Areal Model Research Articles

A Kalman filter modelling of space-time rainfall using radar and raingauge observations

Conventional raingauge networks have proven to be inadequate to describe the temporal and spatial distribution of rainfall, but raingauge data combined with radar data can provide a very useful technique for monitoring the quantitative distribution of rainfall in time and space. A square grid mean areal rainfall model is developed, which utilizes raingauge measurements and a Kalman filter state-space model of the Z–R relationship from radar measurements. The developed method applies in areas where radar information exists, and it has been tested for only one watershed. Key words: space-time rainfall model, radar, raingauge, Kalman filter.

A Material-Balance Method for Deriving Interblock Water/Oil Pseudofunctions for Coarse-Grid Reservoir Simulation

Summary This paper describes a material-balance method for deriving interblock water/oil pseudofunctions for application in coarsely gridded areal reservoir models. The method is shown to be effective in pattern waterflood simulation. This technique differs from previous methods in (1) explicitly accounting for grid orientation by deriving directionally dependent pseudos, (2) combining a dispersion control scheme with the pseudofunction derivation, and (3) matching fractional flow in pattern models through a material balance. The pseudofunctions are derived from fine-grid pattern models and account for various effects, including anisotropic formation properties, dominant displacement forces (viscous or gravitational), and correction for preferential directional flow.

Design Equations for Bod Removal in Facultative Ponds

Facultative pond performance data collected for the US Environmental Protection Agency (USEPA) at four locations throughout the USA and data collected by others were used to evaluate the most frequently used design equations and to develop non-linear design equations. Empirical models were evaluated as well as the classical plug flow and complete mix models. The first order plug flow model gave the best fit of all the rational models. The empirical non-linear models did not fit the data, nor did the other empirical models with the exception being the areal loading and removal model. Attempts to verify the models developed with the USEPA data using data collected by others were not successful with the exception of the areal loading and removal model.

A New Stability Theory For Designing Graded Viscosity Banks

Abstract Immiscible secondary recovery operations, even when successful, may leave behind significant amounts of oil. A possible candidate for recovering more of the oil is a process which uses an injection fluid which is miscible with the oil ill the reservoir. However, if such a process is to be viable, it is important to amid the continuous injection of an expensive solvent. This may be accomplished by injecting a small slug of solvent and then chasing it with a less expensive fluid which is miscible with the solvent. The success of such a scheme depends, to a large extent, upon whether the displacement process is stable or unstable. That is, if the displacement is unstable, viscous fingering greatly increases the amount of solvent required. Thus, when designing a miscible slug process, it is important to know whether the displacement is stable or unstable. Moreover, if the process is unstable, it is important to take into account the effect of viscous fingering. This paper presents a newly-developed theory which is based on the small perturbations method. In particular, a dimensionless number, which is a function of the dimensions of the porous medium as well as the properties of the fluids and the porous medium and its critical value for the onset of viscous fingering, is derived. With the help of this dimensionless number, it is possible to design a graded viscosity solvent bank which is able to resist the adverse effects of viscous fingering. That is, the theory enables the prediction of the minimum size and configuration of a graded viscosity bank so as to maximize recovery by avoiding bank breakdown due to viscous fingering. Introduction Miscible flooding has received widespread attention as an enhanced oil recovery method because of its potential for recovering all of the oil contacted by the solvent. However, it is not usually economically feasible to displace the oil and entirely replace it with a solvent. Therefore, a process, in which only a small slug of solvent chased by a cheaper fluid which is miscible with the solvent, has been devised for use in field operations(1,2). Nevertheless, experiments carried out in areal models supported by field data have revealed that channeling and fingering of both the solvent into the oil and the chase-fluid into the solvent, because of the adverse mobility ratios encountered, reduced the areal sweep efficiency drastically. Thus, lower recoveries than expected were attained(3,4). In an oil reservoir two factors significantly affect the sweep efficiency: in-homogeneity of the reservoir rock and frontal instability. Only the latter will be considered in this paper, Frontal instability can be reduced by controlling the mobility of the displacing fluid. It has been shown experimentally that one way to achieve this goal is to gradually change the viscosity of the injected fluid from that of the displaced fluid to that of the displacing fluid(5,6). Because the viscosity of a mixture depends upon composition, the desired change in viscosity can be achieved by gradually changing the composition of the fluids in the system from that of high viscosity to that of low viscosity.

A block preconditioned conjugate gradient-type iterative solver for linear systems in thermal reservoir simulation

A preconditioned residual-norm-reducing iterative solver is described. Based on a truncated form of the generalized-conjugate-gradient method for nonsymmetric systems of linear equations, the iterative scheme is very effective for linear systems generated in reservoir simulation of thermal oil recovery processes. As a consequence of employing an adaptive implicit finite-difference scheme to solve the model equations, the number of variables per cell-block varies dynamically over the grid. The data structure allows for 5- and 9-point operators in the areal model, 5-point in the cross-sectional model, and 7- and 11-point operators in the three-dimensional model. Block-diagonal-scaling of the linear system, done prior to iteration, is found to have a significant effect on the rate of convergence. Block-incomplete-LU-decomposition (BILU) and block-symmetric-Gauss-Seidel (BSGS) methods, which result in no fill-in, are used as preconditioning procedures. A full factorization is done on the well terms, and the cells are ordered in a manner which minimizes the fill-in in the well-column due to this factorization. The convergence criterion for the linear (inner) iteration is linked to that of the nonlinear (Newton) iteration, thereby enhancing the efficiency of the computation. The algorithm, with both BILU and BSGS preconditioners, is evaluated in the context of a variety of thermal simulation problems. The solver is robust and can be used with little or no user intervention.

