Flow-based Upscaling Research Articles

Machine learning assisted two-phase upscaling for large-scale oil-water system

The computation of two-phase upscaled functions entails solving time-dependent flow and transport equations over target regions, which is usually the most time-demanding component in the overall two-phase upscaling procedure. For large-scale reservoir models with a great number of coarse grid blocks, it can be very computationally expensive to calculate the two-phase upscaled functions for each individual coarse block. To address this problem, we develop a machine learning assisted upscaling (MLAU) approach, in which the two-phase upscaling is only performed for representative coarse blocks selected by a convolutional neural network (CNN) based clustering model, while the two-phase upscaled functions are quickly predicted for the rest of the coarse blocks using a regression algorithm. The performance of MLAU approach was assessed with three cases involving Gaussian, channelized and SPE 10 sector models, respectively. Numerical results have shown that the MLAU approach consistently provides coarse-scale results with close agreement with the results using full flow-based upscaling. Because two-phase numerical upscaling is only applied for representative coarse blocks (about 5% in each case), the speedups relative to the full flow-based upscaling are significant, ranging from 6.2 to 13.5. Compared to the fine-scale simulations, the speedups range from 27.0 to 47.2.

A Hybrid Embedded Discrete Fracture Model and Dual-Porosity, Dual-Permeability Workflow for Hierarchical Treatment of Fractures in Practical Field Studies

Summary Natural fracture systems comprise numerous small features and relatively few large ones. At field scale, it is impractical to treat all fractures explicitly. We represent the largest fractures using an embedded discrete fracture model (EDFM) and account for smaller ones using a dual-porosity, dual-permeability (DPDK) idealized representation of the fracture network. The hybrid EDFM + DPDK approach uses consistent discretization schemes and efficiently simulates realistic field cases. Further speedup can be obtained using aggregation-based upscaling. Capabilities to visualize and post-process simulation results facilitate understanding for effective management of fractured reservoirs. The proposed approach embeds large discrete fractures as EDFM within a DPDK grid (which contains both matrix and idealized fracture continua for smaller fractures) and captures all connections among the triple media. In contrast with existing EDFM formulations, we account for discrete fracture spacing within each matrix cell via a new matrix-fracture transfer term and use consistent assumptions for classical EDFM and DPDK calculations. In addition, the workflow enables coarse EDFM representations using flow-based cell-aggregation upscaling for computational efficiency. Using a synthetic case, we show that the proposed EDFM + DPDK approach provides a close match of simulation results from a reference model that represents all fractures explicitly, while providing runtime speedup. It is also more accurate than previous standard EDFM and DPDK models. We demonstrate that the matrix-fracture transfer function agrees with flow-based upscaling of high-resolution fracture models. Next, the automated workflow is applied to a waterflooding study for a giant carbonate reservoir, with an ensemble of stochastic fracture realizations. The overall workflow provides the computational efficiency needed for performance forecasts in practical field studies, and the 3D visualization allows for the derivation of insights into recovery mechanisms. Finally, we apply a finite-volume tracer-based flux post-processing scheme on simulation results to analyze production allocation and sweep for understanding expected waterflood performance.

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.

Accurate geothermal and chemical dissolution simulation using adaptive mesh refinement on generic unstructured grids

A coupled description of flow and thermal-reactive transport is spanning a wide range of scales in space and time, which often introduces a significant complexity for the modelling of such processes. Subsurface reservoir heterogeneity with complex multi-scale features increases the modelling complexity even further. Traditional multiscale techniques are usually focused on the accuracy of the pressure solution and often ignore the transport. Improving the transport solution can however be quite significant for the performance of the simulation, especially in complex applications related to thermal-compositional flow. The use of an Adaptive Mesh Refinement enables the grid to adapt dynamically during the simulation, which facilitates the efficient use of computational resources. This is especially important in applications with thermal flow and transport where the region requires high-resolution calculations as often localized in space. In this work, the aim is to develop an Adaptive Mesh Refinement framework for geothermal reservoir simulation. The approach uses a multi-level connection list and can be applied to fully unstructured grids. The adaptivity of the grid in the developed framework is based on a hierarchical connectivity list. First, the fine-scale model is constructed, which accurately approximates all reservoir heterogeneity. Next, a global flow-based upscaling is applied, where an unstructured partitioning of the original grid is created. Once the full hierarchy of levels is constructed, the simulation is started at the coarsest grid. Grid space refinement criteria is based on the local changes and can be adjusted for specific models and governing physics. The multi-level connectivity lists are redefined at each timestep and used as an input for the next. The developed Adaptive Mesh Refinement framework was implemented in Delft Advanced Research Terra Simulator which uses the Operator-Based Linearization technique. The performance of the proposed approach is illustrated for several challenging geothermal applications of practical interest.

