Calculation Of Relative Permeability Research Articles

A realistic modeling of fluid infiltration in thin fibrous sheets

In this paper, a modeling study is presented to simulate the fluid infiltration in fibrous media. The Richards’ equation of two-phase flow in porous media is used here to model the fluid absorption in unsaturated/partially saturated fibrous thin sheets. The required consecutive equations, relative permeability, and capillary pressure as functions of medium’s saturation are obtained via fiber-level modeling and a long-column experiment, respectively. Our relative permeability calculations are based on solving the Stokes flow equations in partially saturated three-dimensional domains obtained by imaging the sheets’ microstructures. The Richards’ equation, together with the above consecutive correlations, is solved for fibrous media inclined with different angles. Simulation results are obtained for three different cases of upward, horizontal, and downward infiltrations. We also compared our numerical results with those of our long-column experiment and observed a good agreement. Moreover, we establish empirical coefficients for the semianalytical correlations previously proposed in the literature for the case of horizontal and downward infiltrations in thin fibrous sheets.

Influence of Initial Water Saturation on Recovery by Spontaneous Imbibition in Gas/Water/Rock Systems and the Calculation of Relative Permeability

Summary It has been a challenge to understand why recovery by spontaneous imbibition could both increase and decrease with initial water saturation. To this end, mathematical models were developed with porosity, permeability, viscosity, relative permeability, capillary pressure, and initial water saturation included. These equations foresee that recovery and imbibition rate can increase, remain unchanged, or decrease with an increase in initial water saturation, depending on rock properties, the quantity of residual gas saturation, the range of initial water saturation, and the units used in the definitions of gas recovery and imbibition rate. The theoretical predictions were verified experimentally by conducting spontaneous water imbibition at five different initial water saturations, ranging from 0 to approximately 50%. The effects of initial water saturation on residual saturation, relative permeability, capillary pressure, imbibition rate, and recovery in gas/water/rock systems by cocurrent spontaneous imbibition were investigated both theoretically and experimentally. Water-phase relative permeabilities and capillary pressures were calculated with the experimental data of spontaneous imbibition. Experimental results in different rocks were compared.

Volume Conservation of the Intermediate Phase in Three-Phase Pore-Network Models

Quasi-static rule-based network models used to calculate capillary dominated multi-phase transport properties in porous media employ equilibrium fluid saturation distributions which assume that pores are fully filled with a single bulk fluid with other fluids present only as wetting and/or spreading films. We show that for drainage dominated three-phase displacements in which a non-wetting fluid (gas) displaces a trapped intermediate fluid (residual oil) in the presence of a mobile wetting fluid (water) this assumption distorts the dynamics of three-phase displacements and results in significant volume errors for the intermediate fluid and erroneous calculations of intermediate fluid residual saturations, relative permeabilities and recoveries. The volume errors are associated with the double drainage mechanism which is responsible for the mobilization of waterflood residual oil. A simple modification of the double drainage mechanism is proposed which allows the presence of a relatively small number of partially filled pores and removes the oil volume errors.

Gravity Drainage-Lab Tests, Relative Permeability Calculation, and Field Simulation

Abstract In order to confirm the expected high recoveries obtained in gravity drainage field operations we performed a series of lab tests, developed an oil relative permeability calculation, and used this data to conduct a reservoir simulation for a Brazilian offshore oil field. The lab tests were conducted at ambient temperature and low pressure, and were performed in long cores (90 - 252 cm long) with dead oil and conventional production where the oil was produced at the lower end of the vertically oriented core and air entered the upper end. For one test we performed a counter-current drainage test using live oil where the production was collected at the lower end of the core and the liberated dissolved gas was removed counter-currently from the upper end. The saturation evolution inside the core was monitored by an X-Ray system. Using the same rock and fluids, centrifuge tests were performed using 3 - 4 cm long plugs to compare to the core displacement test results. When the results did not agree, an explanation for the variance using a model of bundles of capillary tubes was postulated. Verification using the principle of minimum energy was performed. A computer program was implemented to calculate the oil relative permeability using the lab data, the results of which were compared with the published literature. The calculated relative permeability was used to simulate the behaviour of a Brazilian offshore oil field that was initially exploited with water injection, and then, due to the appearance of a secondary gas cap, subsequently subjected to gas injection into the crest of the structure. Introduction Numerous papers describe lab test results concerning gravity drainage(2–4, 7, 8, 10, 12, 14, 16, 19, 23, 24, 26, 27). We present the results of gravity drainage long core tests and centrifuge tests using samples with identical rock and fluid properties. In those situations where the results of the two methods did not agree, we calculated the results by considering a bundle of capillary tubes to understand the difference. The long core tests showed, in accordance with the literature, a very low residual oil saturation at the top of the core. The results of these tests were used to calculate the oil relative permeability to be used in to simulate the performance of an offshore Brazilian oil field with a 110.6 MMSTm3 OOIP has been under production for 13 years under peripheral water injection and, due to the appearance of a secondary gas cap, has been supplemented with a crestal gas injection. After a history match, we performed the production forecast for both water and combined water and gas injection. Methodology of the Experimental Tests Long Core Tests After the traditional core mounting process, the sequence of the tests can be summarized in the following steps:Measurement of rock and fluid properties;Water saturation of the sample under vacuum;Displacing the water by oil and establishing the initial water saturation; and,Drainage of oil in the presence of air.

