A Comparison Study Between an Adaptive Quadtree Grid and Uniform Grid Upscaling for Reservoir Simulation

An adaptive quadtree grid generation algorithm is developed and applied for tracer and multiphase flow in channelized heterogeneous porous media. Adaptivity was guided using two different approaches. In the first approach, wavelet transformation was used to generate a refinement field based on permeability variations. The second approach uses flow information based on the solution of an initial-time fine-scale problem. The resulting grids were compared with uniform grid upscaling. For uniform upscaling, two commonly applied methods were used: renormalization upscaling and local-global upscaling. The velocities obtained by adaptive grid and uniformly upscaled grids, were downscaled. This procedure allows us to separate the upscaling errors, on adaptive and uniform grids, from the numerical dispersion errors resulting from solving the saturation equation on a coarse grid. The simulation results obtained by solving on flow-based adaptive quadtree grids for the case of a single phase flow show reasonable agreement with more computationally demanding fine-scale models and local-global upscaled models. For the multiphase case, the agreement is less evident, especially in piston-like displacement cases with sharp frontal movement. In such cases a non-iterative transmissibility upscaling procedure for adaptive grid is shown to significantly reduce the errors and make the adaptive grid comparable to iterative local-global upscaling. Furthermore, existence of barriers in a porous medium complicates both upscaling and grid adaptivity. This issue is addressed by adapting the grid using a combination of flow information and a permeability based heuristic criterion.     

Gravitational forces in dual-porosity systems: I. Model derivation by homogenization

We consider the problem of modeling flow through naturally fractured porous media. In this type of media, various physical phenomena occur on disparate length scales, so it is difficult to properly average their effects. In particular, gravitational forces pose special problems. In this paper we develop a general understanding of how to incorporate gravitational forces into the dual-porosity concept. We accomplish this through the mathematical technique of formal two-scale homogenization. This technique enables us to average the single-porosity, Darcy equations that govern the flow on the finest (fracture thickness) scale. The resulting homogenized equations are of dual-porosity type. We consider three flow situations, the flow of a single component in a single phase, the flow of two fluid components in two completely immiscible phases, and the completely miscible flow of two components.This work was supported in part by the National Science Foundation and by the State of Texas Governor's Energy Office.     

A Sedimentological Approach to Upscaling

Optimised upscaling in reservoir simulations requires the construction of realistic petrophysical properties that are representative of the heterogeneity in the sedimentary deposits. Reservoir heterogeneities are controlled by the arrangement of various hierarchies of sedimentary facies and their internal bounding surfaces. The conventional sedimentological approach to reservoir upscaling involves subdivision and ranking of various hierarchies of architectural units and associated bounding surfaces of the reservoir sequence according to their geological significance. This global upscaling approach produces realistic scaled up models that retain both the structural and non-structural heterogeneities of the original sedimentological models. Analyses of sedimentary sequences from various depositional environments indicate that the fractional Levy model can adequately describe the heterogeneity and scaling characteristics of individual genetic sediment sequences in the clastic sedimentary system without further subdividing and ranking of the heterogeneous sequences. The heterogeneous nature of each sedimentary system can be quantified by the Levy index parameter, whereas the maximum upscaling magnitude (or upscaling index) for a particular sequence can be determined from the Levy width parameter plot. Depositional modelling mimics the sedimentary processes in a range of scales and honours hierarchies of sedimentary facies and their bounding surfaces. It can be used effectively for upgridding and upscaling in accordance with the stratigraphic framework and sedimentological models. Both the fractional Levy model and the depositional modelling provide quantitative alternatives to the conventional global sedimentological upscaling approach.     

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.     

Gravitational forces in dual-porosity systems: II. Computational validation of the homogenized model

Three models are considered for single component, single phase flow in naturally fractured porous media. The microscopic model holds on the Darcy scale, and it is considered to govern the system. The macroscopic, dual-porosity model was derived in Part I of this work from the microscopic model by two-scale mathematical homogenization. In this paper, we show that the dual-porosity model predicts well the behavior of the microscopic model by comparing their computed solutions in certain reasonable test cases. Homogenization gives a complex formula for a key parameter in the dual-porosity model; herein a simple approximation to this formula is presented. The third model considered is a single-porosity model with averaged parameters. It is shown that this type of model cannot predict the behavior of the microscopic flow.This work was supported in part by the National Science Foundation and by the State of Texas Governor's Energy Office.     

