Gradual Deformation and Iterative Calibration of Sequential Stochastic Simulations

Based on the algorithm for gradual deformation of Gaussian stochastic models, we propose, in this paper, an extension of this method to gradually deforming realizations generated by sequential, not necessarily Gaussian, simulation. As in the Gaussian case, gradual deformation of a sequential simulation preserves spatial variability of the stochastic model and yields in general a regular objective function that can be minimized by an efficient optimization algorithm (e.g., a gradient-based algorithm). Furthermore, we discuss the local gradual deformation and the gradual deformation with respect to the structural parameters (mean, variance, and variogram range, etc.) of realizations generated by sequential simulation. Local gradual deformation may significantly improve calibration speed in the case where observations are scattered in different zones of a field. Gradual deformation with respect to structural parameters is necessary when these parameters cannot be inferred a priori and need to be determined using an inverse procedure. A synthetic example inspired from a real oil field is presented to illustrate different aspects of this approach. Results from this case study demonstrate the efficiency of the gradual deformation approach for constraining facies models generated by sequential indicator simulation. They also show the potential applicability of the proposed approach to complex real cases.     

Comparing the Gradual Deformation with the Probability Perturbation Method for Solving Inverse Problems

Inverse problems are ubiquitous in the Earth Sciences. Many such problems are ill-posed in the sense that multiple solutions can be found that match the data to be inverted. To impose restrictions on these solutions, a prior distribution of the model parameters is required. In a spatial context this prior model can be as simple as a Multi-Gaussian law with prior covariance matrix, or could come in the form of a complex training image describing the prior statistics of the model parameters. In this paper, two methods for generating inverse solutions constrained to such prior model are compared. The gradual deformation method treats the problem of finding inverse solution as an optimization problem. Using a perturbation mechanism, the gradual deformation method searches (optimizes) in the prior model space for those solutions that match the data to be inverted. The perturbation mechanism guarantees that the prior model statistics are honored. However, it is shown with a simple example that this perturbation method does not necessarily draw accurately samples from a given posterior distribution when the inverse problem is framed within a Bayesian context. On the other hand, the probability perturbation method approaches the inverse problem as a data integration problem. This method explicitly deals with the problem of combining prior probabilities with pre-posterior probabilities derived from the data. It is shown that the sampling properties of the probability perturbation method approach the accuracy of well-known Markov chain Monte Carlo samplers such as the rejection sampler. The paper uses simple examples to illustrate the clear differences between these two methods     

Optimization with the Gradual Deformation Method

Building reservoir models consistent with production data and prior geological knowledge is usually carried out through the minimization of an objective function. Such optimization problems are nonlinear and may be difficult to solve because they tend to be ill-posed and to involve many parameters. The gradual deformation technique was introduced recently to simplify these problems. Its main feature is the preservation of the spatial structure: perturbed realizations exhibit the same spatial variability as the starting ones. It is shown that optimizations based on gradual deformation converge exponentially to the global minimum, at least for linear problems. In addition, it appears that combining the gradual deformation parameterization with optimizations may remove step by step the structure preservation capability of the gradual deformation method. This bias is negligible when deformation is restricted to a few realization chains, but grows increasingly when the chain number tends to infinity. As in practice, optimization of reservoir models is limited to a small number of iterations with respect to the number of gridblocks, the spatial variability is preserved. Last, the optimization processes are implemented on the basis of the Levenberg–Marquardt method. Although the objective functions, written in terms of Gaussian white noises, are reduced to the data mismatch term, the conditional realization space can be properly sampled.     

Combination of Dependent Realizations Within the Gradual Deformation Method

Gradual deformation is a parameterization method that reduces considerably the unknown parameter space of stochastic models. This method can be used in an iterative optimization procedure for constraining stochastic simulations to data that are complex, nonanalytical functions of the simulated variables. This method is based on the fact that linear combinations of multi-Gaussian random functions remain multi-Gaussian random functions. During the past few years, we developed the gradual deformation method by combining independent realizations. This paper investigates another alternative: the combination of dependent realizations. One of our motivations for combining dependent realizations was to improve the numerical stability of the gradual deformation method. Because of limitations both in the size of simulation grids and in the precision of simulation algorithms, numerical realizations of a stochastic model are never perfectly independent. It was shown that the accumulation of very small dependence between realizations might result in significant structural drift from the initial stochastic model. From the combination of random functions whose covariance and cross-covariance are proportional to each other, we derived a new formulation of the gradual deformation method that can explicitly take into account the numerical dependence between realizations. This new formulation allows us to reduce the structural deterioration during the iterative optimization. The problem of combining dependent realizations also arises when deforming conditional realizations of a stochastic model. As opposed to the combination of independent realizations, combining conditional realizations avoids the additional conditioning step during the optimization process. However, this procedure is limited to global deformations with fixed structural parameters.     