Miscible Flood Forecasting Technique at Judy Creek

Summary A simplified forecasting technique was developed to predict the recovery performance of a hydrocarbon miscible flood in the Judy Creek Beaverhill Lake (BHL) A Pool in Alberta. This technique incorporates the physics of the hydrocarbon miscible-displacement process as well as the complex geological reservoir description across the entire pool. The advantages of the technique over conventional three-dimensional (3D) reservoir simulation include cost-effectiveness, speed, and flexibility of parametric evaluations. A single-layer areal reservoir simulation model described the areal sweep; oil recovery performance as a function of injection and solvent bank size; and solvent, chase gas, and postmiscible water breakthrough times. A cross-sectional reservoir model determined the effect of gravity override. A geological model provided the porosity and permeability distribution for every 0.6-m [2-ft] interval in each injection pattern as well as the continuous PV-i.e., the volume that is available for miscible displacement. These results were combined into a simplified algorithm (the Forecast Model) to predict tertiary recovery. The Forecast Model adds the tertiary recovery forecast to the waterflood forecast developed from decline analysis to generate a total oil forecast under miscible flooding. This miscible flood forecast is physically reasonable and predicts recovery levels consistent with observed performance of large-scale miscible projects. This technique is being extended for use in other reservoirs that are candidates for miscible flooding.

Simulating a Steamflood at the Georgsdorf Field, Federal Republic of Germany (includes associated papers 13965 and 14471 )

Summary This paper describes the use of a black-oil thermal simulator to evaluate the performance of an ongoing steamflood at the Georgsdorf reservoir in the Federal Republic of Germany. An areal model was developed to represent about 1.2 × 10(6) m2 [300 acres] of the steam flood and the adjoining area where wells are located on irregular spacing. Ten years' historical performance, including six years' steam injection, were history matched to calibrate the model. The calibrated model then was used to study the future depletion of the reservoir under several options. This information was used by Gewerkschaften Brigitta und Elwerath Betriebsfuhrungs GmbH (BEB), the operator, to help make decisions about the future operation of the steam flood. Introduction Reservoir engineering was defined by Moore as the "artof developing and producing oil and gas fields in such a way as to obtain a high economic recovery." In practicing this art, the repeated objectivity of reservoir simulation can avoid the subjective distortion of analytical procedures. However, the art becomes increasingly abstract when simulation is applied to reservoir processes involving nonisothermal operations and other complex phenomena. This paper is concerned with the use of a thermal reservoir simulator to assess the historical performance of anon going steam flood at the Georgsdorf field so that possible future operating options can be evaluated. Such information can assist management in decidinghow much steam is required in certain areas,how the steam flood might be expanded, andwhere new injection wells and infill producers might be located. No analytical procedures exist to evaluate the displacement of oil by steam and water adequately in a complex geological setting where wells are drilled on irregular spacing. Thermal reservoir simulation provides this capability, but applications of such simulators can be impractical to justify, especially for smaller projects. The literature is replete with information about simulating conventional fluid displacement in isothermal systems. Substantially less is available for simulating on isothermal, chemical, and miscible processes. Coats cited much of the pertinent simulation technology and put this information into perspective in his 1982state-of-the-art paper. There is little published information on specific applications of thermal simulation. Chu and Trimble used a three-dimensional (3D), three-phasenumerical simulator in the black-oil mode to history match5 1/2 years' performance for a steam stimulation pattern in the Kern River field, CA. The model subsequently was used to optimize operating parameters. Gomaa et al. also used a black-oil thermal simulator to model elements of an inverted five-spot pattern to determine the relative importance of various steam flood and reservoir parameters for the Monarch sand in the Midway-Sunset field, CA. Munoz simulated steam flooding in the Tia Juana M-6Project, Venezuela. In this work, a 3D model was prepared for an element of symmetry in a regular inverted seven-spot. These studies addressed the effects of permeability variations, gravity segregation, and positions of completion intervals. Williams prepared a 3D model with which he determined the response to production by steam stimulation and steam displacement in the steeply dipping North Midway field, CA. Another 3D study of a steeply dipping reservoir was reported by Moughamian et al. This work provided information used to design a steam flood in the heavy-oil bearing Webster sand sequence in the Midway-Sunset field. All the studies mentioned treated a representative segment or an element of a repeated pattern where either steam stimulation or steam displacement was contemplated. All the studies provided important ideas about simulation methodology but none dealt with fieldwide simulation of ongoing steam floods with irregularly spaced wells. The Georgsdorf Field Lillie and Springer discussed the technical and economic aspects of a steam flood in the Georgsdorf field. Somepertinent reservoir description information from this paperis repeated for convenience. The Georgsdorf field was discovered in 1943 and developed largely between 1946 and 1963 by the drilling of about 350 wells. BEB operates the field for aconsortium of companies that includes itself, C. Deilmann A.G., Preussag A.G., and Wintershall A.G. Each company owns 25% of the field. The Valanginian sandstone of lower Cretaceous age is a good quality reservoir throughout much of Georgsdorf. JPT P. 1952^

