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.

Analytical-Numerical Method in Waterflooding Predictions

1965 ◽  
Vol 5 (03) ◽  
pp. 247-258 ◽  
Author(s):  
Hubert J. Morel-Seytoux
Keyword(s):  
Finite Difference ◽  
Exact Solutions ◽  
Numerical Method ◽  
Analytical Method ◽  
Numerical Solutions ◽  
Numerical Technique ◽  
Scale Factor ◽  
Mobility Ratio ◽  
Finite Difference Approximation ◽  
Difference Approximation

Abstract Methods of predicting the influence of pattern geometry and mobility ratio on waterflooding recovery predictions are discussed. Two methods of calculation are used separately or concurrently. The analytical method yields exact solutions in a convenient form for a unit mobility ratio piston-like displacement. A few typical pressure distributions, sweep efficiencies and oil recoveries are presented for various patterns. For non-unit mobility ratio, one may resort to a numerical method, such as that of Sheldon and Dougherty. Because the domains of applicability of the analytical and numerical techniques overlap, the exact solutions provide estimates of the errors in the numerical procedures. The advantages of the analytical and numerical methods can be combined. To develop a numerical technique as independent of geometry as possible, the physical space is transformed into a standard rectangle. The entire effect of geometry is rendered through one term, the "scale-factor", derived from mapping relations. The scale factor can be calculated from the exact unit-mobility ratio solution for the particular pattern of interest. By this means recovery performances for arbitrary mobility ratio can be obtained for many patterns. A sample of results obtained in this manner is presented. Introduction Pattern geometry and mobility ratio are two major factors in making a waterflood recovery prediction. Because assisted recovery has become increasingly important to the oil industry, pattern configuration and mobility ratio also assume a greater significance in the assessment of the economic value of recovery projects. The influence of pattern geometry and mobility ratio in shaping a recovery curve and on the other quantities of interest to the reservoir engineer is the main subject of this paper. Much effort has already been spent on estimating quantitatively the influence of either pattern or mobility ratio or both on oil recovery. The literature reports many investigations of this nature. However, many results or methods of recovery prediction presented in the literature cannot be considered fully satisfactory. Even for unit mobility ratio and piston-like displacement, where analytical solutions are available, the literature shows discrepancies. For non-unit mobility ratio, the divergence in the results is extreme. For infinite mobility ratio in a repeated five-spot, depending on the investigator, the sweep efficiency ranges from 0 per cent to 60 per cent. With respect to the influence of pattern on recovery, only the repeated five-spot has received much attention. Other confined patterns and pilot configurations have received very little attention. Two calculation methods are presented in this paper, either separately or concurrently: the analytical method of potential theory and the numerical method of finite-difference approximation. The analytical method is more restricted in scope than the finite-difference method, but it has the definite advantage of providing exact solutions within its range of applicability. If a unit-mobility ratio piston-like displacement is assumed, the analytical approach is possible. A few typical results are reported in this paper; the detailed description of the general method and of a great variety of results will be the subject of other articles. For non-unity mobility ratio, we must resort to a numerical scheme. The numerical technique is that which was described by Sheldon and Dougherty. It is not limited to piston-like displacement. However, mainly single interface results will be presented here. Because the respective domains of applicability of the analytical and the numerical method overlap, useful comparisons of exact and numerical solutions can be made for a variety of patterns. The advantages of the analytical and numerical approaches can be combined. SPEJ P. 247ˆ

scholarly journals Enhanced Oil Recovery: Chemical Flooding

2021 ◽  
Author(s):  
Ahmed Ragab ◽  
Eman M. Mansour
Keyword(s):  
Interfacial Tension ◽  
Enhanced Oil Recovery ◽  
Oil Recovery ◽  
Recovery Phase ◽  
Mobility Ratio ◽  
Chemical Flooding ◽  
Sweep Efficiency ◽  
Oil Displacement ◽  
Remaining Oil ◽  
Oil Displacement Efficiency

The enhanced oil recovery phase of oil reservoirs production usually comes after the water/gas injection (secondary recovery) phase. The main objective of EOR application is to mobilize the remaining oil through enhancing the oil displacement and volumetric sweep efficiency. The oil displacement efficiency enhances by reducing the oil viscosity and/or by reducing the interfacial tension, while the volumetric sweep efficiency improves by developing a favorable mobility ratio between the displacing fluid and the remaining oil. It is important to identify remaining oil and the production mechanisms that are necessary to improve oil recovery prior to implementing an EOR phase. Chemical enhanced oil recovery is one of the major EOR methods that reduces the residual oil saturation by lowering water-oil interfacial tension (surfactant/alkaline) and increases the volumetric sweep efficiency by reducing the water-oil mobility ratio (polymer). In this chapter, the basic mechanisms of different chemical methods have been discussed including the interactions of different chemicals with the reservoir rocks and fluids. In addition, an up-to-date status of chemical flooding at the laboratory scale, pilot projects and field applications have been reported.

