A Nine-Point, Finite-Difference Reservoir Simulator for Realistic Prediction of Adverse Mobility Ratio Displacements

1979 ◽  
Vol 19 (04) ◽  
pp. 253-262 ◽  
Author(s):  
J.L. Yanosik ◽  
T.A. McCracken
Keyword(s):  
Finite Difference ◽  
Oil Recovery ◽  
Mobility Ratio ◽  
Spot Pattern ◽  
Oil Saturation ◽  
Grid Systems ◽  
Displacement Front ◽  
Grid Orientation ◽  
Recovery Performance ◽  
Reservoir Simulator

Abstract Reservoir simulators based on five-point difference techniques do not predict the correct recovery performance for unfavorable mobility-ratio, piston-type performance for unfavorable mobility-ratio, piston-type displacements. For a developed five-spot pattern, the predicted performance depends on the grid orientation predicted performance depends on the grid orientation (parallel or diagonal) used. This paper discusses the development and testing of a nine-point, finite-difference reservoir simulator. Developed five-spot-pattern flood predictions are presented for piston-type displacements predictions are presented for piston-type displacements with mobility ratios ranging from 0.5 to 50-0. We show that the predicted fronts are realistic and that very little or no difference exists between the results of parallel and diagonal grids. The maximum difference in the recovery curves is less than 1.5 %. The nine-point-difference method is extended to any grid network composed of rectangular elements. Results for two example problems - a linear flood and a direct line-drive flood - indicate the extension is correct. The techniques discussed here can be applied directly in the development of any reservoir simulator. We anticipate that the greatest utility will be in the development of simulators for the improved oil-recovery processes that involve unfavorable mobility ratio processes that involve unfavorable mobility ratio displacements. Examples are miscible flooding, micellar/ polymer flooding (water displacing polymer), and direct polymer flooding (water displacing polymer), and direct steam drive. Introduction Miscible displacement oil-recovery methods often are characterizedby a large viscosity ratio between the oil and its miscible fluid andby a very low immobile oil saturation behind the displacement front. These conditions represent an unfavorable mobility-ratio, piston-type displacement. They differ from a conventional piston-type displacement. They differ from a conventional gas drive, where a substantial mobile oil saturation remains behind the displacement front. Reservoir simulators based on five-point, finitedifference techniques do not predict the correct performance for unfavorable mobility-ratio, piston-type performance for unfavorable mobility-ratio, piston-type displacements. Results of an areal simulation for a developed five-spot flood depend on the grid orientation (diagonal or parallel, Fig. 1). Grid orientation significantly influences the predicted recovery performance and displacement front positions. performance and displacement front positions. A nine-point, finite-difference reservoir simulator is described. Predictions of piston-type displacements in a developed five-spot pattern are presented for mobility ratios ranging from 0.5 to 50. We show that the predicted fronts are realistic and that very little or no predicted fronts are realistic and that very little or no difference exists between the results of parallel and diagonal grid orientations. A formulation of the nine-point, finite-difference technique applicable to any rectangular grid network is presented. Results for two example two-dimensional presented. Results for two example two-dimensional problems, a linear flood, and a direct line-drive flood problems, a linear flood, and a direct line-drive flood indicate that the formulation is correct for nonsquare grid networks. Background Grid-orientation effects for five-point reservoir simulators were demonstrated by Todd et al. They studied two developed five-spot grid systems - a diagonal grid and a parallel grid. These grid systems are shown in Fig. 1. parallel grid. These grid systems are shown in Fig. 1. The diagonal grid represents a quarter of a five-spot pattern, with grid lines at 45 degrees to a line connecting the pattern, with grid lines at 45 degrees to a line connecting the injector and producer. The parallel grid represents one-half of a five-spot pattern, with grid lines either parallel or perpendicular to the lines connecting the parallel or perpendicular to the lines connecting the injector-producer pads. SPEJ P. 253

Optimizing Recovery for Waterflooding Under Dynamic Induced Fracturing Conditions