The Krafla Geothermal Field, Iceland: 3. The generating capacity of the field

This paper presents analytical and numerical studies of the generating capacity of the Krafla field. A general lumped parameter model is developed which can be used to obtain rough estimates of the generating capacity of a geothermal field based on the size of the wellfield, the average formation porosity, and the amount of recharge to the system. The model is applied to the old wellfield at Krafla. More sophisticated calculations of the generating capacity of the Krafla field are also performed using distributed‐parameter models. Two‐dimensional areal models of the various reservoir regions at Krafla are developed and their generating capacities (MWe) evaluated. The results obtained indicate that the old wellfield can sustain steam production of 30 MWe for 30 years. The estimated power potential of the new wellfield is 20 MWe for 30 years. To obtain the required steam production several additional wells may be drilled in the old and new wellfields.

Porosity Distribution in Wolfcamp Strata, Palo Duro Basin, Texas Panhandle: Implications for Deep-Basin Ground-Water Flow: ABSTRACT

Average-porosity distributions in the Wolfcamp deep-basin aquifer are critical to discernment of the geographic trends in effective-porosity in the Palo Duro basin. Precise data are used to improve resolution of porosity values for computer-simulated areal ground-water modeling. Assessing vertical distributions of lithology and porosity in each well studied involves analysis of crossplotted neutron- and density-porosity log responses. This method more accurately identifies lithology and porosity than does the commonly employed crossplotted neutron-porosity and sonic (interval traveltime) responses. Log-derived average-porosity distributions yield information about effective pore volume (i.e., movable water) in the Wolfcamp aquifer and enhance the accuracy of estimates of raveltimes and velocities of brines in basinwide traverses. Mathematical analysis of average traveltime and total effective pore volume yield estimates of the rates of annual discharge from the Wolfcamp aquifer in the Palo Duro basin. Based on average flush rates between 2.2 and 1.5 m.y., annual discharge rates from the Wolfcamp aquifer, across the northern and eastern basin boundaries, are about 3.6 × 105 m3 year-1 to 5.3 × 105 m3year-1. End_of_Article - Last_Page 465------------

Race, Class, and Politics in the Black Community of Boston

We found that there are indeed some slight differences in the electoral behavior of black voters when we separate them into high and low socio-economic areal categories based on the level of median income. These slight differences include turnout rates for primary and general elections and voter registration rates. The higher the socioeconomic status of the black electorate in Boston, the higher the turnout and registration rates for that area tended to be; however, this relationship was not found to be consistent among all the areas selected for study. Despite these differences in electoral behavior or “practices,” our areal model does not suggest any significant political schisms based on the sicioeconomic indicators used. Black voters tended to vote the same way on a number of city and state ballot questions, support the same candidates at both the city and state level, and vote the liberal preference on both issues and candidates regardless of socioeconomic background. In addition, there is some evidence that the higher a black voter is on the socioeconomic scale, the greater his or her tendency will be to vote the liberal position on various issues and candidates. Our model also suggests that black voters will tend to support black candidates regardless of some socioeconomic factors. If this model can be used as an indication of class differences in the black community, then our data show that socioeconomic criteria are not as salient as race in determining the political behavior and preferences of black voters.

G-2 and G-3 Reservoirs, Delta South Field, Nigeria: Part 1-An Engineering Study

Summary This paper describes a two-dimensional, three-phase, black-oil simulation of the G-2 and G-3 reservoirs in the Delta South field offshore Nigeria. The purpose of these studies was to investigate, from an engineering standpoint, various operating schemes for optimizing the oil recovery from each of these highly gravity-segregated reservoirs. Introduction The G-2 and G-3 sands are two major reservoirs in the Delta South field. The field is offshore Nigeria in approximately 10 to 16 ft of water (Fig. 1). Production from these reservoirs began in March 1968. By the end of March 1978, about 61.3 MMSTB of oil and more than 64 Bscf of gas were produced from the G-2 reservoir, and about 92 MMSTB of oil and more than 75 Bscf of gas were produced from the G-3 reservoir. These productions represent about 2 1% of the original oil in place (OOIP) for G-2 and 29% of OOIP for G-3. During this period, the average reservoir pressures for G-2 and G-3 declined from 3,936 to 2,665 psig and 3,829 to 2,343 psig, respectively. The G-2 reservoir is at an average depth of 8,920 ft subsea. It is underlain by the G-3 reservoir, but the two are separated by a shale barrier. The two reservoirs are similar in structure and in rock and fluid properties. The drive mechanism for these reservoirs is primarily gravity segregation. These reservoirs are characterized as massive, clean sands with high porosity (about 28% for G-2 and 26.5% for G-3) and high horizontal and vertical permeabilities (kh=2,000 md and kv=1,000 md). This paper presents the results of the current G-2 and G-3 reservoir studies, which are extensions of previous work. First, a history match of 10 years of production performance was made. Later, several prediction cases were run under a variety of operating conditions. This study was undertaken to update the previous match, including the past performance data through March 1978. We believe that, by including more history match data, we have obtained a better reservoir characterization. Although about 20 prediction cases were run for each study, only four pertinent cases for G-2 and five for G-3 are included. Gulf's two-dimensional, black-oil model in an areal mode has been used for this study. The use of this areal model is believed sufficient in describing the reservoir and fluid flow behavior and is justified since the fluids will be segregated because of the high kv/kh ratio (about 0.5:1 were kh=2,000 md). In this cases, gravity/capillary equilibrium exists throughout the thickness. However, there are some limitations in using an areal model for this study:the well coning effects are represented by well pseudo relative permeability curves and not with grid blocks in the vertical direction,the well completions are represented by modifying the well relative permeability curves, andany well coning effect matched using pseudo relative permeability curves may not be strictly valid for future predictions. These limitations, however, do not seem to be very important for the G-2 and G-3 reservoirs.