Nanofluid Precoating: An Effective Method To Reduce Fines Migration in Radial Systems Saturated With Two Mobile Immiscible Fluids

SPE Journal ◽  
2018 ◽  
Vol 23 (03) ◽  
pp. 998-1018 ◽  
Author(s):  
Bin Yuan ◽  
Rouzbeh Ghanbarnezhad Moghanloo
Keyword(s):  
Oil Recovery ◽  
Analytic Solution ◽  
Numerical Solutions ◽  
Fine Particles ◽  
Water Saturation ◽  
Splitting Method ◽  
Immiscible Fluids ◽  
Mechanistic Modeling ◽  
Sweep Efficiency ◽  
Fines Migration

Summary Prediction of how nanofluid applications can potentially control fines migration in porous media saturated with two immiscible fluids requires a mechanistic modeling approach. We develop analytic solutions to evaluate the efficiency of nanofluid utilization to reduce fines migration in systems saturated with two immiscible fluids. In this study, fines migration in the radial-flow system saturated with two immiscible fluids (oil and water) is considered; two capture mechanisms of fine particles—fines attachment and straining—are incorporated into the modeling work. The analytic solution is derived by implementing the splitting method and stream-function transformation to convert a 2 × 2 (nonhomogeneous) system of equations into an equation with a fine-particle component (nanoparticle effects) and a lifting equation in which only water saturation appears. Through quantitative comparison of suspended fines and water-saturation-profile plots, the accuracy of the analytic solution is verified with finite-difference numerical solutions. Saturation c-shock and saturation s-shock appear in the analytical solutions. The fines migration and consequent phenomena (fines attachment, fines straining, and fines suspension) decelerate the breakthrough of the injected fluids (better sweep efficiency) and increase the corresponding front saturation of the injected fluid near the wellbore—i.e., larger relative permeability (better injectivity). The results suggest that fines attachment onto the grain surface and well injectivity are enhanced after nanofluid pretreatment; moreover, the smallest radius to be pretreated by nanofluid is approximated to maintain its benefits. In practice, our analytic approach provides a valuable mathematical structure to evaluate how nanoparticle usage can enhance performance of water-based enhanced-oil-recovery (EOR) techniques in reservoirs with a fines-migration issue.

General Deformations of Neo-Hookean Membranes

1973 ◽  
Vol 40 (1) ◽  
pp. 7-12 ◽  
Author(s):  
W. H. Yang ◽  
C. H. Lu
Keyword(s):  
Differential Equations ◽  
Strain Energy ◽  
Numerical Solutions ◽  
Nonlinear Partial Differential Equations ◽  
Strain Energy Function ◽  
Finite Deformations ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Special Cases ◽  
Convergent Algorithm

A set of three nonlinear partial-differential equations is derived for general finite deformations of a thin membrane. The material that composes the membrane is assumed to be hyperelastic. Its mechanical property is represented by the neo-Hookean strain-energy function. The equations reduce to special cases known in the literature. A fast convergent algorithm is developed. The numerical solutions to the finite-difference approximation of the differential equations are computed iteratively with a trivial initial iterant. As an example, the problem of inflating a rectangular membrane with fixed edges by a uniform pressure applied on one side is presented. The solutions and their convergence are displayed and discussed.

