Pseudo Relative Permeability Research Articles

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.

Technology Focus: Reservoir Simulation (July 2012)

Technology Focus This year, there were approximately 200 papers on simulation to select from—and that is after a separate feature on history matching. So, the discipline continues to be active. A noticeable feature is the growing number of simulation papers that use different technology, in the broadest sense of the term—for example, using concepts from signal processing and electrical engineering to model subsurface flow or related phenomena. However, the dominant technology remains finite different representations of Darcy’s law, conservation of mass, and a fluid model. What is also heartening is the fact that all significant papers on case studies start with descriptions of the geology and often include detailed description of 3D geological modeling. Any simulation that is based on a physical model of the field must surely depend on the quality of the geological model used as input; yet, not that long ago, it was normal to pay only scant regard to the geology when constructing a model and even less when altering it during history matching. Recent discussion within the SPE Simulation Technical Interest Group has raised the issue of pseudorelative permeabilities, with some arguing that they are obsolete and others strongly disputing the claim. Sadly, there were no papers on the subject that I could include in this feature. The relative permeability curve is where engineering meets geology; anyone involved in complex projects—and who of us is not—knows that it is at the interfaces that complexities arise and are too often ignored. The same is true in our models, so surely relative permeabilities and their multiphase upscaling are topics worth renewed investigation. All three of the case studies I have selected, which are all from very different settings and parts of the world, were studies directed toward making tangible decisions (e.g., selecting well locations and completion intervals). This highlights once again that good simulation studies are directed toward decision making; having a clear sight of the purpose of the work improves the quality of the work and, thus, of the ultimate decision. The converse is also true. Recommended additional reading at OnePetro: www.onepetro.org. SPE 146366 A Capacitance-Resistive Model and InSAR Imagery of Surface Subsidence Explain Performance of a Waterflood Project at Lost Hills by Wenli Wang, The University of Texas at Austin, et al. SPE 146524 Reservoir Simulation: A Reliable Technology? by W.J. Lee, Texas A&M University, et al. SPE 146574 Impacts of the Porosity/Permeability Transform Throughout the Reservoir-Modeling Workflow by R.D. Roadifer, ConocoPhillips Alaska, et al.

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.

Waterflooding Performance in Inclined Communicating Stratified Reservoirs

Summary A mathematical model is developed for performance prediction of waterflooding performance in communicating stratified reservoirs with a dip angle from the horizontal. The effect of the gravitational force is reflected by a dimensionless gravity number in the fractional flow formula. The gravity number accounts for the dip angle and the density difference between the displacing and displaced fluids. The developed fractional flow formula is used to estimate the fractional oil recovery, the dimensionless time, and the injectivity ratio at times of water breakthrough in the successive layers. The developed model allows for each layer to have its own porosity, endpoint saturations, and endpoint relative permeabilities. Solutions for the waterflooding performance in inclined communicating stratified systems with log-normal permeability distribution were obtained and compared with that of the horizontal systems. The effects of the gravity number, the mobility ratio, and the Dykstra-Parsons permeability-variation coefficient VDP on the performance were investigated. The obtained results showed that the gravity effect of the dip angle enhances the performance in terms of delayed water breakthrough, higher fractional oil recovery, and lower water cut. This improved performance is more significant in the cases of unfavorable mobility ratio and of highly heterogeneous reservoirs. Reservoir dipping does not affect the pseudorelative permeability functions but results in a decrease in the injectivity ratio.

Retaining geological realism in dynamic modelling: a channelized turbidite reservoir example from West Africa

ABSTRACT A comprehensive simulation study tested and analysed the sensitivity of dynamic connectivity in turbidite channel reservoirs to a large number of stratigraphic and engineering parameters. The study showed that subseismic shale architecture has a significant effect on reservoir connectivity. However, representing the complete spectrum of fine-scale architectural details in full-field simulation models is beyond the limits of existing computational capabilities. Previous work demonstrated that incorporating geologically based pseudo-relative permeabilities into relatively coarse full-field reservoir models renders practically intractable simulation cases tractable. We developed a methodology for generating pseudo-relative permeabilities at multiple geological scales, incorporating the effect of channel architecture and reservoir connectivity into fast simulation models. We describe a dynamic modelling workflow that integrates geologically based pseudo-relative permeabilities into a two-stage automatic history-matching algorithm. The history-matching problem is posed as one of data conditioning in the Bayesian framework. We show the application of the workflow to a channelized turbidite reservoir in West Africa. It is demonstrated that multiple geologically consistent models that are conditioned to production data can be generated rapidly thanks to optimally coarse simulation models that capture the effect of subseismic channel architecture on recovery behaviour, and run efficiently as the forward model within a Bayesian inference framework. Proof-of-concept tests carried out using field data indicate that the history-matched models predict well-by-well future recovery response with good accuracy.

