Statistical framework for scale-up of dispersivity in multi-scale heterogeneous media

A hierarchical scale-up framework is formulated to study the scaling characteristics of reservoir attributes and input dispersivities at the transport modeling scale, where heterogeneity distribution exhibits both non-stationarity (trend) and sub-scale variability. The proposed method is flexible to handle heterogeneities occurring at multiple scales, without any explicit assumption regarding the multivariate distribution of the heterogeneity. This paper extends our previous work by incorporating the effects of non-stationarity into the modeling workflow. Rock property at a given location is modeled as a random variable, which is decomposed into the sum of a trend (available on the same resolution of the transport modeling scale) and a residual component (defined at a much smaller scale). First, to scale up the residual component to the transport modeling scale, the corresponding volume variance is computed; by sampling numerous sets of “conditioning data” via bootstrapping and constructing multiple realizations of the residual components at the transport modeling, uncertainty due to this scale-up process is captured. Next, to compute the input dispersivity at the transport modeling scale, a flow-based technique is adopted: multiple geostatistical realizations of the same physical size as the transport modeling scale are generated to describe the spatial heterogeneity below the modeling scale. Each realization is subjected to particle-tracking simulation. Effective longitudinal and transverse dispersivities are estimated by minimizing the difference in effluent history for each realization and that of an equivalent average medium. Probability distributions of effective dispersivities are established by aggregating results from all realizations. The results demonstrate that both large-scale non-stationarity and sub-scale variability are both contributing to anomalous non-Fickian behavior. In comparison with our previous work, which ignored large-scale non-stationarity, the non-Fickian characteristics observed in this study is dramatically more pronounced.     

基于非均质系数的裂隙岩体表征单元体研究

表征单元体（REV）是岩体力学中的一个基础性概念,其存在性是确定岩体等效参数和应用连续介质方法进行研究的前提条件。为综合反映裂隙大小、产状和密度对岩体REV的影响,本文提出采用岩体非均质系数HI来确定岩体的REV。首先,详细介绍了非均质系数的概念和含义。利用General Block软件建立了中等间距-中等延展性（MS1-MP1）裂隙的网络模型,并完成20次随机实现。从各模型中选取10个不同尺寸的岩体模型计算非均质系数,结果表明:非均质系数具有明显的尺寸效应,当岩体尺寸不小于8 m时,非均质系数的平均值和标准差分别为0.5和0.14,综合确定该岩体的REV为8 m。采用HI对三峡地下电站厂房围岩的研究表明,该岩体的REV为60 m,可作为非连续介质方法适用性的判断标准。HI是从岩体结构角度提出的新指标,适用于确定岩体的REV和统计范围,对岩体的统计分析和等效参数研究具有重要意义。     

Stochastic simulation of categorical variables using a classification algorithm and simulated annealing

An important step of reservoir characterization is the stochastic modeling of the geometry of lithofacies which control large-scale heterogeneities of petrophysical properties. Although multiple realizations are necessary to appreciate the uncertainty in the spatial distribution of facies, a common short cut consists of retaining the first realization drawn. This paper presents an alternative to this potentially hazardous selection: (1) a categorical map is generated by allocating a single facies to each grid node according to the local probabilities of occurrence of the facies, and (2) the map then is post-processed using a steepest descent-type algorithm so as to improve reproduction of spatial continuity and transition probabilities between facies. The procedure is illustrated using a synthetic dataset. A waterflood simulation shows that retaining a single realization would yield, in average, larger errors in production forecasts (water cuts and recovered oil) than the single postprocessed facies map.     

Simulation of a reactive tracer experiment using stochastic hydraulic conductivity fields

Reactive tracer tests are performed to derive flow, transport and in situ biodegradation parameters. This paper describes the 3D simulation of a reactive tracer test using the transition probability geostatistical approach. Fifty different equally probable aquifer realizations were generated based on the geological information of 107 boreholes in an area of 62,500 m². One realization was chosen for the reactive transport simulation based on the results of groundwater flow modeling and on particle tracking calculations for the site. Field velocities at the site vary between 0.4 and 3 m/d. The transport of the reactive tracers deuterium ring labeled toluene-d₅ and fully deuterated toluene-d₈ was simulated and first-order biodegradation rates of 0.017 d⁻¹ for toluene-d₅ and 0.012 d⁻¹ for toluene-d₈ were determined.     