Gas Injection for Upstructure Oil Drainage

Any attic oil project tries to displace with gas the oil located above the structurally highest well in the reservoir. Results from a numerical simulator show that the most important controllable variable in attic oil recovery is the volume of gas injected during each cycle. A method is presented for calculating the required gas injection volume for each cycle presented for calculating the required gas injection volume for each cycle of an attic oil project. Introduction Oil recovery from high-relief reservoirs can be increased by downstructure gas injection. In this process, often called "attic oil recovery," gas is injected in the structurally highest well in the reservoir. The injected gas will migrate upstructure, forming a secondary gas cap and displacing oil downward.In high-relief reservoirs, which typically occur around "piercement type" salt domes, field development is complicated by the lack of control on the upper limits of the structure. To ensure a commercial completion, operators usually locate development wells a safe distance from the estimated sand/salt interface. In reservoirs with an active bottomwater drive, much of the updip oil will not have been produced when the highest well in the structure waters out. Sidetracking or drilling new wells is expensive and risky. It would be almost impossible to locate new wells to drain all updip oil adequately. Reservoirs with steep dip and high permeability allow injected gas to migrate to the most permeability allow injected gas to migrate to the most inaccessible areas of the pinchout, providing good lateral coverage. Injected gas may be traded economically for oil as long as the price of a unit volume of reservoir oil is sufficiently greater than the price of a corresponding unit volume of reservoir gas.Many authors have reported on field applications of the attic oil recovery process. In 1971, Combs and Knezek published theoretical guidelines and field data concerning the maximum gas/oil segregation rate and the minimum and actual gas requirements to recover 1 bbl oil.This paper presents results of a numerical model study of an attic oil recovery process and defines the variables that control the reservoir performance of a successful project. A method is presented to calculate the required project. A method is presented to calculate the required gas injection volumes. Numerical Model The numerical model used in this study is a conventional beta model that was modified to simulate the attic oil recovery process properly. The model includes these features: three-phase flow of oil, water, and gas; gas/oil solubility; gas/oil and oil/water capillary pressure; three-phase relative permeability with hysteresis effects; variable bubble-point scheme; variable flow rates; gravity effects, both areally and vertically; conformance to any reservoir geometry; and variable rock and fluid properties. properties. Even with the advent of relatively efficient three-dimensional reservoir simulators, computing costs for a meaningful study can become expensive. A two-dimensional areal model can simulate the same problem, provided the distribution and flow of fluids in the vertical dimension is included implicitly in the areal model. Several authors' have reported on the theory and calculation techniques of "pseudofunctions." Jacks et al. presented a technique for calculating the "dynamic pseudofunctions" that could be applied over a wide range of How rates and initial fluid saturations. JPT P. 1323

Reservoir Simulation of the Empire Abo Field: The Use of Pseudos in a Multilayered System

