Nested gridding and streamline-based simulation for fast reservoir performance prediction

Detailed reservoir models routinely contain 10⁶–10⁸ grid blocks. These models often cannot be used directly in a reservoir simulation because of the time and memory required for solving the pressure grid on the fine grid. We propose a nested gridding technique that efficiently obtains an approximate solution for the pressure field. The domain is divided into a series of coarse blocks, each containing several fine cells. Effective mobilities are computed for each coarse grid block and the pressure is then found on the coarse scale. The pressure field within each coarse block is computed using flux boundary conditions obtained from the coarse pressure solution. Streamline-based simulation is used to move saturations forward in time. We test the method for a series of example waterflood problems and demonstrate that the method can give accurate estimates of oil production for large 3D models significantly faster than direct simulation using streamlines on the fine grid, making the method overall approximately up to 1,000 times faster than direct conventional simulation.     

Approximate Derivative Computations for the Gradient-Based Optimization Methods in the Local Gradual Deformation for History Matching

Reservoir characterization needs the integration of various data through history matching, especially dynamic information such as production or four-dimensional seismic data. To update geostatistical realizations, the local gradual deformation method can be used. However, history matching is a complex inverse problem, and the computational effort in terms of the number of reservoir simulations required in the optimization procedure increases with the number of matching parameters. History matching large fields with a large number of parameters has been an ongoing challenge in reservoir simulation. This paper presents a new technique to improve history matching with the local gradual deformation method using the gradient-based optimizations. The new approach is based on the approximate derivative calculations using the partial separability of the objective function. The objective function is first split into local components, and only the most influential parameters in each component are used for the derivative computation. A perturbation design is then proposed to simultaneously compute all the derivatives with only a few simulations. This new technique makes history matching using the local gradual deformation method with large numbers of parameters tractable.     

Neural networks and their derivatives for history matching and reservoir optimization problems

In geosciences, complex forward problems met in geophysics, petroleum system analysis, and reservoir engineering problems often require replacing these forward problems by proxies, and these proxies are used for optimizations problems. For instance, history matching of observed field data requires a so large number of reservoir simulation runs (especially when using geostatistical geological models) that it is often impossible to use the full reservoir simulator. Therefore, several techniques have been proposed to mimic the reservoir simulations using proxies. Due to the use of experimental approach, most authors propose to use second-order polynomials. In this paper, we demonstrate that (1) neural networks can also be second-order polynomials. Therefore, the use of a neural network as a proxy is much more flexible and adaptable to the nonlinearity of the problem to be solved; (2) first-order and second-order derivatives of the neural network can be obtained providing gradients and Hessian for optimizers. For inverse problems met in seismic inversion, well by well production data, optimal well locations, source rock generation, etc., most of the time, gradient methods are used for finding an optimal solution. The paper will describe how to calculate these gradients from a neural network built as a proxy. When needed, the Hessian can also be obtained from the neural network approach. On a real case study, the ability of neural networks to reproduce complex phenomena (water cuts, production rates, etc.) is shown. Comparisons with second polynomials (and kriging methods) will be done demonstrating the superiority of the neural network approach as soon as nonlinearity behaviors are present in the responses of the simulator. The gradients and the Hessian of the neural network will be compared to those of the real response function.     

A surrogate-based simulation–optimization approach application to parameters’ identification for the HydroGeoSphere model

A systematic approach is needed to use water more productively, because water shortages limit socio-economic development in many parts of the world. The aim of this paper is to establish a surrogate-based simulation–optimization approach to identify parameter values for a fully integrated surface water and groundwater flow coupling simulation. A surface water and groundwater flow coupling simulation model was implemented using HydroGeoSphere (HGS) model and the parameter sensitivities in the model were analyzed using local sensitivity analysis method. The parameters that exerted a large influence on the output results of the HGS model were then selected as stochastic variables, and the stochastic variable data sets were generated using the latin hypercube sampling (LHS) method, which, thereby, were used as inputs in HGS model to obtain the corresponding outputs. On the basis of input and output data sets, a kriging surrogate model of the HGS model was then established and verified, and parameter values of HGS model were identified using a surrogate-based simulation–optimization approach. The results of this study show that parameters that exert a large influence on the simulation output results include hydraulic conductivity, porosity, the van genuchten parameter (\(\alpha\)), and channel manning coefficient. The established kriging surrogate model is an ideal alternative to the HGS model for simulating and predicting, while optimal parameter values can be identified effectively and accurately using the established approach. The results of this research reveal that huge computational loads can be mitigated while using the kriging surrogate as an alternative for a simulation model in the solution process of optimization model.     