Connectivity and single/dual domain transport models: tests on a point-bar/channel aquifer analogue

In porous aquifers, groundwater flow and solute transport strongly depend on the sedimentary facies distribution at fine scale, which determines the heterogeneity of the conductivity field; in particular, connected permeable sediments could form preferential flow paths. Therefore, properly defined statistics, e.g. total and intrinsic facies connectivity, should be correlated with transport features. In order to improve the assessment of the relevance of this relationship, some tests are conducted on two ensembles of equiprobable realizations, obtained with two different geostatistical simulation methods—sequential indicator simulation and multiple point simulation (MPS)—from the same dataset, which refers to an aquifer analogue of sediments deposited in a fluvial point-bar/channel association. The ensembles show different features; simulations with MPS are more structured and characterised by preferential flow paths. This is confirmed by the analysis of transport connectivities and by the interpretation of data from numerical experiments of conservative solute transport with single and dual domain models. The use of two ensembles permits (1) previous results obtained for single realizations to be consolidated on a more firm statistical basis and (2) the application of principal component analysis to assess which quantities are statistically the most relevant for the relationship between connectivity indicators and flow and transport properties.     

Stochastic computational modelling of highly heterogeneous poroelastic media with long‐range correlations

The compaction of highly heterogeneous poroelastic reservoirs with the geology characterized by long‐range correlations displaying fractal character is investigated within the framework of the stochastic computational modelling. The influence of reservoir heterogeneity upon the magnitude of the stresses induced in the porous matrix during fluid withdrawal and rock consolidation is analysed by performing ensemble averages over realizations of a log‐normally distributed stationary random hydraulic conductivity field. Considering the statistical distribution of this parameter characterized by a coefficient of variation governing the magnitude of heterogeneity and a correlation function which decays with a power‐law scaling behaviour we show that the combination of these two effects result in an increase in the magnitude of effective stresses of the rock during reservoir depletion. Further, within the framework of a perturbation analysis we show that the randomness in the hydraulic conductivity gives rise to non‐linear corrections in the upscaled poroelastic equations. These corrections are illustrated by a self‐consistent recursive hierarchy of solutions of the stochastic poroelastic equations parametrized by a scale parameter representing the fluctuating log‐conductivity standard deviation. A classical example of land subsidence caused by fluid extraction of a weak reservoir is numerically simulated by performing Monte Carlo simulations in conjunction with finite elements discretizations of the poroelastic equations associated with an ensemble of geologies. Numerical results illustrate the effects of the spatial variability and fractal character of the permeability distribution upon the evolution of the Mohr–Coulomb function of the rock. Copyright © 2004 John Wiley & Sons, Ltd.     

Efficient Simulation of (Log)Normal Random Fields for Hydrogeological Applications

Two methods for generating representative realizations from Gaussian and lognormal random field models are studied in this paper, with term representative implying realizations efficiently spanning the range of possible attribute values corresponding to the multivariate (log)normal probability distribution. The first method, already established in the geostatistical literature, is multivariate Latin hypercube sampling, a form of stratified random sampling aiming at marginal stratification of simulated values for each variable involved under the constraint of reproducing a known covariance matrix. The second method, scarcely known in the geostatistical literature, is stratified likelihood sampling, in which representative realizations are generated by exploring in a systematic way the structure of the multivariate distribution function itself. The two sampling methods are employed for generating unconditional realizations of saturated hydraulic conductivity in a hydrogeological context via a synthetic case study involving physically-based simulation of flow and transport in a heterogeneous porous medium; their performance is evaluated for different sample sizes (number of realizations) in terms of the reproduction of ensemble statistics of hydraulic conductivity and solute concentration computed from a very large ensemble set generated via simple random sampling. The results show that both Latin hypercube and stratified likelihood sampling are more efficient than simple random sampling, in that overall they can reproduce to a similar extent statistics of the conductivity and concentration fields, yet with smaller sampling variability than the simple random sampling.     

