Adaptive Multiscale Permeability Estimation

With multiscale permeability estimation one does not select parameterization prior to the estimation. Instead, one performs a hierarchical search for the right parameterization while solving a sequence of estimation problems with an increasing parameterization dimension. In some previous works on the subject, the same refinement is applied all over the porous medium. This may lead to over-parameterization, and subsequently, to unrealistic permeability estimates and excessive computational work. With adaptive multiscale permeability estimation, the new parameterization at an arbitrary stage in the estimation sequence is such that new degrees of freedom are not necessarily introduced all over the porous medium. The aim is to introduce new degrees of freedom only where it is warranted by the data. In this paper, we introduce a novel adaptive multiscale estimation. The approach is used to estimate absolute permeability from two-phase pressure data in several numerical examples.     

Impact of time-lapse seismic data for permeability estimation

We consider the impact of using time-lapse seismic data in addition to production data for permeability estimation in a porous medium with multiphase fluid flows, such as a petroleum reservoir under water-assisted production. Since modeling seismic wave propagation in addition to modeling fluid flows in the reservoir is quite involved, it is assumed that the time-lapse seismic data have already been inverted into fluid saturation differences (pseudoseismic data). Because an inversion process often leads to considerable error growth, we will consider pseudoseismic data with large uncertainties. The impact of pseudoseismic data is assessed through permeability estimation with and without such data and through application of some uncertainty measures for the estimated parameters. A multiscale algorithm is used for the parameter estimations, so that potential differences in attainable permeability resolution will be easily revealed. The numerical examples clearly indicate that the permeability estimation problem is stabilized at a higher level of resolution when pseudoseismic data are applied in addition to production data, even if the pseudoseismic data have large associated uncertainties. Use of the parameter uncertainty measures confirm these results.     

A multiscale method for distributed parameter estimation with application to reservoir history matching

A method for multiscale parameter estimation with application to reservoir history matching is presented. Starting from a given fine-scale model, coarser models are generated using a global upscaling technique where the coarse models are tuned to match the solution of the fine model. Conditioning to dynamic data is done by history-matching the coarse model. Using consistently the same resolution both for the forward and inverse problems, this model is successively refined using a combination of downscaling and history matching until model-matching dynamic data are obtained at the finest scale. Large-scale corrections are obtained using fast models, which, combined with a downscaling procedure, provide a better initial model for the final adjustment on the fine scale. The result is thus a series of models with different resolution, all matching history as good as possible with this grid. Numerical examples show that this method may significantly reduce the computational effort and/or improve the quality of the solution when achieving a fine-scale match as compared to history-matching directly on the fine scale.     

Assessing adaptive capacity within regional climate change vulnerability studies—an Alpine example

Adaptive capacity represents a crucial component in the assessment of a region’s vulnerability to climate change. The term adaptive capacity is only fuzzily defined, and determining it is difficult and often neglected in previous studies. In this paper, a newly developed adaptive capacity concept is introduced, with a respective indicator/criteria system and simple aggregation methods. The approach allows for adaptive capacity assessments at 3 levels of specificity (impact specific, sector specific and regional generic). The selection of indicators is tailor-made for Alpine regions, where the approach has been widely tested. The presented approach requires extended stakeholder involvement, namely for the evaluation of the indicators. The overall effort needed for its implementation remains reasonable. The outcome of the assessment exercise does not provide precise objective measurements, but remains an indicative estimation due to the fuzziness and complexity of the underlying concept. The conceptual approach is transferable to other mountain areas and beyond, the selection of indicators however is only valid for the Alpine region. The showcase presents results from the adaptive capacity assessment in the region of South Tyrol, where the method was carried out as part of a climate change vulnerability study. The outcomes indicate a number of issues that future actions could address in order to improve adaptive capacity in the region, namely in the field of prevention measures against meteorological extremes and natural hazards.     

Quick and Approximate Estimation of Earthquake Loss Based on Macroscopic Index of Exposure and Population Distribution

In the traditional method of earthquake loss estimation, all exposed facilities are classified according to their structural type and/or occupancy. Inventory data is collected and the total loss is estimated as the aggregate of all facility losses from each facility class separately. For many regions of the world, however, the vast amount of data required for this method is difficult or impossible to obtain. The traditional method is also unable to estimate quickly the loss from an unexpected catastrophic earthquake. It is difficult to give the necessary risk information to help the government with rescue and relief after the earthquake disaster.In this paper, we propose a quick and approximate estimation method of earthquake loss based on a macroscopic index of exposure and population distribution from GIS. This method was applied to analyze several earthquake scenarios with World Bank and CIESIN data. The preliminary analysis and comparison results show that our method is effective and reasonable for quick assessment of earthquake risk.     

