Upscaled Relative Permeability Research Articles

Upscaling of Transport Properties in Complex Hydraulic Fracture Systems

Summary Fluid flow in complex fracture systems near wellbore is influenced by heterogeneous fluid pathway structure, proppant distribution, and stress-induced fracture aperture change. The current physical experiments and pore-scale simulations only study the multiphase flow properties of hydraulic fracture (HF) with no proppant while the multiphase flow properties of induced fracture network (IFN) and HF with proppant are not available. It is well known that the simplified “straightline” relative permeability model does not apply to multiphase flow in IFN and HF with proppant. Consequently, there is no upscaled relative permeability model that works. In this study, we develop the physics-driven level set lattice Boltzmann method (LS-LBM)-coupled model to study multiphase flow properties in complex fractures during injected water flowback and propose the upscaled relative permeability models of IFN and HF with proppant. The imaged HF is applied to generate HFs with different aperture and proppant distributions using morphology operation and discrete element method (DEM). The imaged IFN is further applied to generate IFN with different aperture distributions by image dilation. The oil/water interface at different drainage pressures is tracked by LS, and the resultant fluid distributions are applied to calculate each phase’s effective permeability by LBM. We found that the aperture variation coefficient difference leads to various fluid expansion patterns in IFN and HF. The oil/water interface moving pattern exhibits “face expansion” in IFN and HF while the oil/water interface moving pattern resembles “finger expansion” in HF with embedded proppant with notably larger aperture variation coefficient. The upscaled relative permeability model is further established considering channel tortuosity variation and pore structure difference based on LS-LBM simulation results.

Machine learning assisted relative permeability upscaling for uncertainty quantification

Traditional two-phase relative permeability upscaling entails the computation of upscaled relative permeability functions for each coarse block (or interface). The procedure can be extremely time-consuming especially for cases with multiple (hundreds of) geological realizations as commonly used in subsurface uncertainty quantification or optimization. In this paper, we develop a machine learning assisted relative permeability upscaling method, in which the flow-based two-phase upscaling is performed for only a small portion of the coarse blocks (or interfaces), while the upscaled relative permeability functions for the rest of the coarse blocks (or interfaces) are quickly computed by machine learning algorithms. The upscaling procedure was tested for generic (left to right) flow problems using 2D models for scenarios involving multiple realizations. Both Gaussian and channelized models with standard boundary conditions and effective flux boundary conditions (EFBCs) are considered. Numerical results have shown that the coarse-scale simulation results using the newly developed machine learning assisted upscaling are of similar accuracy to the coarse results using full numerical upscaling at both ensemble and realization-by-realization levels. Because the full flow-based upscaling is only performed for a small fraction of the models, significant speedups are achieved.

Simulating Core Floods in Heterogeneous Sandstone and Carbonate Rocks

AbstractThe characterization of multiphase flow properties is essential for predicting large‐scale fluid behavior in the subsurface. Insufficient representation of small‐scale heterogeneities has been identified as a major gap in conventional reservoir simulation workflows. We systematically evaluated the workflow developed by Jackson et al. (2018), https://doi.org/10.1029/2017wr022282 for use on rocks with complex porosity and capillary heterogeneities. The workflow characterizes capillary heterogeneity at the millimeter scale. The method is a numerical history match of a coreflood experiment with the 3D saturation distribution as a matching target and the capillary pressure characteristics as a fitting parameter. Coreflood experimental datasets of five rock cores with distinct heterogeneities were analyzed: two sandstones and three carbonates. The sandstones exhibit laminar heterogeneities. The carbonates have isotropic heterogeneities at a range of length scales. We found that the success of the workflow is primarily governed by the extent to which heterogeneous structures are resolved in the X‐ray imagery. The performance of the characterization workflow systematically improved with increasing characteristic length scales of heterogeneities. Using the validated models, we investigated the flow rate dependency of the upscaled relative permeability. The findings showed that the isotropic heterogeneity in the carbonate sample resulted in non‐monotonic behavior; initially the relative permeability increased, and then subsequently decreased with increasing flow rate. The work underscores the importance of capturing small‐scale heterogeneities in characterizing subsurface fluid flows, as well as the challenges in doing so.