Geostatistical approach for the assessment of the water reservoir capacity in arid regions: a case study of the Akdarya reservoir,Uzbekistan

The paper presents the results of a new geostatistical approach to generate bathymetric surface models from point measurement converted into continuous contour surfaces of reservoir bottoms in Uzbekistan. Sedimentation of reservoirs raises engineering, environmental and economical issues for the communities around the world in areas affected by a strong water deficit. Because of Uzbekistan’s arid climatic conditions, and uneven spatial and temporal water resources distribution, responsive and innovative water availability assessment surveys of all major water reservoirs are required. Bathymetric surveying is a traditional method that is carried out for the estimation of reservoir volumes and surface areas of the corresponding reservoir stages in order to assess the water availability. Volume and surface area differences derived from multiple surveys of a reservoir provide storage loss estimates over time due to sedimentation. However, two main factors, such as intensive field data measurement and post data-processing, often limit the frequency of these surveys. Alternatively, innovative depth measurement technologies coupled with contouring and surface mapping programs provide automated reservoir volume and surface area calculations. This significantly reduces time, workload and financial burdens for reservoir sedimentation projects. This research proposes the use of geostatistical approach to assess the reservoir sedimentation in the Akdarya reservoir of Uzbekistan. The geostatistical approach includes (semi-) variogram analysis and interpolation (kriging and simulations—turning bands) techniques predicting values at unsampled locations for generating digital bathymetric surface models of reservoir bottom conditions in order to calculate the volume and surface area at a given water elevation. The simulation enables to have range of reservoir volumes and surface areas with the same probability, in comparison to the kriging and traditional methods. This gives a real estimation of the resource availability for water operators to manage natural resources and hydraulic infrastructure in a sustainable manner.     

Identifying influence areas with connectivity analysis – application to the local perturbation of heterogeneity distribution for history matching

Numerical representations of a target reservoir can help to assess the potential of different development plans. To be as predictive as possible, these representations or models must reproduce the data (static, dynamic) collected on the field. However, constraining reservoir models to dynamic data – the history-matching process – can be very time consuming. Many uncertain parameters need to be taken into account, such as the spatial distribution of petrophysical properties. This distribution is mostly unknown and usually represented by millions of values populating the reservoir grid. Dedicated parameterization techniques make it possible to investigate many spatial distributions from a small number of parameters. The efficiency of the matching process can be improved from the perturbation of specific regions of the reservoir. Distinct approaches can be considered to define such regions. For instance, one can refer to streamlines. The leading idea is to identify areas that influence the production behavior where the data are poorly reproduced. Here, we propose alternative methods based on connectivity analysis to easily provide approximate influence areas for any fluid-flow simulation. The reservoir is viewed as a set of nodes connected by weighted links that characterize the distance between two nodes. The path between nodes (or grid blocks) with the lowest cumulative weight yields an approximate flow path used to define influence areas. The potential of the approach is demonstrated on the basis of 2D synthetic cases for the joint integration of production and 4D saturation data, considering several formulations for the weights attributed to the links.     

Disjunctive Kriging with Hard and Imprecise Data

This paper presents a methodology for assessing local probability distributions by disjunctive kriging when the available data set contains some imprecise measurements, like noisy or soft information or interval constraints. The basic idea consists in replacing the set of imprecise data by a set of pseudohard data simulated from their posterior distribution; an iterative algorithm based on the Gibbs sampler is proposed to achieve such a simulation step. The whole procedure is repeated many times and the final result is the average of the disjunctive kriging estimates computed from each simulated data set. Being data-independent, the kriging weights need to be calculated only once, which enables fast computing. The simulation procedure requires encoding each datum as a pre-posterior distribution and assuming a Markov property to allow the updating of pre-posterior distributions into posterior ones. Although it suffers some imperfections, disjunctive kriging turns out to be a much more flexible approach than conditional expectation, because of the vast class of models that allows its computation, namely isofactorial models.     