三维非均匀不稳定渗流方程的二维不均匀粗化算法

在无源汇条件下,根据流过某一个横截面的流体流量等于流过这一横截面内所有精细网格的流体流量之和这一特点提出了粗化网格等效渗透率的计算方法。在粗化区内,利用直接解法求解二维渗流方程,再用这些解合成粗化网格的三维合成解,并由合成解计算粗化网格的等效渗透率。根据精度的要求采用了不均匀网格粗化,在流体流速大的区域采用精细网格。利用所得等效渗透率计算了粗化网格的某三维非均匀不稳定渗流场的压降解,结果表明三维非均匀不稳定渗流方程的二维不均匀粗化解非常逼近采用精细网格的解,但计算的速度比采用精细网格提高了80倍。     

Numerical Homogenization of the Absolute Permeability Using the Conformal-Nodal and Mixed-Hybrid Finite Element Method

The modeling of hydrocarbon reservoirs and of aquifer-aquitard systems can be separated into two activities: geological modeling and fluid flow modeling. The geological model focuses on the geometry and the dimensions of the subsurface layers and faults, and on its rock types. The fluid flow model focuses on quantities like pressure, flux and dissipation, which are related to each other by rock parameters like permeability, storage coefficient, porosity and capillary pressure. The absolute permeability, which is the relevant parameter for steady single-phase flow of a fluid with constant viscosity and density, is studied here. When trying to match the geological model with the fluid flow model, it generally turns out that the spatial scale of the fluid flow model is built from units that are at least a hundred times larger in volume than the units of the geological model. To counter this mismatch in scales, the fine-scale permeabilities of the geological data model have to be upscaled' to coarse-scale permeabilities that relate the spatially averaged pressure, flux and dissipation to each other. The upscaled permeabilities may be considered as complicated averages, which are derived from the spatially averaged flow quantities in such a way that the continuity equation, Darcy's law and the dissipation equation remain valid on the coarse scale. In this paper the theory of upscaling will be presented from a physical point of view aiming at understanding, rather than mathematical rigorousness. Under the simplifying assumption of spatial periodicity of the fine-scale permeability distributions, homogenization theory can be applied. However, even then the spatial distribution of the permeability is generally so intricate that exact solutions of the homogenized permeability cannot be found. Therefore, numerical approximation methods have to be applied. To be able to estimate the approximation error, two numerical methods have been developed: one based on the conventional nodal finite element method (CN-FEM) and the other based on the mixed-hybrid finite element method (MH-FEM). CN-FEM gives an upper bound for the sum of the diagonal components of the homogenized mobility matrix, while MH-FEM gives a lower bound. Three numerical examples are presented.     

A Steady-State Upscaling Approach for Immiscible Two-Phase Flow

The paper presents a model for computing rate-dependent effective capillary pressure and relative permeabilities for two-phase flow, in 2 and 3 space-dimensions. The model is based on solving the equations for immiscible two-phase flow at steady-state, accounting for viscous and capillary forces, at a given external pressure drop. The computational performance of the steady-state model and its accuracy is evaluated through comparison with a commercial simulator ECLIPSE. The properties of the rate-dependent effective relative permeabilities are studied by way of computations using the developed steady-state model. Examples presented show the dependence of the effective relative permeabilities and capillary pressures, which incorporate the effects of fine scale wettability heterogeneity, on the external pressure drop, and thereby on the dimensionless macro-scale capillary number. The effective relative permeabilities converge towards the viscous limit functions as the capillary number tends to infinity. Special cases, when the effective relative permeabilities are rate-invariant, are also studied. The applicability of the steady-state upscaling algorithm in dynamic displacement situations is validated by comparing fine-gridded simulations in heterogeneous reservoirs against their homogenized counterparts. It is concluded that the steady-state upscaling method is able to accurately predict the dynamic behavior of a heterogeneous reservoir, including small scale heterogeneities in both the absolute permeability and the wettability.     

Immiscible Displacement of a Wetting Fluid by a Non-wetting One at High Capillary Number in a Micro-model Containing a Single Fracture

Most reservoirs in Iran are heterogeneous fractured carbonate reservoirs. Heterogeneity causes an earlier breakthrough and an unstable front which leads to a lower recovery. A series of experiments were conducted whereby the distilled water displaced n-Decane in strongly oil-wet glass micro-models containing a single fracture. Experimental data from image analysis of immiscible displacement processes are used to modify the Buckley?CLeverett and fractional flow equations by a heterogeneity factor. It is shown that the heterogeneity factor in the modified equations can be expressed as a function of fracture length and orientation.     

