Experiments in 'wbr'cb calcium chloride displaced sodi~m chloride from four cores showed the extent of asymmetry in the resulting effluent concentration profiles.These results provided a check on bow validly the mixing process is modeled by a differential "(i.e., not finite-stage) capacitance matbernatical model. The effluent concentration profile from two consolidated cores exhibited corisidera ble asymmetry, while two unconsolidated cores yielded nearly symmetrical profiies. All runs resulted in brea~tbrough of the SO per cent concerztratios ignificantly before one pore volume was injected In addition, velocity appreciably afiected the efiluent concentration profile irom a Torpedo sandstone core. The differential capacitance model matched the data significantly better than a simple diffusion 'modeL The capacitance model allows det erminat ion o{ the amount oi dead-end pore space in a porous matrix and tbe eifect of velocity on the rate oi diifusion into tb is SP ace. An experimental program yielding insight into the physical validity oi the capacitance eifect is described.
This paper describes an implicit, three-dimensional formulation for simulating compositional-type reservoir problems. The model treats three-phase flow in Cartesian (x-y-z) or cylindrical (r-theta-z) geometries. Applicability ranges from depletion or cycling of volatile oil and gas condensate to miscible flooding operations involving either outright or multicontact-miscibility.The formulation uses an equation of state for phase equilibrium and property calculations. The equation of state provides consistency and smoothness as gas- and oil-phase compositions and properties converge near a critical point. This avoids computational problems near a critical point associated with use of different correlations for K values as opposed to phase densities. Computational testing with example multicontact-miscibility (MCM) problems indicates stable convergence of this formulation as phase properties converge at a critical point. Results for these MCM problems show significant numerical dispersion, primarily affecting the calculated velocity of the miscible-front advance. Our continuing effort is directed toward reduction of this numerical disperson and comparison of model results with laboratory experiments for both MCM and outright-miscibility cases.We feel that the implicit nature of the model enhances efficiency as well as reliability for most compositional-type problems. However, while we report detailed problem results and associated computing times, we lack similar reported times to compare the overall efficiency of an implicit compositional formulation with that of a semi-implicit formulation. Introduction Many papers have treated increasingly sophisticated or efficient methods for numerical modeling of black-oil reservoir performance. That type of reservoir allows an assumption that reservoir gas and oil have different but fixed compositions, with the solubility of gas in oil being dependent on pressure alone.A smaller number of papers have presented numerical models for simulating isothermal "compositional" reservoirs, where oil and gas equilibrium compositions vary considerably with spatial position and time. With some simplification, the reservoir problems requiring compositional treatment can be divided into two types. The first type is depletion and/or cycling of volatile oil and gas condensate reservoirs. The second type is miscible flooding with MCM generated in situ.A distinction between these types is that the first usually involves phase compositions removed from the critical point, while the second type generally requires calculation of phase compositions and properties converging at the critical point. A compositional model should be capable of treating the additional problem of outright miscibility where the original oil and injected fluid are miscible on first contact.A difficulty in modeling the MCM process is achievement of consistent, stable convergence of gas-and oil-phase compositions, densities, and viscosities as the critical point is approached. A number of studies have reported models that use different correlations for equilibrium K-values as opposed to phase densities. Use of an equation of state offers the advantage of a single, consistent source of calculated K-values, phase densities, and their densities near a critical point. SPEJ P. 363^
This paper discusses the use of the Vertical Equilibrium (VE) concept in simulating heterogeneous reservoirs. Where VE criteria are met, this technique allows two-dimensional (2-D) simulation of three-dimensional (3-D) problems with equivalent accuracy, and with attendant substantial savings in data preparation and machine time. The paper presents the VE concept itself and a new dimensionless group as a possible criterion for the validity of VE as applied to thick reservoirs or to reservoirs where the capillary transition zone is a small fraction of thickness. A description of the generation of the appropriate pseudo relative permeability and capillary pressure curves is permeability and capillary pressure curves is presented. presented. In addition to the dimensionless group criterion, an actual comparison of the results of an x-z cross-section and a one-dimensional (1-D) areal run with VE illustrates the validity of the VE concept. Numerical results of such a comparison along with the attendant machine-time requirements are presented. More than an order of magnitude difference in machine-time requirements was experienced. Finally, an actual field case example shows the utility of VE as applied to a reservoir containing one or multiple gas pools residing on a common aquifer. Introduction Numerical simulation of reservoir performance currently encompasses a wide variety of recovery processes, reservoir types and purposes or questions