Topological analysis of fracture networks integrated with flow simulation models for equivalent fracture permeability estimation

Models that predict permeability by considering analogues of fractured media have been used to enhance the understanding of the relative contribution of fracture networks to reservoir quality. The analogue data can overcome scale limitations (sub-seismic to mm) and/or lack of consistency in subsurface datasets (e.g., seismic data, well logs, and cores). This paper presents a numerical tool (TopoScan), designed to process images of fractured systems based on topological methods to determine fracture attributes and automatically make these available for fracture modeling and numerical simulations. We use this program to carry out fracture modeling considering different topological patterns of two-dimensional synthetic fracture networks, with variable patterns of interconnection, aperture, and spatial organization. We used the finite element method for modeling the fracture system, with the incorporation of discontinuities for the representation of fractures in the medium. The models were used for numerical simulations of fluid flow (flow-based upscaling), and the results demonstrated that the influence of fractures interconnection degree (topological patterns) on the equivalent permeability of the fractured medium.

Inference of Global Reservoir Connectivity from Static Pressure Data with Fast Coarse-Scale Simulation Models

Characterization of field-scale reservoir connectivity is critical for production optimization and field development planning. The information content of the data corresponding to different physical processes, coupled with the discrepancy in production data and geologic model resolutions, complicate the estimation of global field connectivity. Due to its diffusive nature, pressure varies smoothly in the reservoir, a characteristic that can be adequately captured using coarse-scale description of the connectivity in the reservoir model. On the other hand, flowrate and watercut information are relatively more localized and require higher-resolution models to adequately describe their variations. Hence, pressure and flowrate data carry information pertinent to their respective scales. In this paper, a fast workflow is presented for estimating large-scale reservoir connectivity from static reservoir pressure measurements. For this purpose, a coarse-scale grid system with Delaunay triangulation is developed to represent the global reservoir connectivity. Flow-based upscaling is applied to create the initial coarse-scale static simulation models from geological or simulation models. The ensemble Kalman filter is then applied to calibrate the initial models against static pressure data and to identify field-scale connectivity parameters that control the global pressure distribution, such as aquifer strength, continuity/discontinuity in reservoir properties, and fault transmissibilities. This approach enables fast characterization of field-scale reservoir connectivity from pressure data with a coarser description of the reservoir connectivity, which offers a parameterization level that is commensurate with the resolution and information content of the static pressure measurements. The proposed calibration approach can serve as preconditioning to generate initial reservoir models with consistent global connectivity patterns for full-scale history matching. To illustrate the feasibility and performance of the method, examples are drawn from real field cases in which static pressure data are integrated to infer global reservoir connectivity maps. In this paper, the resulting connectivity maps are used to obtain facies probability maps to generate initial facies models for full-scale history matching.

An Improved Workflow for Permeability Estimation from Image Logs with Uncertainty Quantification

Production forecasts are extremely sensitive to the spatial distribution of certain reservoir parameters, especially permeability, which controls fluid flow and efficiency of many recovery methods. Modeling the heterogeneity in permeability distribution is challenging due to various reasons. First, proper sampling is difficult due to biased sampling, dilation effects and sample disturbance. Second, calibration of core-plug measurements with well test data is problematic because of their vast differences in scale; in the case of heavy oil reservoirs, conventional well tests are not feasible due to bitumen’s ultra-low mobility at initial conditions. Therefore, in addition to flow-based measurements, alternative static data including image logs and core photographs, which capture detailed high-resolution information regarding heterogeneities at the millimeter scale, may be integrated for reliable permeability estimation along the wellbore. This paper examines the existing technique for permeability prediction from image logs (or core photographs) and proposes a set of improvements. For example, automatic variogram modeling is performed over a set of moving windows to simulate the sand–shale indicators along the well trajectory, capturing the locally varying anisotropy in variogram model parameters. In addition, it presents a unified approach to properly quantify the uncertainty in the histogram of core measurements and well log data: Joint histogram uncertainty considering the correlations between variables is incorporated into the existing methodology. The millimeter-scale model is constructed and upscaled to the centimeter scale via steady-state flow-based upscaling. This step is repeated with the centimeter-scale model until the final flow-simulation scale is attained. The entire methodology is applied within individual facies with distinct spatial correlations. A field case study involving data from the McMurray Formation (Alberta, Canada) is presented, where the permeability distributions in five different facies are modeled. The results indicate that considering automatic variogram modeling may lead to more realistic representation of the orientation of certain heterogeneous features that are strongly influencing the permeability to flow; furthermore, incorporating histogram uncertainty enables the modeling of porosity and permeability with improved uncertainty.