Generation of Voronoi Grid Based on Vorticity for Coarse-Scale Modeling of Flow in Heterogeneous Formations

We present a novel unstructured coarse grid generation technique based on vorticity for upscaling two-phase flow in permeable media. In the technique, the fineness of the gridblocks throughout the domain is determined by vorticity distribution such that where the larger is the vorticity at a region, the finer are the gridblocks at that region. Vorticity is obtained from single-phase flow on original fine grid, and is utilized to generate a background grid which stores spacing parameter, and is used to steer generation of triangular and finally Voronoi grids. This technique is applied to two channelized and heterogeneous models and two-phase flow simulations are performed on the generated coarse grids and, the results are compared with the ones of fine scale grid and uniformly gridded coarse models. The results show a close match of unstructured coarse grid flow results with those of fine grid, and substantial accuracy compared to uniformly gridded coarse grid model.     

Numerical Simulation of Single-Phase and Multiphase Non-Darcy Flow in Porous and Fractured Reservoirs

A numerical method as well as a theoretical study of non-Darcy fluid flow through porous and fractured reservoirs is described. The non-Darcy behavior is handled in a three-dimensional, multiphase flow reservoir simulator, while the model formulation incorporates the Forchheimer equation for describing single-phase or multiphase non-Darcy flow and displacement. The non-Darcy flow through a fractured reservoir is handled using a general dual-continuum approach. The numerical scheme has been verified by comparing its results against those of analytical methods. Numerical solutions are used to obtain some insight into the physics of non-Darcy flow and displacement in reservoirs. In addition, several type curves are provided for well-test analyses of non-Darcy flow to demonstrate a methodology for modeling this type of flow in porous and fractured rocks, including flow in petroleum and geothermal reservoirs.     

Accurate Representation of Near-well Effects in Coarse-Scale Models of Primary Oil Production

Near-well effects can have a strong impact on many subsurface flow processes. In oil production, because dissolved gas is released from the oil phase when the pressure falls below the bubble point, the detailed pressure field in the immediate vicinity of a production well strongly impacts gas (and thus oil) production. This effect is complicated by the interplay of fine-scale heterogeneity and two-phase flow physics, and can be difficult to capture in coarse-grid simulations. In this article, we develop and apply a new upscaling (coarse-graining) procedure to capture such near-well subgrid effects in coarse-scale flow simulation models. The method entails the use of preprocessing computations over near-well domains [referred to as local well models (LWM)] for the determination of upscaled single-phase and two-phase near-well parameters. These parameters are computed by minimizing the mismatch between fine and coarse-scale flows over the LWM. Minimization is accomplished using a gradient-based optimization procedure, with gradients calculated through solution of adjoint equations. The boundary conditions applied on the LWM can impact the upscaled parameters, but these boundary conditions depend on the global flow and are not, therefore, known a priori. In order to circumvent this difficulty, an adaptive local–global procedure is applied. This entails performing a global coarse-scale simulation with initial estimates for well-block parameters. The resulting pressure and saturation fields are then used to define local boundary conditions for the near-well computations. The overall procedure is applied to several example problems and is shown to provide results in close agreement with reference fine-scale computations. Significant improvement in accuracy over existing near-well upscaling treatments is demonstrated, particularly for a heavy oil case with oil viscosity of ~10⁴ cp.     

Dimensionally Scaled Miscible Displacements in Heterogeneous Permeable Media

Understanding and predicting the performance of solvent drives and remediation of contaminated aquifers in heterogeneous reservoirs is of great importance to the petroleum and environmental industries. In this paper, a general method to scale flow through heterogeneous reservoirs is presented for a miscible displacement of oil by a solvent. Results show that scaling miscible displacements in a two-dimensional, heterogeneous, anisotropic vertical cross section requires the matching of 13 dimensionless scaling groups. These groups were derived using a general procedure of inspectional analysis. A detailed numerical sensitivity study was performed to reveal the relationship between the scaling groups and the fractional oil recovery of miscible displacements in heterogeneous reservoirs. This relationship was then mapped using an artificial neural network, which can be used as a quick prediction tool for the fractional oil recovery for any combinations of the scaling groups, thus eliminating the need for the expensive fine-mesh simulations. These results have potential applications in modeling miscible displacements and in the scaling of laboratory displacements to field conditions.     