2009 ◽  
Vol 12 (05) ◽  
pp. 671-682 ◽  
Author(s):  
Paul J. van den Hoek ◽  
Rashid A. Al-Masfry ◽  
Dirk Zwarts ◽  
Jan-Dirk Jansen ◽  
Bernhard Hustedt ◽  
...  
Keyword(s):  
Oil Recovery ◽  
Water Injection ◽  
Injection Rate ◽  
Reservoir Engineering ◽  
Spot Pattern ◽  
Engineering Parameters ◽  
Induced Fractures ◽  
Modeling Strategy ◽  
Reservoir Simulator ◽  
Fully Coupled

Summary It is well established within the industry that water injection mostly takes place under induced fracturing conditions. Particularly in low-mobility reservoirs, large fractures may be induced during the field life. This paper presents a new modeling strategy that combines fluid flow and fracture growth (fully coupled) within the framework of an existing "standard" reservoir simulator. We demonstrate the coupled simulator by applications to repeated five-spot pattern flood models, addressing various aspects that often play an important role in waterfloods: shortcut of injector and producer, fracture containment to the reservoir layer, and areal and vertical reservoir sweep. We also demonstrate how induced fracture dimensions (length, height) can be very sensitive to typical reservoir engineering parameters, such as fluid mobility, mobility ratio, 3D saturation distribution (in particular, shockfront position), 3D temperature distribution, positions of wells (producers, injectors), and geological details (e.g., layering and faulting). In particular, it is shown that lower overall (time-dependent) reservoir transmissibility will result in larger induced fractures. Finally, it is demonstrated how induced fractures can be taken into account to determine an optimum life-cycle injection rate strategy. The results presented in this paper are expected to also apply to (part of) enhanced-oil-recovery operations (e.g., polymer flooding).

Sensitivity Coefficients for History Matching Oil Displacement Processes

1975 ◽  
Vol 15 (01) ◽  
pp. 39-49 ◽  
Author(s):  
George J. Hirasaki
Keyword(s):  
Relative Permeability ◽  
Oil Recovery ◽  
History Matching ◽  
Mobility Ratio ◽  
Dimensional System ◽  
Relative Significance ◽  
One Dimensional ◽  
Relative Permeability Curve ◽  
Sensitivity Coefficients ◽  
Recovery Performance

Abstract An improved estimate of the reservoir parameters is made during the history-matching phase of a reservoir simulation study by determining the set of parameters that result in the best match of the simulated performance with the observed performance. Often, the process of determining which parameters are to be adjusted is a trial-and-error process. Graphs of the sensitivity coefficients for comparing the cumulative oil recovery with the reservoir parameters are presented to determine the relative significance of parameters are presented to determine the relative significance of the parameters and to provide guidelines for the magnitude of change to the parameters. The sensitivity coefficients are based on a one-dimensional system with dip, incompressible fluids, and polynomial relative-permeability curves. The recovery efficiency can be expressed as a function of the dimensionless cumulative injection with the gravity number (gravity/viscous-forces ratio), mobility ratio, and relative-permeability exponent as parameters. The sensitivity of the cumulative oil recovery (at a given value of cumulative injection) to the movable pore volume, mobility ratio, permeability, and the exponent of the relative-permeability curve permeability, and the exponent of the relative-permeability curve can be calculated from the expression for the recovery efficiency. The graphs of the sensitivity coefficients can be used to determine the relative significance of the parameters, if a unique set of parameters can be determined, and how much they should be adjusted. parameters can be determined, and how much they should be adjusted Introduction When the simulated oil-recovery performance differs from the observed performance history, the engineer must determineif the history match is satisfactory, orif not, which reservoir parameters are to be adjusted and how much. The purpose of this parameters are to be adjusted and how much. The purpose of this discussion is to provide guidelines for the engineer in choosing the parameter(s) to be adjusted and to determine the magnitude and direction of the change. This will be accomplished by first illustrating the sensitivity of water or gas displacement performance to the reservoir parameters so that the critical parameter(s) can be identified, and then graphically presenting the magnitude of the sensitivity coefficients to determine the magnitude of change in the parameter value necessary to achieve a match. The following guidelines will be limited in scope to two-phase displacement processes with negligible interfacial mass transfer (e.g., waterflood, natural water drive, gas injection, or gas-cap expansion). Processes such as solution gas drive or vaporizing gas drive will not be presented. The results will be expressed in terms of the gross fluids produced or injected rather than time. The analysis and results are based on a one-dimensional system. Although the recovery performance of a multidimensional system will be different from that of a one-dimensional system, the relative sensitivity of the recovery performance to the parameters should not differ significantly for most recovery processes. Examples of exceptions that cannot be represented with the one-dimensional system are where well coning is significant or where permeability barriers exist between the injection well and production well. production well. The reservoir parameters that are investigated arethe movable pore volume of the displacement process, SVp;the mobility ratio of the displacing fluid to the displaced fluid, M, where the mobilities are evaluated at the maximum saturation of each phase;the permeability, which is represented as a factor in the gravity number, N(G) if the formation is dipping; andthe shape of the relative permeability curve, expressed in terms of a single parameter, n. ASSUMPTIONS AND MODEL OF THE DISPLACEMENT PROCESS The following assumptions are made about the displacement process. process.The saturations, relative permeabilities, porosity, and permeability are averaged over the reservoir thickness. permeability are averaged over the reservoir thickness.The areal displacement is modeled with a linear system. SPEJ P. 39