Abstract A three-layered model with vertical equlibrium pseudo-relative permeabilities and pseudo-capillary pseudo-relative permeabilities and pseudo-capillary pressures was used to match the results from a pressures was used to match the results from a 22-layered model of a portion of the Empire Abo field. Results of the two models were almost identical in both history match and prediction phases. Directionally dependent pseudo-relative permeabilities were introduced to match the drainage mechanism of the reservoir. Both horizontal displacement and drainage mechanisms can be modeled simultaneously in a multilayered system using these pseudo functions.The excellent comparison between the finely gridded model and the pseudo model allowed the simulation of alternative recovery schemes using the pseudo model with the same confidence as the pseudo model with the same confidence as the original three-dimensional model. In particular, a comparison between pressure maintenance with carbon dioxide and nitrogen was simulated, with a substantial savings in both manpower and computer time. Development of this pseudo model makes the simulation of the entire Empire Abo field feasible. Introduction For several years vertical equilibrium pseudo-relative permeabilities and pseudo-capillary pressures have permeabilities and pseudo-capillary pressures have been used throughout the industry to collapse a three-dimensional model with many layers to a two-dimensional areal model with a single layer. The obvious advantages of pseudo models are reduced computer and manpower costs. The justification for the use of pseudo models has been the comparison of results from one-dimensional models using pseudos with those from two-dimensional, cross-sectional models. Few comparisons have been made between an areal model using pseudos and a three-dimensional simulation.The Empire Abo field offered an excellent example for such a comparison. The field had been simulated in three dimensions, using 22 layers. An analysis of the simulation results showed that conditions were such that vertical equilibrium existed in most columns of the model. By applying pseudos to the three-dimensional model, it appeared possible to reduce the number of layers and associated computer time substantially.In this paper, the techniques for obtaining a match of the three-dimensional models with a pseudo model are described. The original three-dimensional model is described along with a history of the Empire Abo field. Coning correlations and directionally dependent pseudo-relative permeabilities also are described. Results between the pseudo model and the three-dimensional model also are compared. Finally, the use of the pseudo model for predicting alternative recovery schemes is discussed. Field History The Empire Abo field is located in southeastern New Mexico, about 8 miles (13 km) southeast of Artesia, NM (Fig. 1). The Abo reservoir was discovered by Amoco Production Co. in Nov. 1957. Subsequent development and unitization of the Abo reservoir was documented by Christianson. The producing horizon is a carbonate reef, basal Permian (lower Leonard) in age. The productive reef is about 12 1/2 miles (20.12 km) long and 1 1/2 miles (2.47 km) wide (Fig. 2), covering about 11,339 surface acres (45.9 x 10(6) m2). The reef dips gently from southwest to northeast at about 1 degree. JPT p. 279

Geothermal reservoir simulation: 3. Application of liquid‐ and vapor‐dominated hydrothermal modeling techniques to Wairakei, New Zealand

A quasi three‐dimensional, areal model based on finite‐difference approximations is applied to the hydrothermal field at Wairakei, New Zealand. The model simulates heat transport associated with the flow of steam and water through porous media. An analysis is made of the production aquifer under steady‐state and transient flow conditions, allowing vertical flow of heat and fluid through confining beds. Computed steady‐state results correlate well with observed data and indicate that portions of the aquifer had a steam cap prior to exploitation. Computed transient results also match observed data and support the hypothesis that the production aquifer is recharged through underlying confining beds. The limiting factor on production is the amount of mass available, both from storage and leakage. Although simulation results indicate that the field can maintain production rates to the year 2000, such long range predictions are unreliable due to the lack of information on the leakage properties of the confining beds.

Geothermal reservoir simulation: 2. Numerical solution techniques for liquid‐ and vapor‐dominated hydrothermal systems

Two numerical models are introduced for simulating three‐dimensional, two‐phase fluid flow and heat transport in geothermal reservoirs. The first model is based on a three‐dimensional formulation of the governing equations for geothermal reservoirs. Since the resulting two partial differential equations, posed in terms of fluid pressure and enthalpy, are highly nonlinear and inhomogeneous, they require numerical solution. The three‐dimensional numerical model uses finite difference approximations, with fully implicit Newton‐Raphson treatment of nonlinear terms and a block (vertical slice) successive iterative technique for matrix solution. Newton‐Raphson treatment of nonlinear terms permits the use of large time steps, while the robust iterative matrix method reduces computer execution time and storage for large three‐dimensional problems. An alternative model is derived by partial integration (in the vertical dimension) of the three‐dimensional equations. This second model explicitly assumes vertical equilibrium (gravity segregation) between steam and water and can be applied to reservoirs with good vertical communication. The resulting equations are posed in terms of depth‐averaged pressure and enthalpy and are solved by a two‐dimensional finite difference model that uses a stable sequential solution technique, direct matrix methods, and Newton‐Raphson iteration on accumulation and source terms. The quasi‐three‐dimensional areal model should be used whenever possible, because it significantly reduces computer execution time and storage and it requires less data preparation. The areal model includes effects of an inclined, variable‐thickness reservoir and mass and energy leakage to confining beds. The model works best for thin (<500 m) reservoirs with high permeability. It can also be applied to problems with vertical to horizontal anisotropy when permeability is sufficiently high. Comparisons between finite difference and higher‐order finite, element approximations show some advantage in using finite element techniques for single‐phase problems. In general, for nonlinear two‐phase problems the finite element method requires use of upstream weighting and diagonal lumping of accumulation terms. These lead to lower‐order approximations and tend to obviate any advantage of using the finite element method.

