Summary. Three related stream injection problems are presented along with simulation results for them obtained from six organizations. The problem selected for six comparison were intended to exercise many of the problem selected for six comparison were intended to exercise many of the features of thermal models that are of practical and theoretical interest. The first problem deals with three cycles of cyclic steam injection and the other two problems deal with steam displacement in an inverted nine-spot pattern. The first two problems are of "black-oil" type and the third of compositional type. Complete data are presented for these problems. The comparison of solutions indicates good agreement for problems. The comparison of solutions indicates good agreement for most of the results of importance in field operations. Introduction Validation of reservoir simulators for complex recovery, processes is a particularly difficult problem because analytical solutions are available under only a few limiting conditions. While good agreement between the results from different simulators for the same problem does not ensure validity of any of the results. a lack of problem does not ensure validity of any of the results. a lack of agreement does give cause for some concern. Such comparisons can also be useful in the development of new models and in optimizing the performance of existing reservoir simulators. This is the fourth in a series of simulation problems for which results from a number of commercial simulators have been obtained and reported in the SPE literature. The first such study was organized and conducted by Odeh on the simulation of a three-dimensional (3D). two-phase, black-oil case. Seven companies participated in that project. This was followed by a comparative participated in that project. This was followed by a comparative solution project developed b,. the program committee for the 1982 SPE Reservoir Simulation Symposium in New Orleans. The problem selected was a three-phase, single-well radial cross-section coning problem. Eleven companies participated in this study. Chappelear problem. Eleven companies participated in this study. Chappelear and Nolen were responsible for organizing this study and for reporting the results at the symposium. A similar approach was adopted by the program committee for the 1983 SPE Reservoir Simulation Symposium in San Francisco. In this case, the problem selected was to study gas cycling in a rich-gas retrograde condensate reservoir. In the first part of the study, the participants matched their phase-behavior packages to the data supplied, and in the second part they considered two options for the depletion of the reservoir. This study required a 3D, three-phase. multicomponent compositional model. Nine companies participated in this study. Kenyon and Behie organized the project and reported the results at the symposium. The enthusiastic response of industry and the academic community to the three problems encouraged the program committee for the 1985 SPE Reservoir Simulation Symposium to continue the tradition and to develop a set of problems suitable for the comparison of steam injection models. We were given the task of organizing this project. The objective of this paper is to present selected results submitted by the participants in this project with a minimum of commentary. It is worth emphasizing that the type of comparison presented here and in previous reports is different from other kinds of-comparisons discussed in the literature (e.g., Refs. 4 and 5). In the SPE comparisons, the problems are designed by one or more knowledgeable people, and model results are provided directly by those who have built or acquired suitable models. This is different from a study where the person doing the comparison develops new software using published descriptions of several models. It is possible-although we have tried to minimize this-that some of possible-although we have tried to minimize this-that some of the differences in the results compared here could be a result of different interpretations of the problem, while differences in a com-parison of the type discussed in Ref. 4 may be caused by differences in the interpretation of the published procedures. Furthermore, the models used in our comparison are all commercial in nature. Some of these models have been in existence for several years and others are new. Problem Statement Problem Statement We have selected three related but independent problems for the comparison of steam injection models:cyclic steam injection in a nondistillable oil reservoir with a two-dimensional (2D) radial cross-sectional grid,nondistillable oil displacement by steam in an inverted nine-spot pattern by considering one-eighth of the full pattern (see Fig. 1), anddisplacement of an oil consisting of two volatile components and one nonvolatile component in the same pattern as Problem 2. The oil properties were the same in the first two problems. The participants had the option to submit results for one, two, or all three problems. In addition we invited optional runs. A complete statement of the problems as offered to the participants is contained in Appendix A. (We have deleted the section on reporting requirements to save space.)The problems were selected to exercise features of the models that are important in practical applications they do not necessarily represent real field situations. In particular. we wanted to see the influence of grid orientation on the results of the steam displacement problems. The inverted nine-spot appeared to us to be ideal for this purpose. Six companies (see Appendix B) participated in the project with only three submitting results for the compositional case (Problem 3). Four of the other companies contacted indicated an interest in the project during the early stages but were unable to provide results for the comparison for a variety of reasons. In addition to providing numerical results, the participants were also asked to providing numerical results, the participants were also asked to describe their model briefly and to answer a number of questions about the model. These descriptions with only minor editorial changes are given in the next section. Description of Models Used p. 1576