Three-Phase Hydraulic Conductances in Angular Capillaries

SummaryIn this paper, we extend to three fluid phases a prior finite-element study of hydraulic conductance of two-phase creeping flow in angular capillaries. Previously, we obtained analytic expressions for the hydraulic conductance of water in corner filaments. Here, we present the results of a large numerical study with a highresolution finite-element method that solves the three-phase creeping flow approximation of the Navier-Stokes equation. Using the projection-pursuit regression approach, we provide simple analytic expressions for the hydraulic conductance of an intermediate layer of oil sandwiched between water in the corners of the capillary and gas in the center. Our correlations are derived for the oil layers bounded by the concave or convex interfaces that are rigid or allow perfect slip. Therefore, our correlations are applicable to drainage, spontaneous imbibition, and forced imbibition with maximum feasible hysteresis of each contact angle, oil/water (O/W), and gas/oil (G/O). These correlations should be useful in pore-network calculations of three-phase relative permeabilities of spreading oils. Finally, we compare our results with the existing correlations by Zhou et al.18 and Hui and Blunt,19 who assumed thin-film flow with an effective film thickness proportional to the ratio of the flow area to the length of the no-flow boundary. On average, our correlations are two to four times closer to the numerical results than the corresponding correlations by Zhou et al. and Hui and Blunt.

Relative permeability from tomographic images; effect of correlated heterogeneity

We examine the calculation of relative permeability and residual saturation using networks derived from tomographic images of Fontainebleau sandstone previously used to successfully calculate single-phase transport properties. In contrast to electrical conductivity and permeability calculations, we find that that computed relative permeabilities and residual saturations for samples of the same sandstone display a high degree of variability. Randomizing pores and throats to remove all correlations almost completely eliminates the variability between samples and produces smooth numerical data sets. We conclude that correlations in rock microstructure, which appear to have little effect on the calculation of single fluid properties, have a major effect on computed relative permeability and residual saturation.

Application of composite membranes to determine relative permeabilities from displacement experiments

Relative permeabilities are important characteristics of multiphase flow in porous media. Displacement experiments for relative permeabilities are usually interpreted by the JBN method neglecting capillary pressure. The experiments therefore have to be run at higher rates than those experienced during reservoir exploitation. Another disadvantage is that the relative permeabilities can only be determined for the usually small saturation interval outside the shock. We have developed a method to interpret displacement experiments with the capillary pressure included. The data needed are in situ measurements of saturations and phase pressures. The experiments can then be run at low flow rates, and relative permeabilities can be determined for all saturations. When using a short core, or if the saturation profile is spread out too much, the measurements can be affected by the end effect. We investigate using a combined water-wet and oil-wet membrane at the core outlet end to eliminate the end effect. With the membrane, the steady-state saturation profile is fairly uniform, and calculations of steady-state relative permeabilities are improved. However, because of the geometry of the membrane, the phases are separated at the core outlet. The separation results in increased flow resistance, and increased pressure in the core. Calculations of relative permeabilities for the traveling wave from pressure measurements after breakthrough will therefore be erroneous.