An Improved Gradual Deformation Method for Reconciling Random and Gradient Searches in Stochastic Optimizations

Constraining stochastic models of reservoir properties such as porosity and permeability can be formulated as an optimization problem. While an optimization based on random search methods preserves the spatial variability of the stochastic model, it is prohibitively computer intensive. In contrast, gradient search methods may be very efficient but it does not preserve the spatial variability of the stochastic model. The gradual deformation method allows for modifying a reservoir model (i.e., realization of the stochastic model) from a small number of parameters while preserving its spatial variability. It can be considered as a first step towards the merger of random and gradient search methods. The gradual deformation method yields chains of reservoir models that can be investigated successively to identify an optimal reservoir model. The investigation of each chain is based on gradient computations, but the building of chains of reservoir models is random. In this paper, we propose an algorithm that further improves the efficiency of the gradual deformation method. Contrary to the previous gradual deformation method, we also use gradient information to build chains of reservoir models. The idea is to combine the initial reservoir model or the previously optimized reservoir model with a compound reservoir model. This compound model is a linear combination of a set of independent reservoir models. The combination coefficients are calculated so that the search direction from the initial model is as close as possible to the gradient search direction. This new gradual deformation scheme allows us for reducing the number of optimization parameters while selecting an optimal search direction. The numerical example compares the performance of the new gradual deformation scheme with that of the traditional one.     

Refinement Indicators for Optimal Selection of Geostatistical Realizations Using the Gradual Deformation Method

In the analysis of petroleum reservoirs, one of the most challenging problems is to use inverse theory in the search for an optimal parameterization of the reservoir. Generally, scientists approach this problem by computing a sensitivity matrix and then perform a singular value decomposition in order to determine the number of degrees of freedom i.e. the number of independent parameters necessary to specify the configuration of the system. Here we propose a complementary approach: it uses the concept of refinement indicators to select those degrees which have the greatest sensitivity to an objective function quantifying the mismatch between measured and simulated data. We apply this approach to the problem of data integration for petrophysical reservoir charaterization where geoscientists are currently working with multimillion cell geological models. Data integration may be performed by gradually deforming (by a linear combination) a set of these multimillion grid geostatistical realizations during the optimization process. The inversion parameters are then reduced to the number of coefficients of this linear combination. However, there is an infinity of geostatistical realizations to choose from which may not be efficient regarding operational constraints. Following our new approach, we are able through a single objective function evaluation to compute refinement indicators that indicate which realizations might improve the iterative geological model in a significant way. This computation is extremely fast as it implies a single gradient computation through the adjoint state approach and dot products. Using only the most sensitive realizations from a given set, we are able to resolve quicker the optimization problem case. We applied this methodology to the integration of interference test data into 3D geostatistical models.     

Experimental Assessment of Gradual Deformation Method

Uncertainty in future reservoir performance is usually evaluated from the simulated performance of a small number of reservoir realizations. Unfortunately, most of the practical methods for generating realizations conditional to production data are only approximately correct. It is not known whether or not the recently developed method of Gradual Deformation is an approximate method or if it actually generates realizations that are distributed correctly. In this paper, we evaluate the ability of the Gradual Deformation method to correctly assess the uncertainty in reservoir predictions by comparing the distribution of conditional realizations for a small test problem with the standard distribution from a Markov Chain Monte Carlo (MCMC) method, which is known to be correct, and with distributions from several approximate methods. Although the Gradual Deformation algorithm samples inefficiently for this test problem and is clearly not an exact method, it gives similar uncertainty estimates to those obtained by MCMC method based on a relatively small number of realizations.     

多物源条件下的储层地质建模方法

濮城沙三中油藏具有两个主物源,分别为NE向与SE向。油藏数值模拟需要在一套地质网格中对其进行模拟。经典的地质统计学利用变差函数描述区域化变量的空间几何结构特性。变差函数的计算是基于两点进行统计的,对其描述主要涉及方位角、变程、块金值和基台值。为了在一套模拟网格中模拟出多个物源条件下储层的分布特征,必须在不同的位置设置不同的变差函数参数。文中给出了两种方法实现这一目的:一是采用人为分区,把不同物源影响的区域分成不同的区块,分别对不同的区块设置不同的变差函数参数;二是采用变方位角,即根据不同的位置设置不同的变差函数方位角。这两种方法都实现了在一套网格中模拟具有多个物源方向的储层分布,更真实地再现了储层的空间展布特征。     