One Approach to the Grid Orientation Problem in Reservoir Simulation

1981 ◽  
Vol 21 (02) ◽  
pp. 160-161 ◽  
Author(s):  
Paul K. Vinsome ◽  
Anthony D.K. Au
Keyword(s):  
Shock Front ◽  
Single Point ◽  
Pressure Profile ◽  
Mobility Ratio ◽  
Orientation Effect ◽  
Grid Orientation ◽  
Reservoir Simulator ◽  
Phase Mobility ◽  
Line P ◽  
Grid Orientation Effect

Consider a linear, incompressible, two-phase flow in the absence of capillary pressure. Assume that a saturation shock front exists as depicted schematically in Fig. 1, and that the mobility behind the front is extremely high. During any time step of a numerical simulation, the exact position of the shock front generally is not known, so we assume that in an average sense the position of the shock may be placed midway between the nodes. In any numerical reservoir simulator that contains an upstream formulation, the value of Ss on Fig. 1 would be representative of the saturation at which the interblock phase mobility is evaluated. This would result in the calculation of the pressure profile shown by the solid Line P. There is clearly some inconsistency, as the true pressure profile is more like the dashed Line P'. Therefore one possible source of the grid orientation effect is that the upstream finite-difference formulation produces far too small a pressure drop across a shock front, and that the directionally dependent pressure field truncation error is a significant fraction of this pressure drop.Equating fluid velocities on either side of the shock front gives (1) where lambda T represents the total fluid mobility. If the pressure at the shock front p s is eliminated from these equations, we obtain (2) whereby it can be seen that the total effective mobility is given by (3) The harmonic mobility given by Eq. 3 is closer to twice the smaller value of (lambda Tu, lambda Td) than to the upstream value, so it has the capability of producing larger pressure drops across a shock front.On Fig. 1, it can be seen that the fluid fraction flowing at the shock front is more similar to the upstream value. This suggests that for the individual phase mobilities an upstream formulation should be phase mobilities an upstream formulation should be used: (4) with a similar upstream expression for the oil phase.Inspection of Eq. 4 reveals that at unit mobility ratio it reduces to the standard single-point formulation. Also, in a one-dimensional incompressible system with fixed injection rate, upsilon T in Eq. 2 is constant. Under these circumstances, the saturation profiles calculated using the harmonic mobilities are profiles calculated using the harmonic mobilities are identical to the standard single-point upstream, although the pressure profiles are different. This similarity to single-point upstream under many circumstances and the fact that the results of Todd et al. already show marked grid orientation effect at unit mobility ratio for single-point upstream suggest that the single-point harmonic formulation should be extended to two-point harmonic. The simplest way to achieve this is to replace the value of lambda wu in Eq. 4 by the two-point value. The overshoot constraints we have used are that the extrapolated value must not be less than min (lambda wu, lambda wd) nor greater than max (lambda wu, lambda wd). We shall refer to this scheme as "two-point harmonic." The net result is very simple. The normal upstream mobilities lambda in any numerical reservoir simulator are to be modified by the term 2 lambda Td/lambda Tu + lambda Td). P. 160