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

Shock waves analysis of planar and non planar nonlinear Burgers’ equation using Scale-2 Haar wavelets

2017 ◽  
Vol 27 (8) ◽  
pp. 1814-1850 ◽  
Author(s):  
Sapna Pandit ◽  
Manoj Kumar ◽  
R.N. Mohapatra ◽  
Ali Saleh Alshomrani
Keyword(s):  
Nonlinear Term ◽  
Numerical Solutions ◽  
Burgers Equation ◽  
Finite Difference Approximation ◽  
Haar Wavelets ◽  
Difference Approximation ◽  
Content Type ◽  
Gauss Elimination ◽  
Gauss Elimination Method ◽  
Better Than

Purpose This paper aims to find the numerical solution of planar and non-planar Burgers’ equation and analysis of the shock behave. Design/methodology/approach First, the authors discritize the time-dependent term using Crank–Nicholson finite difference approximation and use quasilinearization to linearize the nonlinear term then apply Scale-2 Haar wavelets for space integration. After applying this scheme on partial differential, the equation transforms into a system of algebraic equation. Then, the system of equation is solved using Gauss elimination method. Findings Present method is the extension of the method (Jiwari, 2012). The numerical solutions using Scale-2 Haar wavelets prove that the proposed method is reliable for planar and non-planar nonlinear Burgers’ equation and yields results better than other methods and compatible with the exact solutions. Originality/value The numerical results for non-planar Burgers’ equation are very sparse. In the present paper, the authors identify where the shock wave and discontinuity occur in planar and non-planar Burgers’' equation.

Water Injection, Polymer Injection and Polymer Alternating Water Injection for Enhanced Oil Recovery: A Laboratory Study

2013 ◽  
Author(s):  
Marcelo F. Zampieri ◽  
Rosangela B. Z. L. Moreno
Keyword(s):  
Oil Recovery ◽  
Produced Water ◽  
Water Injection ◽  
Mobility Ratio ◽  
Sweep Efficiency ◽  
Polymer Injection ◽  
Paraffin Oil ◽  
Water Viscosity ◽  
Water Volumes ◽  
Expensive Process

Developing an efficient methodology for oil recovery is extremely important in this commodity industry, which may indeed lead to wide spread profitability. In the conventional water injection method, oil displacement occurs by mechanical behavior between fluids. Nevertheless, depending on mobility ratio, a huge quantity of injected water is necessary. Polymer injection aims to increase water viscosity and improve the water/oil mobility ratio, thus improving sweep efficiency. The alternating banks of polymer and water injection appear as an option for the suitable fields. By doing so, the bank serves as an economic alternative, as injecting polymer solution is an expensive process. The main objective of this study is to analyze and comparison of the efficiency of water injection, polymer injection and polymer alternate water injection. For this purpose, tests were carried out offset in core samples of sandstones using paraffin oil, saline solution and polymer and were obtained the recovery factor and water-oil ratio for each method. The obtained results for the continuous polymer injection and alternating polymer and water injection were promising in relation to the conventional water injection, aiming to anticipate the oil production and to improve the water management with the reduction of injected and produced water volumes.

Toe-To-Heel Waterflooding. Part ll: 3D Laboratory-Test Results

2006 ◽  
Vol 9 (03) ◽  
pp. 202-208 ◽  
Author(s):  
Alexandru T. Turta ◽  
Ashok K. Singhal ◽  
Jon Goldman ◽  
Litong Zhao
Keyword(s):  
Oil Recovery ◽  
3D Model ◽  
Water Saturation ◽  
Mobility Ratio ◽  
Laboratory Model ◽  
Oil Viscosity ◽  
Vertical Wells ◽  
Sweep Efficiency ◽  
Gravity Segregation ◽  
Laboratory Test Results