Fluid Displacement in a Dual-Permeability Medium with Local Capillary Equilibrium

The solution to the Buckley–Leverett problem in fractured porous media is investigated by applying the dual-porosity dual-permeability model. It is shown that the dual-permeability medium is equivalent to an upscaled homogenized medium at long injection times when local capillary equilibrium is established. Simple analytical relationships for the petrophysical properties and saturation functions of the equivalent medium are derived. The upscaled relative permeability and the fractional flow function can exhibit kinks, resulting in more complicated solutions to the Buckley–Leverett problem as compared to the classical case. It is found that immiscible displacement in fractured porous media leads to the appearance of saturation profiles with several displacement fronts and rarefaction waves. Up to four solution types are possible, which are constrained in a solution map constructed in the space of the dimensionless parameters of the problem. It is shown that, for particular parameters, the displacing fluid moves faster through the matrix than through the fractures.

Pseudorelative Permeabilities for Simulation of Unstable Viscous Oil Displacement

Summary Relative permeabilities are well understood for light oils involving stable displacement. However, conflicting arguments have been presented in the literature regarding relative permeabilities for viscous oils. Most nonthermal viscous oil displacements are unstable. Depending on the magnitude of mobility ratio, displacement is influenced by varying degrees of viscous instability, often referred to as fingering. For viscous oils (>500 cp), even a polymer flood must be designed at partially stable conditions (mobility ratio > 1) to maintain an economical processing rate [% pore volume (PV) injected (PVI) per year]. Typically, viscous fingering is difficult to model in full-field simulation because of the large grid sizes used. To design and optimize a partially stable polymer or waterflood, it is critical to correctly upscale the laboratory-generated relative-permeability curves for reservoir simulation. In recent years, such models have been published in the Society of Petroleum Engineers literature. Unfortunately, most of these models require multiple fitting parameters (at least three). In this work, we present a simplified technique that requires systematic change in only one parameter to generate upscaled relative permeability curve for a given viscosity ratio. Using fine-grid simulations, we show that the flow at high-viscosity ratio is channelized even in a core perceived to be homogeneous at laboratory scale. This happens because of small-scale heterogeneities that are present in every rock. Upscaling averages these fine variations in heterogeneities, causing the grids to be overswept, thus overpredicting recovery. To compensate for this shortcoming, it is recommended to upscale the relative-permeability curves in the simulation model.

Multiphase flow model from pores to cores in organic-rich shale

The multiphase flow processes during water injection and flowback in shales are largely affected by its multiscale pore structure, different pore types and complex fluid transport mechanisms. At present, it is difficult to measure shale multiphase flow properties in laboratory. This study presents an upscaling model to derive core-scale relative permeability from pore-scale multiphase simulation using organic matter (OM)/inorganic matter (IOM) pore network models. Gas and water absolute permeabilities on OM/IOM pore network models are determined, and gas-water two-phase imbibition (during water injection) and drainage (during flowback) processes in the IOM pore network model are conducted using the invasion percolation theory. A core-scale model is generated with stochastically distributed IOM/OM patches. The upscaled relative permeabilities are derived by assembling gas-water transport properties in the IOM/OM based on the corresponding pore network simulation results. The upscaling method is validated by comparison with an analytical model during water injection and flowback processes. Furthermore, the impacts of total organic carbon (TOC) content and intrinsic permeability ratio of OM to IOM on the upscaled relative permeabilities are analyzed. The critical TOC content and intrinsic permeability ratio at which OM influences the upscaled relative permeabilities are determined. This study provides an effective upscaling workflow to obtain core-scale gas-water relative permeabilities based on pore-scale modeling considering multiple gas/water transport mechanisms in different pore systems.

