Pseudo Relative Permeability Curves Research Articles

Upscaling Low-Salinity Benefit from Laboratory Scale to Field Scale: An Ensemble of Models with a Relative Permeability Uncertainty Range

Summary The distribution of low-salinity benefit for an ensemble of models is required to evaluate low-salinity enhanced oil recovery (OREC) projects. To enable this, low-salinity pseudorelative permeability curves are required to estimate the benefit of low-salinity waterflooding at the field level. We present how the low-salinity benefit can be propagated through an ensemble of full-field models in which each simulation case could have a set of distinctive high-salinity pseudos. A 0.5-ft vertical resolution sector and its 10-ft upscaled counterpart are used. Relative permeability curves and the low-salinity benefit from corefloods are used in the high-resolution sector to create profiles. Pseudohigh- and low-salinity curves are generated for the upscaled sector by history matching high-salinity and incremental low-salinity profiles. Low-salinity benefit is typically measured from corefloods and the same benefit is often assumed at the field scale. Our results show that generating low-salinity curves for high-salinity pseudos using low-salinity benefit from corefloods slightly underestimates the true low-salinity benefit at field scale estimated from high-resolution models. This conclusion is consistent for two extreme relative permeability scenarios tested (i.e., a high-total-mobility unfavorable fractional flow and low-total-mobility favorable fractional flow). Including capillary pressure in high-resolution models was crucial. We would have come to another conclusion if we had not used capillary pressure in fine-grid simulation as approximately one-third of the benefit of low-salinity waterflooding was attributable to more favorable capillary pressure under low-salinity injection. We demonstrate how a set of high-salinity relative permeability data obtained from corefloods, which encompasses a range for fractional flow and total mobility, can be included in ensemble modeling appropriately and how low-salinity benefit could be estimated for such an ensemble. It is adequate to generate low-salinity curves for bounding high-salinity sets of curves. The bounding low-salinity curves can then be used to estimate low-salinity curve for any interpolated high-salinity curve. This workflow significantly simplifies the process of generating the distribution of low-salinity benefit corresponding to an ensemble of models which may be calibrated to limited history.

Scaling Up Low-Salinity Waterflooding in Heterogenous Reservoirs

SummaryModeling the dynamic fluid behavior of low-salinity waterflooding (LSWF) at the reservoir scale is a challenge that requires a coarse-grid simulation to enable prediction in a feasible time scale. However, evidence shows that using low-resolution models will result in a considerable mismatch compared with an equivalent fine-scale model with the potential of strong, numerically induced pulses and other dispersion-related effects. This work examines two new upscaling methods that have been applied to improve the accuracy of predictions in a heterogeneous reservoir where viscous crossflow takes place.We apply two approaches to upscaling to bring the flow prediction closer to being exact. In the first method, we shift the effective-salinity range for the coarse model using algorithms that we have developed to correct for numerical dispersion and associated effects. The second upscaling method uses appropriately derived pseudorelative permeability curves. The shape of these new curves is designed using a modified fractional-flow analysis of LSWF that captures the relationship between dispersion and the waterfront velocities. This second approach removes the need for explicit simulation of salinity transport to model oil displacement. We applied these approaches in layered models and for permeability distributed as a correlated random field.Upscaling by shifting the effective-salinity range of the coarse-grid model gave a good match to the fine-scale scenario, while considerable mismatch was observed for upscaling of the absolute permeability alone. For highly coarsened models, this method of upscaling reduced the appearance of numerically induced pulses. On the other hand, upscaling by using a single (pseudo)relative permeability produced more robust results with a very promising match to the fine-scale scenario. These methods of upscaling showed promising results when they were used to scale up fully communicating and noncommunicating layers as well as models with randomly correlated permeability.Unlike documented methods in the literature, these newly derived methods take into account the substantial effects of numerical dispersion and effective concentration on fluid dynamics using mathematical tools. The methods could be applied for other models where the phase mobilities change as a result of an injected solute, such as surfactant flooding and alkaline flooding. Usually these models use two sets of relative permeability and switch from one to another as a function of the concentration of the solute.

