This paper (SPE 52607) was revised for publication from paper SPE 36737, first presented at the 1996 SPE Annual Technical Conference & Exhibition, Denver, 6-9 October. Original manuscript received for review 25 October 1996. Revised manuscript received 17 August 1998. Paper peer approved 1 September 1998. Summary In naturally fractured formations such as coal, permeability is sensitive to changes in stress or pore pressure (i.e., changes in effective stress). This paper presents a new theoretical model for calculating pore volume (PV) compressibility and permeability in coals as a function of effective stress and matrix shrinkage, by means of a single equation. The equation is appropriate for uniaxial strain conditions, as expected in a reservoir. The model predicts how permeability changes as pressure is decreased (i.e., drawdown). PV compressibility is derived in this theory from fundamental reservoir parameters. It is not constant, as often assumed. PV compressibility is high in coals because porosity is so small. A rebound in permeability can occur at lower drawdown pressures for the highest modulus and matrix shrinkage values. We have also history matched rates from a boomer well in the fairway of the San Juan basin by use of various stress-dependent permeability functions. The best fit stress/permeability function is then compared with the new theory. P. 539

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CopW!Qhf 1SS6 Sceiaty C4Petrolwm Engineers, k This pcpar was prqmred for p+esentatwn at the 19% SPE Annual Technical Conferorw am Exh!bitmn iwtd m Damfer, Co&ade U S A S-9 OUcWr 199S TM POWwas aaktad for preaanlatmn by an SPE Progrsm Ccfnm!ftee following ravmw of !nformatna mnlmmd m an abstract submmd by tha author(s) ConlonN of lha Pqx!r a$ prosanled, hava not ban r.vmwad by tlw Scaety of Potrolawn Eogmeors and are vA@ 10 ccfroclm+l by tlw author(s) Tha m@l@r@ qs praaonlad do98 not !wco$sar,ty reflect any posilon of tha SOWty of Pelrohum Engmears it ufksrs 0! members Papers WOWWd a! SPE meetngs qm lubIact !0 publtitmn rovmw by Edflorml Ccmmmeaa of the Scaety of POtrOlOUm Engmers PwmIsmc+I to copy IS feslricfod to an abslracl of not more Ihan 30iJ Words. llluWatKnIa may nol ba cxmod Th@ qbstracl should cnnlam consmcuo. ! q duwfwadgmant of wkro mnd by w+wr"fha Papar w-s piesentad Wnlo L!br8rmri 'SPk P O Box 83383S R!chardsm TX 750S3-3S3S U S A, ffix 01-21 4-9S2-9435 AbstractIn naturally fractured formations, such as coal, permeability is sensitive to changes in stress or pore pressure (i.e., effective stress). This paper presents a new theoretical model for calculating pore volume compressibility and permeability in coals as a fun~on of effective stress and matrix shrinkage, using a single equation. The equation is appropriate for uniaxial strain conditions, as expected in a reservoir. The model predicts how permeability changes as pressure is decreased (i.e., drawdown). Pore volume compressibility is derived in this theory from fundamental reservoir parameters. It is not constant, as often assumed. Pore volume compressibility is high in coals because porosity is so small. A rebound in permeability can occur at lower drawdown pressures for the highest modulus and matrix shrinkage values, We have also history matched rates from a "boomer" well in the fairway of the San Juan basin using various stressdependent permeability functions. The best fit stress-permeability function is then compared with the new theory.

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SPE Members Abstract This paper presents the results of a compositional simulation study to investigate the relationship between the displacement mechanism, the minimum enrichment requirement, and the oil recovery efficiency for a rich gas system. The simulation results indicated that the displacement mechanism over a wide range of solvent enrichment was governed by a condensing/vaporizing process; however, the relative importance of the condensing or vaporizing mass transfer processes, and their impact on the overall displacement efficiency, was a function of the enrichment level. For displacements simulated at or above the slim tube minimum enrichment (ME) level (determined from the recovery breakover point on a recovery versus enrichment plot), recovery was high and the process behaved very similar to a classical condensing mechanism. Displacements simulated below the slim tube ME were less efficient; however, both condensing and vaporizing mass transfer processes contributed to the overall recovery processes contributed to the overall recovery efficiency. At enrichment levels considerably above the slim tube ME, the effect of the condensing mass transfer process on the displacement became less significant and the behavior approached that of a classical vaporizing mechanism. The slim tube technique for determining the ME requirement based on simulation results was further compared with two other computational methods: one based on Hutchinson and Braun's multiple contact mixing, and the other based on displacement behavior at the limit of zero dispersion. For this rich gas system, the minimum enrichment level predicted by these methods was significantly higher than the slim tube ME and corresponded to conditions of thermodynamic miscibility development by a classical vaporizing mechanism. Despite such a high enrichment requirement needed to achieve thermodynamic miscibility, the oil recovery did not increase significantly beyond that achieved at the slim tube ME. Simulation results for the condensing/vaporizing system were also compared with those predicted by a classical condensing mechanism. These comparisons showed that:for a classical condensing mechanism, the thermodynamic and the slim tube methods predicted the same minimum enrichment requirement;at low dispersion the two mechanisms showed similar displacement behavior; andhigher values of dispersion had a bigger impact in reducing the displacement efficiency of the condensing/vaporizing