Areal Sweep Efficiency of Pseudoplastic Fluids in a Five-Spot Hele-Shaw Model

1968 ◽  
Vol 8 (01) ◽  
pp. 52-62 ◽  
Author(s):  
K.S. Lee ◽  
E.L. Claridge
Keyword(s):  
Porous Medium ◽  
Oil Recovery ◽  
Polymer Solutions ◽  
Newtonian Fluids ◽  
Mobility Ratio ◽  
Spot Pattern ◽  
Sweep Efficiency ◽  
Flood Water ◽  
Miscible Displacements ◽  
Connate Water

Abstract Areal sweep efficiency of oil displacement by enhanced-viscosity water exhibiting pseudoplastic behavior was measured in a Hele-Shaw model representing one-quarter of a five-spot pattern. The pseudoplasticity of polymer solutions and the velocity distribution in the five-spot pattern produced a condition under which the mobility ratio between the displacing and the displaced fluid could not be assigned a single value. Instead, the movement of the displacement front is governed by local mobility ratios which are also time dependent. The areal sweep at breakthrough with polymer solutions was poorer than the sweep obtained with Newtonian fluids of comparable viscosity. However, the areal sweep and 1 PV throughput was greatly improved as compared to flood water without polymer. It was also demonstrated that, even after the oil-cut had declined to a low value during a regular waterflood, switching to polymer flood efficiently swept out the oil remaining in the model. Introduction The behavior of fluid displacements in isotropic porous media for various patterns of injection and production wells has been extensively investigated. These investigations all concerned Newtonian fluids, i.e., the viscosity of each fluid was constant regardless of flow rate. The generally unfavorable influence on areal sweep efficiency of higher mobility of the displacing fluid as compared to the mobility of the displaced fluid has been established for both miscible and immiscible fluids. The principle was also established that a close correspondence exists between miscible and immiscible flood front behavior, although oil recovery in a waterflood at unfavorable mobility ratio may be less than that observed in a miscible displacement at the same mobility ratio. This is true even when oil recovery is expressed on the basis of movable oil. The reason is that oil saturation only slowly achieves its final value behind the waterflood front in accordance with the Buckley-Leverett simultaneous flow relations. It is convenient to use miscible displacements for laboratory simulation of waterflood frontal advance since the interfacial tension forces which are negligible in proportion to viscous forces on a reservoir scales are thus made nonoperative in the laboratory model. For miscible displacements, the Hele-Shaw type of model adequately represents a porous medium so long as the appropriate scaling rules are observed in its design and operation. During simulation of waterflood front behavior in the laboratory by using miscible displacements, the behavior of connate water may ordinarily be disregarded since it is usually indistinguishable from flood water in this process. However, when the flood water is deliberately thickened to improve the mobility ratio between water and oil, the effect on the sweep efficiency due to generation of a connate water bank during the process must be considered. In a uniform porous medium, such a bank is generated and efficiently displaced by injection of thickened water. The oil originally in-place at the start of the waterflood is then displaced by connate water followed by thickened water. If the flood water must be thickened to obtain a favorable mobility ratio, the mobility of the oil phase is appreciably less than that of the connate water. Hence, the oil phase is inefficiently displaced by the connate water bank, and a considerable proportion of the oil comes in contact with and is displaced by the thickened waterflood front. SPEJ P. 52ˆ

scholarly journals Numerical Simulation Study of Tracking the Displacement Fronts and Enhancing Oil Recovery Based on Ferrofluid Flooding