Theory for the semi-analytical calculation of oil recovery and effective relative permeabilities using streamtubes

A semi-analytical method has been developed for calculating oil recovery in two and three dimensions, and for calculating effective relative permeabilities for coarse grids. The calculations are based on the assumption that the effects of a changing mobility field can be accounted for by using fixed streamtube geometries with flowrates updated to account for the changing mobility distribution. The single-phase pressure distribution from a numerical solution of Laplace's equation is used to calculate the pressure distribution for a two-phase flow based on a mapping of the solution of the Buckley-Leverett equation onto the streamtubes derived from the single-phase solution. The displacement calculations for oil recovery are based on theory previously developed by Dykstra and Parsons, extended to include the effects of spatially varying permeability and continuously changing mobilities, as occurs in solutions of the Buckley-Leverett equation for typical values of the mobility ratio. This idea has also been extended to the calculation of effective relative permeabilities for coarse-grid simulation and finally establishes the proper rules for averaging the results of fine-grid numerical simulations of two-phase flow for the definition of effective two-phase flow properties on coarse grids. These calculations have been generalized to three-dimensional flows by the simple device of conceptually inserting a gridded plane across the flow and defining each streamtube at that location as those streamlines which pass through any one of the grid cells. When combined with time-of-flight calculations from the gridded plane to both the producer and injector, the distribution of pore volume along each streamtube can be calculated. This information, combined with a tabulation of the single-phase, steady-state pressure distribution along each streamtube, provides all of the information needed for the semi-analytical calculation of oil recovery and effective flow properties in three-dimensional flows. © 1997 Elsevier Science Ltd. All rights reserved

Wettability effects on two- and three-fluid relative permeabilities

Specification of relative permeability ( k r)-saturation ( S) relations for all fluid phases is required for the simulation of multiphase flow and transport in porous media. Indirect methods are frequently employed to estimate these k r- S relations owing to the time, expense, and difficulty associated with direct measurements. A common indirect approach uses capillary pressure data in conjunction with a selected pore-size distribution model to estimate k r- S relations. Such methods typically assume perfect wettability of the solid. Natural porous media, however, are composed of a variety of mineral constituents with different adsorptive properties, which can exhibit non-zero contact angles and/or fractional wettability. Consequently, fluid distributions in natural media may be more complex than those predicted by simple pore-size distribution models and, under such conditions, current estimation approaches for k r may be inadequate. In this work, the pore-size distribution model of N.T. Burdine (1953, Relative permeability calculations from pore-size distribution data. Transactions of the American Institute of Mining, Metallurgical and Petroleum Engineers 198, 71–77) is extended to incorporate wettability variations. In this model, wetting and less wetting (non-wetting or intermediate) fluid pore classes are used to calculate k r for water or organic. The wettability of the porous medium is used to determine the contributions of the pore classes to k r. For both two- and three-fluid systems, the model predicts that an increase in the contact angle (measured through water) or organic-wet fraction of a medium will be accompanied by an increase in the water k r and a decrease in the organic k r. In three-fluid media, k r values for water and organic depend on both liquid saturations when the solid is imperfectly wetted. The model assumes that wettability variation has no influence on the air k r. Model predictions are shown to be consistent with available experimental data.

Effect of temperature and pore size on the hydraulic properties and flow of a hydrocarbon oil in the subsurface

Capillary pressure relationships in a porous medium determine the distribution of immiscible fluids under static conditions and can largely influence the movement of the fluids when the system is not at equilibrium. Theory predicts that for a given porous medium, the effect of different fluid properties or changes in temperature on capillary pressure are due to changes in interfacial tension and contact angle of the system. The capillary pressure-saturation curves measured here for hydrocarbon oil-water do not have a constant capillary pressure ratio with saturation when compared to the water-air system, and changes in ratio were found when comparing the water-air curves measured at different temperatures. Thus, the scaling theory based on interfacial tensions and contact angles does not adequately account for differences in capillary pressures due to different fluid pairs or temperatures. Also, the curves show the residual wetting and nonwetting phase saturations are greatly affected by temperature and sometimes by the fluid pair, which is not accounted for in the scaling theory and cannot be predicted. When the capillary pressure-saturation relationship is extended to the calculation of relative permeabilities and the prediction of fluid flow in the subsurface, the differences in measured and scaled capillary pressure curves lead to differences in the predicted flow rates and saturations of the immiscible fluids in the subsurface. Thus, care must be used when applying capillary pressure-saturation data from one fluid system to that of another, or when applying it to different conditions.