随机反演在储层预测中的应用

利用常规地震反演进行储层预测,影响因素诸多。而随机反演将随机模拟技术与常规地震反演相结合,有效地综合地质、测井和三维地震数据,可以弥补井点资料的不足,提高预测精度,在滚动开发阶段可以起到积极的作用。利用随机反演对某地区三维工区Eq41、Eq42、Eq43层段的渗透性砂岩进行预测,取得了较好的效果。     

地学模拟相关技术的研究与进展

地学模拟技术的一个发展方向是与地学过程分析密切结合,另一个发展方向是与数据可视化技术相结合。前者试图通过使用各种数学方法模拟地学随机现象,并对这些不良结构化或半结构化的地学问题进行定量化描述;后者运用计算机的三维可视化功能,在三维环境下将空间信息管理、地学解译、空间分析、地学统计与预测、三维图形可视化等技术相结合,实现计算可视化、分析可视化、过程可视化、结果可视化和决策可视化,并用于地学分析。回顾了地学中计算机三维地学建模技术、地质统计学和地学非线性现象模拟方法,并对该领域的发展进行了展望,认为加强地学模拟的理论体系、方法体系、技术体系的研究和实践既有着重要的理论意义,又有着重要的现实意义。     

地质统计反演在苏里格气田致密薄砂体预测中的应用

苏里格气田盒8段储层岩性结构复杂、砂体薄、孔隙度低,砂、泥岩波阻抗叠置严重,常规的确定性反演结果难以满足识别高孔单砂体的需求.针对这一现状,在详细分析储层敏感参数与反演关键参数基础上,开展了地质统计反演的应用研究.研究表明,地质统计反演方法能有效综合地质、测井与三维地震数据,极大地提高预测结果的纵、横向分辨率,有效地解决了盒8段砂泥岩薄互层的单砂体识别问题;根据纵波阻抗与孔隙度的统计关系能准确刻画了高孔隙度砂体的空间展布,应用效果非常明显.应用实践证明,地质统计反演是陆相致密碎屑岩性油气藏进行储层精细描述的有效方法,具有广阔的应用前景.     

Minimum-variance unbiased quadratic estimation of covariances of regionalized variables

The parameters of covariance functions (or variograms) of regionalized variables must be determined before linear unbiased estimation can be applied. This work examines the problem of minimum-variance unbiased quadratic estimation of the parameters of ordinary or generalized covariance functions of regionalized variables. Attention is limited to covariance functions that are linear in the parameters and the normality assumption is invoked when fourth moments of the data need to be calculated. The main contributions of this work are (1) it shows when and in what sense minimum-variance unbiased quadratic estimation can be achieved, and (2) it yields a well-founded, practicable, and easy-to-automate methodology for the estimation of parameters of covariance functions. Results of simulation studies are very encouraging.     

A Markov Chain Model for Subsurface Characterization: Theory and Applications

This paper proposes an extension of a single coupled Markov chain model to characterize heterogeneity of geological formations, and to make conditioning on any number of well data possible. The methodology is based on the concept of conditioning a Markov chain on the future states. Because the conditioning is performed in an explicit way, the methodology is efficient in terms of computer time and storage. Applications to synthetic and field data show good results.     

基于地统计学烟区土壤养分及pH值空间异质性分析——以重庆市酉阳县岩溶区为例

调查了解土壤养分及pH值空间异质性特征是制定烟区土壤养分管理与施肥决策的前提。本研究基于经典统计学和地统计学方法,分析了酉阳烟区土壤表层pH值及土壤养分要素的空间异质性特征。结果表明:酉阳烟区土壤有机质含量丰富,有效磷较缺乏;土壤养分指标的变异系数在13.11%～71.46%之间,有效磷的变异系数最大,而有机质、全氮、碱解氮在土壤中比较稳定;土壤有机质和全氮、全磷、碱解氮和有效磷呈极显著的正相关,pH值和有效磷和速效钾呈极显著的负相关;有机质、全氮、全磷及速效钾的变异函数曲线理论模型符合指数模型,而pH值和碱解氮则与球状模型拟合较好;全氮、碱解氮、全磷及有效磷表现出中等空间相关性,而有机质、全钾及速效钾的空间相关性表现较弱;分维数大小表现为:全钾的分维数最大,pH值、速效钾、有机质其次,全氮、碱解氮最少;土壤养分及pH值空间分布在东西和南北方向具有显著性差异,且在东西方向上的变异要高于南北方向。酉阳烟区土壤养分空间相关性不强,说明地形、施肥及耕作等随机性因素能够显著影响烟区土壤养分空间异质性。      

Significance of conditioning to piezometric head data for predictions of mass transport in groundwater modeling

Transmissivity and head data are sampled from an exhaustive synthetic reference field and used to predict the arrival positions and arrival times of a number of particles transported across the field, together with an uncertainty estimate. Different combinations of number of transmissivity data and number of head data used are considered in each one of a series of 64 Monte-Carlo analyses. In each analysis, 250 realizations of transmissivity fields conditioned to both transmissivity and head data are generated using a novel geostatistically based inverse method. Pooling the solutions of the flow and transport equations in all 250 realizations allows building conditional frequency distributions for particle arrival positions and arrival times. By comparing these fresquency distributions, we can assess the incremental gain that additional head data provide. The main conclusion is that the first few head data dramatically improve the quality of transport predictions.     