Summary A new waterflooding process, toe-to-heel waterflooding (TTHW), was developed, based partly on a recently developed thermal TTH displacement process, TTH air injection (THAI). TTHW is a novel oil-recovery process that uses a horizontal producer (HP) and a vertical injector (VI). The HP has its horizontal leg located at the top of formation, while its toe is close to the VI, which is perforated at the lower part of the formation. TTHW realizes a gravity-stable displacement, in which the water/oil mobility ratio becomes less important and its detrimental effect on sweep efficiency is diminished; the injected water always breaks through at the toe, after which water cut gradually increases. A systematic investigation of the TTHW process in a Hele-Shaw laboratory model mimicking a simulated porous medium showed that the process substantially improved the vertical sweep efficiency as compared to conventional waterflooding. Following these semiquantitative tests, a more comprehensive 3D-model testing was undertaken to investigate the overall sweep efficiency of the process. The 3D model consists of a metal box filled with glass beads and saturated with oil at connate-water saturation. Oil was displaced with high-salinity brine, either in a TTH configuration or in a conventional array, using only vertical wells. A staggered line drive was used by injecting water in two vertical wells located at one side of the box and producing oil by using either an HP with its toe close to the injection line or a vertical producer located at the HP's heel position. Several TTHW tests were carried out at different injection rates. For a given injection rate, the TTHW results were compared to those of conventional-waterflooding tests. For the same amount of water injected, the ultimate oil recovery increased by a factor of up to 2, as compared to that for conventional waterflooding. All in all, the results of these investigations show that the novel TTHW process is sound and can be optimized further. Introduction Conventional waterfloods in heavy-gravity-oil reservoirs (oil viscosity higher than 100 mPa.s) are limited by three main factors:• Reservoir heterogeneity, leading to water channelling.• Gravity segregation (caused by oil/water density contrast), leading to underriding of the injected water.• Highly unfavorable water/oil mobility ratio, which aggravates the adverse effects of the first two factors. Usually, the heterogeneity is caused by pronounced vertical stratification, manifested by a relatively large contrast in the horizontal permeability of different layers. On the other hand, the negative effect of gravity segregation is felt mainly when the stratification is not very pronounced and effective vertical permeability of the pay zone is relatively high. The effect of gravity segregation on waterflood performance was reported in 1953 when the first mathematical model of water tonguing (underriding) was published by Dietz (1953). Initially, Dietz's theory was believed to be applicable mostly to thick formations. However, subsequently, Outmans showed that this theory was equally applicable to thin oil formations (Sandrea and Nielsen 1974).

scholarly journals The Investigation of Silica Nanoparticles-CO2 Foam Stability for Enhancing Oil Recovery Purpose

2020 ◽  
Vol 9 (1) ◽  
pp. 36-45
Author(s):  
David Maurich
Keyword(s):  
Silica Nanoparticles ◽  
Oil Recovery ◽  
Volume Ratio ◽  
Mobility Ratio ◽  
Phase Volume ◽  
Sweep Efficiency ◽  
Co2 Gas ◽  
Phase Volume Ratio ◽  
Co2 Foam ◽  
Brine Salinity

Carbon dioxide (CO2) gas injection is one of the most successful Enhanced Oil Recovery (EOR) methods. But the main problem that occurs in immiscible CO2 injection is the poor volumetric sweep efficiency which causes large quantities of the oil to be retained in pore spaces of reservoir. Although this problem can be improved through the injection of surfactant with CO2 gas where the surfactant will stabilize CO2 foam, this method still has some weaknesses due to foam size issue, surfactants compatibility problems with rocks and reservoir fluids and are less effective at high brine salinity and reservoir temperature such as typical oil reservoirs in Indonesia. This research aims to examine the stability of the foams/emulsions, compatibility and phase behavior of suspensions generated by hydrophobic silica nanoparticles on various salinity of formation water as well as to determine its effect on the mobility ratio parameter, which correlate indirectly with macroscopic sweep efficiency and oil recovery factor. This research utilizes density, static foam, and viscosity test which was carried out on various concentrations of silica nanoparticles, brine salinity and phase volume ratio to obtain a stable foam/emulsion design. The results showed that silica nanoparticles can increase the viscosity of displacing fluid by generating emulsions or foams so that it can reduce the mobility ratio toward favorable mobility, while the level of stability of the emulsion or foam of the silica nanoparticles suspension is strongly influenced by concentration, salinity and phase volume ratio. The high resistance factor of the emulsions/foams generated by silica nanoparticles will promote better potential of these particles in producing more oil.

scholarly journals Time-space dependent fractional viscoelastic MHD fluid flow and heat transfer over accelerating plate with slip boundary

2017 ◽  
Vol 21 (6 Part A) ◽  
pp. 2337-2345
Author(s):  
Shengting Chen ◽  
Liancun Zheng ◽  
Chunrui Li ◽  
Jize Sui
Keyword(s):  
Heat Transfer ◽  
Numerical Solutions ◽  
Fluid Model ◽  
Mhd Flow ◽  
Constitutive Relationship ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Slip Boundary ◽  
Flow And Heat Transfer ◽  
Time Space

The MHD flow and heat transfer of viscoelastic fluid over an accelerating plate with slip boundary are investigated. Different from most classical works, a modified time-space dependent fractional Maxwell fluid model is proposed in depicting the constitutive relationship of the fluid. Numerical solutions are obtained by explicit finite difference approximation and exact solutions are also presented for the limiting cases in integral and series forms. Furthermore, the effects of parameters on the flow and heat transfer behavior are analyzed and discussed in detail.

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