An Uncertainty Quantification Framework for Studying the Effect of Spatial Heterogeneity in Reservoir Permeability on CO2 Sequestration

A new uncertainty quantification framework is adopted for carbon sequestration to evaluate the effect of spatial heterogeneity of reservoir permeability on CO₂ migration. Sequential Gaussian simulation is used to generate multiple realizations of permeability fields with various spatial statistical attributes. In order to deal with the computational difficulties, the following ideas/approaches are integrated. First, different efficient sampling approaches (probabilistic collocation, quasi-Monte Carlo, and adaptive sampling) are used to reduce the number of forward calculations, explore effectively the parameter space, and quantify the input uncertainty. Second, a scalable numerical simulator, extreme-scale Subsurface Transport Over Multiple Phases, is adopted as the forward modeling simulator for CO₂ migration. The framework has the capability to quantify input uncertainty, generate exploratory samples effectively, perform scalable numerical simulations, visualize output uncertainty, and evaluate input-output relationships. The framework is demonstrated with a given CO₂ injection scenario in heterogeneous sandstone reservoirs. Results show that geostatistical parameters for permeability have different impacts on CO₂ plume radius: the mean parameter has positive effects at the top layers, but affects the bottom layers negatively. The variance generally has a positive effect on the plume radius at all layers, particularly at middle layers, where the transport of CO₂ is highly influenced by the subsurface heterogeneity structure. The anisotropy ratio has weak impacts on the plume radius, but affects the shape of the CO₂ plume.     

On the effective thermal and hydraulic conductivity of binary heterogeneous sediments

Heterogeneity is present in geological sedimentary structures at all scales from pore to basin, and its distribution has an impact on transport processes such as heat and fluid flow. The rock masses at any scale need to be characterized by their effective properties at that scale, based on the individual characteristics of the heterogeneous porous medium. The effective thermal and hydraulic conductivity of sediments characterized by a stochastic distribution of heterogeneity is studied using an inverse approach based on numerical experiments. The simulations, covering a large range of conductivity contrasts, are carried out for actual core-scale cases of shale clasts in a sand matrix, and on a diagrammatic cross-section through a clastic sedimentary group at the basin scale.

The effective conductivity depends primarily on the heterogeneity fraction and on the conductivity contrast between heterogeneities and the embedding matrix, a dependency which can be described by a generalized weighted mean model. This model is better suited to estimate the effective conductivity at any scale than other models like the self-consistent, or any of the arithmetic, geometric or harmonic averages. The effective conductivity has an anisotropic character although the individual components are homogeneous and isotropic. The variation in effective conductivity is significant even for small conductivity contrasts, like in heat flow processes, and exhibits an asymptotic behaviour for large conductivity contrasts characteristic of fluid flow processes. The effective conductivity has a second-order dependence on such heterogeneity characteristics as shape, aspect ratio, orientation, and distribution. Depending on these characteristics, the bounds of effective conductivity values can be narrowed further from the extreme bounds expressed by the arithmetic and harmonic averages.     

Conditioning Simulations of Gaussian Random Fields by Ordinary Kriging

Conditioning realizations of stationary Gaussian random fields to a set of data is traditionally based on simple kriging. In practice, this approach may be demanding as it does not account for the uncertainty in the spatial average of the random field. In this paper, an alternative model is presented, in which the Gaussian field is decomposed into a random mean, constant over space but variable over the realizations, and an independent residual. It is shown that, when the prior variance of the random mean is infinitely large (reflecting prior ignorance on the actual spatial average), the realizations of the Gaussian random field are made conditional by substituting ordinary kriging for simple kriging. The proposed approach can be extended to models with random drifts that are polynomials in the spatial coordinates, by using universal or intrinsic kriging for conditioning the realizations, and also to multivariate situations by using cokriging instead of kriging.     

Structural controls on fluid flow and related mineralization in the Xiangshan uranium deposit, Southern China

The Xiangshan uranium deposit of Southern China is a Mesozoic volcanic uranium deposit where mineralization is closely associated with regional rock deformation and fluid flow. To better understand the genesis of this deposit, we have carried out coupled deformation and fluid flow modeling to explore the relationship between mineralizing fluid flow and structural deformation processes. The numerical modeling results demonstrate that fluid transport and focusing are predominantly controlled by deposit scale structures and associated rock deformation.     