Explicit Fracture Network Modelling: From Multiple Point Statistics to Dynamic Simulation

This paper addresses the problem of explicit fractured media modelling in an operational case. On one side, realistic fracture models are mainly used for research purposes in order to investigate better the flow behaviour impacted by the complex multi-scale fracture network. Often, a very fine grid and hence an increased computation time are needed. On the other hand, an operational fractured reservoir is still generally modelled using an implicit fracture media representation. The upscaled petrophysical properties and dual media are defined on a coarse grid to limit the computational time of dynamic simulation. The challenge of this work is to demonstrate that an explicit fracture modelling is not reserved only for the research domain, but can be applied to an operational case study. The static model is constructed using a multiple point statistics approach in order to represent complex interaction patterns of fractures and faults observed at the analogue outcrop. The dynamic behaviour is simulated based on this spatial fracture network representation.     

Geostatistical approach for identification of transmissivity structure at Dulliu area in Taiwan

 A thorough understanding of the characteristics of transmissivity makes groundwater deterministic models more accurate. These transmissivity data characteristics occasionally possess a complicated spatial variation over an investigated site. This study presents both geostatistical estimation and conditional simulation methods to generate spatial transmissivity maps. The measured transmissivity data from the Dulliu area in Yun-Lin county, Taiwan, is used as the case study. The spatial transmissivity maps are simulated by using sequential Gaussian simulation (SGS), and estimated by using natural log ordinary kriging and ordinary kriging. Estimation and simulation results indicate that SGS can reproduce the spatial structure of the investigated data. Furthermore, displaying a low spatial variability does not allow the ordinary kriging and natural log kriging estimates to fit the spatial structure and small-scale variation for the investigated data. The maps of kriging estimates are smoother than those of other simulations. A SGS with multiple realizations has significant advantages over ordinary kriging and even natural log kriging techniques at a site with a high variation in investigated data. These results are displayed in geographic information systems (GIS) as basic information for further groundwater study. Received: 27 August 1999 · Accepted: 22 February 2000     

Compensating for estimation smoothing in kriging

Smoothing is a characteristic inherent to all minimum mean-square-error spatial estimators such as kriging. Cross-validation can be used to detect and model such smoothing. Inversion of the model produces a new estimator—compensated kriging. A numerical comparison based on an exhaustive permeability sampling of a 4-ft² slab of Berea Sandstone shows that the estimation surface generated by compensated kriging has properties intermediate between those generated by ordinary kriging and stochastic realizations resulting from simulated annealing and sequential Gaussian simulation. The frequency distribution is well reproduced by the compensated kriging surface, which also approximates the experimental semivariogram well—better than ordinary kriging, but not as well as stochastic realizations. Compensated kriging produces surfaces that are more accurate than stochastic realizations, but not as accurate as ordinary kriging.     

A Functional Data Analysis Approach to Surrogate Modeling in Reservoir and Geomechanics Uncertainty Quantification

Uncertainty quantification for geomechanical and reservoir predictions is in general a computationally intensive problem, especially if a direct Monte Carlo approach with large numbers of full-physics simulations is used. A common solution to this problem, well-known for the fluid flow simulations, is the adoption of surrogate modeling approximating the physical behavior with respect to variations in uncertain parameters. The objective of this work is the quantification of such uncertainty both within geomechanical predictions and fluid-flow predictions using a specific surrogate modeling technique, which is based on a functional approach. The methodology realizes an approximation of full-physics simulated outputs that are varying in time and space when uncertainty parameters are changed, particularly important for the prediction of uncertainty in vertical displacement resulting from geomechanical modeling. The developed methodology has been applied both to a subsidence uncertainty quantification example and to a real reservoir forecast risk assessment. The surrogate quality obtained with these applications confirms that the proposed method makes it possible to perform reliable time–space varying dependent risk assessment with a low computational cost, provided the uncertainty space is low-dimensional.     