Numerical investigation of fracture-rock matrix ensemble saturation functions and their dependence on wettability

Quantification of multiphase flow in fractured permeable rocks is vital for hydrocarbon recovery, gas storage, and water resource management. Where fractures provide the permeability and the rock matrix provides the storage for oil and gas, fracture-matrix transfer has a decisive impact on recovery and capillary-driven transfer. However, while most modelling efforts assume a constant fracture aperture and fracture capillary pressure, in reality, fracture aperture and capillary pressure, pcf vary among fractures. Here we contrast and compare relative permeability curves obtained from constant aperture or constant capillary pressure, discrete fracture and matrix (DFM) models with more realistic ones taking variations into account. These models are constructed from line drawings of pervasively fractured layered rock mapped in meter-to kilometer-scale outcrops. Fracture aperture is obtained by mechanical modelling. Capillary pressure is computed from fracture aperture taking into account the wettability of the rock matrix.An unsteady state approach is applied to the analysis of fracture-matrix ensemble relative permeability and ultimate recovery. Imbibition simulations are performed with the Finite Element-Centered Finite Volume Method (FECFVM). Globally flow-based upscaling is used to establish ensemble relative permeability curves between capillary and viscous limits.The obtained fracture-matrix ensemble relative permeability curves show that ultimate recovery is 2–3 times higher in the water-wet than in the oil-wet case. With increasing wetting angle, counter-current-imbibition (CCI) rate decreases because the capillary pressure difference between small fractures that contribute the most to the fracture-matrix transfer area is very small. As the rock becomes oil wet, ensemble relative permeability curves steepen and matrix imbibition proceeds only over a very narrow saturation range.

Well-data-based discrete fracture and matrix modelling and flow-based upscaling of multilayer carbonate reservoir horizons

Abstract Discrete fracture and matrix (DFM) homogenization, simultaneously capturing reservoir layers and contained fractures, is an alternative to discrete fracture network (DFN) upscaling. Here, the DFM approach was applied to a fractured carbonate reservoir. Honouring geostatistical data from well logs, near-well multilayer reservoir models were constructed and analysed. Fracture aperture variations were modelled with a new semi-analytical model including a special treatment of layer-restricted fractures. Important results concern both pre-processing of stochastically generated DFMs for finite-element meshing, and the ensemble permeability values obtained by numerical homogenization of single v. multilayer models, respectively. Upscaling by volume averaging of vertically stacked single-layer DFMs results only in a fraction of the equivalent horizontal permeability that is obtained by homogenization of the multilayer models. Inspection of the flow patterns shows that this discrepancy arises because many fractures contact each other at layer boundaries fostering cross-flow. This effect is further enhanced where fractures intersect multiple layers. Compared to earlier DFN models for this reservoir, the DFM-derived fracture and matrix ensemble permeabilities are up to four times higher, highlighting how important it is to include the rock matrix into equivalent permeability calculations.

Dynamic impact and flow-based upscaling of the estuarine point-bar stratigraphic architecture

Abstract The estuarine point-bar architecture – one of the main sub-depositional environments of tide-dominated shallow-marine and fluvial (SM&F) systems – constitutes one of the most shale-prone clastic reservoir architectures potentially posing risks to the production performance. A set of simulation-based sensitivity studies quantify the effects of a number of stratigraphic and reservoir engineering parameters on the dynamic connectivity of the confined estuarine point-bar architecture. Water injection recovery mechanism is utilized for polarizing the stratigraphic and reservoir engineering parameters in terms of their effects on hydrocarbon recovery. The point-bar shale-drape architecture and abandonment channel attributes (e.g., permeability, facies composition, and shale-drape coverage) dominate the dynamic reservoir connectivity, hence the recovery factor and water breakthrough time/profile. In terms of reservoir engineering parameters, well spacing and, to some degree, relative mobility ratio constitute the factors of importance for the investigated model parameter ranges. The dynamic effects of intra-facies permeability heterogeneity are evaluated using a number of realizations comparing them to those stemming from facieswise-homogeneous realizations. Results indicate that intra-facies permeability heterogeneity influences the connectivity factor for low shale-drape coverage levels. For high coverage levels, presence of shale drapes almost completely overshadows the influences of intra-facies permeability heterogeneity. A novel multiphase flow-based upscaling method, specifically developed for clastic reservoirs containing long-correlation-length permeability heterogeneity, is evaluated first-time on the scale-up of the estuarine point-bar architecture. Resulting coarse-scale models capture the fine-scale dynamic behavior with an accuracy level that is significantly higher than those obtained by use of single-phase flow-based upscaling.