On the prediction of pseudo relative permeability curves: meta-heuristics versus Quasi-Monte Carlo

This article reports the first application of the Quasi-Monte Carlo (QMC) method for estimation of the pseudo relative permeability curves. In this regards, the performance of several meta-heuristics algorithms have also been compared versus QMC, including the Genetic Algorithm (GA), Particle Swarm Optimization (PSO), and the Artificial Bee Colony (ABC). The mechanism of minimizing the objective-function has been studied, for each method. The QMC has outperformed its counterparts in terms of accuracy and efficiently sweeping the entire search domain. Nevertheless, its computational time requirement is obtained in excess to the meta-heuristics algorithms.

An improved procedure for generating pseudorelative permeabilities for water flooding in stratified reservoirs

This paper presents an improved procedure for generating pseudorelative permeability curves for stratified water flooding using either constant pressures or constant flux at the reservoir/grid block boundaries. The concept of pseudorelative permeability reflects the generation of a relative permeability curve that can be used to represent the entire reservoir thickness, rather than a specific layer during reservoir simulation, thus saving computational time. In this paper, fractional flow theory is applied to the generation of pseudorelative permeability curves for (1) constant flow rate and (2) constant pressure boundary conditions. Previously, pseudorelative permeability curves were generated for constant flow rate only, since the analytical solutions for constant pressure boundaries were non-existant. In this paper, this restriction has been removed based on novel analytical solutions for constant pressure boundaries. The method within this paper also differs from previous methodologies and studies, which are all based on an approximation using piston-like displacement for water flooding. Instead, this new model uses fractional flow theory to its fullest extent to generate pseudorelative permeability curves that are physically more realistic. The solution is extended to generate pseudorelative permeability curves for waterflooding of a reservoir under the assumption of constant pressure boundaries which is an equally realistic assumption in comparison to constant flow rate. The generated pseudorelative permeability curves are used in a 2D areal reservoir model in a standard reservoir simulator to predict the behavior of the fully layered 3D reservoir model. It is found that there is considerably better agreement between the results obtained with this new method and the fully layered reservoir model compared to previous methods.

Flow Simulation Using Local Grid Refinements to Model Laminated Reservoirs

Super-giant carbonate fields, such as Ghawar, in Saudi Arabia, and Lula, at the Brazilian pre-salt, show highly heterogeneous behavior that is linked to high permeability intervals in thin layers. This article applies Local Grid Refinements (LGR) integrated with upscaling procedures to improve the representation of highly laminated reservoirs in flow simulation by preserving the static properties and dynamic trends from geological model. This work was developed in five main steps: (1) define a conventional coarse grid, (2) define LGR in the conventional coarse grid according to super-k and well locations, (3) apply an upscaling procedure for all scenarios, (4) define LGR directly in the simulation model, without integrate geological trends in LGR and (5) compare the dynamic response for all cases. To check results and compare upscaling matches, was used the benchmark model UNISIM-II-R, a refined model based on a combination of Brazilian Pre-salt and Ghawar field information. The main results show that the upscaling of geological models for coarse grid with LGR in highly permeable thin layers provides a close dynamic representation of geological characterization compared to conventional coarse grid and LGR only near-wells. Pseudo-relative permeability curves should be considered for (a) conventional coarse grid or (b) LGR scenarios under dual-medium flow simulations as the upscaling of discrete fracture networks and dual-medium flow models presents several limitations. The conventional approach of LGR directly in simulation model, presents worse results than LGR integrated with upscaling procedures as the extrapolation of dynamic properties to the coarse block mismatch the dynamic behavior from geological characterization. This work suggests further improvements for results for upscaling procedures that mask the flow behavior in highly laminated reservoirs.