Direct Sequential Simulation and Cosimulation

Sequential simulation of a continuous variable usually requires its transformation into a binary or a Gaussian variable, giving rise to the classical algorithms of sequential indicator simulation or sequential Gaussian simulation. Journel (1994) showed that the sequential simulation of a continuous variable, without any prior transformation, succeeded in reproducing the covariance model, provided that the simulated values are drawn from local distributions centered at the simple kriging estimates with a variance corresponding to the simple kriging estimation variance. Unfortunately, it does not reproduce the histogram of the original variable, which is one of the basic requirements of any simulation method. This has been the most serious limitation to the practical application of the direct simulation approach. In this paper, a new approach for the direct sequential simulation is proposed. The idea is to use the local sk estimates of the mean and variance, not to define the local cdf but to sample from the global cdf. Simulated values of original variable are drawn from intervals of the global cdf, which are calculated with the local estimates of the mean and variance. One of the main advantages of the direct sequential simulation method is that it allows joint simulation of N _v variables without any transformation. A set of examples of direct simulation and cosimulation are presented.     

Conditional Spectral Simulation with Phase Identification

Spectral simulation is used widely in electrical engineering to generate random fields with a given covariance spectrum. The algorithms used are fast particularly when based on Fast Fourier Transform (FFT). However, because of lack of phase identification, spectral simulation only generates unconditional realizations. Local data conditioning is obtained typically by adding a simulated kriging residual. This conditioning process requires an additional kriging at each simulated node thus forfeiting the speed advantage of FFT. A new algorithm for conditioning is proposed whereby the phase values are determined iteratively to ensure approximative data reproduction while reproducing the frequency spectrum, that is, the covariance model. A case study is presented to demonstrate the algorithm.     

Mortar Upscaling for Multiphase Flow in Porous Media

In mortar space upscaling methods, a reservoir is decomposed into a series of subdomains (blocks) in which independently constructed numerical grids and possibly different physical models and discretization techniques can be employed in each block. Physically meaningful matching conditions are imposed on block interfaces in a numerically stable and accurate way using mortar finite element spaces. Coarse mortar grids and fine subdomain grids provide two-scale approximations. In the resulting effective solution flow is computed in subdomains on the fine scale while fluxes are matched on the coarse scale. In addition the flexibility to vary adaptively the number of interface degrees of freedom leads to more accurate multiscale approximations. This methodology has been implemented in the Center for Subsurface Modeling's multiphysics multiblock simulator IPARS (Integrated Parallel Accurate reservoir Simulator). Computational experiments demonstrate that this approach is scalable in parallel and it can be applied to non-matching grids across the interface, multinumerics and multiphysics models, and mortar adaptivity. Moreover unlike most upscaling approaches the underlying systems can be treated fully implicitly.     

Estimation or simulation? That is the question

The issue of smoothing in kriging has been addressed either by estimation or simulation. The solution via estimation calls for postprocessing kriging estimates in order to correct the smoothing effect. Stochastic simulation provides equiprobable images presenting no smoothing and reproducing the covariance model. Consequently, these images reproduce both the sample histogram and the sample semivariogram. However, there is still a problem, which is the lack of local accuracy of simulated images. In this paper, a postprocessing algorithm for correcting the smoothing effect of ordinary kriging estimates is compared with sequential Gaussian simulation realizations. Based on samples drawn from exhaustive data sets, the postprocessing algorithm is shown to be superior to any individual simulation realization yet, at the expense of providing one deterministic estimate of the random function.     

煤层气藏数值模拟垂向网格优化

数值模拟是研究煤层气藏工程的一种常规方法,建立模型时人们常忽略垂向网格精细程度对模拟的影响,无法准确反映出垂向上流体流动规律、气水分异现象以及压降漏斗展布等,对模拟结果有很大影响。为了研究垂向网格划分精度对煤层气藏数值模拟过程的影响,采用不同模拟器对垂向网格的精细程度进行模拟计算,运用渗透率等效方法、局部网格加密方法和模拟器自带压裂方法模拟煤层压裂缝,总共模拟了3种方法15套方案。结果表明,垂向网格划分精度对煤层气生产影响较大,当网格步长达到1.5m时计算结果较为精确,可以满足模拟需求。网格数量及网格步长的合理划分,能够更好地呈现煤层中气水分异现象,有助于分析气水流动状态,便于历史拟合和产量预测。      