Simplified Modeling of Turbidite Channel Reservoirs

Summary Effective properties can represent fine-scale geologic heterogeneities in simple full-field reservoir models without having to model them explicitly. A comprehensive simulation study tests the sensitivity of dynamic connectivity in turbidite channel reservoirs to a large number of stratigraphic and engineering parameters. Simulations performed using geologically realistic sector models at multiple levels of stratigraphic resolution show that dynamic connectivity is governed by large-scale architectural parameters, such as meander-belt size, net-to-gross ratio, degree of depositional-storey amalgamation, and stratigraphic parameters that describe the shale architecture at multiple scales (e.g., shale-drape coverage and shale-drape frequency of occurrence). We demonstrate how to rapidly generate effective properties at multiple geologic scales, incorporating the effect of channel architecture and reservoir connectivity into simple dynamic models. Use of simple dynamic models in conjunction with effective properties, principally geologically based pseudorelative permeabilities, accelerates the simulation workflow significantly. We show that a statistical distribution of the recovery factor (RF) can be produced within hours instead of days by the combined use of Monte Carlo simulation and a simple dynamic model with effective properties. Recovery factors estimated with our simplified modeling method agree well with observed RF distributions of turbidite channel reservoirs with significant production history.

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.

Immiscible Displacement of Non-Newtonian Fluids in Communicating Stratified Reservoirs

Summary The displacement of non-Newtonian power-law fluids in communicating stratified reservoirs with a log-normal permeability distribution is studied. Equations are derived for fractional oil recovery, water cut, injectivity ratio, and pseudorelative permeability functions, and the performance is compared with that for Newtonian fluids. Constant-injection-rate and constant-total-pressure-drop cases are studied. The effects of the following factors on performance are investigated: the flow-behavior indices, the apparent mobility ratio, the Dykstra-Parsons variation coefficient, and the flow rate. It was found that fractional oil recovery increases for nw>no and decreases for nw<no, as compared with Newtonian fluids. For the same ratio of nw/no, oil recovery increases as the apparent mobility ratio decreases. The effect of reservoir heterogeneity in decreasing oil recovery is more apparent for the case of nw>no. Increasing the total injection rate increases the recovery for nw>no, and the opposite is true for nw<no. It also was found that the fractional oil recovery for the displacement at constant total pressure drop is lower than that for the displacement at constant injection rate, with the effect being more significant when nw<no.

Efficient Incorporation of Global Effects in Upscaled Models of Two-Phase Flow and Transport in Heterogeneous Formations

Geological variability occurring over multiple length scales can significantly affect fluid flow in subsurface formations. In this paper we explore the impact of large scale (global) information on the accuracy of coarse scale models for two-phase flow and transport. Based on these findings, a new methodology for generating coarse models is introduced. This approach, which avoids any global fine scale calculations, entails adaptive local-global single-phase parameter upscaling coupled with subgrid models for two-phase parameters. Two related subgrid treatments for two-phase flow effects---a pseudorelative permeability model and a generalized convection-diffusion model---are investigated and applied. The upscaled single-phase parameters (transmissibilities), computed using the adaptive local-global procedure, account explicitly for global boundary conditions as they are adapted for a specific flow scenario. The two-phase coarse scale functions are computed from local simulations using effective flux boundary conditions (EFBCs), which account approximately for global effects in the local computations. The advantages of the adaptive local-global upscaled single-phase parameters, as well as the superior accuracy of EFBCs relative to standard treatments for two-phase parameters, are demonstrated for several challenging two-dimensional example cases. The overall method is also applied to highly heterogeneous systems with global boundary conditions that vary in time. Using standard upscaling methods, errors for some of these cases are very large, though the new methodology is shown to consistently provide reasonably accurate coarse models.

