Upscaling facies models to preserve connectivity of designated facies

An upscaling algorithm has been developed that generates an irregular coarse grid that preserves flow connectivity by applying a rule-based upscaling algorithm to a fine-scale facies distribution. The algorithm is demonstrated using stochastically generated paleo-fluvial facies distributions. First, an irregular grid honoring the channel facies is created, followed by computation of effective anisotropic parameters for all coarse-grid cells. For the apparent layer-cake geometry of overbank deposits seen in outcrop, two local upscaling methods are compared: (1) the layered system approximation and (2) the mode. To assess upscaling performance, flow simulations for the original and upscaled grids are compared. The horizontal layered approximation (arithmetic mean) performs poorly, over-predicting lateral connectivity where even infrequent disconnection becomes important. Performance of the mode as an upscaling algorithm depends on the probability that a coarse-grid cell will be dominated by a single facies, and it performs surprisingly well because the upscaled grid-generation algorithm honors the channels, informing the upscaling process. Lastly, the irregular coarse grid was compared to a uniform coarse grid, showing superior performance with the irregular grid. The reduction in grid size achieved by irregular-grid generation will be a function of the geometrical complexity of the geologic objects to be honored.     

Mixed multiscale finite element methods using approximate global information based on partial upscaling

The use of limited global information in multiscale simulations is needed when there is no scale separation. Previous approaches entail fine-scale simulations in the computation of the global information. The computation of the global information is expensive. In this paper, we propose the use of approximate global information based on partial upscaling. A requirement for partial homogenization is to capture long-range (non-local) effects present in the fine-scale solution, while homogenizing some of the smallest scales. The local information at these smallest scales is captured in the computation of basis functions. Thus, the proposed approach allows us to avoid the computations at the scales that can be homogenized. This results in coarser problems for the computation of global fields. We analyze the convergence of the proposed method. Mathematical formalism is introduced, which allows estimating the errors due to small scales that are homogenized. The proposed method is applied to simulate two-phase flows in heterogeneous porous media. Numerical results are presented for various permeability fields, including those generated using two-point correlation functions and channelized permeability fields from the SPE Comparative Project (Christie and Blunt, SPE Reserv Evalu Eng 4:308–317, 2001). We consider simple cases where one can identify the scales that can be homogenized. For more general cases, we suggest the use of upscaling on the coarse grid with the size smaller than the target coarse grid where multiscale basis functions are constructed. This intermediate coarse grid renders a partially upscaled solution that contains essential non-local information. Numerical examples demonstrate that the use of approximate global information provides better accuracy than purely local multiscale methods.     

FV-MHMM method for reservoir modeling

The present paper proposes a new family of multiscale finite volume methods. These methods usually deal with a dual mesh resolution, where the pressure field is solved on a coarse mesh, while the saturation fields, which may have discontinuities, are solved on a finer reservoir grid, on which petrophysical heterogeneities are defined. Unfortunately, the efficiency of dual mesh methods is strongly related to the definition of up-gridding and down-gridding steps, allowing defining accurately pressure and saturation fields on both fine and coarse meshes and the ability of the approach to be parallelized. In the new dual mesh formulation we developed, the pressure is solved on a coarse grid using a new hybrid formulation of the parabolic problem. This type of multiscale method for pressure equation called multiscale hybrid-mixed method (MHMM) has been recently proposed for finite elements and mixed-finite element approach (Harder et al. 2013). We extend here the MH-mixed method to a finite volume discretization, in order to deal with large multiphase reservoir models. The pressure solution is obtained by solving a hybrid form of the pressure problem on the coarse mesh, for which unknowns are fluxes defined on the coarse mesh faces. Basis flux functions are defined through the resolution of a local finite volume problem, which accounts for local heterogeneity, whereas pressure continuity between cells is weakly imposed through flux basis functions, regarded as Lagrange multipliers. Such an approach is conservative both on the coarse and local scales and can be easily parallelized, which is an advantage compared to other existing finite volume multiscale approaches. It has also a high flexibility to refine the coarse discretization just by refinement of the lagrange multiplier space defined on the coarse faces without changing nor the coarse nor the fine meshes. This refinement can also be done adaptively w.r.t. a posteriori error estimators. The method is applied to single phase (well-testing) and multiphase flow in heterogeneous porous media.     