Tensor Analysis of the Relative Permeability in Naturally Fractured Reservoirs

Summary Fluid evidence shows that prediction of water breakthrough and oil recovery from fractured reservoirs cannot be performed accurately without upscaled relative permeability functions. Relative permeability is commonly assumed to be a scalar quantity, although the justification of that—specifically for naturally fractured reservoirs (NFRs)—is rarely attempted. In this study, we investigate the validity of this scalar-quantity assumption and how it affects fracture/matrix equivalent relative permeabilities, kri(Sw), achieved by a numerical simulation of unsteady-state waterflooding of discrete-fracture/matrix models (DFMs). Numerical determination of relative permeability requires a realistic model, a spatially adaptive simulation approach, and a sophisticated analysis procedure. To fulfil these requirements, we apply the discrete-fracture/matrix modeling to well-characterized outcrop analogs at the hectometer to kilometer scale. These models are parameterized with aperture and capillary entry pressure data, taking into account variations from fracture segment to segment, trying to emulate in-situ conditions. The finite-element-centered finite-volume method is used to simulate two-phase flow in the fractured rock, while also considering a range of wettability conditions from water-wet to oil-wet. Our results indicate that the fracture/matrix equivalent relative permeability is a weakly anisotropic property. The tensors are not necessarily symmetric, and the absolute-permeability tensor is the most influential factor, determining the level of anisotropy of kri. The anisotropy ratio (AR) changes with saturation, is influenced by the fracture/matrix-interface wetted area (Awf), and differs for each phase. In addition, the diagonal terms of the equivalent relative permeability tensor (krii), determined using our novel approach, can be different from those obtained using the assumption that kri is scalar. The magnitude of the difference is controlled by the absolute permeability, wettability, flow rate, and orientation of the fractures in the model. It is worth mentioning that the type and direction of imbibition can be determined by off-diagonal terms of the kri tensor. Furthermore, krii largely depends on the direction of the waterflood along the i-axis.

3D numerical and experimental study on upscaling two phase relative permeability curve of naturally fractured reservoirs

In this paper, an integrated approach of upscaling of laboratory derived relative permeability to reservoir grid block scale under poroelastic frame work is presented. In this approach, capillary and viscous forces are assumed to be dominating which allows us to obtain a reasonable up-scaled relative permeability curves. In order to improve the computation efficiency first, the reservoir is divided into a number of grid blocks (20 m × 20 m × 30 m) and then each block is further divided into laboratory glass bead scale blocks (20 cm × 10 m × 2 cm). Next, relative permeability correlation generated based on glass bead model is used to upscale the laboratory estimate of the relative permeability curve to each fine scale grid (Fahad et al., 2017). Levenberg-Marquardt inversion algorithm is used in the upscaling process in order to match the oil recovery or water breakthrough time. The upscaling approach is applied to a reservoir domain of a typical fractured basement reservoir in Vietnam offshore. [Received: August 20, 2017; Accepted: February 1, 2018]

3D numerical and experimental study on upscaling two phase relative permeability curve of naturally fractured reservoirs

In this paper, an integrated approach of upscaling of laboratory derived relative permeability to reservoir grid block scale under poroelastic frame work is presented. In this approach, capillary and viscous forces are assumed to be dominating which allows us to obtain a reasonable up-scaled relative permeability curves. In order to improve the computation efficiency first, the reservoir is divided into a number of grid blocks (20 m × 20 m × 30 m) and then each block is further divided into laboratory glass bead scale blocks (20 cm × 10 m × 2 cm). Next, relative permeability correlation generated based on glass bead model is used to upscale the laboratory estimate of the relative permeability curve to each fine scale grid (Fahad et al., 2017). Levenberg-Marquardt inversion algorithm is used in the upscaling process in order to match the oil recovery or water breakthrough time. The upscaling approach is applied to a reservoir domain of a typical fractured basement reservoir in Vietnam offshore. [Received: August 20, 2017; Accepted: February 1, 2018]

Upscaling of CO2 injection into brine with capillary heterogeneity effects

The large-scale simulation of CO2 storage operations can be expensive computationally, particularly when the effects of fine-scale capillary pressure heterogeneity are included. The application of upscaling techniques could lead to substantial reductions in computational cost. This work targets the development of an upscaling procedure that is applicable for the CO2 injection stage, which corresponds to a drainage process. Specifically, we develop and apply a new upscaling technique for two-phase flow in heterogeneous formations with capillary heterogeneity effects. The procedure entails first upscaling capillary pressure in the capillary limit, and then computing coarse-scale relative permeability functions using a global dynamic upscaling procedure. An iterative method is applied to enhance the accuracy of the upscaled capillary pressure functions. The new dynamic upscaling approach is applied to a synthetic heterogeneous two-dimensional aquifer model that involves injection of CO2 into brine. Fine-scale simulation results are compared with coarse-scale results generated using both the dynamic upscaling approach and a simpler method that entails the use of upscaled capillary pressure in conjunction with rock relative permeability curves. It is shown that the dynamic upscaling procedure provides results that are close to those from the fine-scale simulation and are consistently more accurate than results from the simpler method. The upscaling procedure is tested over a range of injection rates spanning three orders of magnitude. We show that, although the upscaled functions are in general rate dependent, accurate coarse-scale results can be obtained using upscaled relative permeability functions computed at only two different flow rates. We also observe that the model constructed using our method retains reasonable accuracy even when the flow problem differs from that used to compute the upscaled functions.