Relative permeabilities of a near critical binary fluid

Relative permeabilities were measured at very low interfacial tensions (IFT) for two-phase mixtures of methanol and hexane flowing through Clashach sandstone. These two components pass from a two- to a single-phase system as the temperature is increased above the critical solution temperature (CST). The interfacial tension between the coexisting phases approaches zero as the solution reaches miscibility. The phase behaviour of methanol and hexane mixtures has been well characterised allowing the calculation of relative permeabilities, saturations and capillary numbers. Flow data are reported for four different temperatures in the two-phase region (i.e., four values of IFT and capillary number). The capillary desaturation curve (CDC) for the strongly wetting methanol rich phase is also presented. In addition to the novel technique presented for measurement of relative permeability, the results indicate that relative permeabilities approach straight line functions very near the critical point. Furthermore, desaturation of the wetting phase was found to be dependent on the capillary number which, in turn, depends on the location of the mixture on the fluid phase diagram and the proximity to the critical temperature.

Prediction of relative permeability in simple porous media.

We present a predictive calculation of two-phase relative permeabilities in granular porous media formed from a dense random packing of equal spheres. The spatial coordinates of every sphere in the pack have been measured, enabling the microstructure of the medium to be completely determined. From these data we extract a network model that replicates the pore space. By compacting the packing or swelling individual spheres, we may generate model porous media of different porosities whose microstructure is also completely determined. We simulate both viscous- and capillary-dominated invasion of a nonwetting fluid into a wetting fluid contained in these media. During invasion we calculate the average hydraulic conductance of each phase to obtain the relative permeabilities as a function of fluid saturation. Because the microstructure is known, the calculations do not involve any adjustable parameters or supplementary measurements of pore structure. The computed relative permeabilities are successfully compared with experimental values previously measured on sand packs, bead packs, and a simple sandstone.

Measurement of Rock Properties in Coal for Coal-Bed Methane Production

Production of methane from coal seams has resulted in increased interest in the laboratory measurement of rock properties (porosity, permeability, and gas-water relative permeability) in coal. Unlike most other porous media, fluid flow in coal occurs only in naturally occurring microfractures of cleats. Standard techniques for rock property measurements are stretched to the limit when applied to coal. Since drying alters the cleat system in coal, standard porosity measurements cannot be made. Permeability measurements are hampered by coal fines migration. Other studies in the literature have not addressed these unique limitations on the measurement of rock properties in coal. For this work a semiautomated instrumented for measurement of permeability and unsteady-state gas-water relative permeability in coal has been built and modifications have been made to the standard JBN relative permeability calculations which address the limitation of core holder dead volume. With this instrument, the unsteady-state technique is accurate and reproducible, and it has been used to measure representative gas-water relative permeability curves on coal cores from the San Juan and Warrior basins with brine permeabilities as low as 0.1 md. The relative permeability curves for the San Juan and Warrior basins have been successfully used in simulation studies without modification.more » A semiautomated tracer test has been developed to determine porosity in coal. The tracer test has been used to show that within experimental error unsteady-state and steady-state gas-water relative permeability measurements in coal produce the same results. The tracer test further indicates that all water in the cleat system of a coal core is displaceable by gas.« less

Relative Permeability From Unsteady-State Displacements With Capillary Pressure Included

Summary A semianalytic method for calculation of relative permeabilities from unsteady-state displacement data is developed. The method is based on the simultaneous solution of the fractional-flow equation and an integro-differential equation, derived from the material-balance and Darcy equations for immiscible displacement of two incompressible phases, including the capillary pressure by use of various volumetric relationships. This method for calculating two-phase relative permeabilities from unsteady-state displacement data is not restricted to the high-flow-rate experimental conditions used to overcome capillary end effects. Removal of this restriction allows the analysis of low-permeability cores with water and oil where the flow rates are low and capillary end effects are important. Our method allows direct calculation from the experimental data without intermediate interpretations of graphs.

Percolation model of equilibrium three-phase flow through porous media

Percolation models of one-phase and two-phase flow through porous media are extended to the three-phase case. The characteristic regions of realization of one-phase, two-phase and three-phase flow are determined, Relative phase permeability calculations are presented for a model capillary radial density function. The theory is compared with the available experimental data.