造缝产烃还是改质造烃？——论含油气源岩层系的储集层属性和烃源岩属性

源岩层系油气已成为世界油气工业体系中举足轻重的重点领域，未来发展潜力很大。本文通过深入解读含油气源岩层系的地质形成条件，发现储集层属性和烃源岩属性在根本上决定了源岩层系油气成功开发的技术路径。储集层属性是指储集层储存和渗滤油气的物理性质及其相互关系，地史时期微—纳米级孔- 喉- 缝系统充注和聚集了大面积连续型油气资源，人工压裂形成缝网系统突破了致密储层的连通属性短板，成功开发了规模油气资源，突破流动属性短板可能是另一个发展方向。烃源岩属性是指烃源岩滞留、转化和排出油气的物理化学性质及其相互关系，包括已转化的滞留油气和未转化的残留有机质，滞留及潜在烃类资源规模巨大，地下人工加热等方式理论上可突破烃源岩的有机质数量和成熟度属性短板，可能是实现成功开发的有效路径。储集层属性和烃源岩属性是实现源岩层系油气规模发展的内在潜质基础，公共属性参数是外部环境基础。造缝产烃还是改质造烃？前提是准确研判优选何种内在属性参数，基础是系统整合优化所有外在属性参数，推动实现源岩层系油气商业化可持续发展，未来中低熟富有机质页岩及油页岩、中高熟富气态烃页岩层系、低变质程度富油煤岩、深层可气化煤炭等是值得期待的战略发展领域。     

Estimating regional-scale permeability–depth relations in a fractured-rock terrain using groundwater-flow model calibration

The trend of decreasing permeability with depth was estimated in the fractured-rock terrain of the upper Potomac River basin in the eastern USA using model calibration on 200 water-level observations in wells and 12 base-flow observations in subwatersheds. Results indicate that permeability at the 1–10 km scale (for groundwater flowpaths) decreases by several orders of magnitude within the top 100 m of land surface. This depth range represents the transition from the weathered, fractured regolith into unweathered bedrock. This rate of decline is substantially greater than has been observed by previous investigators that have plotted in situ wellbore measurements versus depth. The difference is that regional water levels give information on kilometer-scale connectivity of the regolith and adjacent fracture networks, whereas in situ measurements give information on near-hole fractures and fracture networks. The approach taken was to calibrate model layer-to-layer ratios of hydraulic conductivity (LLKs) for each major rock type. Most rock types gave optimal LLK values of 40–60, where each layer was twice a thick as the one overlying it. Previous estimates of permeability with depth from deeper data showed less of a decline at <300 m than the regional modeling results. There was less certainty in the modeling results deeper than 200 m and for certain rock types where fewer water-level observations were available. The results have implications for improved understanding of watershed-scale groundwater flow and transport, such as for the timing of the migration of pollutants from the water table to streams.     

Fast wavelet-based stochastic simulation using training images

Spatial uncertainty modelling is a complex and challenging job for orebody modelling in mining, reservoir characterization in petroleum, and contamination modelling in air and water. Stochastic simulation algorithms are popular methods for such modelling. In this paper, discrete wavelet transformation (DWT)-based multiple point simulation algorithm for continuous variable is proposed that handles multi-scale spatial characteristics in datasets and training images. The DWT of a training image provides multi-scale high-frequency wavelet images and one low-frequency scaling image at the coarsest scale. The simulation of the proposed approach is performed on the frequency (wavelet) domain where the scaling image and wavelet images across the scale are simulated jointly. The inverse DWT reconstructs simulated realizations of an attribute of interest in the space domain. An automatic scale-selection algorithm using dominant mode difference is applied for the selection of the optimal scale of wavelet decomposition. The proposed algorithm reduces the computational time required for simulating large domain as compared to spatial domain multi-point simulation algorithm. The algorithm is tested with an exhaustive dataset using conditional and unconditional simulation in two- and three-dimensional fluvial reservoir and mining blasted rock data. The realizations generated by the proposed algorithm perform well and reproduce the statistics of the training image. The study conducted comparing the spatial domain filtersim multiple-point simulation algorithm suggests that the proposed algorithm generates equally good realizations at lower computational cost.     