计算剥蚀厚度的优化孔隙度法：程序及应用

碎屑岩（泥岩和砂岩）在正常压实埋藏过程中，原始孔隙度的演化随埋深的增加呈指数减小，利用在深度Zi处的孔隙度实测值Φi及其理论值Φ，考虑m次观察中的误差平方和，找到最有可能使得该误差平方和达到最小的原始孔隙度Φ0及地层的剥蚀厚度h，对地层剥蚀厚度的一种新的计算方法一优化孔隙度法的原理作了简单介绍。并给出了实用的计算机应用程序和应用实例，还就本方法应用中的问题进行了简单的讨论。     

Multiscale model reduction for shale gas transport in fractured media

In this paper, we develop a multiscale model reduction technique that describes shale gas transport in fractured media. Due to the pore-scale heterogeneities and processes, we use upscaled models to describe the matrix. We follow our previous work (Akkutlu et al. Transp. Porous Media 107(1), 235–260, 2015), where we derived an upscaled model in the form of generalized nonlinear diffusion model to describe the effects of kerogen. To model the interaction between the matrix and the fractures, we use Generalized Multiscale Finite Element Method (Efendiev et al. J. Comput. Phys. 251, 116–135, 2013, 2015). In this approach, the matrix and the fracture interaction is modeled via local multiscale basis functions. In Efendiev et al. (2015), we developed the GMsFEM and applied for linear flows with horizontal or vertical fracture orientations aligned with a Cartesian fine grid. The approach in Efendiev et al. (2015) does not allow handling arbitrary fracture distributions. In this paper, we (1) consider arbitrary fracture distributions on an unstructured grid; (2) develop GMsFEM for nonlinear flows; and (3) develop online basis function strategies to adaptively improve the convergence. The number of multiscale basis functions in each coarse region represents the degrees of freedom needed to achieve a certain error threshold. Our approach is adaptive in a sense that the multiscale basis functions can be added in the regions of interest. Numerical results for two-dimensional problem are presented to demonstrate the efficiency of proposed approach.     

Multilevel Monte Carlo methods using ensemble level mixed MsFEM for two-phase flow and transport simulations

In this paper, we propose multilevel Monte Carlo (MLMC) methods that use ensemble level mixed multiscale methods in the simulations of multiphase flow and transport. The contribution of this paper is twofold: (1) a design of ensemble level mixed multiscale finite element methods and (2) a novel use of mixed multiscale finite element methods within multilevel Monte Carlo techniques to speed up the computations. The main idea of ensemble level multiscale methods is to construct local multiscale basis functions that can be used for any member of the ensemble. In this paper, we consider two ensemble level mixed multiscale finite element methods: (1) the no-local-solve-online ensemble level method (NLSO); and (2) the local-solve-online ensemble level method (LSO). The first approach was proposed in Aarnes and Efendiev (SIAM J. Sci. Comput. 30(5):2319-2339, 2008) while the second approach is new. Both mixed multiscale methods use a number of snapshots of the permeability media in generating multiscale basis functions. As a result, in the off-line stage, we construct multiple basis functions for each coarse region where basis functions correspond to different realizations. In the no-local-solve-online ensemble level method, one uses the whole set of precomputed basis functions to approximate the solution for an arbitrary realization. In the local-solve-online ensemble level method, one uses the precomputed functions to construct a multiscale basis for a particular realization. With this basis, the solution corresponding to this particular realization is approximated in LSO mixed multiscale finite element method (MsFEM). In both approaches, the accuracy of the method is related to the number of snapshots computed based on different realizations that one uses to precompute a multiscale basis. In this paper, ensemble level multiscale methods are used in multilevel Monte Carlo methods (Giles 2008a, Oper.Res. 56(3):607-617, b). In multilevel Monte Carlo methods, more accurate (and expensive) forward simulations are run with fewer samples, while less accurate (and inexpensive) forward simulations are run with a larger number of samples. Selecting the number of expensive and inexpensive simulations based on the number of coarse degrees of freedom, one can show that MLMC methods can provide better accuracy at the same cost as Monte Carlo (MC) methods. The main objective of the paper is twofold. First, we would like to compare NLSO and LSO mixed MsFEMs. Further, we use both approaches in the context of MLMC to speedup MC calculations.     

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.     