Relative Permeabilities at Simulated Reservoir Conditions

Summary Relative permeability was determined at conditions from ambient to those simulating a depth of 1200 m [3,940 ft] from laboratory displacement data. The data were analyzed with the semianalytical interpretation method of Civan and Donaldson, which includes capillary pressure. The unsteady-state displacement method requires that the data be collected at high flow rates to diminish the influence of capillary end effects. including capillary pressure removes this constraint but results in a more complex mathematical model. The results show the capillary end effects and the error introduced by not including them in the calculation of relative permeability. When capillary pressure was ignored in our model, it reproduced the literature values. Relative permeability measurements at elevated temperature show the characteristic temperature effect of higher oil relative permeability and lower residual oil saturation (ROS). These effects result from a change of wettability to a more water-wet condition at the higher temperatures. The equipment used for relative permeability measurement can easily be constructed; in fact, some commercial designs can be used without modification. The results represent data at reservoir conditions; the data are taken on cores that are recompressed, eliminating the adverse influence of microfractures and changes of pore-size distribution frequently present in field cores run at ambient conditions. The displacement data are not restricted to high-flow-rate experimental conditions; removal of this restriction allows the analysis of low-permeability cores when flow rates are necessarily low.

Calculation of Parachors for Compositional Simulation

Summary Linear regression analysis has been used to develop an expression for the calculation of parachor values from the critical temperature and critical volume of a fluid. The expression has applicability in compositional calculations where parachor values for components or pseudocomponents are needed. Introduction One of the advantages of compositional simulation relative to black-oil simulation is the ability of the compositional simulator to model the flow performance of the hydrocarbon components that constitute the hydrocarbon liquid and vapor phases. On the other hand, additional computational labor is required. A set of pseudocomponents is used commonly to minimize the additional labor. pseudocomponents is used commonly to minimize the additional labor. The use of pseudocomponent data sets is illustrated in the SPE comparative compositional problem. Each set of pseudocomponents should be designed to be equivalent to a full compositional analysis of the problem, but should require less computational labor. There is no "best" method for constructing a reliable pseudocomponent data set. Some different pseudoization techniques pseudocomponent data set. Some different pseudoization techniques have been described by Hong. Among the problems of pseudoization is the determination of fluid properties. This paper focuses on the calculation of parachors for use with pseudocomponent data sets. Parachors are used in the Macleod-Sugden correlation to Parachors are used in the Macleod-Sugden correlation to calculate the interfacial tension (IFT) between the hydrocarbon liquid and vapor phases. This is important in a compositional simulation of a miscible or partially miscible displacement process. Actual implementation of the IFT effect is through the calculation of relative permeabilities. (For an example, see Coats) First, however, the parachors for each component or pseudocomponent must be known. In the following, the results of a regression analysis provide a relatively simple equation for the calculation of provide a relatively simple equation for the calculation of parachors from known component or pseudocomponent properties. parachors from known component or pseudocomponent properties. Parachor Calculation Parachor Calculation Amyx el al. present the parachors for a number of pure substances. Attempts to correlate the parachor with fluid properties have been made. These methods were primarily properties have been made. These methods were primarily applicable to relatively small molecules and generally are not suited to complex hydrocarbon systems. To avoid this difficulty, regression analysis was used to develop a relationship between parachor and known fluid properties. Among the fluid properties that are parachor and known fluid properties. Among the fluid properties that are known, either before or after pseudoization, are the critical volume (V), critical temperature (T), and critical pressure (p) of individual components or pseudocomponents. After trying a number of functional relationships, I found that the parachor values listed by Amyx et al. and shown in Table 1 are represented best by the following expression: (1) where H denotes Herzog's parameter and is defined by (2) The regression coefficients a ... a have the values These equations require values of V and T with units of liters per gram mole and Kelvin, respectively. Results of the regression per gram mole and Kelvin, respectively. Results of the regression analysis and regression statistics are shown in Table 2. Significance of regression statistics is discussed generally in Ref. 6. For this case, a reasonably good match of the actual parachors is obtained with Eqs. 1 and 2, and the regression coefficients. The magnitude of the maximum deviation of the calculated parachor from the actual parachor is less than 1%, and the average of the absolute values of the deviation is approximately 0.3%. Discussion The use of pseudocomponents in a compositional simulation presupposes the selection of a mixing rule-such as mole-fraction presupposes the selection of a mixing rule-such as mole-fraction weighting. Mixing rules are prescriptions for combining pure component properties to determine pseudocomponent properties, such as V and T. The pseudocomponent values of V and T should be substituted directly into Eqs. 1 and 2. JPT P. 2049

Graphical Techniques for Determining Relative Permeability From Displacement Experiments