Simulation of Water Injection in a Barrier-Bar-Type, Oil-Rim Reservoir in Nigeria

Weber, K.J., SPE-AIME, Koninklijke/Shell Exploratie en Produktie Laboratorium Klootwijk, P.H., SPE-AIME, Shell Internationale Petroleum Mij. B.V. Konieczek, J., Shell Internationale Petroleum Mij. B.V. van der Vlugt, W.R., Shell Internationale Petroleum Mij. B.V. A three-dimensional model study was conducted on a typical Niger Delta reservoir in a low-relief, rollover structure associated with growth faulting. Detailed description of sedimentological characteristics and subsequent reservoir subdivision were essential for successful well-by-well history matching, which accounted for cusping and coning of gas and water. Predictions will almost double recovery. Predictions will almost double recovery. Introduction The Niger Delta is a large arcuate delta of a highly destructive, wave-dominated type and has been growing gradually since the Eocene. A sequence of undercompacted marine clays, overlain by deltaic deposits in turn covered by continental sands, is present throughout. The deltaic interval was formed by imbricated superposition of numerous regressive cycles that built out toward the sea. In the deltaic interval, growth-fault-associated rollover structures trapped hydrocarbons. The reservoirs are the sandy parts of sedimentary cycles ranging from 50 to 300 ft thick. The thinnest, most simple cycles are those composed of a marine clay overlain by delta fringe sediments. The laminated sand-silt-shale sediments deposited by longshore currents in water ranging from 30 to about 50 ft deep are called "barrier-face" deposits here. In shallower water, the current, tidal, and wave energy result in the deposition of long bars of clean, well sorted sand, called "barrier bars." The thickest cycles usually are composed mainly of fluviatile deposits - e.g., a series of superimposed "point bars." The barrier bar is the most common reservoir type, and the Obigbo North D1.30A reservoir, the subject of this study, is typical. The Obigbo North Field lies about 18 km (11 miles) northeast of Port Harcourt. The field was discovered in 1963. In A-Fault Block, 11 oil reservoirs can be differentiated. Reservoir D1.30A at about 7,250 ft subsea with 90.6 million STB oil initially in place (STOIIP) is the second largest. A-Block is a simple, unfaulted, rollover anticline. The low-relief, dome-shaped Reservoir D1.30A (Fig. 1) initially contained an oil ring of 130-ft column (25 deg. API gravity) between gas and water; gas volume in the primary gas cap was 33.8 Bscf. Gross reservoir thickness primary gas cap was 33.8 Bscf. Gross reservoir thickness varies from 40 to 85 ft. The reservoir seems to pinch out toward the south - i.e., toward the sea, away from the growth fault. Permeability is high, but variable. Production from D1.30A began in Oct. 1965. So far, the reservoir has been produced mainly by gas-cap expansion and (segregated) solution gas drive, with limited natural water influx. Its depletion character, demonstrated by an average reservoir pressure decline of 55 psi/% STOIIP produced (at 10 million bbl cumulative psi/% STOIIP produced (at 10 million bbl cumulative production) and the absence of pressure buildup during a production) and the absence of pressure buildup during a closed-in period in 1967-68, indicated the need for some form of pressure maintenance. In 1971, the first reservoir study was conducted with a two-dimensional, areal model to investigate the merits of alternative pressure-maintenance schemes. Based on this study, a peripheral waterflood was recommended, and a pilot water injection subsequently was initiated in April pilot water injection subsequently was initiated in April 1974 to determine injectivity and method of water injection (Well 22). The more comprehensive study discussed here was conducted to provide the technical basis for project optimization when extending the pilot test to a project optimization when extending the pilot test to a full-scale injection project. P. 1555

Pseudofunctions for Water Coning in a Three-Dimensional Reservoir Simulator

Abstract Three-dimensional numerical models of bottom-water-drive reservoirs show delayed water breakthrough into individual wells when compared with observed well performance and individual-well coning models. This reservoir-model behavior results from masking of the well coning effect by volume-averaging pressure and saturation profiles around a well over a grid block with a large volume. The reservoir-simulator prediction of well performance can be improved by mathematically performance can be improved by mathematically transforming the production performance of a detailed well-coning model into a set of time-independent pseudorelative-permeability and capillary-pressure curves that then can be used in the reservoir model. A procedure for obtaining the required pseudofunctions is described and the results of their application in simple models and in a large reservoir-simulator model are shown. Introduction The prohibitive cost of numerical reservoir simulation with fine-grid definition models of large reservoirs has led to development of techniques whereby vertical saturation distribution and/or localized flow conditions in the vicinity of individual wells can be approximately accounted for in relatively coarse-grid models at an acceptable incremental cost. In particular, vertical cross-section models under capillary and gravity equilibrium have been used to derive pseudorelative permeabilities and capillary pressures for use in two-dimensional, areal models to simulate the average vertical distribution of flow without having to pay the computing price of a full three-dimensional model. Coats et al. described the use of the vertical equilibrium concept for developing pseudorelative-permeability and capillary-pressure pseudorelative-permeability and capillary-pressure functions for simulating the vertical dimension in a two-dimensional, areal simulator model This method assumes gravity-capillary equilibrium in the vertical direction. Also, Coats et al. developed a dimensionless parameter for estimating when these conditions are valid. Martin formed a mathematical basis for pseudofunctions by reducing the equations for pseudofunctions by reducing the equations for three-phase, three-dimensional, compressible flow to two-dimensional relations by partial integration of the equations of flow. Hearn extended the pseudorelative-permeability concept by adapting it pseudorelative-permeability concept by adapting it to stratified reservoirs where viscous rather than gravity and capillary forces dominate the vertical sweep efficiency. Hawthorne studied the effects of capillary pressure on pseudorelative permeability derived from the Hearn stratified model. Jacks et al. further enlarged thepseudorelative-perrneability concept by developing dynamic pseudorelative permeabilities. (Dynamic pseudos, denoting pseudos permeabilities. (Dynamic pseudos, denoting pseudos determined under flowing rather than static conditions, allow one to account for the interaction between viscous and gravity forces resulting from rate variation in the vertical plane.) Kyte and Berry generalized the work of Jacks et al. by introducing the concept of pseudocapillary pressures and improving dynamic pseudofunction calculations to include varying flow potential gradients. Emanual and Cook expanded the concept of vertical cross-section, pseudorelative permeabilities to include the vertical performance of individual wells. Their procedure combines the effect of coning and well pseudorelative permeabilities for use in a two-dimensional, areal model. Chappelear and Hirasaki used a different approach to including of coning effects in a two-dimensional, areal simulator by developing a functional relationship among water cut, average oil-column thickness, and total rate based on an analytical incompressible, steady-state model. The most sophisticated approach to representing well-coning effects in a reservoir simulator has been taken by Mrosovsky and Ridings and Akbar et al. They incorporated detailed numerical well models into the reservoir-model grid. SPEJ P. 251