Development and application of near-well multiphase upscaling for forecasting of heavy oil primary production

Abstract Near-well effects can have a strong impact on reservoir flow. Current reservoir modeling practice often uses coarse-scale flow simulation models, which may lead to biased results, compared with fine-scale models. In this work, we extend and apply a recently developed near-well multiphase flow upscaling technique to the coarse-scale simulation of heavy-oil primary production. For heavy oils, oil viscosity is a strong function of pressure when the pressure is below the bubble point. Therefore, the upscaled mobility functions (from near-well multiphase upscaling) depend on both pressure and saturation, which cannot be directly input to general purpose reservoir simulators. This is very different than the upscaled mobility functions for typical black-oil fluids, in which oil viscosity does not vary significantly with pressure. Accordingly, the upscaled mobility functions are often equivalent to upscaled relative permeabilities (as functions of saturation only). In this work, we develop two procedures to derive either the upscaled relative permeability or viscosity functions from the phase mobility functions, thus decoupling the dependency on pressure and saturation. It is found that the upscaled oil viscosity provides more accurate predictions than the upscaled relative permeabilities, especially at early time. This is because that the rapid change of pressure at the early stage of production is captured sufficiently in the upscaled oil viscosity (as a function of pressure). The use of upscaled viscosity function in multiphase upscaling is new, and has not been presented in previous studies. We also introduce a grouping technique to reduce the number of upscaled flow functions in coarse-scale models. This is based on an observation that there is a strong correlation between the upscaled flow functions and the coarse-scale well-block permeabilities. The proposed methods are applied to realistic models from heavy-oil fields. For cases considered, the near-well multiphase flow upscaling considerably improves upon the standard coarse-scale models. The use of upscaled relative permeability and viscosity functions, as well as the grouping of upscaled flow functions, provides practical applicability of the proposed method.

Hysteretic upscaled constitutive relationships for vertically integrated porous media flow

Subsurface two-phase flow in porous media often takes place in reservoirs with a high ratio between the associated lateral and vertical extent and the lateral and vertical flow time scales. This allows for a two-scale approach with effective quantities for two-dimensional horizontal flow equations obtained from reconstructed hydrostatic vertical pressure and saturation distributions. Here, we derive explicit expressions for the two dimensional constitutive relationships for a play-type hysteretic Brooks---Corey capillary pressure function with a pore-size distribution index of 2 and quadratic relative permeabilities. We obtain an explicit hysteretic parametrization for the upscaled capillary pressure function and the upscaled relative permeabilities. The size of the hysteresis loop depends on the ratio between buoyancy and the entry pressure, i.e. it scales with the reservoir height and the ratio between drainage and imbibition capillary pressure. We find that the scaling for the relative permeability is non-monotonic and hysteresis vanishes for both small and large reservoirs.

Accurate coarse modeling of well-driven, high-mobility-ratio displacements in heterogeneous reservoirs

High mobility ratios are often encountered in improved-oil-recovery processes because the displaced oil can be much more viscous than the injected water or gas. In this work, a new two-phase upscaling approach for modeling high-mobility-ratio displacements is developed and applied. For the near-well region, a specialized upscaling procedure, which includes single-phase and two-phase upscaling components, is presented. In this upscaling, local regions around each well are considered and the coarse-scale well indices, wellblock transmissibilities, and relative permeabilities are determined such that the fine- and coarse-scale flow rates are in agreement. Away from wells, the upscaled relative permeabilities for each coarse block are computed by imposing effective flux boundary conditions, which have been shown to provide better accuracy than standard procedures. The performance of these techniques is demonstrated by considering multiple realizations of synthetic 3D models with varying correlation structures and degrees of spatial variability, as well as different fluid mobility contrasts and production scenarios (involving horizontal wells and five-spot patterns). By quantifying the upscaling errors using well-defined metrics, the contribution of each component of the overall procedure to the accuracy of the coarse models is assessed. Application of the overall upscaling methodology is shown to provide significantly more accurate coarse models than those generated using standard procedures.