Summary The vaporizing gas drive (VGD) process was modeled with the Peng-Robinson equation of state (PR-EOS) and a compositional simulator. The comparison of numerical results with available experimental data has shown that the PR-EOS overestimates the minimum miscibility pressure (MMP), but it correctly shows that the length required to achieve miscibility is different for N2 and methane. Experimental data and some simulation runs have been used to develop a simple and reliable correlation for the prediction of MMP. Introduction For some high-pressure oil reservoirs, N2or a lean hydrocarbon gas may be suitable for achieving miscibility conditions. These gases are particularly attractive because of the ease with which they can be handled and the potential they offer for establishing gravity-stabilized displacement in thick oil columns. Two field projects involving N2 and lean-gas injection are discussed. In the Devonian reservoir of Block 31 field in west Texas,1–4 the world's first large-scale high-pressure gas injection project has been under way since 1949. The reservoir rock consists of about 65 % tripolitic chert and 20% fine crystalline, sucrosic limestone. The remainder is variable amounts of lime mud, skeletal material, pellets, and quartz silt. The porosity is intercrystalline and averages 15%. The permeability averages 1 md, but is about 10 times this amount in the hairline fractures that are present. Lean-gas injection in Block 31 field started 3 1/2 years after field discovery, and the reservoir pressure was raised to the miscibility pressure in 1952. Since 1966, flue gas (88% N2 and 12% CO2) has been injected in one part of the field and N2-contaminated lean hydrocarbon gas has been reinjected in the rest of the field. Because of the high solubility of CO2 in the interstitial water, the displacing fluid was essentially N2. Even though the miscible wne seems to have difficulty in maintaining its integrity because of reservoir stratification, fractures, and an unfavorable (10:1)gas-to-oil mobility ratio, the ultimate recovery is expected to be greater than 65 % of the original oil in place (OOIP). A large part of Algeria's Hassi Messaoud field5is undergoing lean-gas miscible drive. The reservoir is made up of a highly siliceous, cemented quartzitic sandstone and is highly heterogeneous. Water is also injected in some parts of the field, and in other parts, water is alternated with gas. Ultimate recovery is expected to be about 50% of OOIP in the miscible gas injection area, about 33% of OOIP in the water injection area, and about 11% for naturally depleting areas. Stalkup6 provides details of several other VGD field projects. On the whole, the reservoirs undergoing high-pressure miscible drive have been rated successful;the recoveries usually exceed 50% of OOIP. In deep, volatile oil reservoirs, N2 and lean-gas miscible drive have the potential to recover oil that is unrecoverable by water injection alone. The most important parameter required for the design and evaluation of N2 or lean-gas miscible drive is the MMP. The literature does not contain any general correlation that can be used to provide an estimate of MMP for N2 or lean gases. In this paper we explore the potential of the PR-EOS to predict the MMP for the VGD process, discuss the effect of N2and lean gases on the MMP, and present a simple correlation for the estimation of VGD MMP. Review of Experimental Data Experimental data reported in the literature for VGD MMP are rather limited. Information on only eight reservoir fluids with known compositions was located. These data are briefly reviewed below.

Summary A semianalytical method is presented for the approximate modeling of the productivity of nonconventional wells in heterogeneous reservoirs. The approach is based on Green's functions and represents an extension of a previous model applicable for homogeneous systems. The new method, referred to as the s-k* approach, models permeability heterogeneity in terms of an effective skin s that varies along the well trajectory and a constant background permeability k*. The skin is computed through local, weighted integrations of the permeability in the near-well region and is then incorporated into the semianalytical solution method. The overall method, which can also model effects due to wellbore hydraulics, is quite efficient in comparison to detailed finite-difference calculations. Results for the performance of nonconventional wells in three-dimensional heterogeneous reservoirs are computed using the s-k* approach and compared to finite-difference calculations resolved on the geostatistical fine grid. The new method is shown to provide an accurate estimate of wellbore pressure and production rate, as a function of position along the wellbore, for various well configurations and heterogeneous permeability fields. The possible use of the overall approach in a simulation while drilling (SWD) tool, in which the well path and trajectory are "optimized" using real-time data, is also discussed. Introduction Nonconventional wells (e.g., horizontal, deviated, or multilateral) have become quite common throughout the oil industry. In designing or optimizing the length and placement of such wells, it is important to estimate accurately the well productivity. One approach for determining this well productivity is to simulate the reservoir performance using a finite-difference simulator. This is the most rigorous approach available, though it is also the most demanding in terms of