Reservoir Stimulation Studies Reveal Key to Maximizing Waterflood Oil Recovery of Virginia Hills Beaverhill Lake "A" Pool

Abstract Two comprehensive simulation studies were performed on this reservoir to evaluate waterflood performance to determine the long-term production and the injection policy for the field. The results of these studies revealed that the key to maximizing economic oil recovery from the pool is to improve the vertical conformance of the waterflood. The major biohermal reservoir is found at an average depth of 5600 ft subsea and contains under-saturated crude oil in two partially overlying, non-communicating zones. Fifteen of the field's 117 wells were converted to water injectors in an irregular peripheral water/load pressure maintenance scheme. The simulation models history matched 5 years of primary performance and 10 years of waterflood performance. As it was not practical to simulate the entire field in three dimensions, a four-layer, two-phase, three-dimensional (3-D) reservoir simulation study was done on all area approximately representing the area of influence of the largest infection well. Results indicated that there was a degree of sub-layering within the layers described in this model, Flood-front saturation profiles from the history- matched three-dimensional model were employed to derive pseudo relative permeability relationships for the two-dimensional (2-D) areal model and an excellent history match of the performance of the entire pool was achieved. The study recommended a program of production logging to establish actual production/injection profiles. Results of subsequent attempts to workover wells to improve the flood profiles are discussed. Introduction It has been common practice in the oil industry for a number of yearsto conduct comprehensive reservoir engineering studies to design enhanced recovery schemes for oil reservoirs. A more recent very valuable use for in-depth reservoir studies is to optimize the performance of existing enhanced recovery schemes. Reservoir model studies on reservoirs which have considerable production history can be used to determine ways to improve performance through changes in production and injection policy, to justify remedial action to correct adverse performance und to investigate the ramifications of alternate ourses of action. Reservoir models can also be used as effective performance surveillance tools. The opportunity to improve performance of an enhanced recovery scheme is greatest when the pool is unrestricted by market demand or other arbitrary rate restrictions, because the full benefits of the changes can be realized as soon as he field responds to the change. Whenever the demand for crude oil approaches or exceeds the productive capacity of a reservoir, opportunity to optimize performance by onducting a comprehensive engineering study presents itself. To be most effective, these studies should be interdisciplinary efforts including input from reservoir petrophysical, geological and production engineering disciplines. The studies described in this paper were performed as a cooperative effort involving all of the above disciplines and the results illustrate some of the advantages gained by this cooperative approach.

