Statistical assignment of upscaled flow functions for an ensemble of geological models

Upscaled flow functions are often needed to account for the effects of fine-scale permeability heterogeneity in coarse-scale simulation models. We present procedures in which the required coarse-scale flow functions are statistically assigned to an ensemble of upscaled geological models. This can be viewed as an extension and further development of a recently developed ensemble level upscaling (EnLU) approach. The method aims to efficiently generate coarse-scale flow models capable of reproducing the ensemble statistics (e.g., cumulative distribution function) of fine-scale flow predictions for multiple reservoir models. The most expensive part of standard coarsening procedures is typically the generation of upscaled two-phase flow functions (e.g., relative permeabilities). EnLU provides a means for efficiently generating these upscaled functions using stochastic simulation. This involves the use of coarse-block attributes that are both fast to compute and correlate closely with the upscaled two-phase functions. In this paper, improved attributes for use in EnLU, namely the coefficient of variation of the fine-scale single-phase velocity field (computed during computation of upscaled absolute permeability) and the integral range of the fine-scale permeability variogram, are identified. Geostatistical simulation methods, which account for spatial correlations of the statistically generated upscaled functions, are also applied. The overall methodology thus enables the efficient generation of coarse-scale flow models. The procedure is tested on 3D well-driven flow problems with different permeability distributions and variable fluid mobility ratios. EnLU is shown to capture the ensemble statistics of fine-scale flow results (water and oil flow rates as a function of time) with similar accuracy to full flow-based upscaling methods but with computational speedups of more than an order of magnitude.     

Deformation band populations in fault damage zone—impact on fluid flow

Fault damage zones in highly porous reservoirs are dominated by deformation bands that generally have permeability-reducing properties. Due to an absence of sufficiently detailed measurements and the irregular distribution of deformation bands, a statistical approach is applied to study their influence on flow. A stochastic model of their distribution is constructed, and band density, distribution, orientation, and flow properties are chosen based on available field observations. The sensitivity of these different parameters on the upscaled flow is analyzed. The influence of a heterogeneous permeability distribution was also studied by assuming the presence of high permeability holes within bands. The fragmentation and position of these holes affect significantly the block-effective permeability. Results of local upscaling with a diagonal and full upscaled permeability tensor are compared, and qualitatively similar results for the flow characteristics are obtained. Further, the procedure of iterative local–global upscaling is applied to the problem.     

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.     

Relative permeability of CBM reservoirs: Controls on curve shape

Relative permeability to gas and water for 2-phase flow coalbed methane (CBM) reservoirs has long been known to exhibit a strong control on (gas and water) production profile characteristics. Despite its important control on both primary and enhanced recovery of CBM for coal seams that have not been fully dewatered, relative permeability in coal has received little attention in the literature in the past decade. There are few published laboratory-derived curves; these studies and their resulting data represent a small subset of the commercial CBM reservoirs and do not allow for a systematic investigation of the physical controls on relative permeability curve shape. Other methods for estimation of relative permeability curves include derivation from simulation history-matching, and production data analysis. Both of these methods will yield pseudo-relative permeability curves whose shapes could be affected by several dynamic CBM reservoir and operating characteristics.The purpose of the current work is to perform a systematic investigation of the controls on CBM relative permeability curve shape, including non-static fracture permeability and porosity, multi-layer effects and transient flow. To derive the relative permeability curves, effective permeability to gas and water are obtained from flow equations, flow rates and pressure data. Simulated cases are analyzed so that derived and input curves may be compared allowing for investigation of CBM reservoir properties on curve shape. One set of relative permeability curves that were input into the simulator were obtained from pore-scale modeling. Field cases from two basins are also examined and controls on derived relative permeability curve shape inferred. The results of this work should be useful for future CBM development and greenhouse gas sequestration studies, and it is hoped that it will spark additional research of this critical CBM flow property.     

Adaptive hierarchical upscaling of flow in heterogeneous reservoirs based on an a posteriori error estimate

This paper treats the upscaling of the absolute permeability in a heterogeneous reservoir. By replacing the fine scale permeability tensor with an upscaled, or effective permeability tensor, a modelling error is introduced. An a posteriori error estimate on this modelling error is formulated and tested. An implementation of the theory, based on domain decomposition coupled with a hierarchical representation of the absolute permeability field, is given. As hierarchical basis functions we have chosen the Haar system, which leads to a wavelet representation of the permeability. The wavelet representation offers a natural upscaling technique which resembles the highcut filters commonly used in signal analysis. This procedure represents an adaptive upscaling method. The numerical results show that this method conserves both the dissipation and the mean velocity in the problem fairly well. The a posteriori error estimate on the modelling error coupled with domain decomposition methods constitutes a powerful modelling tool.     

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.     