time and data requirements. An alternate approach for modeling the productivity of nonconventional wells operating under primary production is to employ a semianalytical solution technique. Early work along these lines included single horizontal wells (of infinite conductivity) aligned parallel to one side of a box-shaped reservoir. Solution methods were successive integral transforms1,2 and the use of instantaneous Green's functions,3–6 resulting in infinite series expressions. More complex geometries were considered later7–9 as numerical integration became more feasible. A number of works (see Ouyang,10 and citations therein) include coupling of wellbore hydraulics (i.e., finite-conductivity wells) with reservoir flow. The method we apply in this study9 has one of the most general treatments of wellbore hydraulics. All of the semianalytical techniques mentioned above have the advantage of limited data requirements and high degrees of computational efficiency. These techniques are, however, limited to homogeneous systems or at most strictly layered systems.11,12 This represents a substantial limitation because the productivity of nonconventional wells can be significantly impacted by fine-scale heterogeneities in the near-well region. Fine-scale heterogeneity can be incorporated into detailed simulation models, though the resulting models are complex to build and require substantial computation time to run. The purpose of this paper is to extend an existing semianalytical approach to approximately account for heterogeneity in the near-well region. This will enable us to apply the semianalytical approach to more realistic heterogeneous systems. We accomplish this by introducing an effective skin s into the semianalytical model and then estimating this effective skin as a function of position along the wellbore. The skin is computed via local, weighted integrations of the permeability field in the near-well region. This skin differs significantly from skin in the usual sense, as it is here due to intrinsic heterogeneity in the permeability field rather than from formation damage or stimulation. Away from the wellbore, the reservoir is modeled in terms of the large-scale effective permeability k* The overall method is highly efficient and approximates both near-well effects (through s) and global effects (through k*) with reasonable accuracy. The approach presented here combines and extends formulations developed in two separate earlier studies. These studies addressed the development of a semianalytical well model9 and the approximation of the effects of heterogeneity in the region near a vertical well.13 The semianalytical well model is applicable for very general well configurations and also accounts for pressure drop in the wellbore due to friction, gravitational, and acceleration effects. These can be important in long horizontal wells. The approximate heterogeneity model applied here was developed for the modeling of vertical wells in heterogeneous, two-dimensional areal systems. Both single-well and two-well systems were considered. The basic approach was shown to provide accurate estimates for well productivity, relative to fine-grid simulation results, for many geostatistical realizations over a range of geostatistical parameters. As will be shown below, our new method successfully builds upon both the semianalytical well model and the approximate heterogeneity model. Another technique for approximately modeling the effects of heterogeneity on horizontal wells was previously developed.14 This method, based on a network modeling type of approach, differs considerably from the procedure presented here in that our methodology has as its basis a semianalytical solution technique. The earlier method does, however, display accurate results for a range of problems similar to those considered here. This paper proceeds as follows. We first describe the overall method in some detail. Then, we present numerical results for horizontal and multilateral wells in heterogeneous three-dimensional systems. These results are in many cases compared with detailed finite-difference calculations to assess their level of accuracy. Our new description is shown to provide an accurate estimate of production rate, as a function of position along the wellbore, for a variety of well configurations and for different heterogeneous permeability fields. Finally, we discuss how the overall approach could represent a component of a simulation while drilling (SWD) capability, in which the well path and trajectory are "optimized" using real-time data coupled with our new well model.

To determine the range of validity of pseudo functions for upscaling of multiphase flow with gravity and capillary pressure, the performance of different kinds of pseudo functions is evaluated under different gravity numbers, capillary numbers and upscaling levels. The performance of pseudo functions can be measured as the percentage error between the results of fine grid simulation and the results of coarse grid simulation using pseudo functions. Furthermore, the performance can be shown as the bar graphs of percentage error for varying gravity number, capillary number and upscaling levels. A two-dimensional homogeneous water flooding case and a three-dimensional heterogeneous case have been studied using pseudo functions (Jacks et al.2, Kyte and Berry 3, Pore Volume Weighted 7, Flux Weighted Potential 10, Modified Stone*4, Streamline 5), and pseudo function performance has been evaluated under different conditions. Some guidelines for the use of pseudo functions are provided. The results show that (1) all of the pseudo functions have similar performance for the homogeneous case considered; (2) only the Jacks et al. pseudo function can be used in all cases; (3) using pseudo functions (except Jacks et al.) is unreliable, especially for the highly heterogeneous cases.

