Pilot Block Method Methodology to Calibrate Stochastic Permeability Fields to Dynamic Data

In the present paper, a new geostatistical parameterization technique is introduced for solving inverse problems, either in groundwater hydrology or petroleum engineering. The purpose of this is to characterize permeability at the field scale from the available dynamic data, that is, data depending on fluid displacements. Thus, a permeability model is built, which yields numerical flow answers similar to the data collected. This problem is often defined as an objective function to be minimized. We are especially focused on the possibility to locally change the permeability model, so as to further reduce the objective function. This concern is of interest when dealing with 4D-seismic data. The calibration phase consists of selecting sub-domains or pilot blocks and of varying their log-permeability averages. The permeability model is then constrained to these fictitious block-data through simple cokriging. In addition, we estimate the prior probability density function relative to the pilot block values and incorporate this prior information into the objective function. Therefore, variations in block values are governed by the optimizer while accounting for nearby point and block-data. Pilot block based optimizations provide permeability models respecting point-data at their locations, spatial variability models inferred from point-data and dynamic data in a least squares sense. A synthetic example is presented to demonstrate the applicability of the proposed matching methodology.     

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.     

用于拟合生产动态数据的现有储层模型重构(英文)

计算机能力的提升和历史拟合方面的最新进展促进了对先前建立的储层模型的重新检验。为了节省工程师和CPU的时间,我们开发了4种独特的算法,来允许无需重新进行储层研究而重建现有模型。这些算法涉及的技术包括:优化、松弛、Wiener滤波或序贯重构。基本上,它们被用来确定一个随机函数和一系列随机数。给定一个随机函数,一族随机数将产生一个实现,这个实现和现有的储层模型十分接近。一旦随机数已知,现有的储层模型将被提交到一个历史拟合过程中,以此来改进数据拟合度或者考虑新收集到的数据。我们关注的是先前建立的相储层模型。虽然我们对模型模拟的方式一无所知,但是我们可以确定一系列随机数,再用多点统计模拟方法来建造一个和现有储层模型十分接近的实现。然后运行一种新的历史拟合程序来更新现有的储层模型,使其拟合两口新生产井的流量数据。     

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.