双河油田储层地质模型研究

在建立双河油田高精度的三维地质模型中,首先进行了基础地质研究,主要包括岩石相的识别、沉积微相的划分、基准面旋回的划分与对比。在详细地质研究基础上,以24个短期旋回为单位采用截断高斯模拟方法建立沉积微相三维模型,采用相控物性参数建模技术,利用顺序高斯模拟方法建立孔隙度、渗透率的三维模型。最后对模型进行了粗化,直接提供给油藏数模使用。     

Ensemble level upscaling for compositional flow simulation

Uncertainty quantification is typically accomplished by simulating multiple geological realizations, which can be very expensive computationally if the flow process is complicated and the models are highly resolved. Upscaling procedures can be applied to reduce computational demands, though it is essential that the resulting coarse-model predictions correspond to reference fine-scale solutions. In this work, we develop an ensemble level upscaling (EnLU) procedure for compositional systems, which enables the efficient generation of multiple coarse models for use in uncertainty quantification. We apply a newly developed global compositional upscaling method to provide coarse-scale parameters and functions for selected realizations. This global upscaling entails transmissibility and relative permeability upscaling, along with the computation of a-factors to capture component fluxes. Additional features include near-well upscaling for all coarse parameters and functions, and iteration on the a-factors, which is shown to improve accuracy. In the EnLU framework, this global upscaling is applied for only a few selected realizations. For 90 % or more of the realizations, upscaled functions are assigned statistically based on quickly computed flow and permeability attributes. A sequential Gaussian co-simulation procedure is incorporated to provide coarse models that honor the spatial correlation structure of the upscaled properties. The resulting EnLU procedure is applied for multiple realizations of two-dimensional models, for both Gaussian and channelized permeability fields. Results demonstrate that EnLU provides P10, P50, and P90 results for phase and component production rates that are in close agreement with reference fine-scale results. Less accuracy is observed in realization-by-realization comparisons, though the models are still much more accurate than those generated using standard coarsening procedures.     

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.     

A multi-resolution workflow to generate high-resolution models constrained to dynamic data

Distance-based stochastic techniques have recently emerged in the context of ensemble modeling, in particular for history matching, model selection and uncertainty quantification. Starting with an initial ensemble of realizations, a distance between any two models is defined. This distance is defined such that the objective of the study is incorporated into the geological modeling process, thereby potentially enhancing the efficacy of the overall workflow. If the intent is to create new models that are constrained to dynamic data (history matching), the calculation of the distance requires flow simulation for each model in the initial ensemble. This can be very time consuming, especially for high-resolution models. In this paper, we present a multi-resolution framework for ensemble modeling. A distance-based procedure is employed, with emphasis on the rapid construction of multiple models that have improved dynamic data conditioning. Our intent is to construct new high-resolution models constrained to dynamic data, while performing most of the flow simulations only on upscaled models. An error modeling procedure is introduced into the distance calculations to account for potential errors in the upscaling. Based on a few fine-scale flow simulations, the upscaling error is estimated for each model using a clustering technique. We demonstrate the efficiency of the method on two examples, one where the upscaling error is small, and another where the upscaling error is significant. Results show that the error modeling procedure can accurately capture the error in upscaling, and can thus reproduce the fine-scale flow behavior from coarse-scale simulations with sufficient accuracy (in terms of uncertainty predictions). As a consequence, an ensemble of high-resolution models, which are constrained to dynamic data, can be obtained, but with a minimum of flow simulations at the fine scale.     

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。      

Modified adaptive local–global upscaling method for discontinuous permeability distribution

The paper is devoted to the upscaling method appropriate for single-phase flow in media with discontinuous permeability distribution. The suggested algorithm is a modification of the iterative adaptive local–global upscaling developed by Chen and coauthors. The key feature of this method is a consistency between local and coarse global calculated characteristics. In this work, we apply a modified procedure to determine the boundary conditions used in the local fine-scale computation. To increase the accuracy of these boundary conditions on each iteration, we involve an additional preliminary step based on the results of coarse scale calculations from the previous iteration. Numerical tests show an essential improvement of the accuracy of upscaled flow rates for most of the realizations of statistical permeability distribution. Although the developed method is universal, its efficiency increases with increasing of permeability contrast.     

Modeling core-scale permeability anisotropy in highly bioturbated “tight oil” reservoir rocks

A high-resolution simulation model of a heterogeneous low-permeability rock sample is used to investigate the effects of physical and biogenic sedimentary structures on scaling and anisotropy of absolute permeability at the core scale. Several simulation sub-samples with random locations and volumes were also selected for evaluation of the effects of scale and lithological composition on the calculated permeability. Vertical and horizontal permeability values (from whole core simulation) are in good agreement with routine core analysis (RCA) measurements from offsetting cores. Despite relatively good reservoir quality associated with geobodies of biogenic and relic bedding structures, results from the full diameter core simulation demonstrate that their limited volumetric abundance and restricted connectivity prevent these features from controlling fluid flow in these rocks. In fact, permeability seems to be dominated by the tighter encasing matrix, which exhibits average permeability values very close to those reported from RCA. Geometric averaging offers a better representation for the upscaling of horizontal permeability datasets; whereas, both geometric and harmonic averaging work similarly well for the vertical measurements. The methodology used in this work is particularly applicable to the detailed characterization of reservoir rocks with a high degree of heterogeneity caused by biological reworking and diagenesis.     