mechanism compared with the classical condensing process. Introduction Hydrocarbon miscible gas flooding has long been recognized as a viable enhanced recovery method by the petroleum industry. To improve the economic and technical success for projects involving rich gas injection, it is essential to have a good understanding of the displacement mechanism and minimum enrichment requirement for achieving an efficient displacement. Traditionally, the displacement mechanism of rich gas drives has been assumed to be via a condensing process in which the intermediate components of the solvent enriched the reservoir oil toward the point of miscibility. However, the work of Stalkup point of miscibility. However, the work of Stalkup in 1965 and recent investigations have challenged this traditional concept for some rich gas displacements. Zick showed that a very efficient oil recovery can be achieved through a combined condensing/vaporizing mass transfer interaction between the reservoir fluid and the injected solvent. His study was based on limited phase behavior experiments and equation of state (EOS) calculations for a rich gas system. P. 35

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Summary A technique is presented that optimizes the accuracy of numerical solutions to convection-diffusion equations while eliminating the detrimental effects of numerical dispersion. Such equations arise in the simulation of EOR processes where front smearing as a result of numerical dispersion can distort the sharp displacement fronts that are such an important part of the processes. Such front smearing can also lead to false predictions processes. Such front smearing can also lead to false predictions of breakthrough times. More accurate, and hence more reliable, simulation of oil recovery processes should facilitate the assessment of the economics of EOR processes. Our technique uses a parameterized, third order-correct algorithm and Fourier analysis. parameterized, third order-correct algorithm and Fourier analysis. The Fourier analysis allows for analytical solutions of both the differential equation and the algorithm. This enables the formulation of a concise method for requiring that the numerical solution conforms as closely as possible to the correct solution. The method is illustrated by performing the optimization on several examples with different degrees of convection and dispersion. The solutions have no numerical dispersion and little truncation error. Our method can serve as a starting point for generalization to the solution of more complicated equations that occur in multicomponent, multiphase reservoir simulators. Suggestions for such generalizations are presented. Introduction Virtually all reservoir simulators obtain solutions to fluid flow equations, usually nonlinear partial-differential equations, by making discrete approximations to derivatives. Whether finite-difference or finite-element methods are used, these approximations always introduce truncation errors that often can distort the accuracy and stability of the solution. The truncation error is often referred to as numerical dispersion because, to lowest order, it can be represented as a second spatial derivative term added to any true dispersion term in the problem. Distortion of the numerical solution is most significant in the simulation of EOR processes where sharp displacement, concentration, and/or temperature fronts are an important part of the efficiency of the processes, and artificial smearing as a result of numerical dispersion can render the simulation meaningless. For processes involving CO2, surfactant, polymer, or steam, it is also necessary to perform calculations of phase behavior, relative permeabilities, viscosities, interfacial tension, heat and mass transfer, and even chemical reactions at each mesh point and timestep. Therefore, resolving the sharp fronts merely by resorting to very fine meshes is impractical. Hence it is essential to be able to derive solution algorithms that have minimal truncation error for a given mesh size. A simple equation that exhibits many of the features of reservoir simulator equations is the one-dimensional (1D) linear convection-diffusion equation (1) where v is the convection rate (velocity) and D is the diffusion coefficient. This equation can be viewed as a good approximation to oil recovery by miscible displacement, immiscible displacement, or thermal methods. Therefore, most discussions of the effects of numerical dispersion have used Eq. 1 as a prototype, although some have preferred to work with specific simulator equations. Lantz was the first to quantify numerical dispersion in ID equations like Eq. His formulation was used by Peaceman to characterize numerical solutions to such equations and proved useful in interpreting the results of a simple chemical-flood model, where the predicted oil recovery was strongly dependent on gridblock size. Fanchi generalized Lantz's formulation to three dimensions. Larson, Jensen and Finlayson, and Laprea-Bigott and Morse used different forms of front-tracking schemes to resolve sharp fronts. Jensen and Finlayson actually used a moving coordinate system and finite-element methods in tracking fronts; the resultant moving boundary conditions were the price paid for the effective front tracking. price paid for the effective front tracking. SPERE p. 373