Unstructured Cut-Cell Grids for Modeling Complex Reservoirs

SummaryEffective workflows for translating Earth models into simulation models require grids that preserve geologic accuracy, offer flexible resolution control, integrate tightly with upscaling, and can be generated easily. Corner-point grids and pillar-based unstructured grids fail to satisfy these objectives; hence, a truly 3D unstructured approach is required. This paper describes unstructured cut-cell gridding tools that address these needs and improve the integration of our overall reservoir-modeling workflows.The construction of simulation grids begins with the geologic model: a numerical representation of the reservoir structure, stratigraphy, and properties. Our gridding uses a geochronological (GeoChron) map from physical coordinates to an unfaulted and unfolded depositional coordinate system. The mapping is represented implicitly on a tetrahedral mesh that conforms to faults, and it facilitates accurate geostatistical modeling of static depositional properties. In the simplest use case, we create an explicit representation of the geologic model as an unstructured polyhedral grid. Away from faults and other discontinuities, the cells are hexahedral, highly orthogonal, and arranged in a structured manner. Geometric cutting operations create general polyhedra adjacent to faults and explicit contact polygons across faults. The conversion of implicit models to explicit grids is conceptually straightforward, but the implementation is nontrivial because of the limitations of finite precision arithmetic and the need to remove small cells formed in the cutting process.In practice, simulation grids are often constructed at coarser resolutions than Earth models. Our implementation of local grid coarsening and refinement exploits the flexibility of unstructured grids to minimize upscaling errors and to preserve critical geologic features. Because the simulation grid and the geologic model are constructed by use of the same mapping, fine cells can be nested exactly inside coarse cells. Therefore, flow-based upscaling can be applied efficiently without resampling onto temporary local grids.This paper describes algorithms and data structures for constructing, storing, and simulating cut-cell grids. Examples illustrate the accurate modeling of normal faults, y-faults, overturned layers, and complex stratigraphy. Flow results, including a field-sector model, show the suitability of cut-cell grids for simulation.

Convective mixing in formations with horizontal barriers

It has been shown that convective mixing in porous media flow is important for applications such as saltwater intrusion and geological storage of carbon dioxide. In the latter case, dissolution from the injected phase to the resident brine is assisted by convective mixing, which leads to enhanced storage security through reduced buoyancy. Here, we focus on the effect of horizontal barriers on the efficiency of convective mixing. Previous investigations of the effect of heterogeneity on mixing efficiency have focused on random permeability fields or barriers of small extent compared to the intrinsic finger wavelength. The effect of horizontal barriers of larger extent, such as mudstone inclusions or thin shale deposits, has not been given sufficient attention. We perform detailed numerical investigations to represent the continuous solution of this problem in semi-infinite domains with barriers arranged in a periodic manner. The results show that mass flux into the domain, which is a measure of the efficiency of redistribution of the solute, is inversely proportional to the barrier length and proportional to the horizontal and vertical aperture between the barriers, for the cases studied. The flow structure is complex, and it depends not only on the total area of barriers but also largely on the distribution of barriers. Therefore, neither simple analytical models nor simple upscaling methods that lack information about the flow paths, can be used to predict the behavior. However, we compute the effective vertical permeability by flow-based upscaling and show that it can be used to directly obtain a first-order approximation to the mass flux into the domain.

Calculation of Permeability Tensors for Unstructured Gridblocks

Summary Geostatistical models of reservoir properties are high resolution with many grid cells; it is impractical to use them directly in flow simulation because of computational costs. Upscaling techniques are applied to average small-scale permeability values to larger flow simulation blocks. In cases where unstructured grids are used or the geological features inside the gridblock are not aligned with the block geometry, symmetric or full-permeability tensors arise instead of a simple diagonal tensor. A method is presented to calculate the permeability tensor for an unstructured gridblock to account for the small-scale heterogeneity information inside the gridblock. Single-phase flow-based upscaling is performed based on random boundary conditions and the effective tensor permeability is found through an optimization technique. Full, symmetric and diagonal permeability tensors are calculated for 2D and 3D blocks and sensitivity analysis is performed.