The use of 3D streamline methodologies as an alternative to finite difference simulation has become more common in the oil industry. When the assumptions for its application are satisfied, results from streamline simulation compare excellently with those from finite difference and typically require less than 10% of the CPU resources. The speed of 3D streamline simulation lends itself not just to simulation but also to other components of the reservoir simulation work process. This is particularly true of history matching. History matching is frequently the most tedious and time-consuming part of a reservoir simulation study. In a previous paper (SPE# 49000), we described a novel method that uses 3D streamline paths to assist in history matching finite difference models. We designated this technique Assisted History Matching (AHM) to distinguish it from automated history matching techniques. In this manuscript, we describe this technique in more detail through its application in three reservoir simulation studies. The example models range in size from 105 to 106 grid blocks and contain several dozen to several hundred wells. These applications have led to refinements of the AHM methodology, the incorporation of several new algorithms, and some insights into the processes typically employed in history matching. Introduction The advent of powerful geostatistical modeling techniques have led to the development of very large >107 cells) geocellular reservoir models. These models capture, in greater detail then before, the heterogeneity in porosity, permeability, lithology, etc. that are critical to accurate simulation of reservoir performance. 3D streamline simulation (3DSM) has received considerable attention over the past several years because of its potential as an alternative to traditional methods for the simulation of these very large models. While 3DSM is a powerful simulation tool, it also has a number of other uses. The speed of 3D streamline simulation is ideal for such applications as geologic/geostatistical model screening1, reservoir scoping and, the focus of this paper, history matching. In a previous paper2, we described a methodology to use 3D streamline information to aid in the process of history matching. In this manuscript, we briefly review that technique and present three example reservoir applications that demonstrate its utility. Assisted History Matching The models that are used in reservoir simulation today contain details of structure and heterogeneity that are orders of magnitude greater than that used just 10 years ago. However, there is still (and probably will always be) a large degree of uncertainty in the property descriptions. Geologic data is typically scattered and imprecise. Laboratory measurements of core properties, for example, often show an order of magnitude variation in permeability for any given porosity and several orders of magnitude variation over the data set. It is unlikely that any geologic model will perfectly match the observed reservoir performance without adjustments. History matching will continue to be the technique by which the adjustments are made to the geologic model in order to achieve a match between model and historical reservoir performance. Assisted History Matching The models that are used in reservoir simulation today contain details of structure and heterogeneity that are orders of magnitude greater than that used just 10 years ago. However, there is still (and probably will always be) a large degree of uncertainty in the property descriptions. Geologic data is typically scattered and imprecise. Laboratory measurements of core properties, for example, often show an order of magnitude variation in permeability for any given porosity and several orders of magnitude variation over the data set. It is unlikely that any geologic model will perfectly match the observed reservoir performance without adjustments. History matching will continue to be the technique by which the adjustments are made to the geologic model in order to achieve a match between model and historical reservoir performance.

Summary This paper describes the planning, implementation, and evaluation of anN2-foam field trial at the Painter reservoir in Wyoming. Foam properties of aproprietary surfactant were measured in corefloods at reservoir conditions andmodeled with an empirical foam simulator. Foam injection into a dualinjector/producer temporarily reduced injectivity but was ineffective incontrolling N2 channeling. Introduction The laboratory objectives of this project were to determine experimentallythe effectiveness of foam in controlling N2 mobility at reservoir conditionsand to develop a simulator for predicting foam behavior in a reservoir. Theobjectives of the field trial were to determine the handling characteristics ofsurface-generated foam, to provide field data for validating the simulator, andto see whether foam could control N2 channeling at a dual injector/producer. The dual well selected for the field trial had an N2 cut of 50% and was instratigraphic communication with two offset wells where the N2 cuts weregreater than 20%. Laboratory work included surfactant screening, displacementexperiments to characterize foam properties, and development of a foamsimulator. Screening procedures identified a proprietary surfactant as anappropriate foaming agent. The surfactant produced a stable, low-mobility foamin the presence of Painter reservoir fluids. A permeability reduction factormeasured in the displacement experiments was used in the simulator to predictfoam injectivity and transport behavior