New Pseudo Functions To Control Numerical Dispersion

Abstract This paper presents an improved procedure for calculating dynamic pseudo junctions that may be used in two-dimensional, areal reservoir simulations to approximate three-dimensional reservoir behavior. Comparison of one-dimensional areal and two-dimensional vertical cross-sectional results for two example problems shows that the new pseudos accurately transfer problems shows that the new pseudos accurately transfer the effects of vertical variations in reservoir properties, fluid pressures, and saturations from the properties, fluid pressures, and saturations from the cross-sectional model to the areal model. The procedure for calculating dynamic pseudo-relative permeability accounts for differences in computing block lengths between the areal and cross-sectional models. Dynamic pseudo-capillary pressure transfers the effects of pseudo-capillary pressure transfers the effects of different pressure gradients in different layers of the cross-sectional model to the areal model. Introduction Jacks et al. have published procedures for calculating dynamic pseudo-relative permeabilities fro m vertical cross-section model runs. Their procedures for calculating pseudo functions are procedures for calculating pseudo functions are more widely applicable than other published approaches. They demonstrated that, in some cases, the derived pseudo functions could be used to simulate three-dimensional reservoir behavior using two-dimensional areal simulators. For our purposes, an areal simulator is characterized by purposes, an areal simulator is characterized by having only one computing block in the vertical dimension. The objectives of this paper are to present an improved procedure for calculating dynamic pseudo functions, including a dynamic pseudo-capillary pressure, and to demonstrate that the new procedure pressure, and to demonstrate that the new procedure generally is more applicable than any of the previously published approaches. The new pseudos previously published approaches. The new pseudos are similar to those derived by jacks et al. in that they are calculated from two-dimensional, vertical cross-section runs. They differ because (1) they account for differences in computing block lengths between the cross-sectional and areal models, and (2) they transfer the effects of different flow potentials in different layers of the cross-sectional potentials in different layers of the cross-sectional model to the areal model. Differences between cross-sectional and areal model block lengths are sometimes desirable to reduce data handling and computing costs for two-dimensional, areal model runs. For very large reservoirs, even when vertical calculations are eliminated by using pseudo functions, as many as 50,000 computing blocks might be required in the two-dimensional areal model to minimize important errors caused by numerical dispersion. The new pseudos, of course, cannot control numerical pseudos, of course, cannot control numerical dispersion in the cross-sectional runs. This is done by using a sufficiently large number of computing blocks along die length of the cross-section. The new pseudos then insure that no additional dispersion will occur in the areal model, regardless of the areal computing block lengths. Using this approach, the number of computing blocks in the two-dimensional areal model is reduced by a factor equal to the square of the ratio of the block lengths for the cross-sectional and areal models. The new pseudos do not prevent some loss in areal flow-pattern definition when the number of computing blocks in the two-dimensional areal model is reduced. A study of this problem and associated errors is beyond the scope of this paper. Our experience suggests that, for very large reservoirs with flank water injection, 1,000 or 2,000 blocks provide satisfactory definition. Many more blocks provide satisfactory definition. Many more blocks might be required for large reservoirs with much more intricate areal flow patterns. The next section presents comparative results for cross-sectional and one-dimensional areal models. These results demonstrate the reliability of the new pseudo functions and illustrate their advantages pseudo functions and illustrate their advantages over previously derived pseudos for certain situations. The relationship between two-dimensional, vertical cross-sectional and one-dimensional areal reservoir simulators has been published previously and will not be repeated here in any detail. Ideally, the pseudo functions should reproduce two-dimensional, vertical cross-sectional results when they are used in the corresponding one-dimensional areal model. SPEJ P. 269

The Use of Numerical Simulation To Design a Carbon Dioxide Miscible Displacement Project

The prediction of a single-layered, two-dimensional areal model was compared with results from cross-sectional, two-dimensional vertical models. Effects of gravity segregation and existing reservoir heterogeneity were not significant enough to justify use of a three-dimensional model for predicting field behavior of the North Cross (Devonian) Unit CO2 miscible displacement project in West Texas. project in West Texas. Summary A numerical simulation method was used to design a carbon dioxide (CO2) miscible displacement project in a low-permeability carbonate reservoir. A field history match was obtained as a check on the model's validity. Two-dimensional, vertical (cross-section) models were used to determine the potential effects of a high gas saturation at the top of the formation on CO2 injection profiles and to evaluate the effect of reservoir heterogeneity on sweep efficiency. A twodimensional areal model was used to describe the field's response to CO2 injection. Using an areal model, various CO2 injection patterns were simulated and compared, and an inverted patterns were simulated and compared, and an inverted nine-spot was found to be the optimum pattern for this field. Further simulation predictions were conducted to evaluate various schemes of handling produced CO2-contaminated gas after CO2 breakthrough. produced CO2-contaminated gas after CO2 breakthrough. It was found that direct reinjection of the contaminated gas was preferable to separating the CO2 from the gas or venting the contaminated gas. The project - the North Cross (Devonian) Unit in West Texas - is unlike the classical miscible project in that continuous injection of CO2 and residue project in that continuous injection of CO2 and residue gas is planned rather than injection of miscible slugs followed by or alternating with water or dry gas. Injection of CO2 commenced in April 1972. Actual project performance to date has been encouraging. project performance to date has been encouraging. Reservoir Description The Crossett field is located at the southern edge of the central basin platform of West Texas. The reservoir is a chalky, siliceous carbonate having 21 percent porosity and 3 md permeability. Fig. 1 shows percent porosity and 3 md permeability. Fig. 1 shows the gross Devonian reservoir thickness above the water-oil contact. There is no significant water drive present, so the primary recovery mechanism is solution gas drive. The reservoir is limited on the west by truncation caused by the post-Devonian unconformity. The eastern limit of the reservoir is provided by a water-oil contact, and the north and provided by a water-oil contact, and the north and south boundaries are defined by permeability and porosity deterioration. The reservoir is unusually uniform for a carbonate. Fig. 2 shows a plot of core permeability and porosity vs well depth. Properties of the reservoir fluid are listed in Table 1. The field contained 53 million STB of oil in place when discovered in 1944. The Crossett field was unitized in 1964 and renamed the North Cross (Devonian) Unit. The purpose of unitization was to implement a gas-injection purpose of unitization was to implement a gas-injection pressure maintenance project. All residue casinghead pressure maintenance project. All residue casinghead gas was injected into the three updip gas injection wells shown on Fig. 1. Fig. 3 shows the Unit's production history since unitization. production history since unitization. Residue gas injection would only increase the field's recovery to 19 percent from the expected primary efficiency of 13 percent. With so much oil being left in the reservoir, further supplemental recovery schemes were considered. P. 1327