Accurate local upscaling with variable compact multipoint transmissibility calculations

We propose a new single-phase local upscaling method that uses spatially varying multipoint transmissibility calculations. The method is demonstrated on two-dimensional Cartesian and adaptive Cartesian grids. For each cell face in the coarse upscaled grid, we create a local fine grid region surrounding the face on which we solve two generic local flow problems. The multipoint stencils used to calculate the fluxes across coarse grid cell faces involve the six neighboring pressure values. They are required to honor the two generic flow problems. The remaining degrees of freedom are used to maximize compactness and to ensure that the flux approximation is as close as possible to being two-point. The resulting multipoint flux approximations are spatially varying (a subset of the six neighbors is adaptively chosen) and reduce to two-point expressions in cases without full-tensor anisotropy. Numerical tests show that the method significantly improves upscaling accuracy as compared to commonly used local methods and also compares favorably with a local–global upscaling method.     

Upscaling Hydraulic Conductivities in Cross-Bedded Formations

Modern geostatistical techniques allow the generation of high-resolution heterogeneous models of hydraulic conductivity containing millions to billions of cells. Selective upscaling is a numerical approach for the change of scale of fine-scale hydraulic conductivity models into coarser scale models that are suitable for numerical simulations of groundwater flow and mass transport. Selective upscaling uses an elastic gridding technique to selectively determine the geometry of the coarse grid by an iterative procedure. The geometry of the coarse grid is built so that the variances of flow velocities within the coarse blocks are minimum. Selective upscaling is able to handle complex geological formations and flow patterns, and provides full hydraulic conductivity tensor for each block. Selective upscaling is applied to a cross-bedded formation in which the fine-scale hydraulic conductivities are full tensors with principal directions not parallel to the statistical anisotropy of their spatial distribution. Mass transport results from three coarse-scale models constructed by different upscaling techniques are compared to the fine-scale results for different flow conditions. Selective upscaling provides coarse grids in which mass transport simulation is in good agreement with the fine-scale simulations, and consistently superior to simulations on traditional regular (equal-sized) grids or elastic grids built without accounting for flow velocities.     

Multiscale mixed/mimetic methods on corner-point grids

Multiscale simulation is a promising approach to facilitate direct simulation of large and complex grid models for highly heterogeneous petroleum reservoirs. Unlike traditional simulation, approaches based on upscaling/downscaling, multiscale methods seek to solve the full flow problem by incorporating subscale heterogeneities into local discrete approximation spaces. We consider a multiscale formulation based on a hierarchical grid approach, where basis functions with subgrid resolution are computed numerically to correctly and accurately account for subscale variations from an underlying (fine-scale) geomodel when solving the global flow equations on a coarse grid. By using multiscale basis functions to discretise the global flow equations on a (moderately sized) coarse grid, one can retain the efficiency of an upscaling method and, at the same time, produce detailed and conservative velocity fields on the underlying fine grid. For pressure equations, the multiscale mixed finite-element method (MsMFEM) has been shown to be a particularly versatile approach. In this paper, we extend the method to corner-point grids, which is the industry standard for modelling complex reservoir geology. To implement MsMFEM, one needs a discretisation method for solving local flow problems on the underlying fine grids. In principle, any stable and conservative method can be used. Here, we use a mimetic discretisation, which is a generalisation of mixed finite elements that gives a discrete inner product, allows for polyhedral elements, and can (easily) be extended to curved grid faces. The coarse grid can, in principle, be any partition of the subgrid, where each coarse block is a connected collection of subgrid cells. However, we argue that, when generating coarse grids, one should follow certain simple guidelines to achieve improved accuracy. We discuss partitioning in both index space and physical space and suggest simple processing techniques. The versatility and accuracy of the new multiscale mixed methodology is demonstrated on two corner-point models: a small Y-shaped sector model and a complex model of a layered sedimentary bed. A variety of coarse grids, both violating and obeying the above mentioned guidelines, are employed. The MsMFEM solutions are compared with a reference solution obtained by direct simulation on the subgrid.     

Numerical Homogenization of Two-Phase Flow in Porous Media

Homogenization has proved its effectiveness as a method of upscaling for linear problems, as they occur in single-phase porous media flow for arbitrary heterogeneous rocks. Here we extend the classical homogenization approach to nonlinear problems by considering incompressible, immiscible two-phase porous media flow. The extensions have been based on the principle of preservation of form, stating that the mathematical form of the fine-scale equations should be preserved as much as possible on the coarse scale. This principle leads to the required extensions, while making the physics underlying homogenization transparent. The method is process-independent in a way that coarse-scale results obtained for a particular reservoir can be used in any simulation, irrespective of the scenario that is simulated. Homogenization is based on steady-state flow equations with periodic boundary conditions for the capillary pressure. The resulting equations are solved numerically by two complementary finite element methods. This makes it possible to assess a posteriori error bounds.     