in the reservoir. For the field trial,20,400 res bbl [3244 res m3] of 60%-quality foam was injected over a 7-weekperiod. N2 was mixed with a brine solution containing 0.5- to 1.5-wt%surfactant before going downhole. The resulting high-density foam reduced N2injectivity by a factor of 10, as predicted by the simulator. N2 injectivityrecovered rapidly after foam injection was completed. Well performance wasessentially unchanged after the foam treatment; there was neither a reductionin N2 cut nor an increase in oil production. More N2 channeling was evident inthe postfoam injection profile, possibly from overpressuring the well. Analysisof postfoam pressure-falloff tests showed that the effects of foam diminishedgradually over several months. The systematic approach taken in this projectcorrelated laboratory and field data with an empirical foam simulator. Thisapproach yielded a better understanding of the foam process, improvedinjectivity predictions, and a more complete evaluation of the field trial. Themethods described in this paper provide the basis for designing future foamtreatments. Background Discovered in 1977, the Painter reservoir is located in the Overthrust Beltof southwestern Wyoming. Hydrocarbon introduction is from the Jurassic Nuggetsandstone, which is approximately 1.000 ft [305 m] thick. The upper two-thirdsof the reservoir is gas condensate, and the lower portion is a light, paraffinic (44 API [0.81-g/cm3]) oil. Table 1 lists average reservoirproperties. We started pressure maintenance by N2, injection in 1980. The plan was toincrease reservoir pressure from 4,200 to 4,700 psig [29 to 32 MPa] so thatcondensate would miscibly displace the oil. N2 was injected in dual-completion(injection/production) wells along the crest of the formation. N2 broke throughmore rapidly than expected in the dual injector/producers and in offsetproducers that are in stratigraphic communication with injection wells. The N2channeling at Painter is attributed to stratigraphic communication betweeninjection and production wells, coning in dual injector/producers, and complexreservoir geology. Gas flows preferentially along the bedding planes ratherthan atop down or perpendicular to the bedding planes, as postulated forefficient miscible displacement. In the dual wells, large pressure gradientscombined with high N2 mobility cause gas coning, which shuts off oilproduction. This effect is accelerated if vertical fractures exist or ifproblems with the well completion are experienced. The reservoir geologyconsists of sand dunes that have directional permeability and uncertain arealextent. Permeability also varies with the grain size of the laminae that makeup a dune set. A foam-mobility-control project was initiated to evaluate thefoam's potential for controlling N2 channeling. JPT P. 504⁁

This paper applies material presented by Chen et al. and by Chavent et al to practical reservoir problems. The pressure history-matching algorithm used is initially based on a discretized single-phase reservoir model. Multiphase effects are approximately treated in the single-phase model by multiplying the transmissibility and storage terms by saturation-dependent terms that are obtained from a multiphase simulator run. Thus, all the history matching is performed by a "pseduo" single-phase model. The multiplicative factors for transmissibility and storage are updated when necessary. The matching technique can change any model permeability thickness or porosity thickness value. Three field examples are given. Introduction History matching using optimal-control theory was introduced by two sets of authors. Their contributions were a major breakthrough in attacking the long-standing goal of automatic history matching. This paper extends the work presented by Chen et al. and Chavent et al. Specifically, we focus on three areas.We derive the optimal-control algorithm using a discrete formulation. Our reservoir simulator, which is a set of ordinary differential equations, is adjoined to the function to be minimized. The first variation is taken to yield equations for computing Lagrange multipliers. These Lagrange multipliers are then used for computing a gradient vector. The discrete formulation keeps the adjoint equations consistent with the reservoir simulator.We include the effects of saturation change in history-matching pressures. We do this in a fashion that circumvents the need for developing a full multiphase optimal-control code.We show detailed results of the application of the optimal-control algorithm to three field examples. DERIVATION OF ADJOINT EQUATIONS Most implicit-pressure/explicit-saturation-type, finite-difference reservoir simulators perform two calculation stages for each time step. The first stage involves solving an "expansivity equation" for pressure. The expansivity equation is obtained by summing the material-balance equations for oil, gas, and water flow. Once the pressures are implicitly obtained from the expansivity equation, the phase saturations can be updated using their respective balance equations. A typical expansivity equation is shown in Appendix B, Eq. B-1. When we write the reservoir simulation equations as partial differential equations, we assume that the parameters to be estimated are continuous functions of position. The partial-differential-equation formulation is partial-differential-equation formulation is generally termed a distributed-parameter system. However, upon solving these partial differential equations, the model is discretized so that the partial differential equations are replaced by partial differential