An upscaling approach using adaptive multi-resolution upgridding and automated relative permeability adjustment

The upscaling process of a high-resolution geostatistical reservoir model to a dynamic simulation grid model plays an important role in a reservoir study. Several upscaling methods have been proposed in order to create balance between the result accuracy and computation speed. Usually, a high-resolution grid model is upscaled according to the heterogeneities assuming single phase flow. However, during injection processes, the relative permeability adjustment is required. The so-called pseudo-relative permeability curves are accepted, if their corresponding coarse model is a good representation of the fine-grid model. In this study, an upscaling method based on discrete wavelet transform (WT) is developed for single-phase upscaling based on the multi-resolution analysis (MRA) concepts. Afterwards, an automated optimization method is used in which evolutionary genetic algorithm is applied to estimate the pseudo-relative permeability curves described with B-spline formulation. In this regard, the formulation of B-spline is modified in order to describe the relative permeability curves. The proposed procedure is evaluated in the gas injection case study from the SPE 10th comparative solution project’s data set which provides a benchmark for upscaling problems [1]. The comparisons of the wavelet-based upscaled model to the high-resolution model and uniformly coarsened model show considerable speedup relative to the fine-grid model and better accuracy relative to the uniformly coarsened model. In addition, the run time of the wavelet-based coarsened model is comparable with the run time of the uniformly upscaled model. The optimized coarse models increase the speed of simulation up to 90% while presenting similar results as fine-grid models. Besides, using two different production/injection scenarios, the superiority of WT upscaling plus relative permeability adjustment over uniform upscaling and relative permeability adjustment is presented. This study demonstrates the proposed upscaling workflow as an effective tool for a reservoir simulation study to reduce the required computational time.

Estimation of pseudo-relative permeability curves through an analytical approach for steam assisted gravity drainage (SAGD) and solvent aided-steam assisted gravity drainage

Fluid flow through the reservoir is defined on the basis of the relative permeability which essentially captures the preferential flow of any specific fluid. Utilization of steam assisted gravity drainage (SAGD) presents challenges to the conventional understanding of relative permeability due to the introduction of heat to the reservoir. The presence of three distinct fluid phases (steam, water, and oil) in conjunction with a temperature dependency has many implications for the modelling of complex fluid dynamics. However, effective characterization of relative permeability curves is integral to successful simulation and so a model capable of representing three-phase flow is required. By implementing a simplified fractional flow analysis, this study is able to regress three-phase flow in the laboratory to a pseudo two-phase relative permeability. Modifications of conventional fractional flow theory allow for the extension of the analysis to the SAGD and solvent aided-SAGD (S-SAGD) process. Overall displacement is defined on the basis of both a microscopic component represented by modified capillary number and a macroscopic component represented by mobility ratio. Negation of the liquid-gas interaction through the assumption of waterflooding enables the generation of pseudo two-phase relative permeability curves that result in comparable performance to lab scale experiments. Because the experimentally obtained SAGD and S-SAGD results are used to construct the pseudo-relative permeability curves, our model includes the many complex fundamental changes occur in viscosity and relative permeability during SAGD and S-SAGD in the newly constructed relative permeability curves. Thus, our results offer a simplistic way to ease the compositional simulation of SAGD and S-SAGD through waterflooding approach.

Estimation of pseudo relative permeability curves for a heterogeneous reservoir with a new automatic history matching algorithm

Abstract Reliable relative permeability curves are important for successful reservoir simulation and modeling. These curves may be available on fine scale models. However, detailed descriptions demand a huge amount of memory and processing time. Therefore, it is needed to upscale core-size relative permeability curves measured by laboratory experiments to a scale suitable for the various types of simulation. This upscaling is accomplished with pseudo functions. In this work, a new automatic history matching algorithm based on B-spline representation was applied to estimate pseudo relative permeability curves. The performance of the generated pseudo curves were compared with four dynamic pseudo functions for a 3D heterogeneous water-flooding case. In addition, the effect of the pseudo function estimation uncertainty on the cumulative oil production was investigated after 10 years of production. Results revealed that relative permeability curves obtained from history matching gives the best performance among other pseudos and this methodology can be used for reservoir simulation over periods beyond that of the original match.