Large-scale averaging analysis of multiphase flow in fractured reservoirs

The method of large-scale averaging is applied to derive and analyze a dual-porosity model of multiphase flow in naturally fractured reservoirs. The dual-porosity model contains the usual equations based on Darcy's law, and the coupling terms representing the fluid transfer between the matrix and the fractures. Both quasisteady and transient closure schemes are considered to obtain and analyze the fracture and matrix permeability tensors and the fluid transfer terms. The techniques developed here are not restricted to regular geometric fractures. Computational work aimed at showing the implications of the theory behind the derivation of the present dual-porosity model is also described. In particular, comparisons among the dual-porosity model, the single porosity model, and other dual-porosity models are presented through numerical experiments.     

页岩气藏的双重介质-离散裂缝模型

考虑页岩气藏开发中渗流的多尺度效应,提出了一个基于裂缝-孔隙双重介质的离散裂缝模型．在该模型中,基质、天然裂缝和人工压裂裂缝采用各自控制方程独立计算,不同介质之间通过流量交换相互关联．为分析模型可靠性,分别和基于渗透率粗化及压裂裂缝导流能力无穷大的模型对比．数值算例显示,伴随着网格细分,该模型与精确渗透率粗化模型具有相同计算精度,两者收敛速度均较快,但该模型易推广到多相流动问题,而等压模型对产量将有所高估．研究了地质参数和工艺参数对气井产量的影响规律．计算结果表明天然裂缝渗透率及基质孔隙扩散系数对产气速率有着重要影响,产气速率伴随着人工压裂裂缝导流能力、长度以及数目的增加而增加,但是增加幅度会逐步趋缓．     

A unified approach to simulation and upscaling of single‐phase flow through vuggy carbonates

Numerical modeling of flow through vuggy porous media, mainly vuggy carbonates, is a challenging endeavor. Firstly, because the presence of vugs can significantly alter the effective porosity and permeability of the medium. Secondly, because of the co‐existence of porous and free flow regions within the medium and the uncertainties in defining the exact boundary between them. Traditionally, such heterogeneous systems are modeled by the coupled Darcy–Stokes equations. However, numerical modeling of flow through vuggy porous media using coupled Darcy–Stokes equations poses several numerical challenges particularly with respect to specification of correct interface condition between the porous and free‐flow regions. Hence, an alternative method, a more unified approach for modeling flows through vuggy porous media, the Stokes–Brinkman model, is proposed here. It is a single equation model with variable coefficients, which can be used for both porous and free‐flow regions. This also makes the requirement for interface condition redundant. Thus, there is an obvious benefit of using the Stokes–Brinkman equation, which can be reduced to Stokes or Darcy equation by the appropriate choice of parameters. At the same time, the Stokes–Brinkman equation provides a smooth transition between porous and free‐flow region, thereby taking care of the associated uncertainties. A numerical treatment for upscaling Stokes–Brinkman model is presented as an approach to use Stokes–Brinkman model for multi‐phase flow. Numerical upscaling methodology is validated by analyzing the error norm for numerical pressure convergence. Stokes–Brinkman single equation model is tested on a series of realistic data sets, and the results are compared with traditional coupled Darcy–Stokes model. Copyright © 2011 John Wiley & Sons, Ltd.     

Multiblock Pore-Scale Modeling and Upscaling of Reactive Transport: Application to Carbon Sequestration

In order to safely store CO₂ in depleted reservoirs and deep saline aquifers, a better understanding of the storage mechanisms of CO₂ is needed. Reaction of CO₂ with minerals to form precipitate in the subsurface helps to securely store CO₂ over geologic time periods, but a concern is the formation of localized channels through which CO₂ could travel at large, localized rates. Pore-scale network modeling is an attractive option for modeling and understanding this inherently pore-level process, but the relatively small domains of pore-scale network models may prevent accurate upscaling. Here, we develop a transient, single-phase, reactive pore-network model that includes reduction of throat conductivity as a result of precipitation. The novelty of this study is the implementation of a new mortar/transport method for coupling pore networks together at model interfaces that ensure continuity of pressures, species concentrations, and fluxes. The coupling allows for modeling at larger scales which may lead to more accurate upscaling approaches. Here, we couple pore-scale models with large variation in permeability and porosity which result in initial preferential pathways for flow. Our simulation results suggest that the preferential pathways close due to precipitation, but are not redirected at late times.