Application of multiple-point geostatistics on modelling groundwater flow and transport in a cross-bedded aquifer (Belgium)

Sedimentological processes often result in complex three-dimensional subsurface heterogeneity of hydrogeological parameter values. Variogram-based stochastic approaches are often not able to describe heterogeneity in such complex geological environments. This work shows how multiple-point geostatistics can be applied in a realistic hydrogeological application to determine the impact of complex geological heterogeneity on groundwater flow and transport. The approach is applied to a real aquifer in Belgium that exhibits a complex sedimentary heterogeneity and anisotropy. A training image is constructed based on geological and hydrogeological field data. Multiple-point statistics are borrowed from this training image to simulate hydrofacies occurrence, while intrafacies permeability variability is simulated using conventional variogram-based geostatistical methods. The simulated hydraulic conductivity realizations are used as input to a groundwater flow and transport model to investigate the effect of small-scale sedimentary heterogeneity on contaminant plume migration. Results show that small-scale sedimentary heterogeneity has a significant effect on contaminant transport in the studied aquifer. The uncertainty on the spatial facies distribution and intrafacies hydraulic conductivity distribution results in a significant uncertainty on the calculated concentration distribution. Comparison with standard variogram-based techniques shows that multiple-point geostatistics allow better reproduction of irregularly shaped low-permeability clay drapes that influence solute transport.     

An integrated approach to shallow aquifer characterization: combining geophysics and geostatistics

We present a method of aquifer characterization that is able to utilize multiple sources of conditioning data to build a more realistic model of heterogeneity. This modeling approach (InMod) uses geophysical data to delineate bounding surfaces within sedimentary deposits. The depositional volumes between bounding surfaces are identified automatically from the geophysical data by a region growing algorithm. Simple geometric rules are used to constrain the growth of the regions in 3-D. The nodes within the depositional volume are assigned to categorical lithologies using geostatistical realizations and a dynamic lookup routine that can be conditioned to field data. The realizations created with this method preserve geologically expected features and produces sharp juxtapositions of high and low hydraulic conductivity lithologies along bounding surfaces. The realizations created with InMod also have higher variance than models created only with geostatistics and honor the volumetric distribution of sediments measured from field data.     

Stochastic analysis of the effect of spatial variability of diffusion parameters on radionuclide transport in a low permeability clay layer

Most studies that incorporate subsurface heterogeneity in groundwater flow and transport models only analyze and simulate the spatial variability of hydraulic conductivity. Heterogeneity of the other flow and transport parameters are usually neglected. This approach is often justified, but there are, however, cases in which disregarding the heterogeneity of the other flow and transport parameters can be questionable. In low permeability media, for instance, diffusion is often the dominant transport mechanism. It therefore seems logical to incorporate the spatial variability of the diffusion parameters in the transport model. This study therefore analyses and simulates the spatial variability of the effective diffusion coefficient and the diffusion accessible porosity with geostatistical techniques and incorporates their heterogeneity in the transport model of a low permeability formation. The formation studied was Boom clay (Belgium), a candidate host rock for the deep geological disposal of high-level radioactive waste. The calculated output radionuclide fluxes of this model are compared with the fluxes calculated with a homogeneous model and a model with a heterogeneous hydraulic conductivity distribution. This analysis shows that the heterogeneity of the diffusion parameters has a much larger effect on the calculated output radionuclide fluxes than the heterogeneity of hydraulic conductivity in the low permeability medium under study.     

Transitive Geostatistics for Stepwise Modeling Across Boundaries between Rock Regions

A single intrinsic stationary random field may not account for transitional heterogeneity and abrupt dissimilarity of geological properties across boundaries between rock type regions. This paper proposes the stepwise construction of transitive covariance models for modeling continuous properties correlated across boundaries of multiple disjoint physical domains such as rock type bodies. Modeling in geology is usually simplified by splitting the geological space into rock type geo-domains (e.g., strata, sedimentary facies, soil series, diagenetic regions and alteration zones). Due to the limitations of simultaneous solutions, a simplification is to model each domain independently at the cost of losing the conditioning of properties across domains. This paper proposes to organize the modeling process in a triangular array which follows events in the geological time domain; for example, the younger formations are at the top of the pyramid and the older formation at the base. The estimation may go from top to base by assuming that younger events have perturbed older formations. Geology shows the scars of events that cumulate in rock formations before they are finally eroded. In some cases, older formations may be parent material for younger formations. The continuous property within each geo-domain has a conditional covariance in the main diagonal of the array which may belong to a specific event in the geological time. This sequence leads to transitive estimation and simulations in the physical space. If a simultaneous solution is sought (i.e., the future and past are correlated both ways), the complex covariance functions can be constructed stepwise from conditional spectra.