An Efficient Numerical Model for Immiscible Two-Phase Flow in Fractured Karst Reservoirs

AbstractNumerical simulation of two-phase flow in fractured karst reservoirs is still a challenging issue. The triple-porosity model is the major approach up to now. However, the triple-continuum assumption in this model is unacceptable for many cases. In the present work, an efficient numerical model has been developed for immiscible two-phase flow in fractured karst reservoirs based on the idea of equivalent continuum representation. First, based on the discrete fracture-vug model and homogenization theory, the effective absolute permeability tensors for each grid blocks are calculated. And then an analytical procedure to obtain a pseudo relative permeability curves for a grid block containing fractures and cavities has been successfully implemented. Next, a full-tensor simulator has been designed based on a hybrid numerical method (combining mixed finite element method and finite volume method). A simple fracture system has been used to demonstrate the validity of our method. At last, we have used the fracture and cavity statistics data from TAHE outcrops in west China, effective permeability values and other parameters from our code, and an equivalent continuum simulator to calculate the water flooding profiles for more realistic systems.

Relative permeability of CBM reservoirs: Controls on curve shape

Abstract Relative permeability to gas and water for 2-phase flow coalbed methane (CBM) reservoirs has long been known to exhibit a strong control on (gas and water) production profile characteristics. Despite its important control on both primary and enhanced recovery of CBM for coal seams that have not been fully dewatered, relative permeability in coal has received little attention in the literature in the past decade. There are few published laboratory-derived curves; these studies and their resulting data represent a small subset of the commercial CBM reservoirs and do not allow for a systematic investigation of the physical controls on relative permeability curve shape. Other methods for estimation of relative permeability curves include derivation from simulation history-matching, and production data analysis. Both of these methods will yield pseudo-relative permeability curves whose shapes could be affected by several dynamic CBM reservoir and operating characteristics. The purpose of the current work is to perform a systematic investigation of the controls on CBM relative permeability curve shape, including non-static fracture permeability and porosity, multi-layer effects and transient flow. To derive the relative permeability curves, effective permeability to gas and water are obtained from flow equations, flow rates and pressure data. Simulated cases are analyzed so that derived and input curves may be compared allowing for investigation of CBM reservoir properties on curve shape. One set of relative permeability curves that were input into the simulator were obtained from pore-scale modeling. Field cases from two basins are also examined and controls on derived relative permeability curve shape inferred. The results of this work should be useful for future CBM development and greenhouse gas sequestration studies, and it is hoped that it will spark additional research of this critical CBM flow property.

Upscaling uncertainty analysis in a shallow-marine environment

Geological models are often created at a scale finer than is suitable for flow simulation and also ignore the effects of sub-cellular heterogeneities. Upscaling of static and dynamic reservoir properties is an important process that captures the impact of smaller scales, ensuring that both heterogeneity and the flow physics are represented more accurately. A Geopseudo upscaling approach for shallow-marine reservoirs is presented, which captures the essential flow characteristics across a range of scales from laminae to the simulation grid. Starting with a base-case set of minimum assumptions enables generation of one set of pseudo-relative permeability and capillary pressure curves per facies. This is then expanded to investigate the limitations of these assumptions and compare their impact against variations in large-scale geological and structural parameters. For the analysis, two-level full factorial experimental design is used to determine important parameters. A comparison of upscaling effects is also performed. The most important upscaling and fine-scale parameters identified by the analysis are the shape of the capillary pressure curve, lamina-scale permeability variation and upscaling flow speed. Of similar importance are the sedimentological parameters for shoreline aggradation angle and curvature. Fault direction (perpendicular and parallel to the shoreline) and the fine-scale upscaling method are of moderate to low importance. The shallow-marine parameter for clinoform barrier strength and the direction of flow considered when upscaling are unimportant. Analysis of upscaling effects suggests that the algorithm used at the intermediate scale is not important, while the assumed flow speed is very important, typically resulting in a 10% maximum variation in cumulative recovery. Fine-scale properties and upscaling methods affect recovery mostly due to increased initial water saturations but also because of early breakthrough.