This paper presents graphical constructions that simplify the calculation of relative permeability from displacement data. These constructions convert raw data to relative permeability in a less tedious, more accurate manner than the usual computations. Fractional-flow saturation curves derived from waterflood displacements are always concave downward and never yield multiple-value saturations. Introduction To find oil and water relative permeabilities by the displacement or unsteady-state method, a small linear core usually is saturated with water, then oilflooded to irreducible water saturation. Subsequently, the core is waterflooded, and during the process, pressure drop (either constant or variable) across the entire core and water injection rate (constant or variable) are determined. Effluent fractions are collected and the amount of water and oil in each is measured. Augmented by the absolute permeability and pore volume of the core and by oil and permeability and pore volume of the core and by oil and water viscosities, these data are sufficient to develop relative permeability curves. The average saturation in the core at any time in the flood can be found from an over-all material balance. However, to calculate relative permeability, the saturation history at some point in the core must be determined, not the average saturation history. The Welge equation yields saturations at the effluent end of the core when the average saturation history is known. Similarly, to compute relative permeability, the point pressure gradient per unit injection rate is needed, not the pressure gradient per unit injection rate is needed, not the average. The equation developed by Johnson et al. converts average relative injectivity to a point value, accomplishing the required task. While the equations of Welge and Johnson et al. have been used successfully for years, they require tedious computation and are subject to error because of the evaluation of derivatives. The graphical techniques presented in this study are equivalent to these equations, but are easier to use and can give a more accurate evaluation of relative permeability. Lefebvre du Prey has presented graphical constructions based on curves of volume of oil produced vs time and pressure drop vs time to develop the required point functions. These constructions are limited to constant rate displacements. The constructions presented here are general and apply to constant rate, constant pressure, or variable rate-pressure displacements. Constant-rate and constant-pressure examples are given to help clarify the methods. The graphical techniques make it easy to see that double or triple saturation values, so extensively discussed in the past simply do not result from the fractional flow curve generated by a single displacement, such as a waterflood or an oilflood. Theory Ignoring gravity effects and capillary pressure, water and oil relative permeabilities (expressed as functions of saturation) are (1) (2) To use these equations, the fractional flow of water or oil and effective viscosity, lambda-1, must be determined as functions of saturation. JPT P. 807

Linear Scaling in Slug-Type Processes Application to Micellar Flooding

Abstract A relaxed scaling technique, "linear scaling," has been employed in such single-fluid injection processes as waterflooding. This paper develops processes as waterflooding. This paper develops theory and applications of linear scaling for such multiple-fluid injection processes as micellar/ polymer flooding. polymer flooding. Linear scaling demands that saturation profiles and related properties be completely defined by specifying the number of pore volumes of each fluid injected at any time in the injection history. Mobility and saturation profiles can be constructed from over-all pressure measurements and production data, respectively, from linear cores. Detailed micellar flooding experiments, in which saturation profiles, differential pressures, and production histories were measured, demonstrate production histories were measured, demonstrate that these laboratory multiple-fluid injection displacements are linearly scalable to a good approximation. We employed two micellar slugs that were used in field tests of the Maraflood oil recovery process. Introduction Rigorous mathematical scaling of displacement processes and relaxed scaling techniques have processes and relaxed scaling techniques have been discussed extensively. "Linear scaling" specifies direct correspondence among experiments when equal pore volumes of fluids have been injected, and is a special relaxed-scaling situation. It requires geometric similarity and identical rock and fluid properties for both model and prototype, but assumes that most of the dimensionless groups normally employed for scaling are unimportant for describing a displacement process. Calculation of relative permeability as presented by Welge and Johnson et al. are examples of single-fluid linear scaling applications. In multifluid chemical injection schemes, linear scaling will have limited application and must be employed with caution. In particular, fluid dispersion may not be expected to follow a linear law; this would be important in field applications. Nonlinear gravity effects could also be influential in certain geometries. In organizing saturation and pressure data from several micellar flooding experiments, it became apparent that these systems might be linearly scalable. This observation prompted clarification and better definition of linear scaling relationships for multiple-fluid injection processes. This paper is an outgrowth of that work. Its purpose is to develop linear scaling theory for multiple-fluid injection processes and demonstrate that our laboratory micellar floods scale linearly to a good approximation. THEORY Basic Considerations For a given porous medium and fluid system, a displacement process is said to be linearly scalable if fluid concentration (saturation) at any point and time is a function of only the number of pore volumes of each fluid injected with respect to that point. This is a fundamental definition. The key to applying linear scaling is understanding the concept of relative pore volumes injected with respect to a point. In one-dimensional systems (linear, radial, or spherical), this is easy to define and the practical consequences of being able to do so will be expanded later. For experimental models having heterogeneity, anisotropy, and/or non-one-dimensional boundaries (such as five-spot patterns), it is impossible to assign a fixed pore volume to a given point, since streamlines change configuration during nonunit mobility ratio displacements. Even so, by assigning an appropriate pore volume history to each point, the basic definition will still hold. One consequence of the linear scaling definition is that, in geometrically similar rocks of different size but at the same point in their relative injection history (same number of pore volumes of each given fluid injected into each rock), the saturation or concentration profiles (or contours) in all the rocks will be similar different by only a linear magnification or reduction. SPEJ p. 11