Summary This paper discusses the treatment of wells in a flexible Voronoi grid. A new model problem is proposed to evaluate exact well indices for multiwell configurations and homogeneous reservoirs. A simplified model also is proposed and discussed. New exact and simplified well models for heterogeneous reservoirs are presented. Introduction An exact well index can be derived by comparing the solution of the differential equation (obtained analytically or numerically) for a given problem with the numerical solution of the difference equation (exact well model). Peaceman1 first published this approach, and well models based on it are defined here as Peaceman-type models. In this sense, well models described by Kuniansky and Hillestad,2 Abou-Kassem and Aziz,3 and Babu et al.4 also are considered to be Peaceman-type models because, although these authors used different reference solutions, the well models were based on Peaceman's1 concept. In this paper, we propose new exact Peaceman-type well models for Voronoi grid and multiwell configurations in homogeneous and heterogeneous reservoirs and propose and discuss a simplified model for this grid. The exact well model can be used easily to investigate the effect of different well configurations (location and rates) over the value of the well index. Because the well index is assumed to be constant during the simulation process, a good grid geometry should result in a constant (within a tolerance) value for each well, regardless of the well configuration. Therefore, this procedure is an additional tool that can be used to select the appropriate grid size for the problem of interest. Current Wen Models Although van Poolen et al.5-type models were extensively used in the past, Peaceman1 showed that this approach is incorrect. For this reason, only Peaceman-type models are considered here. The relationship between wellblock pressure and bottomhole flowing pressure (BHFP) is a function of fluid rates, rock and fluid properties, and grid geometry:Equations 1 and 2 where Iw=well index, ?=angle open to flow, and req=equivalent wellblock radius. Because only single-phase cases are discussed, the subscript P (phase index) will be dropped from all remaining equations. Peaceman1,6 proposed the following simplified model based on the comparison between numerical and analytical solutions for the total pressure drop in a homogeneous, isotropic, repeated five-spot pattern under steady-state, single-phase flow conditions.Equation 3 The conditions that must be satisfied to apply this model safely may be as important as the model itself and are discussed by Peaceman.7,8 In his first paper, Peaceman1 showed that the block pressures increase logarithmically with the radial distance measured from the gridpoint to the well when a regular Cartesian grid with square blocks (?x=?y) is used to discretize an isolated one-quarter of a five-spot pattern. The extension of this concept led to the development of Kuniansky and Hillestad's2 and Abou-Kassem and Aziz's3 analytical well model. This model does not produce good results for wellblocks with large grid-aspect ratios (Ry=?y/?x) and may not produce good results for wells close to reservoir boundaries.7 However, it does yield good results for isolated wells in blocks with small Ry. This model's main advantage is that it can be applied to any kind of grid geometry and discretization scheme. Kuniansky and Hillestad2 and Peaceman7 proposed exact well models for multiwell configurations. They used different model problems for a reservoir with constant pressure at the external boundaries. In principle, these models can be used to derive well indices for grids of any geometry. However, conventional simulators assume closed boundaries, and the exact representation of constant pressure boundaries in these codes may be very tedious, although it is possible. Babu et al.4 proposed a well model based on the analytical solution presented by Babu and Odeh9 for a single well producing at constant and uniform. flux from a closed, box-shaped (3D) drainage volume under pseudo-steady-state conditions. The solution is valid for a well that partially penetrates the reservoir length. On the basis of a series of numerical experiments, they also presented a simplified model for regular Cartesian grids. Although their analytical solution is valid for 3D configurations, the actual well model was restricted to 2D (areal or cross-sectional) cases. Further research on fully 3D models is needed to investigate such factors as the effect of boundary conditions at the well (uniform flux, uniform pressure, or mixed boundary conditions) and partial penetration. All the authors mentioned above have focused their attention on the treatment of wells in a Cartesian grid, which is a special case of the Voronoi grid10,11,12 discussed in this paper (Fig. 1). For this reason, currently used well models were used as a foundation to develop the well models presented here for the Voronoi grid and heterogeneous reservoirs. Exact Well Model for Homogeneous Reservoirs By definition, the use of an exact well index in numerical simulation yields the same well pressure, pw, as that calculated analytically for a given model problem. Because this approach was used first by Peaceman,1 this condition characterizes the well model as Peaceman-type. A model problem consists of defining the reservoir geometry and its external and internal (well) boundary conditions, The linear, single-phase flow equation then is solved in the proposed domain to obtain the analytical solution for pressure at the well of interest, pw. The same problem is solved numerically to compute the pressure of the block containing the well, po. The equivalent radius, req, is evaluated by arranging Eq. 1 asEquation 4 This approach is also applicable to Voronoi grids because there is no restriction on grid geometry. The results presented in the literature show that different model problems produce similar well indices provided there are "enough" gridblocks between wells (or images). While the best model problem is the one that is closest to the actual field configuration, its choice should be based on ease of use, flexibility to represent flow geometries and boundary effects, and capability of conventional simulators to model the same problem. p. 15–21