Upscaling Immiscible Gas Displacements: Quantitative Use of Fine-Grid Flow Data in Grid-Coarsening Schemes

Summary This paper presents and assesses grid-coarsening schemes based on the quantitative use of fine-scale two-phase-flow information. The basic approach is motivated from a volume-average analysis of the fine-scale saturation equation including gravitational effects. Extensive results for layered systems are presented. We show that coarse-grid simulation error correlates closely with specific subgrid quantities involving higher moments of fine-grid variables, which can be computed from the fine-scale simulations. With formation of a coarse grid that minimizes the appropriate subgrid property, optimal coarse-scale descriptions can be generated. The overall approach is shown to be applicable to coarse-scale descriptions with either rock or pseudorelative permeability curves. The accuracy of the coarse-grid calculations is, however, significantly better when pseudofunctions are used. The method is applied to determine the optimal number and configuration of coarse-grid layers in more general cases and it is shown that coarse-grid results do not always improve as the number of coarse layers is increased.

Reservoir Characterization of Ekofisk Field: A Giant, Fractured Chalk Reservoir in the Norwegian North Sea-History Match

SummaryThe Ekofisk reservoir is a high porosity, low matrix permeability naturally fractured chalk. Fluid flow is largely governed by the distribution, orientation, and interconnectivity of the natural fracture system associated with complex structure and reservoir distribution. Because of the impact heterogeneity has on preferential fluid-flow direction, significant attention was given to capturing as much of the intrinsic heterogeneity as possible, both laterally and vertically, in the new three-dimensionally geological model. A non-uniform simulation mesh was defined for the fluid-flow model with a grid size of 450 ft × 450 ft in the crestal area of the field and increasing towards the flanks. A flow-based upscaling technique was then applied to preserve the heterogeneity from the geological to the fluid-flow model. Because of the complexity of the Ekofisk field, with its numerous faults and fracture networks, anisotropy was one of the primary attributes calibrated to achieve an individual well and field history match. However, faults and fault sealing factors, vertical permeability, pseudorelative permeability curves, bubblepoint pressure correlations, local permeability, and rock compressibility were also key parameters in the history match, and are presented in this paper. A brief discussion on the preliminary implementation of water-induced compaction is included.

A Case Study in Scaleup for Multicontact Miscible Hydrocarbon Gas Injection

SummaryWe describe the strategy and results of scaleup done to simulate a multicontact miscible hydrocarbon water alternating gas (WAG) injection process. To adequately model both oil recovery and solvent retention in WAG, one must model three-phase flow including gas trapping. Scaleup of the multicontact miscible gas process is particularly difficult because of the very fine-scale structure of the gas fingers and the miscible front. The case studied is a heterogeneous mixed wet reservoir with a transition zone down to an underlying aquifer. The objective was to develop pseudo relative permeability curves and other parameters that are suitable for running in a full-field limited compositional model with three hydrocarbon components.Both history-matching and systematic approaches were used to generate pseudo relative permeability curves that reproduced results of high-resolution, fully compositional (FC) reference simulations. Dynamic pseudoization techniques were used to derive first guesses at pseudos, but required further calibration to reproduce reference simulations successfully. In matching incremental miscible gas/oil recovery timing and solvent retention, varying three-phase water relative permeability was much more effective than varying the mixing parameter.The predictive capability of pseudos was tested for changes with respect to slug size, WAG ratio, and solvent enrichment. Pseudos derived for one pattern or cross section were tested in other patterns or cross sections. Pseudos worked well with respect to changes in WAG ratio, fairly well with respect to changes in solvent enrichment, and moderately well for changes in slug size. They were less robust with respect to changes in description.