Accurate Subgrid Models for Two-Phase Flow in Heterogeneous Reservoirs

Summary Subgrid effects can have a strong influence on flow and transport in oil reservoirs. In this work, a new model for the representation of subgrid terms is introduced and applied to two-phase reservoir flows. The model entails a generalized convection-diffusion treatment of the saturation equation as well as an extended representation for total mobility in the pressure equation. Motivation for the form of the model is provided through a consideration of volume averaging and homogenization results. The numerical computation of the subgrid terms and the implementation of the overall method are described. The accuracy of the new subgrid representation is compared to that of coarse-scale models with no subgrid treatment and to coarse models based on pseudorelative permeabilities. The new model consistently provides more accurate overall coarse-scale predictions, relative to reference fine-scale results, than the other coarse-scale models, particularly in cases when the global boundary conditions vary in time.

Gas-Condensate Reservoir Pseudo-relative Permeability Derived by Pressure-Transient Analysis. Incorporation of Near-Wellbore Effects

Gas-condensate reservoirs exhibit a near-wellbore complex behavior due to the existence of a two-phase system when pressure drops below dew point. In analyzing well-testing data, a single-phase approach is only valid for well condition above dew point, whereas a multi-phase pseudo-pressure approach is widely adopted for a gas reservoir operating below dew-point. This article utilizes the pseudo-pressure approach, for analyzing well-test data in gas-condensate reservoirs producing below the dew-point pressure, to generate Corey exponents for relative permeability in order to determine near-wellbore representative relative permeability that history matches the actual well-testing data.

The representation of two phase fault-rock properties in flow simulation models

Faults are represented conventionally in production flow simulation models using transmissibility multipliers which capture the single phase, but not the two phase, fault-rock properties. Available data indicate that fault-rocks have similar two phase properties to sediments of the same permeability, hence existing methods can be applied to estimate two phase fault-rock properties from their intrinsic permeabilities. Two methods of representing the two phase fault-rock properties implicitly in the flow simulator are compared, using one-dimensional numerical flow models containing water-wet faults with imbibition capillary pressure curves. The method which is the closer two phase analogue of the single phase transmissibility multiplier is inappropriate, as the implementation is unreasonably unwieldy. A simpler implementation is to derive pseudo-relative permeability functions including the fault-rock properties in the upstream grid block; these properties are then incorporated directly in the simulator. Relative transmissibility multiplier functions can be back-calculated from the pseudo-relative permeability functions, and indicate how closely the single phase multiplier approximates two phase flow through the fault. Implementation in a 3D model with complex fault juxtapositions validates the approach, and a practical workflow for the routine inclusion of two phase fault-rock properties in conventional faulted flow simulation models is outlined.

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.

An assessment of steady-state scale-up for small-scale geological models

The calculation of pseudo-relative permeabilities can be speeded up considerably by using steady-state methods. The capillary equilibrium limit may be assumed at small scales (30 cm or less), when the flood rate is low. At high flow rates and larger distance scales, we may use a viscous-dominated steady-state method which assumes constant fractional flow. Steady-state pseudos may also be calculated at intermediate flow rates using fine-scale simulations, and allowing the flood to come into equilibrium at different fractional flow levels. The aim of this paper is to assess the accuracy of steady-state scale-up for small-scale sedimentary structures. We have tested steady-state scale-up methods using a variety of small-scale geological models. The success of steady-state scale-up depends not only on the flow rate, but also on the nature of the heterogeneity. If high permeability zones are surrounded by low permeability ones (e.g. low permeability laminae or bed boundaries), oil trapping may occur in a water-wet system. In this case pseudo-oil-relative permeabilities are very sensitive to flow rate, and care must be taken to upscale using the correct viscous/capillary ratio. However, in permeability models, where phase trapping may not occur (unconnected low permeability regions), the pseudos are similar, whatever the viscous/capillary ratio. The disadvantage of steady-state scale-up is that it cannot take account of numerical dispersion, in the manner in which dynamic methods can. However, we show examples of coarse-scale simulations with viscous-dominated steady-state pseudos which agree favourably with fine-scale simulations. Provided there are sufficient grid blocks in the coarse-scale model, the smearing of the flood front due to numerical effects is not serious.