Constraint energy minimizing generalized multiscale finite element method in the mixed formulation

This paper presents a novel mass-conservative mixed multiscale method for solving flow equations in heterogeneous porous media. The media properties (the permeability) contain multiple scales and high contrast. The proposed method solves the flow equation in a mixed formulation on a coarse grid by constructing multiscale basis functions. The resulting velocity field is mass-conservative on the fine grid. Our main goal is to obtain first-order convergence in terms of the mesh size which is independent of local contrast. This is achieved, first, by constructing some auxiliary spaces, which contain global information that cannot be localized, in general. This is built on our previous work on the generalized multiscale finite element method (GMsFEM). In the auxiliary space, multiscale basis functions corresponding to small (contrast-dependent) eigenvalues are selected. These basis functions represent the high-conductivity channels (which connect the boundaries of a coarse block). Next, we solve local problems to construct multiscale basis functions for the velocity field. These local problems are formulated in the oversampled domain, taking into account some constraints with respect to auxiliary spaces. The latter allows fast spatial decay of local solutions and, thus, allows taking smaller oversampled regions. The number of basis functions depends on small eigenvalues of the local spectral problems. Moreover, multiscale pressure basis functions are needed in constructing the velocity space. Our multiscale spaces have a minimal dimension, which is needed to avoid contrast dependence in the convergence. The method’s convergence requires an oversampling of several layers. We present an analysis of our approach. Our numerical results confirm that the convergence rate is first order with respect to the mesh size and independent of the contrast.     

A method for automatic history matching of a compositional reservoir simulator with multipoint flux approximation

A method for history matching of an in-house petroleum reservoir compositional simulator with multipoint flux approximation is presented. This method is used for the estimation of unknown reservoir parameters, such as permeability and porosity, based on production data and inverted seismic data. The limited-memory Broyden–Fletcher–Goldfarb–Shanno method is employed for minimization of the objective function, which represents the difference between simulated and observed data. In this work, we present the key features of the algorithm for calculations of the gradients of the objective function based on adjoint variables. The test example shows that the method is applicable to cases with anisotropic permeability fields, multipoint flux approximation, and arbitrary fluid compositions.     

Multiscale estimation of the Freundlich adsorption isotherm

Adsorption plays an important role in water and wastewater treatment. The analysis and design of processes that involve adsorption rely on the availability of isotherms that describe these adsorption processes. Adsorption isotherms are usually estimated empirically from measurements of the adsorption process variables. Unfortunately, these measurements are usually contaminated with errors that degrade the accuracy of estimated isotherms. Therefore, these errors need to be filtered for improved isotherm estimation accuracy. Multiscale wavelet based filtering has been shown to be a powerful filtering tool. In this work, multiscale filtering is utilized to improve the estimation accuracy of the Freundlich adsorption isotherm in the presence of measurement noise in the data by developing a multiscale algorithm for the estimation of Freundlich isotherm parameters. The idea behind the algorithm is to use multiscale filtering to filter the data at different scales, use the filtered data from all scales to construct multiple isotherms and then select among all scales the isotherm that best represents the data based on a cross validation mean squares error criterion. The developed multiscale isotherm estimation algorithm is shown to outperform the conventional time-domain estimation method through a simulated example.     

Uncertainty quantification of two-phase flow problems via measure theory and the generalized multiscale finite element method

The problem of multiphase phase flow in heterogeneous subsurface porous media is one involving many uncertainties. In particular, the permeability of the medium is an important aspect of the model that is inherently uncertain. Properly quantifying these uncertainties is essential in order to make reliable probabilistic-based predictions and future decisions. In this work, a measure-theoretic framework is employed to quantify uncertainties in a two-phase subsurface flow model in high-contrast media. Given uncertain saturation data from observation wells, the stochastic inverse problem is solved numerically in order to obtain a probability measure on the space of unknown permeability parameters characterizing the two-phase flow. As solving the stochastic inverse problem requires a number of forward model solves, we also incorporate the use of a conservative version of the generalized multiscale finite element method for added efficiency. The parameter-space probability measure is used in order to make predictions of saturation values where measurements are not available, and to validate the effectiveness of the proposed approach in the context of fine and coarse model solves. A number of numerical examples are offered to illustrate the measure-theoretic methodology for solving the stochastic inverse problem using both fine and coarse solution schemes.     

3‐D mesoscale simulation of crack‐permeability coupling in the Brazilian splitting test

In this paper, mesoscale hydromechanical simulations are performed to study (1) fracture features and (2) crack‐gas permeability coupling evolution in the context of the tensile splitting test. The mesostructure is based on a 2‐phase 3‐D representation of heterogeneous materials, such as concrete, where stiff aggregates are embedded into a mortar matrix. To take into account these heterogeneities without any mesh adaptation, a weak discontinuity is introduced into the strain field. In addition, a strong discontinuity is also added to take into account microcracking. This mechanical model is cast into the framework of the enhanced finite element method. Concerning the coupling with gas permeability, a double‐porosity method is used to simulate the flow through the cracks and the porosity. The apparent gas permeability is afterwards evaluated by a homogenization method. On the basis of finite element simulations, influence of aggregate size on ultimate crack opening, macroscopic ultimate tensile stress, total dissipated energy, and gas permeability evolution is numerically investigated. Furthermore, gas permeability evolution is also compared with experimental results from the literature. In addition, in the spirit of a sequential multiscale approach, macroscale gas permeability equations are identified from the hydromechanical results coming from the mesoscale computations. These equations lead to a relation between macroscale gas permeability evolution and crack opening. Besides, we show how the aggregate size influences the percolation threshold and that after this threshold, a cubic relation between macroscale gas permeability and crack opening is obtained.     