equations are replaced by sets of ordinary differential equations, and the parameters that were continuous functions of parameters that were continuous functions of position become specific values. Eq. B-1 is a position become specific values. Eq. B-1 is a set of ordinary differential equations that reflects lumping of parameters. Each cell has three associated parameters: a right-side permeability thickness, a bottom permeability thickness, and a pore volume. pore volume.Once the discretized model is written and we have one or more ordinary differential equations per cell, we can then adjoin these differential equations to the integral to be minimized by using one Lagrange multiplier per differential equation. The ordinary differential equations for the Lagrange multipliers are now derived as part of the necessary conditions for stationariness of the augmented objective function. These ordinary differential equations are termed the adjoint system of equations. A detailed example of the procedure discussed in this paragraph is given in Appendix A. The ordinary-differential-equation formulation of the optimal-control algorithm is more appropriate for use with reservoir simulators than the partial-differential-equation derivation found in partial-differential-equation derivation found in Refs. 1 and 2. SPEJ P. 347

Summary This paper describes a method for blending fractal statistics, detailed geologic data, finite-difference simulation, and streamtube models into a systematic approach for prediction of reservoir performance. The objective is to make accurate predictions for large-scale projects by detailed accounting of reservoir heterogeneity with reduced history-matching effort at a low overall cost. The method has been tested for waterflood and miscible gas injection projects with balanced injection/production volumes. Example applications are shown for four field cases. Introduction Recent papers1,2 have described methods for applying the concepts of geostatistics to reservoir modeling. Ref. 2, in particular, has shown how fractal distributions can be used to describe reservoir heterogeneity in simulation models. These papers demonstrate that detailed descriptions of reservoir heterogeneity can improve the accuracy of fluid-flow models. Because of computer limitations, however, increased detail limits the applicability of these methods for performance predictions of large-scale projects. Ref. 3 describes a hybrid finite-difference/streamtube model for calculating the performance of large-scale chemical flood projects. The concept of the method is to use a finite-difference model to represent displacement efficiency and vertical sweep. A stream-tube model completes the calculation for areal conformance. The work described in this paper generalizes the hybrid model approach by incorporating fractal geostatistics to improve the estimation of vertical sweep efficiency and an improved streamtube model formulated for more accurate calculation of areal conformance. This paper also describes procedures for coupling these methodologies into a system for large-scale project performance predictions and example applications of the procedures. The theoretical bases are presented in a brief overview, with detailed explanations of the underlying principles included in the references. Example applications are presented for three CO2 flood projects and a mature waterflood. Procedure for Calculating Reservoir Performance The procedure for performance calculation is based on the following steps.Establish the porosity/permeability character of the reservoir from well logs and cores and determine the statistical structure with the concept of random fractals.Use a random fractal-interpolation scheme based on the fractal characteristics determined from the well logs to project well data to the interwell region.Establish fluid-flow and displacement parameters from PVT, relative permeability, and, if available, coreflood data.Assemble geologic and fluid data into a highly detailed finite-difference cross-sectional model representing reservoir flow between a typical injector/producer pair. The cross-sectional model is highly detailed in the vertical direction. This model represents the geology on a scale much more detailed than conventional methods. The intent is to model heterogeneity near the same level of detail for which the data are available. A typical well-log resolution or sampling frequency of laboratory core measurement is 1 to 2 ft [0.3 to 0.6 m]. Therefore, this scale should be approached for the grid-block size in the vertical direction.Run the finite-difference model for projected flood conditions and develop a dimensionless characteristic solution that relates phase fractional flow at the producer to PV of fluid injected.Develop a streamtube model of the reservoir to represent areal conformance. The formulation of the streamtubes should incorporate variable mobility ratios, permeability trends, no-flow boundaries, etc.Couple the streamtube model with the characteristic solution to estimate field-wide project performance. Adjust gross fluid voidage to history value to check the model against known reservoir performance. Impose planned injection rate to forecast future performance. Each of these steps is described in detail below. Fractal Distributions. Petroleum engineers have long recognized the need to represent heterogeneity in reservoir fluid-flow calculations. Dykstra and Parsons4 described the reservoir as distinct layers of varying permeability. More recent papers have treated permeability distributions1 and discontinuous shales.5 This work