Much attention has been given to the simulation of naturally fractured reservoirs in the recent literature. The most prevalent approach is a dual-porosity (or dual-porosity/dual-permeability) formulation with computation blocks that may represent several individual matrix blocks. In models of this type, the processes of gas/oil drainage and water/oil imbibition have caused particular difficulties. Some authors have attempted to represent the correct behavior through a gravity term that assumes a simplified fluid distribution in the matrix. Others have used pseudo-capillary-pressure functions for the matrix, the fracture, or both. These functions also assume a simplified matrix fluid distribution or are obtained by "history matching" with a fine-grid model of a single matrix block. Still others have introduced refinement of the matrix into multiple blocks.In this paper, the authors examine the mechanisms involved in gas/oil drainage and water/oil imbibition and propose a simple way to represent that behavior in a dual-porosity simulator. Basically, the formulation uses pseudo-capillary-pressure curves for both the matrix and fracture. The fracture curve can be determined directly from rock properties and matrix-block dimensions, while the matrix curve can be obtained from the results of a single simulation of a fine-grid model of a single matrix block. The approach is less costly than matrix subdomain and history-matching alternatives and is often more accurate than solutions that rely on a simplified gravity term.

A mathematical model is presented for analyzing the effect of small, discontinuous shales on oil recovery. The model was validated by calculations with a fine-grid computer model. The study illustrates the importance of teamwork between geologists and engineers when describing a reservoir and when predicting its performance. Introduction A complex problem when predicting reservoir performance is assessment of permeability of a formation performance is assessment of permeability of a formation perpendicular to bedding planes. Vertical permeabilities, perpendicular to bedding planes. Vertical permeabilities, which may be less than horizontal permeabilities because of orientation of rock particles and cementing material, can be measured on core samples. It also is necessary to determine distribution of nonpay intervals because small amounts of impermeable rock can profoundly affect vertical permeabilities, even if the rock is discontinuous and randomly distributed. Lateral dimensions of shale laminae, shale bodies, or other impervious materials and their distribution are difficult to describe solely from data on cores and logs. Proper description requires knowledge of depositional environment and detailed information from geologic studies of outcrops and recent sediments. Correct assessment of vertical permeability is particularly important in gas drive operations because high oil particularly important in gas drive operations because high oil recovery depends on effective oil drainage by gravity. For example, shale bodies may be large enough laterally in some reservoirs to prevent coning and yet be few enough to permit good vertical oil drainage. A laminate barrier in a pinnacle reef (carbonate reservoir) was extensive enough to cause loss of miscibility by collecting oil and part of an LPG bank above it. Recent studies indicate the presence of frequent, small, randomly distributed shale bodies in the upper formation of a large sandstone reservoir, which reduce vertical permeability in that interval. Fortunately, conceptual models based on geologic studies of recent sediments, outcrops, cores, and logs are available for describing the distribution and dimensions of pay and nonpay in both sandstone and carbonate reservoirs. Also, reservoir simulators are available for solving increasingly complex problems in practical times at reasonable costs. Construction of simulators is relatively straightforward when a reservoir is divided into separate zones by continuous barriers. Usually, at least three layers are provided in a computer model for each zone. Thin layers provided in a computer model for each zone. Thin layers are used above impervious boundaries to represent flow from gas-invaded regions. An alternative procedure is to generate pseudorelative permeabilities for layers above boundaries. Layers beneath impervious boundaries in water-invaded regions require similar treatment for accurate simulation of reservoir performance. However, no standard treatment is available for modeling reservoirs containing many small, discontinuous barriers. Adequate simulation of performance in large regions of a reservoir could requite many more small computer blocks than the user could afford. This paper illustrates how geologists and engineers can cooperate in describing and predicting oil recovery from complex distributions of permeable and impermeable intervals. A brief overview is given on geologic modeling of different types of deposits. A simple mathematical model is presented for assessing the impact of a given sized impervious body on oil recovery. JPT P. 1531

