This paper describes the development of a three-dimensional (3D), three-phase model for simulating the flow of water, oil, and gas in a naturally fractured reservoir. A dual porosity system is used to describe the fluids present in the fractures and matrix blocks. Primary flow present in the fractures and matrix blocks. Primary flow in the reservoir occurs within the fractures with local exchange of fluids between the fracture system and matrix blocks. The matrix/fracture transfer function is based on an extension of the equation developed by Warren and Root and accounts for capillary pressure, gravity, and viscous forces. Both the fracture flow equations and matrix/fracture flow are solved implicitly for pressure, water saturation, gas saturation, and saturation pressure. We present example problems to demonstrate the utility of the model. These include a comparison of our results with previous results: comparisons of individual block matrix/fracture transfers obtained using a detailed 3D grid with results using the fracture model's matrix/fracture transfer function; and 3D field-scale simulations of two- and three-phase flow. The three-phase example illustrates the effect of free gas saturation on oil recovery by waterflooding. Introduction Simulation of naturally fractured reservoirs is a challenging task from both a reservoir description and a numerical standpoint. Flow of fluids through the reservoir primarily is through the high-permeability, low-effective-porosity fractures surrounding individual matrix blocks. The matrix blocks contain the majority of the reservoir PV and act as source or sink terms to the fractures. The rate of recovery of oil and gas from a fractured reservoir is a function of several variables, included size and properties of matrix blocks and pressure and saturation history of the fracture system. Ultimate recovery is influenced by block size, wettability, and pressure and saturation history. Specific mechanisms pressure and saturation history. Specific mechanisms controlling matrix/fracture flow include water/oil imbibition, oil imbibition, gas/oil drainage, and fluid expansion. The study of naturally fractured reservoirs has been the subject of numerous papers over the last four decades. These include laboratory investigations of oil recovery from individual matrix blocks and simulation of single- and multiphase flow in fractured reservoirs. Warren and Root presented an analytical solution for single-phase, unsteady-state flow in a naturally fractured reservoir and introduced the concept of dual porosity. Their work assumed a continuous uniform porosity. Their work assumed a continuous uniform fracture system parallel to each of the principal axes of permeability. Superimposed on this system was a set of permeability. Superimposed on this system was a set of identical rectangular parallelopipeds representing the matrix blocks. Mattax and Kyte presented experimental results on water/oil imbibition in laboratory core samples and defined a dimensionless group that relates recovery to time. This work showed that recovery time is proportional to the square root of matrix permeability divided by porosity and is inversely proportional to the square of porosity and is inversely proportional to the square of the characteristic matrix length. Yamamoto et al. developed a compositional model of a single matrix block. Recovery mechanisms for various-size blocks surrounded by oil or gas were studied. SPEJ P. 42

This paper describes the development of a three-dimensional, three-phase model for simulating the flow of water, oil and gas in a naturally fractured reservoir. A dual porosity system is used to describe the fluids present in the fractures and matrix blocks. Primary flow in the reservoir occurs within the fractures with local exchange of fluids between the fracture system and matrix blocks. The matrix-fracture transfer function is based on an extension of the equation developed by Warren and Root and accounts for capillary pressure, gravity, and viscous forces. Both the fracture flow equations and matrix-fracture flow are solved implicitly for pressure, water saturation, gas saturation and saturation pressure. Example problems are presented to demonstrate the utility of the model. These include a comparison of the results from this paper with previous results; comparisons of individual block matrix-fracture transfer, obtained using a detailed three-dimensional grid, with results using the fracture model's matrix-fracture transfer function; and three-dimensional field scale simulations of two-and three-phase flow. The three-phase example illustrates the effect of free gas saturation on oil recovery by water flooding.