Use of a Reservoir Simulator To Interpret Laboratory Waterflood Data

Abstract Relative permeability curves calculated from laboratory waterflood history by the method of Johnson, Bossler and Naumann (JBN) are often poorly defined or anomalous at low and intermediate poorly defined or anomalous at low and intermediate water saturations. Poor definition can be encountered with strongly water-wet homogeneous cores when the displacement is piston-like. Anomalous curve shapes are associated with laboratory-observed water breakthrough ahead of the main flood front and are common in cores that have contrasting permeability streaks. The JBN technique, although permeability streaks. The JBN technique, although valid for the conditions assumed in its development, is unsatisfactory for the conditions specified above. A reservoir simulator has been used to model laboratory tests and thereby provide an alternative interpretation procedure. The simulation uses core properties and trial-and-error relative permeabilities. properties and trial-and-error relative permeabilities. The shapes of the relative permeability curves are adjusted until calculated oil recovery and relative injectivity curves match those obtained from the laboratory displacement tests. The technique has been used successfully to obtain meaningful relative permeability curves for piston-like displacement, mixed wettability systems, piston-like displacement, mixed wettability systems, and heterogeneous carbonates. The technique has also been used in evaluating empirical equations for calculating relative permeability. Introduction Numerical reservoir simulators are finding increasing application in production history matching and performance predictions. Because of the degree of sophistication reached with these models, it is mandatory that the fluid flow properties be of the highest possible quality. Of all the rock and fluid properties required in predicting performance, it is properties required in predicting performance, it is often the relative permeability characteristics that are the most critically important. These data are usually obtained from laboratory waterflood tests using reservoir core samples. The laboratory waterflood test is an attempt to represent the linear displacement behavior of the oil/water/reservoir-rock system. The wettability properties of the rock system should be preserved properties of the rock system should be preserved in the laboratory core sample if reliable results are to be obtained. Furthermore, the viscosity ratio and surface tension of the oil/water system in the laboratory test should ideally be made the same as those in the reservoir. In interpreting laboratory waterflood tests the unsteady-state equations are usually solved by methods of Buckley-Leverett, Welge and Johnson, Bossler and Neumann (JBN). These interpretations are sometimes inadequate for defining relative permeability curves for heterogeneous reservoir permeability curves for heterogeneous reservoir rock systems or for water displacing a very light oil in a homogeneous sandstone. For example, a number of writers have observed anomalous changes in the relative permeability to water during the flooding of heterogeneous carbonate core samples. The relative permeability to water does not increase smoothly with increasing water saturation, but increases stepwise or even humps. Such behavior appears to reflect small-scale local heterogeneity in the core sample and is likely to be insignificant on a field scale. The heterogeneity is often indicated in the laboratory by an observed water breakthrough at the core-sample production face ahead of the main flood front. The time of water breakthrough is an important measurement used in the calculation of relative permeability by the JBN method. If the breakthrough permeability by the JBN method. If the breakthrough time observed is not that of the main flood front but is a little early, then the relative permeabilities calculated will not represent the properties of the bulk of the core sample. It is under these conditions that anomalous relative permeability curves usually occur. We suggest in this paper that, in many cases, the small changes in pressure and in oil and water production rate that accompany anomalous relative production rate that accompany anomalous relative permeability curves can be smoothed to reflect permeability curves can be smoothed to reflect properties more consistent with the bulk behavior properties more consistent with the bulk behavior of the core sample. In essence, we are saying that together the smoothed oil and water production history and the pressure history of the laboratory core sample represent a unique property of that sample.