The final blowdown of a Gulf Coast water-drive gas reservoir at a reserves/ production ratio of less than 2 provided an increase in gas recovery. production ratio of less than 2 provided an increase in gas recovery. Pressure/production performance did not conform to conventional tank-type Pressure/production performance did not conform to conventional tank-type material balance predictions because of a large pressure gradient in the water-invaded region. The material balance equation was altered to account for the pressure behind the front and was adapted to an economic optimizer program. Introduction The added recovery benefits attributable to accelerated blowdown of strong water-drive gas reservoirs are well known. The planning for effective blowdown of such reservoirs requires the accurate prediction of reservoir pressure and production performance. Deliverability pressure and production performance. Deliverability maintenance investments must be scheduled well in advance and economic incentives depend on accurate forecasts of gas recovery. Beginning in late 1969, a strong water-drive gas reservoir in the Katy field was blown down at a reserves/ production ratio (R/P) of less than 2. Although gas production ratio (R/P) of less than 2. Although gas recovery during blowdown was more than 30 percent greater than that obtained at a low production rate equivalent to an R/P of 15, reservoir pressure/production performance varied considerably from predictions made performance varied considerably from predictions made using a conventional van Everdingen-Hurst unsteady-state material balance. Reservoir pressures declined more rapidly than predicted, and gas recovery was less than expected. Performance of the reservoir suggested the presence of a substantial pressure gradient in the presence of a substantial pressure gradient in the waterinvaded region, and the conventional material balance used was not capable of modeling it. To obtain accurate predictions, the van Everdingen-Hurst unsteady-state material balance equations were modified to account for higher pressures in the water-invaded region. Using the modified equations, it was possible to match accurately the pressure/production possible to match accurately the pressure/production performance for the Katy reservoir. performance for the Katy reservoir. Katy V-C Reservoir Description and History The Katy V-C reservoir is a uniformly developed sand in the Yequa formation, approximately 40 ft thick, having an original productive area of 7,300 acres. Structurally, the reservoir is an elongate, north-south trending, unfaulted anticline with 105 ft of structural closure above the original gas-water contact at -7,240 ft. Dip on the flanks is a fairly uniform 180 ft/mile. Original gas in place was 330 Bcf. Reservoir rock and fluid data developed from laboratory analysis are shown in Table 1. Early development of the reservoir was designed to supply gas for a small sale and as make-up for fuel and shrinkage incurred in cycling other zones. Fig. 1 shows the completion history. Cycling of the V-C reservoir began in 1950. The cycle program included overinjection during the early years. program included overinjection during the early years. This contributed to a substantial repressuring. The cycle pattern was end-to-end with injection confined to Well pattern was end-to-end with injection confined to Well 4302, completed below the original gas-water contact. A bottom-hole pressure measured in this well, outside the original productive limits and before injection, was 270 psi above the pressure in the uninvaded gas zone as psi above the pressure in the uninvaded gas zone as measured uniformly in three wells active at the time. After cycling was completed, limited gas production was resumed until Sept. 1969 when accelerated blow-down was started. Before blowdown, the production rate had declined below 15 MMcf/D as water influx began repressuring the reservoir (Fig. 2). Cumulative production to Sept. 1969 was 151 Bcf while the reservoir production to Sept. 1969 was 151 Bcf while the reservoir pressure was 2,830 psi. pressure was 2,830 psi. JPT P. 1533

Conventional reservoir simulation techniques prove to be inadequate when applied directly to the prove to be inadequate when applied directly to the study of fractured reservoir systems. Such systems are characterized by extremes in porosity, permeability, and saturation. The vast bulk of the permeability, and saturation. The vast bulk of the reservoir volume is occupied by relatively low-permeability, disjoint matrix blocks of various sizes surrounded by a small volume of high-permeability, interconnected fracture space. Our approach to this complex problem bas been to treat the matrix blocks as source and sink terms in an otherwise conventional simulation that models only the fracture system. The source/sink terms are functions of matrix rock and fluid properties with fracture saturation and pressure defining the boundary conditions. These functions are derived either by history-matching simulations or independently by laboratory experiments or single matrix-block simulation. This basic concept of a source/sink treatment is not unique to this work. However, the numerical formulation and implementation of these terms in the fracture simulation offers significant advantages over existing modeling procedures. The fundamental advantage of our approach is that these source terms are handled semi-implicitly in both the pressure and saturation calculations involved in pressure and saturation calculations involved in the fracture simulation. This avoids instability problems that are inherent in a sequential problems that are inherent in a sequential fracture-matrix solution and links more closely the behavior of the matrix and the fracture. Special techniques are developed for modeling the effects of fluid contact movement within a large simulation grid block and for treating receding gas-oil and water-oil contacts. Hysteresis effects are included in matrix blocks that begin to imbibe oil after drainage has begun. Introduction Conventional reservoir simulation techniques are not capable of adequately modeling large, naturally fractured reservoir systems. Extreme discontinuities in porosity, permeability, and saturation exist throughout the reservoir. Most of the fluids are found in very low-permeability, disjoint matrix blocks of various sizes, while most of the fluid mobility is in a small volume of high-permeability, interconnected