Upscaling relative permeabilities in a structured porous medium

Upscaling of multi-phase flow problems for a heterogeneous porous medium requires modification of constitutive functions at the grid-block scale. A particular type of heterogeneity that has important environmental consequences involves thin, continuous streaks of high permeability through lower-permeability background rocks. These streaks, which may correspond to features like abandoned wells in mature sedimentary basins, can become preferential flow paths for an invading fluid. Quantification of flow through these types of heterogeneities in deep, geological formations is necessary for estimates of migration and possible leakage of injected fluids such as hazardous liquid wastes, municipal liquid wastes, and, possibly, carbon dioxide. One of the important constitutive functions for proper estimation of flow through these flow paths is the relative permeability function. In the simple case of a single high-permeability streak in a uniform rock matrix, with both materials having identical (local) relative permeability functions, the upscaled relative permeability must be changed significantly to capture the proper leakage. Standard petroleum reservoir pseudo-functions for relative permeability capture the general features of the upscaled function, but they still produce errors of several hundred percent in the leakage estimation. Detailed three-dimensional numerical simulations and associated upscaled calculations demonstrate the proper form for the upscaled relative permeability, and provide a modified derivation of pseudo-functions to capture the leakage behavior in upscaled models.

Application of Effective Flux Boundary Conditions to Two-Phase Upscaling in Porous Media

A new algorithm is introduced for upscaling relative permeabilities, and tested in simulations of two-dimensional reservoir displacement processes. The algorithm is similar to existing algorithms for computing upscaled relative permeabilities from subgrid simulations, but uses new boundary conditions for the pressure field. The new 'effective flux boundary conditions' were introduced in a previous paper and provide a more accurate estimate of flux through high permeability channels. The algorithm was tested in conjunction with uniform grid coarsening and upscaled absolute permeabilities for a broad range of coarsenings. The permeability fields were highly heteroge-neous and layered, and were obtained from synthetic data and from conditioned realizations of actual oil reservoirs. The algorithm was tested for a wide variety of grid aspect ratios, and for both viscous-and gravity-dominated flow. Typical fine grids were of the order of 100×100 cells; the coarsest scaled-up grids were on the order of 5×5 cells. The quality of scale up was evaluated by comparing oil cut curves for the fine and coarse grid simulations. We consistently obtained excellent agreement, even at the coarsest levels of scale up.

Upscaling, Gridding, and Simulating Using Streamtubes

Summary This paper presents an innovative upscaling methodology based on a semianalytical streamtube technique. The methodology generates a coarse grid based on streamtubes and isobars, whose up-scaled properties are accurately calculated using the properties of each streamtube that constitutes the coarse block. Beyond the calculation of upscaled static properties of the grid, the methodology also calculates upscaled relative permeability curves, or pseudofunctions, using a semianalytical streamtube method to perform this without requiring any significant additional time. Tests showed that the results from the streamtube coarse grid had a good agreement with the fine grid solution. In contrast, simulations with a coarse grid upscaled with conventional techniques had a bad performance in many situations. Although this upscaling methodology is heavily based on a fixed streamtube simulation method, it provided good results even for situations in which streamtube simulation techniques are not supposed to work well, such as displacements that consider gravity and compressibility.

Use of Higher Moments for the Description of Upscaled, Process Independent Relative Permeabilities

Summary Traditional approaches for the scale up of highly detailed reservoir descriptions to the coarser scales more appropriate for reservoir simulation often rely on the use of pseudo relative permeabilities. Two of the major limitations in the use of pseudo relative permeabilities are the process dependency inherent in the resulting curves and the need for a different set of curves in each coarse scale grid block. In this paper, these limitations are illustrated and quantified through numerical simulations of viscous dominated displacements in fine scale, heterogeneous, two dimensional systems. It is concluded that the functional form of traditional pseudo relative permeability descriptions is too limited to capture a wide variety of flow behavior and that an ‘extended description,’ in which the upscaled relative permeabilities depend on variables in addition to the phase saturations, is required. The additional variables we propose are higher moments of the fine scale variables, specifically the variance of saturation, the variance of pressure and the velocity-saturation covariance. It is then shown that these new variables are capable of accurately correlating upscaled relative permeabilities for a variety of displacements in a process independent manner; i.e., the extended description can account for varying inlet conditions and permeability realizations. The practical use of this methodology in an actual reservoir simulation will, however, require the derivation and solution of coarse scale evolution equations for the moments.