A system-theoretical approach to selective grid coarsening of reservoir models

From a system-theoretical point of view and for a given configuration of wells, there are only a limited number of degrees of freedom in the input–output dynamics of a reservoir system. This means that a large number of combinations of the state variables (pressure and saturation values) are not actually controllable and observable from the wells, and accordingly, they are not affecting the input–output behavior of the system. In an earlier publication, we therefore proposed a control-relevant upscaling methodology that uniformly coarsens the reservoir. Here, we present a control-relevant selective (i.e. non-uniform) coarsening (CRSC) method, in which the criterion for grid size adaptation is based on ranking the grid block contributions to the controllability and observability of the reservoir system. This multi-level CRSC method is attractive for use in iterative procedures such as computer-assisted flooding optimization for a given configuration of wells. In contrast to conventional flow-based coarsening techniques our method is independent of the specific flow rates or pressures imposed at the wells. Moreover the system-theoretical norms employed in our method provide tight upper bounds to the ‘input–output energy’ of the fine and coarse systems. These can be used as an a priori error-estimate of the performance of the coarse model. We applied our algorithm to two numerical examples and found that it can accurately reproduce results from the corresponding fine-scale simulations, while significantly speeding up the simulation.     

Grid adaptation for the Dirichlet–Neumann representation method and the multiscale mixed finite-element method

A Dirichlet–Neumann representation method was recently proposed for upscaling and simulating flow in reservoirs. The DNR method expresses coarse fluxes as linear functions of multiple pressure values along the boundary and at the center of each coarse block. The number of flux and pressure values at the boundary can be adjusted to improve the accuracy of simulation results and, in particular, to resolve important fine-scale details. Improvement over existing approaches is substantial especially for reservoirs that contain high-permeability streaks or channels. As an alternative, the multiscale mixed finite-element (MsMFE) method was designed to obtain fine-scale fluxes at the cost of solving a coarsened problem, but can also be used as upscaling methods that are flexible with respect to geometry and topology of the coarsened grid. Both methods can be expressed in mixed-hybrid form, with local stiffness matrices obtained as “inner products” of numerically computed basis functions with fine-scale sub-resolution. These basis functions are determined by solving local flow problems with piecewise linear Dirichlet boundary conditions for the DNR method and piecewise constant Neumann conditions for MsMFE. Adding discrete pressure points in the DNR method corresponds to subdividing faces in the coarse grid and hence increasing the number of basis functions in the MsMFE method. The methods show similar accuracy for 2D Cartesian cases, but the MsMFE method is more straightforward to formulate in 3D and implement for general grids.     

Fast 3D Reservoir Simulation and Scale Up Using Streamtubes

This paper presents an implementation of a semianalytical method for oil recovery calculation in heterogeneous reservoirs that is both fast and accurate. The method defines streamline paths based on a conventional single-phase incompressible flow calculation. By calculating the time-of-flight for a particle along a streamline and assigning a volumetric flux to each streamline, the cumulative pore volume of a streamtube containing the streamline can be calculated. Subsequently, the streamtube geometries are kept constant and the effects of the time varying mobility distribution in two-phase flow are accounted for by varying the flow rate in each streamtube, based on fluid resistance changes along the streamtube. Oil recovery calculations are then done based on the 1D analytical Buckley–Leverett solution. This concept makes the method extremely fast and easy to implement, making it ideal to simulate large reservoirs generated by geostatiscal methods. The simulation results of a 3D heterogeneous reservoir are presented and compared with those of other simulators. The results shows that the new simulator is much faster than a traditional finite difference simulator, while having the same accuracy. The method also naturally handles the upscaling of absolute and relative permeability. We make use of these upscaling abilities to generate a coarse curvilinear grid that can be used in conventional simulators with a great advantage over conventional upscaled Cartesian grids. This paper also shows an upscaling example using this technique.     