2021 ◽  
Vol 9 ◽  
Author(s):  
Tao Huang ◽  
Fuquan Song ◽  
Renyi Wang ◽  
Xiaohe Huang
Keyword(s):  
Magnetic Field ◽  
Numerical Simulation ◽  
Oil Recovery ◽  
Water Flooding ◽  
Oil Saturation ◽  
Reservoir Heterogeneity ◽  
Displacement Front ◽  
Injection Process ◽  
Volume Method ◽  
Reservoir Identification

Water flooding is crucial means to improve oil recovery after primary production. However, the utilization ratio of injected water is often seriously affected by heterogeneities in the reservoir. Identification of the location of the displacement fronts and the associated reservoir heterogeneity is important for the management and improvement of water flooding. In recent years, ferrofluids have generated much interest from the oil industry owning to its unique properties. First, saturation of ferrofluids alters the magnetic permeability of the porous medium, which means that the presence of ferrofluids should produce magnetic anomalies in an externally imposed magnetic field or the local geomagnetic field. Second, with a strong external magnetic field, ferrofluids can be guided into regions that were bypassed and with high residual oil saturation. In view of these properties, a potential dual-application of ferrofluid as both a tracer to locate the displacement front and a displacing fluid to improve recovery in a heterogeneous reservoir is examined in this paper. Throughout the injection process, the magnetic field generated by electromagnets and altered by the distribution of ferrofluids was calculated dynamically by applying a finite element method, and a finite volume method was used to solve the multiphase flow. Numerical simulation results indicate that the displacement fronts in reservoirs can indeed be detected, through which the major features of reservoir heterogeneity can be inferred. After the locations of the displacement fronts and reservoir heterogeneities are identified, strong magnetic fields were applied to direct ferrofluids into poorly swept regions and the efficiency of the flooding was significantly improved.

Grid-Orientation Effects and the Use of Orthogonal Curvilinear Coordinates in Reservoir Simulation

1978 ◽  
Vol 18 (01) ◽  
pp. 13-19 ◽  
Author(s):  
G.E. Robertson
Keyword(s):  
Oil Recovery ◽  
Reservoir Simulation ◽  
Oil Field ◽  
Orientation Effect ◽  
Spot Pattern ◽  
Direct Flow ◽  
Parallel Orientation ◽  
Orientation Effects ◽  
Grid Orientation ◽  
Grid Orientation Effect

Robertson, G.E., Woo, P.T., Members SPE-AIME, Chevron Oil Field P.T., Members SPE-AIME, Chevron Oil Field Research Co., La Habra, Calif. Abstract Grid orientation effects in reservoir simulation recently have received considerable attention. Several authors have proposed methods to reduce or to eliminate these effects. However, the proposed methods require reprogramming of simulators based on standard techniques. The reprogramming effort can be considerable if the numerical model is highly complex, such as in steamflood simulation. An orthogonal curvilinear coordinate system that essentially eliminates the problem of grid orientation was investigated. With no reprogramming, this computing grid can be used readily in existing simulators. Such grids were used to study pattern steamfloods and pattern waterfloods. The results are compared in detail with those obtained by using conventional Cargesian grids. Grid orientation is shown to have a more pronounced effect on the saturation fronts than oil recovery, whether in a steamflood or a waterflood. It is concluded that orthogonal curvilinear grids can be used easily to estimate pattern flood performance without modification of the solution method. Introduction In a simulation study of unfavorable mobility displacement, the areal orientation of the grid system, with respect to the location of the injectors and producers, can affect the calculated results. This is referred to as the "grid-orientation" effect. Fig. 1 illustrates two Cartesian grid orientations for one-eighth of a repeated five-spot pattern. The "parallel" orientation has a direct-flow pattern. The "parallel" orientation has a direct-flow path along the grid lines between the injector and path along the grid lines between the injector and producer, whereas the path is indirect in the producer, whereas the path is indirect in the "diagonal" system. Using a repeated five-spot pattern, Todd et al., Holloway et al., and Yanosik and McCracken investigated grid-orientation effects in gas or waterfloods. Coats et al. reported grid-orientation effects in steamfloods. In all cases, the investigators found that grid orientation significantly affected the movement of saturation fronts and the breakthrough time. Oil recovery was affected to a lesser extent. The effect appears to increase as the mobility ratio becomes more unfavorable and as the transition zone across the saturation front shortens. Todd et al. presented the two-point mobility approximation to alleviate the grid-orientation effect. Holloway et al. introduced a transmissibility modification scheme to reduce this effect further. While these two schemes can be introduced readily into simulators using explicit relative permeability, it is difficult to implement the schemes in simulators using implicit or semi-implicit relative permeability. Yanosik and McCracken observed that the grid-orientation effect probably was caused by the lack of flow from a grid block to its diagonal neighbors and introduced a nine-point finite-difference formula to account for this flow. SPEJ p. 13