Summary This paper presents an experimental study that uses many slim-tube displacements to characterize the effect of hydrocarbon solvent composition onmiscibility development. It was found that the minimum enrichment requirement can be estimated reliably with a previously proposed pseudocritical temperature, T, concept. previously proposed pseudocritical temperature, T, concept. Tuning equation-of-state (EOS) parameters to match these results required incorporating rate and recoveries from selected slim-tube displacement tests in addition to conventional PVT data. Introduction The operation of an enriched-gas injection project often involves blending one or more natural gas liquid (NGL) streams with dry gases to obtain an optimum solvent. The composition of such NGL's (ethane, propane, and some higher-carbon-number components) is subject to frequent change over the life of the flood because of source and supply availability. Therefore, a procedure is needed to estimate miscibility conditions between the potential injection solvents and the reservoir fluid. Benham et al. 1 developed a graphical technique where the minimum enrichment requirement can be estimated from the temperature, pressure, average molecular weight of intermediate hydrocarbons in the displacing fluid, and C5+ molecular weight of the reservoir oil. This approach, however, often results in minimum miscibility requirements that are too conservative. Rutherford conducted a series of displacements in sandpacks and found that miscibility development with a reservoir oil at constant pressure was a function only of the injected gas T . Rutherford studied a limited range of NGL fluids, consisting primarily of ethane and propanemixtures. His results were primarily of ethane and propane mixtures. His results were extended by Jacobson to systems containing CO2 and H2S.Rutherford's TPI method is used in this study to develop a miscibility correlation over a broad range of hydrocarbon species and enrichment levels. Accurate EOS oil recovery predictions require a fluid description with parameters tuned to match certain laboratory data. Typically, these experiments include such routine measurements as differential vaporization analysis (DVA)for black-oil properties and static PVT tests on selected mixtures. In some applications, the resulting fluid description may be adequate for predicting the performance of dynamic slim-tube and coreflood tests. Kremesec and performance of dynamic slim-tube and coreflood tests. Kremesec and Sebastian, however, showed that fluid descriptions based on static PVT data alone were inadequate for predicting the results of PVT data alone were inadequate for predicting the results of dynamic tests for a vaporizing CO2/Oil system. They concluded that matching the multiple-contact minimum miscibility pressure (MMP)can improve the accuracy of such predictions significantly. Negahban and Kremesec observed that this EOS tuning strategy can correctly identify miscible and immiscible regimes and slim-tube rates and recoveries over a wide range of displacement pressures. One objective of this study is to extend theseinvestigators' findings to rich-gas condensing drives by investigating whether fluid descriptions based on only static PVT tests can predict multiple-contact displacement behavior in these systems. A further objective is to determine the additional information required to improve fluid descriptions if the PVT-based descriptions are inadequate. Experimental Studies and Simulation Requirements To develop a miscibility correlation based on the T concept for the reservoir oil in this study, we conducted 25 slim-tube tests at 135F and 2,000psi using an apparatus previously described. Table 1 shows reservoir oil properties and NGL mixture compositions with as few as one hydrocarbon component (NGL 1) and with multiple hydrocarbon components and additional species (NGL's 2 through 5). These NGL's were diluted with methane to give injection solvents with enrichment levels ranging from 35 to 51 mol % C21. The hydrocarbon components in these solvents ranged from two to seven, and some contained up to 5 mol% CO2 or nitrogen. Sight-glass observations were recorded in all slim-tube tests to provide supporting information for the displacement mechanism. Slimtube tests to measure MMP also were conducted with Solvent A(Table 1) over a 1,650- to 3,000-psi pressure range. For fluid description development, DVA, saturation pressure, and p-x diagram measurements and constant-composition-expansion (CCE) tests were conducted with the reservoir oil and Solvent A. In addition, we compared actual and predicted displacement mechanisms using Tiffin et al.'s results from a coreflood experiment conducted with the same reservoir oil and Solvent A to verify the fluid description further. Amoco's version of the Redlich-Kwong EOS was used for fluid property and phase-equilibrium calculations. Slim-tube and property and phase-equilibrium calculations. Slim-tube and coreflood simulations were performed with a previously described generalized compositional model. Other details of the slim-tube and coreflood displacement modeling were also reported previously. previously. Results and Discussion Miscibility Correlation. Using Rutherford's method, we plotted slim-tube oilrecovery measured at 1.2 HCPV of solvent injection vs. solvent T for all NGL streams (Fig. 1). T is defined as the mole-weighted average of critical temperature T of all drive gas components corrected to CO2 and nitrogen T as discussed later. The general behavior in these experiments suggested that the data could be fitted reasonably well with two straight lines intersecting at T- 464R. We observed that displacement efficiency initially increased sharply with increasing T, followed by a gradual leveling off at higher enrichment levels. For solvents with T greater than 464R, slim-tube recoveries were highand sight-glass observations confirmed a miscible condensing-gas-drive mechanism. These tests were characterized by an early gas breakthrough, followed by a gradual color change in the two-phase transition zone. For solvents with T less than 464R, injected PV at gas breakthrough decreased and the transition zone became broader, resulting in lower oil recoveries. These displacements showed immiscible behavior despite a prolonged oil production period after 1.2 HCPV of solvent injection. Fig. 1 also shows data from all other supporting slim-tube experiments conducted with the same recombined reservoir oil. The results further substantiate the choice of TPC as a correlating parameter.