a structure map with sea level as a reference plane. For example, the subsea depth may be shown as a contour map on the reference plane with a Cartesian coordinate grid superimposed on the reference plane as shown on Fig. 1. The Cartesian coordinates, plane as shown on Fig. 1. The Cartesian coordinates, (y1, y2), have been defined as the coordinates for the reference plane. If the reservoir surfaces are parallel planes, Cartesian coordinates may be used. The Cartesian coordinate may be rotated such that the coordinate surfaces coincide with the reservoir surfaces. SPEJ P. 393
Variational principles stated by, Biot have been applied to obtain a two-parameter (approximation for heat losses to cap and base rock from a reservoir undergoing thermal recovery. The approximation predicts heat losses to within a few percent of the predicts heat losses to within a few percent of the exact value when the beat losses result from one-dimensional conduction into cap and base rock in the direction normal to the reservoir boundary surfaces. Conduction in the longitudinal direction is neglected. Therefore, the approximate temperature distribution is valid only when the temperature gradient in this direction is small. But because the Peclet number (ratio of convective to conductive heat transport) is high in most reservoir thermal processes, the horizontal temperature gradient will processes, the horizontal temperature gradient will be small everywhere except in the vicinity of a thermal front, and the approximation will be valid. Comparison with a finite-difference solution in cap and base rock shows that reasonable accuracy is obtained when the Peclet number is 100 or greater. The variation solution has been incorporated into our thermal simulator and yields a considerable sailings in core storage. It is no longer necessary to store grid-block temperatures for cap and base rock nor to solve the finite-difference form of the energy balance in this region. Instead a system of two nonlinear ordinary differential equations must be solved for each grid block at the interface of the reservoir and the cap rock. In addition to savings in core storage, a reduction in computation time is achieved because fewer finite-difference grid blocks are needed. Introduction Heat losses to cap and base rock must be considered in modeling thermal processes in petroleum reservoirs. Since there is no mass petroleum reservoirs. Since there is no mass transport in the cap and base rock, the only mechanism for heat transfer is conduction. One of the most obvious ways of determining heat losses from the reservoir is to solve the energy equation in the cap- and base-rock region by finite differences. To do this, the reservoir finite-difference grid must be extended into the cap- and base-rock region. This can consume a good deal of computer core storage - at a time when all available core storage is needed to adequately model mass and energy transport in the reservoir region. Furthermore, since there is no mass transport in the cap and base rock, one would like to eliminate having to solve the conservation-of-mass equations in this region, but to do so requires a special computer code. Hence, a finite-difference solution can be costly. It does, however, have the advantage of generality in that a minimum of assumptions is involved in formulating the conservation equations. There are ways of calculating heat losses to cap and base rock other than by finite differences. However, for a method to be competitive with the finite-difference method, it must offer some advantage such as accuracy, reduced computer core storage, or lower computation time. One alternative to finite differences is the use of superposition to couple an analytic solution for the cap and base-rock temperature distribution with the finite-difference solution of the reservoir energy balance. But, during the course of the simulation, the superposition principle would necessitate having temperature data for all previous time steps for each grid block adjacent to the cap and base rock. This requires an appreciable amount of computer core storage, perhaps even more than would be required for a complete finite-difference solution. Hence, this method does not seem attractive. The use of variational principles appeared to offer the advantages of both reduced core storage and lower computation time and was therefore considered as a means of treating heat losses to cap and base rock. The advantage of the variational method is that a priori knowledge of the approximate shape of the temperature profile can be used to choose the functional form of the temperature distribution. The chosen functional form will contain several free parameters. SPEJ P. 200
Operators of steamflood projects seem to prefer low pressure steam zones in their operations. The difference between steam zone temperature and initial reservoir temperature drives all energy requirements, which are usually the largest single cost in a steamflood project. However, lower steam zone pressure implies lower drawdown available for production. And lower steam zone temperature implies higher oil viscosity and therefore lower oil production rate. The influence of steam zone pressure and temperature on oil production rate is large. Theoretically, the influence on ultimate recovery (residual oil saturation) is moderate to none. Since costs encourage lower steam zone pressure and productivity encourages higher steam zone pressure, there should be an economic optimum.The rate of change of energy requirements and oil production rate with respect to changes in steam zone temperature and pressure are determined analytically. If the reservoir geology is conducive to gravity drainage, even by a very tortuous path, then low steam zone pressure is highly favored. Lowering steam zone pressure usually comes at the cost of increased withdrawals, so it is important to carefully consider the requirements, consequences, and benefits on a case-by-case basis. A recommended method for this analysis is discussed. Significant project improvement after a pressure reduction has been reported. These production improvements and steam zone pressure in major steamflood projects are discussed. Industry ExperienceOperators prefer to keep steam zone pressure as low as possible since pressure and temperature have a one-to-one relationship, temperature level above initial temperature drives all energy requirements (energy for steam zone growth, energy for heat losses, and energy produced and lost from wellbores and pipelines), and energy required for a steamflood project is usually the largest single operating cost. Some documented projects have steam zone pressure lower than tire pressure. A sampling of industry experience is given in Table 1.In addition to the cost side, there is an incentive to develop as large a steam zone as possible. Several papers show pre-steam and post-steam core analyses with noted oil saturation changes. In Kern River Field, for example, "oil saturation is significantly reduced wherever a steam zone develops and is only slightly reduced in an underlying hot water zone" (reference 2).
TX 75083-3836, U.S.A., fax 01-972-952-9435. AbstractHow to ensure that miscibility of oil and gas is achieved in each reservoir is a fundamental issue for miscible gasfloods involving different oil reservoirs with varying fluid properties. This paper reports on all the work done to help decide on how to optimally blend available gas such that miscibility can be achieved in all reservoirs with appropriate focus on the first reservoirs to be flooded. These studies have resulted in an investment decision to undertake a miscible gasflood already in 2005, whereas initial production from four reservoirs had only started in March 2004. Main components of this paper are: (1) Design of experiments for a wide spectrum of fluids (from near-critical systems to black oil systems) using miscible sour gas-blends while minimizing cost and the time spent on the experiments. (2) Acquisition, interpretation of the data (3) Utilization of the data for reservoir engineering/design calculations using a consistent approach for a cluster of sour reservoir fluids, (4) Recommendations based on the experimental data and calibrated simulation models.
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