油藏精细地质模型网格粗化算法及其效果

在前人研究基础上, 根据DP(Dykstra-Parsons)系数能定量评价储层非均质性, 微网格块的渗透率值粗化后, 其等效渗透率的上、下限(C_min、C_max)能反映渗透率的各向异性的特点, 提出了一种运算速度快和相对有效的网格粗化算法。该算法能考虑到储层非均质性对不同方向渗透率值的影响, 且求解过程相对简单。应用该方法对鄂尔多斯盆地中部某油藏的陆相储层的精细地质模型进行了网格粗化计算, 然后在粗化后的模型上进行油藏数值模拟研究, 同时针对研究区地质背景和产出流体微可压缩的物性特征, 首次利用流线模拟器对精细地质模型进行了油藏数值模拟研究, 并以此结果为标准, 对该网格粗化算法时效性进行了系统评价。分析表明, 该算法具有较快的计算速度和较高的可靠性, 是解决储层非均质强、物性差的陆相成因油藏精细油藏数值模拟的一种行之有效的手段。      

A comparison of multiscale methods for elliptic problems in porous media flow

We review and perform comparison studies for three recent multiscale methods for solving elliptic problems in porous media flow; the multiscale mixed finite-element method, the numerical subgrid upscaling method, and the multiscale finite-volume method. These methods are based on a hierarchical strategy, where the global flow equations are solved on a coarsened mesh only. However, for each method, the discrete formulation of the partial differential equations on the coarse mesh is designed in a particular fashion to account for the impact of heterogeneous subgrid structures of the porous medium. The three multiscale methods produce solutions that are mass conservative on the underlying fine mesh. The methods may therefore be viewed as efficient, approximate fine-scale solvers, i.e., as an inexpensive alternative to solving the elliptic problem on the fine mesh. In addition, the methods may be utilized as an alternative to upscaling, as they generate mass-conservative solutions on the coarse mesh. We therefore choose to also compare the multiscale methods with a state-of-the-art upscaling method – the adaptive local–global upscaling method, which may be viewed as a multiscale method when coupled with a mass-conservative downscaling procedure. We investigate the properties of all four methods through a series of numerical experiments designed to reveal differences with regard to accuracy and robustness. The numerical experiments reveal particular problems with some of the methods, and these will be discussed in detail along with possible solutions. Next, we comment on implementational aspects and perform a simple analysis and comparison of the computational costs associated with each of the methods. Finally, we apply the three multiscale methods to a dynamic two-phase flow case and demonstrate that high efficiency and accurate results can be obtained when the subgrid computations are made part of a preprocessing step and not updated, or updated infrequently, throughout the simulation. The research is funded by the Research Council of Norway under grant nos. 152732 and 158908.     

Nearly perfectly matched layer boundary conditions for operator upscaling of the acoustic wave equation

Acoustic imaging and sensor modeling are processes that require repeated solution of the acoustic wave equation. Solution of the wave equation can be computationally expensive and memory intensive for large simulation domains. One scheme for speeding up solution of the wave equation is the operator-based upscaling method. The algorithm proceeds in two steps. First, the wave equation is solved for fine grid unknowns internal to coarse blocks assuming the coarse blocks do not need to communicate with neighboring blocks in parallel. Second, these fine grid solutions are used to form a new problem which is solved on the coarse grid. Accurate and efficient wave propagation schemes also must avoid artificial reflections off of the computational domain edges. One popular method for preventing artificial reflections is the nearly perfectly matched layer (NPML) method. In this paper, we discuss applying NPML to operator upscaling for the wave equation. We show that although we only apply NPML to the first step of this two step algorithm (directly affecting the fine grid unknowns only), we still see a significant reduction of reflections back into the domain. We describe three numerical experiments (one homogeneous medium experiment and two heterogeneous media examples) in which we validate that the solution of the wave equation exponentially decays in the NPML regions. Numerical experiments of acoustic wave propagation in two dimensions with a reasonable absorbing layer thickness resulted in a maximum pressure reflection of 3–8%. While the coarse grid acceleration is not explicitly damped in our algorithm, the tight coupling between the two steps of the algorithm results in only 0.1–1% of acceleration reflecting back into the computational domain.     

Integration of local–global upscaling and grid adaptivity for simulation of subsurface flow in heterogeneous formations