基于微观力学特性的脆性岩石变形过程模拟

为了建立能够描述岩石宏观变形及其组成部分的变形模拟方法,考虑微缺陷的不均匀性对岩石宏观变形产生的影响,将岩石视为由空隙部分材料和骨架部分材料两部分组成。首先,基于岩石各组成部分材料变形分析,利用真应变描述方法及非线性变形的特点建立空隙部分材料的变形力学分析方法,同时,引入统计损伤以及固体力学理论建立骨架部分材料的变形力学分析方法。然后,通过岩石及其组成部分的变形力学分析建立岩石宏观变形力学分析方法,从而获得能够模拟岩石变形破坏全过程的统计损伤本构模型,并给出了模型参数的确定方法。最后,将文中模型与既有模型的理论曲线与试验曲线进行了对比分析,且基于文中模型对岩石及其组成部分变形进行了讨论。研究结果表明：文中模型不仅能反映既有模型所描述的主要变形特征,还能克服其难以较好地反映初期宏观变形非线性的不足;同时,该模型还能描述岩石组成部分的变形过程,不仅揭示了岩石宏观变形与其组成部分变形之间的关系,也揭示了空隙部分材料变形是引起岩石宏观变形非线性的根本原因。表明本文模型与方法具有一定的合理性与优越性。     

沉积体系非均质性对致密油气储集层控制及其定量评价方法

以鄂尔多斯盆地延长组长8-长6致密储集层为例,采用地质统计数学(Levy Sim)和沉积地层正演模拟对沉积非均质性进行研究。结果表明,地质统计学Levy函数能够有效地表征非均质性参数,不仅可以表征油藏尺度(岩心、露头和测井曲线尺度),而且也可以有效刻画岩石微观尺度的非均质性。而三维正演地层模拟方法 Sedsim则能够有效地模拟浅水三角洲砂体的空间展布。沉积地层正演模拟与非传统地质统计学相结合,能够更真实地定量刻画不同尺度的沉积非均质性,该方法不仅有效地解决了不同尺度沉积储集层、烃源岩及盖层非均质性定量表征的难题,而且为非常规油气成藏研究提供了新思路与新的定量评价方法。     

Quantitative Analysis of Porosity Heterogeneity: Application of Geostatistics to Borehole Images

High levels of heterogeneity in many carbonate reservoirs have raised concerns about the validity and relevance of small-scale measurements from core plugs and high-resolution logs. While the measurements themselves may be accurate, they may not be representative of the average formation properties. A related question is one of reconciling the measurements made in small volume of investigation data (e.g., core plugs), with the measurements from relatively large volume of investigation data (e.g., wireline logs). This paper presents a technique to quantitatively describe the porosity heterogeneity in a borehole at the scale of several tenths of an inch. The method involves treating high-resolution borehole imagery as a 2D sample from a 3D data volume, and applying geostatistical analysis to these data. We compute the experimental semi-variogram and upscale its range and sill to larger (several inches) scales of measurement to predict the impact of heterogeneity on conventional core plug and logging tool porosity measurements. The resulting dispersion variance between the different measurement scales support the interpretation, application and comparison of these porosity measurements. This technique was applied to an Early Cretaceous carbonate reservoir in Abu Dhabi. We found that the scale of the heterogeneity is typically less than 1– 2in., so that while significant heterogeneity is observed at the core plug and smaller scales of measurement, the larger-volume logging tool measurements smooth out the heterogeneity and show considerably less variability. The differences between porosity measured in core plugs can be completely accounted for by this upscaling effect.     

不同土壤有机质组分对憎水有机物的吸附机理研究

采用完全混合一步平衡反应实验方法，选择菲、萘、三氯苯和二甲苯作为憎水有机物(HOCs)的探针，研究了它们在土壤及其中的胡敏酸、胡敏素等有机质组分中的等温平衡吸附行为，探讨了非线性吸附的机制。结果表明，HOCs在水与土壤体系中表现为非线性吸附，其非线性的吸附行为主要受到土壤有机质(SOMs)的控制。SOMs具有高度不均匀的性质，HOCs表现出非线性吸附行为是其在不同的SOMs相扩散转移以及在相对聚结的胡敏素和“黑炭”的表面吸附的结果。