uses fractal statistics to represent reservoir heterogeneity between wells as a random fractal variation superimposed on a smooth interpolation of correlated well-log values. The characteristics of the random fractal variation are determined from an analysis of the well logs or core properties used as the staring points for the interpolation. This amounts to a smooth interpolation witha superimposed texture. Fractals are characterized by the fact that they exhibit variations at all scales of observation and have partial correlations over all scales. Every attempt to divide such a geometry into smaller, more uniform regions results in the resolution of even more structure or roughness; the closer you look, the more detail you see. The variation of properties of many natural systems has been shown to be fractal in character. For instance, varve thickness in lake sediments and the flooding cycles of the Nile River have been shown to have fractal variation. The assumption of the method in this work is that the natural processes that created oil reservoirs yielded porosity/permeability distributions with a fractal character. The geometries of fractal distributions are characterized by their intermittent or "spotty" nature. This characteristic is quantified by a parameter called the intermittency exponent, H. Ref. 2 describes the theory and application of fractal statistics with particular reference to reservoir description. Step 1 - Analyzing Data for Statistical structure. To construct a heterogeneous reservoir cross-sectional model, the well-log and/or core data are analyzed for their intermittency exponent. This can be accomplished by testing the well-log or core data for their degree of correlation using the rescaled range (R/S) procedure.6 Log analysis using the R/S procedure typically indicates an average exponent, H, of 0.6 to 0.9. A value of H=0.5 indicates a totally random structure, while a value close to 1.0 implies a highly correlated structure. A layered sandstone might have an H between 0.85 and 0.9. Fractal Distributions. Petroleum engineers have long recognized the need to represent heterogeneity in reservoir fluid-flow calculations. Dykstra and Parsons4 described the reservoir as distinct layers of varying permeability. More recent papers have treated permeability distributions1 and discontinuous shales.5 This work uses fractal statistics to represent reservoir heterogeneity between wells as a random fractal variation superimposed on a smooth interpolation of correlated well-log values. The characteristics of the random fractal variation are determined from an analysis of the well logs or core properties used as the staring points for the interpolation. This amounts to a smooth interpolation witha superimposed texture. Fractals are characterized by the fact that they exhibit variations at all scales of observation and have partial correlations over all scales. Every attempt to divide such a geometry into smaller, more uniform regions results in the resolution of even more structure or roughness; the closer you look, the more detail you see. The variation of properties of many natural systems has been shown to be fractal in character. For instance, varve thickness in lake sediments and the flooding cycles of the Nile River have been shown to have fractal variation. The assumption of the method in this work is that the natural processes that created oil reservoirs yielded porosity/permeability distributions with a fractal character. The geometries of fractal distributions are characterized by their intermittent or "spotty" nature. This characteristic is quantified by a parameter called the intermittency exponent, H. Ref. 2 describes the theory and application of fractal statistics with particular reference to reservoir description. Step 1 - Analyzing Data for Statistical structure. To construct a heterogeneous reservoir cross-sectional model, the well-log and/or core data are analyzed for their intermittency exponent. This can be accomplished by testing the well-log or core data for their degree of correlation using the rescaled range (R/S) procedure.6 Log analysis using the R/S procedure typically indicates an average exponent, H, of 0.6 to 0.9. A value of H=0.5 indicates a totally random structure, while a value close to 1.0 implies a highly correlated structure. A layered sandstone might have an H between 0.85 and 0.9.

TX 75083-3836, U.S.A., fax 01-972-952-9435. AbstractIn the past several years, 3D streamline simulation (3DSL) software has become available to the industry through commercial vendors, university coop projects, and DOE sponsored research. This paper describes the use of 3D streamlines to assist in history matching conventional finite difference reservoir simulation models.One of the most difficult tasks in history matching individual well performance in a finite difference model is the identification of the grid blocks that affect the performance of a well. The task is exacerbated by the progression to large, complex, geostatistical models. This paper describes a technique that integrates, in a simple computer program, streamline and finite difference simulation to identify the grid blocks that affect each well and to effect changes to the model for history matching. Two case studies are presented: case 1 is a thick, stratified 60 well eolian sandstone producing under pattern waterflood and case 2 is a 70 well carbonate anticline producing under partial aquifer drive. The two cases illustrate the efficiency of the streamline-based code and the potential of the process to greatly simplify history matching.