processes, reservoir types and purposes or questions to which answers are sought. A feature common to virtually all reservoir simulation studies, however, is the choice of simulation in one, two or three dimensions. Most frequently this choice is one between an areal (x-y) study and a 3-D study. While the areal study is considerably cheaper than a 3-D simulation, the validity or accuracy of the former is often questioned in light of its apparent inability to simulate flow and fluid saturation distributions in the vertical direction. Areal studies are frequently performed with little attention to or understanding of the extent to which the x-y calculations do or can be made to account for this vertical flow and fluid distribution. Previous papers describe a VE assumption or concept which leads to the definition of pseudo relative permeability and capillary pressure curves to be used in areal studies to simulate 3-D flow. A dimensionless group proposed as a criterion for the assumptions validity primarily treats the case of a reservoir where the capillary transition zone is an appreciable fraction of reservoir thickness. This paper neats the case of a reservoir where the capillary capillary transition zone is a small fraction of reservoir thickness (e.g., less than 10 percent). We propose to describe the VE concept as percent). We propose to describe the VE concept as applied to thick reservoirs or to reservoirs where capillary transition zone is a small fraction of thickness; to describe the generation of appropriate relative permeability and capillary pressure curves for such reservoirs to represent 3-D performance by 2-D areal calculations; to propose a new dimensionless group as a criterion for the VE assumptions' validity, obtained from an analysis of countercurrent gravity segregation; and finally, to present a cross-sectional vs 1-D (VE) comparison and a 2-D areal field case study. THE VERTICAL EQUILIBRIUM CONCEPT Most oil and gas reservoirs extend distances areally which are at least two orders of magnitude greater than reservoir thickness. Viewed in perspective, these reservoirs appear as "blankets" perspective, these reservoirs appear as "blankets" For a variety of reasons, some valid and some invalid, numerical simulations of such reservoirs are performed occasionally in three dimensions as opposed to only two areal (x-y) dimensions. SPEJ P. 63
Summary An equation-of-state (EOS)-based PVT program was applied to match laboratory PVT data for three published and nine additional reservoir fluid samples. This paper includes laboratory test data for the nine samples and describes PVT program features, especially regression, that we find conducive to rapid determination of EOS parameter values needed to match data. With regression, both the parameter values needed to match data. With regression, both the Peng-Robinson (PR) and Zudkevitch-Joffe-Redlich-Kwong (ZJRK) EOS Peng-Robinson (PR) and Zudkevitch-Joffe-Redlich-Kwong (ZJRK) EOS give comparable and generally good agreement with laboratory data. Without regression or significant adjustment of EOS parameters, neither EOS adequately predicts observed reservoir fluid PVT behavior. Our EOS tuning uses a small degree of C7+ fraction splitting. The agreement of these EOS results with data compares favorably with that obtained in previously published studies that used extensive C7+ splitting. Introduction A recent trend in compositional simulation is the use of an EOS, as opposed to independent correlations, to calculate K-values and equilibrium-phase properties. An important prerequisite in meaningful use of the EOS-based prerequisite in meaningful use of the EOS-based compositional model is satisfactory agreement between EOS results and laboratory PVT test data relevant to the reservoir fluid and recovery process. A number of studies report comparisons of cubic EOS and laboratory PVT results for a wide variety of reservoir fluids and conditions. Most of these studies emphasize the C7+ characterization as the key element in attaining agreement between EOS and laboratory results. Some studies use more than 40 components that result from splitting the C7+ fraction. Some authors imply a predictive EOS capability provided one EOS parameter predictive EOS capability provided one EOS parameter is adjusted to match the reservoir fluid saturation pressure. The work reported here reflects our experience that the EOS is generally not predictive and extensive splitting of the C7+ fraction to match laboratory data is generally unnecessary. We indicate that more of the available laboratory data than were frequently used (or reported) in past studies should be used in evaluating and tuning an EOS. The reservoir fluid studies presented illustrate the capability and efficiency of multivariable, nonlinear regression in seeking agreement between EOS and observed PVT results. PVT results. We do not dismiss "proper" C7+ characterization as a necessary element in tuning an EOS. Rather, we support a philosophy of minimal splitting followed by adjustment, using regression, of the heaviest (plus) fraction's two EOS parameters, generally denoted by and . We describe regression-based PVT program features that we feel contribute to time-efficient tuning of an EOS, which is necessary before its use in field-scale simulation. Laboratory data given for six oil and three retrograde gas condensate samples include reservoir temperature expansions, surface separations, N2 reservoir fluid behavior, and one set of multiple-contact data. Results are presented for three additional fluids with data reported in the literature. Generalizations regarding the regression procedure and results, based on these 12 fluid systems and a larger number of unreported fluid studies, are stated where possible or warranted.
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