We propose a methodology, called multilevel local–global (MLLG) upscaling, for generating accurate upscaled models of permeabilities or transmissibilities for flow simulation on adapted grids in heterogeneous subsurface formations. The method generates an initial adapted grid based on the given fine-scale reservoir heterogeneity and potential flow paths. It then applies local–global (LG) upscaling for permeability or transmissibility [7], along with adaptivity, in an iterative manner. In each iteration of MLLG, the grid can be adapted where needed to reduce flow solver and upscaling errors. The adaptivity is controlled with a flow-based indicator. The iterative process is continued until consistency between the global solve on the adapted grid and the local solves is obtained. While each application of LG upscaling is also an iterative process, this inner iteration generally takes only one or two iterations to converge. Furthermore, the number of outer iterations is bounded above, and hence, the computational costs of this approach are low. We design a new flow-based weighting of transmissibility values in LG upscaling that significantly improves the accuracy of LG and MLLG over traditional local transmissibility calculations. For highly heterogeneous (e.g., channelized) systems, the integration of grid adaptivity and LG upscaling is shown to consistently provide more accurate coarse-scale models for global flow, relative to reference fine-scale results, than do existing upscaling techniques applied to uniform grids of similar densities. Another attractive property of the integration of upscaling and adaptivity is that process dependency is strongly reduced, that is, the approach computes accurate global flow results also for flows driven by boundary conditions different from the generic boundary conditions used to compute the upscaled parameters. The method is demonstrated on Cartesian cell-based anisotropic refinement (CCAR) grids, but it can be applied to other adaptation strategies for structured grids and extended to unstructured grids.     

Equivalent Permeability and Simulation of Two-Phase Flow in Heterogeneous Porous Media

The problem of calculating equivalent grid block permeability tensors for heterogeneous porous media is addressed. The homogenization method used involves solving Darcy's equation subject to linear boundary conditions with flux conservation in subregions of the reservoir and can be readily applied to unstructured grids. The resulting equivalent permeability tensor is stable as defined relative to G-convergence. It is proposed to use both conforming and mixed finite elements to solve the local problems and compute approximations from above and below of the equivalent permeability, respectively. Comparisons with results obtained using periodic, pressure and no-flux boundary conditions and the renormalization method are presented. A series of numerical examples demonstrates the effectiveness of the methodology for two-phase flow in heterogeneous reservoirs.     

A coarse-scale compositional model

In subsurface flow modeling, compositional simulation is often required to model complex recovery processes, such as gas/CO ₂ injection. However, compositional simulation on fine-scale geological models is still computationally expensive and even prohibitive. Most existing upscaling techniques focus on black-oil models. In this paper, we present a general framework to upscale two-phase multicomponent flow in compositional simulation. Unlike previous studies, our approach explicitly considers the upscaling of flow and thermodynamics. In the flow part, we introduce a new set of upscaled flow functions that account for the effects of compressibility. This is often ignored in the upscaling of black-oil models. In the upscaling of thermodynamics, we show that the oil and gas phases within a coarse block are not at chemical equilibrium. This non-equilibrium behavior is modeled by upscaled thermodynamic functions, which measure the difference between component fugacities among the oil and gas phases. We apply the approach to various gas injection problems with different compositional features, permeability heterogeneity, and coarsening ratios. It is shown that the proposed method accurately reproduces the averaged fine-scale solutions, such as component overall compositions, gas saturation, and density solutions in the compositional flow.     

Equivalent Permeability and Simulation of Two-Phase Flow in Heterogeneous Porous Media

The problem of calculating equivalent grid block permeability tensors for heterogeneous porous media is addressed. The homogenization method used involves solving Darcy's equation subject to linear boundary conditions with flux conservation in subregions of the reservoir and can be readily applied to unstructured grids. The resulting equivalent permeability tensor is stable as defined relative to G-convergence. It is proposed to use both conforming and mixed finite elements to solve the local problems and compute approximations from above and below of the equivalent permeability, respectively. Comparisons with results obtained using periodic, pressure and no-flux boundary conditions and the renormalization method are presented. A series of numerical examples demonstrates the effectiveness of the methodology for two-phase flow in heterogeneous reservoirs.     

Using Multiple Grids in Markov Mesh Facies Modeling

A multigrid Markov mesh model for geological facies is formulated by defining a hierarchy of nested grids and defining a Markov mesh model for each of these grids. The facies probabilities in the Markov mesh models are formulated as generalized linear models that combine functions of the grid values in a sequential neighborhood. The parameters in the generalized linear model for each grid are estimated from the training image. During simulation, the coarse patterns are first laid out, and by simulating increasingly finer grids we are able to recreate patterns at different scales. The method is applied to several tests cases and results are compared to the training image and the results of a commercially available snesim algorithm. In each test case, simulation results are compared qualitatively by visual inspection, and quantitatively by using volume fractions, and an upscaled permeability tensor. When compared to the training image, the method produces results that only have a few percent deviation from the values of the training image. When compared with the snesim algorithm the results in general have the same quality. The largest computational cost in the multigrid Markov mesh is the estimation of model parameters from the training image. This is of comparable CPU time to that of creating one snesim realization. The simulation of one realization is typically ten times faster than the estimation.