Simulation of Residual Oil Saturation in Near-Miscible Gasflooding Through Saturation-Dependent Tuning of the Equilibrium Constants

2015 ◽  
Vol 18 (03) ◽  
pp. 28-302 ◽  
Author(s):  
Leonardo Patacchini ◽  
Sébastien Duchenne ◽  
Marcel Bourgeois ◽  
Arthur Moncorgé ◽  
Quentin Pallotta
Keyword(s):  
Oil Recovery ◽  
Equilibrium Constants ◽  
K Value ◽  
Oil Saturation ◽  
Water Films ◽  
Dead End ◽  
Reservoir Conditions ◽  
Production Stages ◽  
Light Components ◽  
Reservoir Simulator

Summary Conventional miscible or near-miscible gasflooding simulation often overestimates oil recovery, mostly because it does not capture a series of physical effects tending to limit interphase compositional exchanges. Those can include microscopic bypassing of oil situated in dead-end pores or blocked by water films, as well as macroscopic bypassing caused by subgrid-size heterogeneities or fingering. We here present a new engineering solution to this problem in the near-miscible case, relying on our in-house research reservoir simulator. The principle is, while using a black-oil or an equation-of-state description, to dynamically decrease the K-value of heavy components and possibly increase the K-value of light components as the oil saturation reaches the desired residual limit; this enables changing the phase boundaries when needed while preserving the original fluid behavior during the initial production stages. The benefits of the proposed solution are demonstrated on a reservoir-conditions tertiary-gas-injection experiment, performed in our laboratories, for which residual saturations as well as oil-phase and individual-component production rates have easily and successfully been history matched. Results are then compared with matches obtained by use of saturation exclusion and α-factors methods. As a proof of concept, the suitability of the new method to simulate incomplete revaporization of condensate during gas cycling is also illustrated, on the third SPE comparative-solution-project case.

scholarly journals Photon-magnon response to fluid variability and saturation levels in sandstone reservoirs

2020 ◽  
Vol 17 (6) ◽  
pp. 1065-1074
Author(s):  
Abdullah Musa Ali ◽  
Amir Rostami ◽  
Noorhana Yahya
Keyword(s):  
Oil Recovery ◽  
Relative Permittivity ◽  
Wave Interaction ◽  
Heterogeneous Porous Media ◽  
Injection System ◽  
Bragg Grating ◽  
Fibre Bragg Grating ◽  
Core Flooding ◽  
Oil Saturation ◽  
Fluid Saturation