Methods of nonlinear regression theory were applied to the reservoir history-matching problem to determine the effect of erroneous problem to determine the effect of erroneous parameter estimates obtained from well testing parameter estimates obtained from well testing on the future prediction of reservoir pressures. Two examples were studied: well testing in a radial one-dimensional slightly compressible reservoir and in an undersaturated, two-dimensional, heterogeneous oil field. The reservoir parameters of permeability, porosity, external radius, and pore volume were considered, and the effects of pore volume were considered, and the effects of measurement error, test time, and flow rate on the confidence limits were computed. Introduction The operation of a reservoir simulator requires accurate estimates of the reservoir properties. However, the simulation parameters, such as permeability, porosity, and reservoir geometry, are permeability, porosity, and reservoir geometry, are usually unknown unless coring and physical property analysis have been undertaken. Because of the cost of these procedures, it is more desirable to use the pressures measured at the well during a well test pressures measured at the well during a well test and indirectly compute the important parameters of the system. By using history matching of the test data to obtain the system parameters, the future pressure behavior of the reservoir can be predicted pressure behavior of the reservoir can be predictedSeveral studies on history matching have indicated that the welltest approach for determining the reservoir parameters often suffers from incorrect and nonunique parameter estimates. The factors that affect the parameter estimation can be classified as model errors, observability, measurement errors or noise, history time, test procedure, and optimization procedure. Model errors arise from the inaccuracy of the model and the numerical integration. For example, a reservoir simulator is only a reasonable approximation for flow through porous media. Solution of a model equation by numerical means also introduces roundoff and discretization errors. Observability of the system plays an important role in estimating the reservoir parameters. Depending on the location of the well and the number of data points, it may not be possible to determine uniquely all reservoir parameters from the measurements made at that well. Observability is strictly a function of the reservoir model used. At a given well, pressure measurements may only reflect the values of the parameters in specific zones of the reservoir. If a specific zone away from the well does not affect the measured pressure, then the system is not observable at that particular location. A rigorous definition of observability can be found in other papers. Measurement errors in the pressures and flow rates are another source of unrealistic parameter estimates. Longer history times always give more information about the reservoir as long as the system remains in a dynamic state. The nature of the system input (well flow rate) also affects the accuracy of the estimates and predictions. The final source of incorrect parameter estimates arises because the history-matching problem, posed mathematically, is usually a nonlinear programming problem that must be solved computationally. Such problem that must be solved computationally. Such a problem yields multiple extrema that often can lead to a relative minimum (rather than a global minimum) in the numerical search for the smallest matching error. Also, the magnitude of the objective function can be quite insensitive to the parameters selected, thus causing the optimization procedure to terminate prematurely. The above factors control the history-matching process; with actual data, it is usually impossible process; with actual data, it is usually impossible to identify the exact contributions of each factor to the errors in the parameter estimates. Since a certain amount of error will be introduced into the estimated parameters from the history-matching process, it is parameters from the history-matching process, it is useful to study the magnitude of this error resulting from various sources under controlled simulation conditions. Also, it is important to determine how the errors in the parameters are reflected in the future predictions of the pressures. SPEJ P. 42

Thomas, L.K.; SPE, Phillips Petroleum Co. Phillips Petroleum Co. Dixon, T.N.; SPE, Phillips Petroleum Co. Phillips Petroleum Co. Evans, C.E.; SPE, Phillips Petroleum Co. Phillips Petroleum Co. Vienot, M.E.; SPE, Phillips Petroleum Co. Phillips Petroleum Co. Copyright 1987 Society of Petroleum Engineers Summary. This paper describes the evaluation of a waterflood pilot in the highly fractured Maastrichtian reservoir of the Ekofisk field in the Norwegian sector of the North Sea. A four-well pilot consisting of one water injector and three producers was initiated in Spring 1981 and was concluded in mid-1984. A total of 21 × 106 bbl [3.3 × 106 m3] of water was injected, and water breakthrough occurred in two of the production wells. Simulation of waterflood performance in the pilot was conducted with a three-dimensional (3D), three-phase dual-porosity model. Initial and boundary conditions were taken from a full 3D single-porosity model of the reservoir. The pilot was conducted to determine the following information for the Maastrichtian: water-cut performance vs. time, water imbibition characteristics, and anisotropy. Results from this work have been incorporated into a full-field waterflood study. Reservoir description included the determination of fractured areas, matrix block sizes, water/oil capillary imbibition, matrix permeability and porosity, and effective permeability. These data were derived from porosity, and effective permeability. These