fracture space. The distribution of fluids within the fracture is usually governed primarily by gravity segregation while the behavior of the individual matrix blocks depends on pressure, fluid environment, and matrix fluid saturation. Fluids can move readily throughout the reservoir in the fracture space, but fluids that reside in matrix rock must enter the fracture to move any great distance. The behavior of individual matrix blocks in response to various drive mechanisms has been studied experimentally by Crawford and Yazdil and has been simulated in two dimensions by Kleppe and Morse and Yamamoto et al. Other investigators have studied the single-phase pressure behavior of fractured reservoirs and its effect on pressure buildup curves. Kazemi investigated single-phase flow in a radial reservoir dominated by horizontal fractures. Closman simultaneously solved equations for flow between matrix and fracture and for flow along the fracture planes for a radial aquifer to develop relationships similar to those developed by van Everdingen and Hursts for water influx. Simulation of an entire reservoir system with multiple phases further complicates the problem and makes additional simulator modifications necessary. Asfari and Witherspoon have developed a modeling approach for reservoirs with a regular pattern of noncommunicating vertical fractures by pattern of noncommunicating vertical fractures by assigning constant pressures along each fracture. Several investigators have applied finiteelement techniques to the fracture-matrix flow problem. problem. Our approach to this complex problem has been to model the flow in the fracture system and to treat fluid transfer to and from the matrix much as injection and production are modeled in conventional simulators. Transfer of fluid into the fracture will be represented by a "source" term and transfer from the fracture to the matrix will be represented by a "sink" (or negative source) term. SPEJ P. 201

Extrapolating the early portion of a plot of p/Z vs volume of gas produced is not always adequate for reservoirs characterized by abnormal pressure or substantial aquifer support. The procedure outlined in this paper applies nonlinear regression analysis to observed pressure and production history to obtain reliable estimates of gas in place and other reservoir properties. properties. Introduction Estimation of the volume of gas in place (GEP) plays an important role in the evaluation and analysis of gas reservoirs. Soon after the discovery of a new gas field, a reliable estimate of the volume of gas available for production is needed for planning long-range contracts and production is needed for planning long-range contracts and commitments to supply gas users. In addition, knowledge of the GIP, as well as other reservoir properties, is necessary for scheduling well drilling, compression installation, and other investments that will maximize profit over the life of a field. profit over the life of a field.The principal methods of predicting GIP are the volumetric method and the material-balance method. The volumetric method requires geologic data defining the physical limits of the reservoir and core or log data physical limits of the reservoir and core or log data describing the distribution of fluids within the reservoir rock. These data can be quite sketchy, especially in the early history of a field, and the estimate of GIP derived from them can be in error by as much as a factor of 2. The material-balance method uses pressure data in the early production period to estimate GIP. Under ideal conditions, a plot of p/Z vs gas production will be a straight line that can be extrapolated to p=0 to determine GIP, as illustrated in Fig. 1. There are, however, several phenomena that prevent the data falling on a straight line. phenomena that prevent the data falling on a straight line. Common examples are illustrated in Figs. 2 through 4. First, if the gas reservoir is in contact with a sizable aquifer, reduction of the reservoir pressure may be accompanied by influx of water and, consequently, the rate of pressure decline may decrease, resulting in a leveling off of the p/Z curve. The rate of water influx and, thus, the shape of the p/Z curve, depend on the physical properties of the aquifer. In addition, extrapolation of p/Z to properties of the aquifer. In addition, extrapolation of p/Z to zero is not valid since the field will be depleted and wells will be watered out at some positive reservoir pressure. Second, if the reservoir initially has abnormally high formation compressibility, as observed in high-pressure reservoirs, the rate of pressure decline might increase with time because the compaction of the reservoir will provide pressure support at the higher pressure level. provide pressure support at the higher pressure level. Finally, if the reservoir gas is near its dew point, retrograde condensation may occur; and because the liquid density is higher than the gas density, the pressure decline may accelerate. In such instances, extrapolation of p/Z data can give erroneous results, sometimes in error by more than 100 percent. The volumetric method and the material-balance method are the two most prevalent procedures for estimating GIP, but there is no guarantee that the estimate generated using either procedure will be even close to the correct value. A more reliable method is needed for determining the volume of gas existing in real reservoirs that may be influenced by one or more of the physical phenomena mentioned above and that may exhibit highly phenomena mentioned above and that may exhibit highly nonlinear p/Z behavior. Fortunately, the effects of these factors on the reservoir pressure can be estimated quantitatively and can be included in a mathematical model of the system. P. 1283