Factors affecting pseudo relative permeability curves

A large number of grid cells are needed to simulate very large field reservoirs such as those in the Middle East. Pseudo relative permeability curves are used to reduce the dimensionality of reservoir simulation models and also to account for intra-cell rock property variations. Identifying the parameters affecting pseudo relative permeability curves is very important so that the model cells can be grouped in different categories in which appropriate pseudo relative permeabilities are generated for each group. The objective of this study is to identify the factors affecting pseudo relative permeability curves. Different reservoir rock and fluid properties were studied for an homogeneous 2D reservoir model. Validation of some curves is also done. The most pronounced effect on pseudo relative permeability curves was found to be caused by the reservoir dip angle. Layer thickness and PVT properties have some effect. Production rate and vertical to horizontal permeability ratio have no significant effect within the parameter ranges studied. While absolute permeability on low dip angle cases has no effect, absolute permeability effects on high dip angle cases needs consideration.

Full-Field Reservoir Modeling of Central Oman Gas-Condensate Fields

Summary Gas reserves sufficient for a major export scheme have been found in Central Oman. To support appraisal and development planning of the gas and condensate fields, a dedicated, multidisciplinary study team comprising both surface and subsurface engineers was assembled. The team fostered a high level of awareness of cross-disciplinary needs and challenges, resulting in timely data acquisition and a good fit between the various work activities. A field-development plan was completed in March 1994. The foundation of the subsurface contributions was a suite of advanced full-field reservoir models that were used to do the following. Provide production and well requirement forecasts. Quantify the impact of uncertainties on field performance and project costs. Support the appraisal campaign. Optimize the field-development plan. Derive recovery factor ranges for reserves estimates. The models were constructed early during the study, initially with general data with large uncertainty ranges and gradually refined with new appraisal data. This ultimately resulted in more than 20 full scenarios, quantifying and ranking the remaining uncertainty ranges. Geological/petrophysical uncertainties were quantified by use of newly developed, 3D probabilistic modeling tools. An efficient computing environment allowed a large number of sensitivities to be run in a timely, cost-effective manner. The models also investigated a key concern in gas-condensate fields, well impairment caused by near-well condensate precipitation. Its impact was assessed by use of measured, capillary-number-dependent, relative permeability curves. Well-performance ranges were established on the basis of equation-of-state (EOS) single-well simulations and translated into the volatile-oil full-field models with pseudorelative permeability curves for the wells. The models used the sparse available data in an optimal way and, as part of the field-development plan, sustained confidence in the reserves estimates and the project, which is currently in the project specification phase.

A Laboratory Investigation of the Pseudo Relative Permeability Characteristics of Unstable Immiscible Displacements