Steady-State Upscaling: From Lamina-Scale to Full-Field Model

Summary In recent years, the impact of small-scale permeability structure on hydrocarbon recovery has been demonstrated, and upscaling procedures, such as the geopseudo method, have been developed to scale-up from the lamina scale using the hierarchy of geological length scales. However, upscaling is very time consuming, so many engineers still input rock curves into large-scale simulations. In this study, we show how the process may be speeded up using steady-state methods for calculating the pseudofunctions. Two examples are used to demonstrate the method. The first is a three-stage scale-up of a water flood in a fluvio-aeolian model. Capillary equilibrium was assumed for the first two stages, and viscous-dominated steady state for the third stage. Two different wettability cases were examined—water-wet and intermediate-wet. The effect of using the small-scale pseudo relative permeabilities depends on both the nature of the heterogeneities and on the wettability. In this study the recovery was reduced when small-scale pseudos were included, especially in the water-wet case. The second case study involved gas injection into the oil leg of a tidal deltaic reservoir. Scale-up was performed in two stages: (a) from the lithofacies scale to the geological model, and (b) from the geological model to the full-field simulation model. The viscous-dominated steady-state method was used in both cases. The results showed that the effect of the fine-scale heterogeneities was of the same order as the effect of the coarse-scale heterogeneity, indicating that (even in a system where capillary pressure is negligible) the fine-scale structure can be important.

Waterflooding Performance of Communicating Stratified Reservoirs With Log-Normal Permeability Distribution

Summary An analytical solution is developed for waterflooding performance of layered reservoirs with a log-normal permeability distribution with complete crossflow between layers. The permeability distribution is characterized by the Dykstra-Parsons (DP) variation coefficient VDP or the standard deviation of the distribution σk. The performance is expressed in terms of vertical coverage as function of the producing water-oil ratio. Also an expression for the dimensionless time (pore volumes of injected water) at a given water-oil ratio is derived. Expressions are also derived for pseudorelative permeability functions and fractional flow curves that can be used in reservoir simulation. Correlation charts are also presented to enable graphical determination of the performance. The variables are combined in such a way that a single chart is constructed for the entire range of water-oil ratio, mobility ratio and permeability variation. Analogy to the Buckley-Leverett (BL) multiple-valued saturation profile is found to occur at low mobility ratios (M<1) where a multiple-valued displacement front is formed. A procedure similar to the BL discontinuity is suggested to handle this situation. Successive layers with different permeabilities are allowed to move with the same velocity resulting in a single-valued profile with a discontinuity. No such behavior is observed for mobility ratios greater than unity. A criterion for the minimum mobility ratio at which this behavior occurs is presented as a function of the variation coefficient VDP.

An analysis of dynamic pseudo-relative permeability methods for oil-water flows

The properties and limitations of six widely used dynamic pseudo-relative permeability methods are analysed for the case of incompressible, immiscible, two-phase flow. The example proposed by Stone is used to illustrate our findings. The analytical results are confirmed by numerical simulation.

Evaluation of Dynamic Pseudofunctions for Reservoir Simulation

Summary We present a detailed investigation on the reliability of some of the dynamic pseudofunctions used to upscale flow properties in reservoir simulation. A one-dimensional example (1D) and a real field application are used to evaluate methods developed by Kyte and Berry and Stone, and a new flux weighted potential (FWP) method. A derivation of Stone's method (based on the description given by Stone above) is presented, which is found to give an inconsistent set of equations. Stone's analytical example was used to illustrate how pseudorelative permeabilities that exhibit non-physical behavior may still give acceptable results, but this success can disappear with changes in boundary conditions. The pseudofunctions from a field application were not able to match the 2D simulations from which they were calculated, even when a different pseudofunction was used for each coarse grid block. Improvements were obtained when directional pseudofunctions were used, but still the results were not satisfactory. Similar results were found when comparing fine and coarse grid 3D simulations for a quarter of a five-spot pattern in this field. The results presented in this article suggest that dynamic pseudofunctions, as applied here and as commonly used in industry, may not be an adequate approach to up-scaling. The possibility of large errors and the difficulty in predicting when they may occur make the use of pseudofunctions examined in this paper unreliable.

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.