Detecting hotspots of urban residents’ behaviours based on spatio-temporal clustering techniques

Hotspots are regions where the number of spatial objects is obviously high within the time intervals. As the behaviours of urban residents are considered as a typical kind of spatio-temporal pattern, the detection of hotspots of urban residents’ behaviours appears necessary, since important information might be discovered in hotspots. In this paper, we propose an approach to detect the spatio-temporal hotspots of urban residents’ behaviours. This approach is validated based on the GPS data of floating cars of a city in southern China. The approach consists of four main steps: first, the effective spatio-temporal trajectories and the important characteristic points contained in each trajectory are extracted from the GPS data; second, the spatio-temporal clusters are generated by clustering the characteristic points based on a kernel density estimation algorithm; third, the spatio-temporal hotspots are detected by filtering the spatio-temporal clusters with high densities; last, the detected hotspots are analysed and interpreted. The results show that the proposed approach is effective and useful in detecting hotspots of urban residents’ behaviours.     

Identifying the Representative Elementary Volume for Permeability in Heterolithic Deposits Using Numerical Rock Models

Using a range of realistic 3D numerical lithofacies (dm-scale) models of ripple laminated sandstone intercalated with mudstone we evaluate how single-phase permeability varies as a function of sample support. The models represent a range of mudstone content which is typical for tidal deposits. Furthermore, the spatial distribution of flow barriers (i.e. mudstone) is not random, but governed by sedimentological rules giving a variable anisotropy ratio as a function of mudstone content. Both vertical and horizontal permeability are found to vary at small sample volumes, but these fluctuations reduce as the sample volume increases. The vertical permeability increases while the horizontal permeability is nearly constant as a function of sample support for small mudstone contents. For higher mudstone content, the horizontal permeability decreases while the vertical permeability is nearly constant as a function of sample support. We propose a criterion, based on a normalised standard deviation, to determine the Representative Elementary Volume (REV). The size of the REV is dependent on both the property measured (vertical and horizontal permeability) and the correlation lengths of the lithological elements (i.e. lithofacies). Based on this we identify three flow upscaling regimes that each require a different method for upscaling: (1) layered systems where the arithmetic and harmonic averages are appropriate, (2) systems close to the percolation threshold where a percolation model should be used, and (3) discontinuous systems where an effective medium method provides the best estimate of permeability. The work gives, by using numerical experiments on a range of heterogeneous systems, a new insight in determination of the REV for permeability at the lithofacies scale and its relation to sedimentological parameters.     

Conditioning Fractal (fBm/fGn) Porosity and Permeability Fields to Multiwell Pressure Data

Stochastic fractal (fGn and fBm) porosity and permeability fields are conditioned to given variogram, static (or hard), and multiwell pressure data within a Bayesian estimation framework. Because fGn distributions are normal/second-order stationary, it is shown that the Bayesian estimation methods based on the assumption of normal/second-order stationary distributions can be directly used to generate fGn porosity/permeability fields conditional to pressure data. However, because fBm is not second-order stationary, it is shown that such Bayesian estimation methods can be used with implementation of a pseudocovariance approach to generate fBm porosity/permeability fields conditional to multiwell pressure data. In addition, we provide methods to generate unconditional realizations of fBm/fGn fields honoring all variogram parameters. These unconditional realizations can then be conditioned to hard and pressure data observed at wells by using the randomized maximum likelihood method. Synthetic examples generated from one-, two-, and three-dimensional single-phase flow simulators are used to show the applicability of our methodology for generating realizations of fBm/fGn porosity and permeability fields conditioned to well-test pressure data and evaluating the uncertainty in reservoir performance predictions appropriately using these history-matched realizations.     

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.     

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.     

基于集合卡尔曼滤波的多相流模型参数估计——以室内二维砂箱中重质非水相污染物入渗为例

重质非水相有机污染物(DNAPL)泄漏到地下后,其运移与分布特征受渗透率非均质性影响显著。为刻画DNAPL污染源区结构特征,需进行参数估计以描述水文地质参数的非均质性。本研究构建了基于集合卡尔曼滤波方法(EnKF)与多相流运移模型的同化方案,通过融合DNAPL饱和度观测数据推估非均质介质渗透率空间分布。通过二维砂箱实际与理想算例,验证了同化方法的推估效果,并探讨了不同因素对同化的影响。研究结果表明:基于EnKF方法同化饱和度观测资料可有效地推估非均质渗透率场;参数推估精度随观测时空密度的增大而提高;观测点位置分布对同化效果有所影响,布置在污染集中区域的观测数据对于参数估计具有较高的数据价值。