A Simple Approach to Account for Radial Flow and Boundary Conditions When Kriging Hydraulic Head Fields for Confined Aquifers

The estimation and mapping of realistic hydraulic head fields, hence of flow paths, is a major goal of many hydrogeological studies. The most widely used method to obtain reliable head fields is the inverse approach. This approach relies on the numerical approximation of the flow equation and requires specifying boundary conditions and the transmissivity of each grid element. Boundary conditions are often unknown or poorly known, yet they impose a strong signature on the head fields obtained by inverse analysis. A simpler alternative to the inverse approach is the direct kriging of the head field using the measurements obtained at observation wells. The kriging must be modified to incorporate the available information. Use of the dual kriging formalism enables simultaneously estimating the head field, the aquifer mean transmissivity, and the regional hydraulic gradient from head data in steady or transient state conditions. In transient state conditions, an estimate of the storage coefficient can be obtained. We test the approach on simple analytical cases, on synthetic cases with solutions obtained numerically using a finite element flow simulator, and on a real aquifer. For homogeneous aquifers, infinite or bounded, the kriging estimate retrieves the exact solution of the head field, the exact hydrogeological parameters and the flow net. With heterogeneous aquifers, kriging accurately estimates the head field with prediction errors of the same magnitude as typical head measurement errors. The transmissivities are also accurately estimated by kriging. Moreover, if inversion is required, the kriged head along boundaries can be used as realistic boundary conditions for flow simulation.     

Generalized sequential kriging (GSK) for heterogeneous cell size: property modeling without matrices

Sequential kriging avoids the use of matrices and resolves the issue of unstable solutions. It allows for stepwise ways to get joint estimations and cosimulations that are equivalent to the simultaneous solution. The approach is proposed as the solution for geocellular modeling with variable cell size from heterogeneous structural properties (HSPs) as required for modeling with structural constraints. Rock properties are controlled by structural domains, regions, and structural geology parameters. In some cases, rock properties are cross-correlated to formation thickness, curvature of structures, and other structural attributes. Cell thickness may be proportional to formation thickness and may enter as a conditioning property in the estimation of rock property parameters for simulation. In addition, cell volume controls the upscaling of covariance structures (i.e., regularized variograms). Structural properties are priorly modeled. Perturbation response functions (PRFs) are computed for each cell vs all possible sample point locations to facilitate sequential kriging. Upscaled PRFs are modified following conditional updating after each new data value is included in the estimation of parameters. Generalized sequential kriging is expected to become the main tool for real-time spatial modeling of 3D cellular models with HSP. In addition, some new developments related to the sequential kriging algorithm are included. Sequential kriging can be used for the estimation of parameters for simulation in the so-called unstructured grids.     

Approximate local confidence intervals under change of support

Approximate local confidence intervals are constructed from uncertainty models in the form of the conditional distribution of the random variable Z given values of variables [Z_i, i=1,...,n]. When the support of the variable Z is any support other than that of the data, the conditional distributions require a change of support correction. This paper investigates the effect of change of support on the approximate local confidence intervals constructed by cumulative indicator kriging, class indicator kriging, and probability kriging under a variety of conditions. The conditions are generated by three simulated deposits with grade distributions of successively higher degree of skewness; a point support and two different block supports are considered. The paper also compares the confidence intervals obtained from these methods using the most used measures of confidence interval effectiveness.     

Kriging in a global neighborhood

The kriging estimator is usually computed in a moving neighborhood; only the data near the point to be estimated are used. This moving neighborhood approach creates discontinuities in mapping applications. An alternative approach is presented here, whereby all points are estimated using all the available data. To solve the resulting large linear system the kriging estimator is expressed in terms of the inverse of the covariance matrix. The covariance matrix has the advantage of being positive definite and the size of system which can be solved without encountering numerical instability is substantially increased. Because the kriging matrix does not change, the estimator can be written in terms of scalar products, thus avoiding the more time-consuming matrix multiplications of the standard approach. In the particular case of a covariance which is zero for distances greater than a fixed value (the range), the resulting banded structure of the covariance matrix is shown to lead to substantial computational savings in both run time and storage space. In this case the calculation time for the kriging variance is also substantially reduced. The present method is extended to the nonstationary case.