Abstract Enhanced oil recovery operations often involve immiscible displacement of the more viscous oil by a less viscous fluid. This often leads to an unstable and inefficient displacement process because of fingering of the more mobile displacing fluid through the more viscous oil. Although, with application of reported stability theories, it is possible to roughly predict whether or not a field displacement is likely to be unstable, rigorous mathematical models to predict the recovery performance of such unstable displacements are not available. The usual practice has been to employ a Buckley-Leverett type analysis, which is based on the assumption of a stable and stabilized displacement, and to account for the presence of instabilities by using empirical sweep efficiency corrections. The relative permeabilities used in this type of analysis are usually obtained from laboratory tests which pay little or no attention to possible viscous instabilities. A more direct approach to account for the presence of viscous instabilities in immiscible displacements would be to use pseudo-relative permeabilities which are modified true relative permeability curves (modified to account for the instabilities). However, before development of such an approach is attempted, one needs to examine whether or not the conventional Buckley-Leverett model, with modified relative permeability curves, can be used to describe the recovery and pressure drop performance of unstable displacements. This was the main objective of this study. Several unstable immiscible displacement experiments were carried out in a rectangular model. The effects of different parameters on the oil recovery were examined and pseudo-relative permeability curves were generated for each set of conditions. The dependence of these pseudo-relative permeability curves on displacement rate, viscosity ratio, gravity, and the presence or absence of initial water saturation were examined. The performance of current stability theories in predicting the breakthrough recovery was tested against the results of these displacements. It was found that the "wettability number" proposed by Peters and Flock was not a constant for a given rock fluid system but rather a function of the flood velocity. Introduction Recovery of oil by waterflooding is an efficient process, provided the water advances as a stable front and sweeps the reservoir uniformly. This occurs when the oil is more mobile than the displacing water and the reservoir is not too heterogeneous. However, when the oil mobility is low as would be the case in medium gravity and heavy oils, the displacing water has a tendency to finger through the oil and this leads to poor recovery efficiency. The factors which control this type of flow instability include the displacement velocity, oil/water viscosity ratio, relative permeability characteristics and the interfacial tension between oil and water. Although, with application of reported stability theories, it is possible to roughly predict whether or not a field displacement is likely to be unstable, rigorous mathematical models to predict the recovery performance of such unstable displacements are not available. The usual practice has been to employ a Buckley-Leverett type analysis(1,2). The Buckley-Leverett theory applies to a linear displacement process involving two immiscible fluids with no mass transfer taking place between them.

Two Phase Flow in Nodular Systems: Laboratory Experiments

Heterogeneity effects within the grid block of a reservoir numerical model are generally taken into account by using pseudo-relative permeability and capillary pressure curves. Some insight into the physics of these pseudo-functions can be obtained from multiple scale analysis such as largescale averaging. Previous theoretical results have shown specific behaviour associated with nodular systems. Laboratory scale experiments were performed that emphasize the difference in behaviour of nodular systems when the properties of the different regions are interchanged. Sandstone nodules embedded in a more permeable continuous region made of sand showed a quasi-static behaviour under experimental conditions while the reverse situation, i. e. sand nodules in a sandstone continuous region, exhibited a totally different behaviour in terms of final saturation and recovery curves. These results are discussed with respect to the large-scale averaging objective.

Waterflood simulation of stratified sandstone reservoirs using dynamic pseudo curves

Many studies have shown that pseudo relative permeability curves can be used in one-dimensional (1-D) linear models to successfully match simulation performance of two-dimensional (2-D) cross-sectional models. Additional studies have shown that pseudo relative permeability curves can be used in 2-D areal models to match 3-D model performance. The techniques, however, used in the latter studies were laborious and complex and often required multiple sets of pseudo relative permeability curves. This study shows that a 2-D areal model that uses only one set of pseudo relative permeability curves (generated from a cross-sectional model) allows accurate simulation of waterflood performance for highly stratified reservoirs that would otherwise require 3-D simulation or 2-D simulation with multiple sets of curves. The cases examined are waterfloods in reservoirs with repeating 5-spot patterns for several sandstone stratifications from beach, fluvial channel, and delta margin environments of deposition. Reservoir simulation in this study makes use of lab-derived capillary pressure curves and the method of Jacks et al. to compute pseudo relative permeability curves. Gravity numbers were developed to characterize cross-sectional and 5-spot (areal) displacements. These gravity numbers were correlated for several geological stratifications to minimize the need for iterative simulations of each particular cross-sectional model to obtain pseudo relative permeability curves. Two of these correlations were applied to field cases to demonstrate their utility.

Novel Relations for Drainage and Imbibition Relative Permeabilities

Abstract Exponential four- and five-parameter equations are proposed for gas/oil drainage and water/oil imbibition relative permeability curves. These equations match the experimentally determined curves, in particular at and near their initial points and endpoints, better than standard Corey et al.1 and polynomial approximations. Some of these parameters have a physical meaning; the others can be determined by nonlinear regression on the experimental data points, and can be adjusted to represent pseudorelative permeability curves. The proposed equations are particularly suitable to describe gas percolation in numerical model simulation of dissolved-gas-drive reservoirs.