data were derived from fracture core analysis, pressure transient tests, laboratory water/oil imbibition studies, repeat formation pressure test results, and open- and cased-hole logs. An excellent match of waterflood performance was obtained with the dual-porosity model. Of particular interest are the imbibition characteristics of the Maastrichtian in the Ekofisk field and the character of the water-cut performance of the producing wells following injector shutdowns and startups. Introduction The Ekofisk field was discovered in Nov. 1969 in Block 2/4 of the Norwegian sector of the North Sea. The field is a north/south-trending anticline located about 160 miles [257 km] from land in about 240 ft [73 m] of water. In July 1971, production began from four subsea wells. These were later abandoned in 1974 when production began through permanent facilities. Field production peaked in Oct. 1976 at about 350,000 STB/D [55 600 stock-tank m3/d] and currently averages 110,000 STB/D [17 500 stock-tank m3/d]. Original oil in place (OOIP) in Ekorisk is estimated to be 6.7 × 109 bbl [1.1 × 109 m3]. The reservoir consists of about 600 ft [180 m] of productive limestone that can be divided into the Ekofisk productive limestone that can be divided into the Ekofisk formation (Danian Age), approximately 400 ft [120 m] thick, 50 to 90 ft [15 to 30 m] of dense limestone and a 200-ft [60-m] -thick section of highly fractured Tor formation (Maastrichtian Age). The reservoir rock is naturally fractured, with fracture intensity increasing with depth. The reservoir was overpressured initially and contained an undersaturated oil at an initial pressure of 7,120 psig at 10,400 ft [50 MPa at 3170 m] subsea. The psig at 10,400 ft [50 MPa at 3170 m] subsea. The bubblepoint pressure was approximately 5,545 psig [38 MPa] at a reservoir temperature of 268 deg. F [131 deg. C]. Initial solution GOR at producing separator conditions was 1,530 scf/STB [276 std m3/stock-tank m3]. Table 1 presents a summary of the Ekofisk reservoir parameters. The field was developed with three production platforms. Produced gas in excess of sales gas has been platforms. Produced gas in excess of sales gas has been reinjected into the Danian formation in the crest of the field. Oil produced from the field is sent by pipeline to Teesside, England, and gas production is transported by pipeline to Emden, Germany. As of Jan. 1, 1984, a total pipeline to Emden, Germany. As of Jan. 1, 1984, a total of 690 × 106 bbl [110 × 106 m3] of stock-tank oil and 2,263 Bcf [64 × 109 m3] of gas have been produced. Gas reinjection totals 621 Bcf [17.6 × 109 m3]. Primary oil recovery with excess gas injection is forecast to be about 1.2 × 109 bbl [190 × 106 m3] or 18% of the OOIP. A Maastrichtian pilot waterflood was initiated in the Ekofisk field in April 1981 to evaluate the performance of water injection in this highly fractured formation. The four wells that make up the heart of the pilot are B-16, of water injection well, and B-19, B-22, and B-24, the three closest Maastrichtian-only producers. Both model and laboratory studies were undertaken to assist in the evaluation and interpretation of waterflood results. The model study of water injection into the Ekofisk Pilot, which is located in the Platform B area of the field, Pilot, which is located in the Platform B area of the field, was conducted with a dual-porosity model. An analysis of available data was made to determine fractured zones in the pilot area, and only those areas were assigned dual porosities. History for this study consists of the period from Jan. 1, 1978, to April 1984 and includes a total pilot water injection of 21 × 106 bbl [3.3 × 106 m3]. pilot water injection of 21 × 106 bbl [3.3 × 106 m3]. During the injection period, 107 STB [1.5 × 106 stock-tank m3] of oil and 38.2 Bcf [1.1 × 109 m3] of gas were produced from the three pilot producers. produced from the three pilot producers. Initial conditions for the study were taken from a 3D history match of the field. The area selected for inclusion in the study is about 1,100 acres [445 ha] and includes five edge wells-B08, B-14, B-18, B-21, and B-23- in addition to the primary pilot wells. JPT P. 221

This paper discusses specific issues encountered when pressure tests are analyzed in reservoirs with complex geological properties. These issues relate to questions concerning the methodology of scaleup, the degree of aggregation, and the reliability of conventional methods of analysis. The paper shows that if we desire to use pressure-transient analysis to determine more complex geological features such as connectivity and widths of channels, we need a model that incorporates reservoir heterogeneity. This complexity can lead to significantly more computational effort in the analysis of the pressure transient.The paper demonstrates that scaleup criteria, based on steadystate procedures, are inadequate to capture transient pressure responses. Furthermore, the number of layers needed to match the transient response may be significantly greater than the number of layers needed for a reservoir-simulation study. The use of models without a sufficient number of layers may lead to interpretations that are in significant error.The paper compares various vertical aggregation methods to coarsen the fine-grid model. The pressure-derivative curve is used as a measure of evaluating the adequacy of the scaleup procedure. Neither the use of permeability at a wellbore nor the average layer permeability as criteria for the aggregation was adequate to reduce the number of layers significantly. 10th SPE Comparative Solution Project. 2 It represents two formations-the Tarbert and the Upper Ness, 70 ft (35 layers) and 100 ft (50 layers) thick, respectively-that form a part of the Brent (Middle and Upper Jurassic) sequence in the North Sea. The Tar-