Abstract The need to recover high viscosity heavy oil from the residual phase of reservoirs has raised interest in the use of electromagnetics (EM) for enhanced oil recovery. However, the transformation of EM wave properties must be taken into consideration with respect to the dynamic interaction between fluid and solid phases. Consequently, this study discretises EM wave interaction with heterogeneous porous media (sandstones) under different fluid saturations (oil and water) to aid the monitoring of fluid mobility and activation of magnetic nanofluid in the reservoir. To achieve this aim, this study defined the various EM responses and signatures for brine and oil saturation and fluid saturation levels. A Nanofluid Electromagnetic Injection System (NES) was deployed for a fluid injection/core-flooding experiment. Inductance, resistance and capacitance (LRC) were recorded as the different fluids were injected into a 1.0-m long Berea core, starting from brine imbibition to oil saturation, brine flooding and eventually magnetite nanofluid flooding. The fluid mobility was monitored using a fibre Bragg grating sensor. The experimental measurements of the relative permittivity of the Berea sandstone core (with embedded detectors) saturated with brine, oil and magnetite nanofluid were given in the frequency band of 200 kHz. The behaviour of relative permittivity and attenuation of the EM wave was observed to be convolutedly dependent on the sandstone saturation history. The fibre Bragg Grating (FBG) sensor was able to detect the interaction of the Fe3O4 nanofluid with the magnetic field, which underpins the fluid mobility fundamentals that resulted in an anomalous response.

Unit Mobility Ratio Displacement Calculations for Pattern Floods in Homogeneous Medium

1966 ◽  
Vol 6 (03) ◽  
pp. 217-227 ◽  
Author(s):  
Hubert J. Morel-Seytoux
Keyword(s):  
Oil Recovery ◽  
Numerical Solutions ◽  
Mobility Ratio ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Numerical Techniques ◽  
Sweep Efficiency ◽  
Closed Form Solutions ◽  
Regular Patterns ◽  
Quantities Of Interest

Abstract The influence of pattern geometry on assisted oil recovery for a particular displacement mechanism is the object of investigation in this paper. The displacement is assumed to be of unit mobility ratio and piston-like. Fluids are assumed incompressible and gravity and capillary effects are neglected. With these assumptions it is possible to calculate by analytical methods the quantities of interest to the reservoir engineer for a great variety of patterns. Specifically, this paper presentsvery briefly, the methods and mathematical derivations required to obtain the results of engineering concern, andtypical results in the form of graphs or formulae that can be used readily without prior study of the methods. Results of this work provide checks for solutions obtained from programmed numerical techniques. They also reveal the effect of pattern geometry and, even though the assumptions of piston-like displacement and of unit mobility ratio are restrictive, they can nevertheless be used for rather crude but quick, cheap estimates. These estimates can be refined to account for non-unit mobility ratio and two-phase flow by correlating analytical results in the case M=1 and the numerical results for non-Piston, non-unit mobility ratio displacements. In an earlier paper1 it was also shown that from the knowledge of closed form solutions for unit mobility ratio, quantities called "scale factors" could be readily calculated, increasing considerably the flexibility of the numerical techniques. Many new closed form solutions are given in this paper. INTRODUCTION BACKGROUND Pattern geometry is a major factor in making water-flood recovery predictions. For this reason many numerical schemes have been devised to predict oil recovery in either regular patterns or arbitrary configurations. The numerical solutions, based on the method of finite difference approximation, are subject to errors often difficult to evaluate. An estimate of the error is possible by comparison with exact solutions. To provide a variety of checks on numerical solutions, a thorough study of the unit mobility ratio displacement process was undertaken. To calculate quantities of interest to the reservoir engineer (oil recovery, sweep efficiency, etc.), it is necessary to first know the pressure distribution in the pattern. Then analytical procedures are used to calculate, in order of increasing difficulty: injectivity, breakthrough areal sweep efficiency, normalized oil recovery and water-oil ratio as a function of normalized PV injected. BACKGROUND Pattern geometry is a major factor in making water-flood recovery predictions. For this reason many numerical schemes have been devised to predict oil recovery in either regular patterns or arbitrary configurations. The numerical solutions, based on the method of finite difference approximation, are subject to errors often difficult to evaluate. An estimate of the error is possible by comparison with exact solutions. To provide a variety of checks on numerical solutions, a thorough study of the unit mobility ratio displacement process was undertaken. To calculate quantities of interest to the reservoir engineer (oil recovery, sweep efficiency, etc.), it is necessary to first know the pressure distribution in the pattern. Then analytical procedures are used to calculate, in order of increasing difficulty: injectivity, breakthrough areal sweep efficiency, normalized oil recovery and water-oil ratio as a function of normalized PV injected.

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