Nodal and Mixed Finite Elements for the Numerical Homogenization of 3D Permeability

The aim of upscaling is to determine equivalent homogeneous parameters at a coarse-scale from a spatially oscillating fine-scale parameter distribution. To be able to use a limited number of relatively large grid-blocks in numerical oil reservoir simulators or groundwater models, upscaling of the permeability is frequently applied. The spatial fine-scale permeability distribution is generally obtained from geological and geostatistical models. After upscaling, the coarse-scale permeabilities are incorporated in the relatively large grid-blocks of the numerical model. If the porous rock may be approximated as a periodic medium, upscaling can be performed by the method of homogenization. In this paper the homogenization is performed numerically, which gives rise to an approximation error. The complementarity between two different numerical methods – the conformal-nodal finite element method and the mixed-hybrid finite element method – has been used to quantify this error. These two methods yield respectively upper and lower bounds for the eigenvalues of the coarse-scale permeability tensor. Results of 3D numerical experiments are shown, both for the far field and around wells.     

Coarse-scale data assimilation as a generic alternative to localization

Coarse-scale data assimilation (DA) with large ensemble size is proposed as a robust alternative to standard DA with localization for reservoir history matching problems. With coarse-scale DA, the unknown property function associated with each ensemble member is upscaled to a grid significantly coarser than the original reservoir simulator grid. The grid coarsening is automatic, ensemble-specific and non-uniform. The selection of regions where the grid can be coarsened without introducing too large modelling errors is performed using a second-generation wavelet transform allowing for seamless handling of non-dyadic grids and inactive grid cells. An inexpensive local-local upscaling is performed on each ensemble member. A DA algorithm that restarts from initial time is utilized, which avoids the need for downscaling. Since the DA computational cost roughly equals the number of ensemble members times the cost of a single forward simulation, coarse-scale DA allows for a significant increase in the number of ensemble members at the same computational cost as standard DA with localization. Fixing the computational cost for both approaches, the quality of coarse-scale DA is compared to that of standard DA with localization (using state-of-the-art localization techniques) on examples spanning a large degree of variability. It is found that coarse-scale DA is more robust with respect to variation in example type than each of the localization techniques considered with standard DA. Although the paper is concerned with two spatial dimensions, coarse-scale DA is easily extendible to three spatial dimensions, where it is expected that its advantage with respect to standard DA with localization will increase.     

煤储层注入/压降试井中注入段曲线的分析及应用

注入/压降试井是目前煤层气井获取煤储层渗透率、储层压力及井筒参数的主要手段。一般情况下,关井阶段压力变化平稳,排量稳定,因此注入/压降试井报告中采用压降阶段的压力数据进行分析。但在一些低渗储层使用压降段曲线进行分析时,能表征储层重要特征的径向流段往往不能出现。注入测试的实质是一次负产量的压降试井,在排量稳定、压力曲线光滑的前提下,可以对注入曲线进行分析,求取储层参数,并以A井为例进行了分析说明,分析结果显示,注入曲线与压降曲线分析所得渗透率一致。     

Estimate CO2 storage capacity of the Johansen formation: numerical investigations beyond the benchmarking exercise

In this paper, Shell’s in-house reservoir simulator MoReS is applied to a recently introduced CO₂ sequestration benchmark problem entitled “Estimation of the CO₂ Storage Capacity of a Geological Formation” (Class et al. 2008). The principal objective of this benchmark is the simulation of CO₂ distribution within a modeling region, and leakage of CO₂ outside of it, for a period of 50 years. This study goes beyond the benchmarking exercise to investigate additional factors with direct relevance to CO₂ storage capacity estimations: water and gas relative permeabilities, permeability anisotropy, presence of sub-seismic features (conductive fractures, thin shale layers), regional hydrodynamic gradient, CO₂-enriched brine convection (due to brine density differences), and injection rates. The effects of hydrodynamic gradients and gravitationally induced convection only become significant over 100 s of years. This study has thus extended simulation time to 1,000 years. It is shown that grid resolution significantly impacts results. Vertical-grid refinement results in larger and thinner CO₂ plumes. Lateral-grid refinement delays leakage out of the model domain and reduces injection pressure for a given injection rate. Sub-seismic geological features such as fractures/faults and shale layers are demonstrated to have impact on CO₂ sequestration. Fractures located up-dip from the injector may lead to more leakage while the opposite may happen in the presence of fractures perpendicular to the dip. Thin shale layers produce stacked CO₂ blankets. They should be explicitly represented instead of being upscaled using a reduced vertical to horizontal permeability ratio. Results are seen to be far more sensitive to gas relative permeability and hysteresis than to variations in the water relative permeability models used. For a multi-injectors project, there is scope to optimize the phasing of injections to avoid potential fracturing near injectors.