Sensitivity Coefficients for Three-Phase Flow History Matching

To estimate the uncertainty of the neutronic parameters it is required to calculate sensitivity coefficients to cross section changes. The sensitivity coefficients are usually calculated based on the diffusion theory. However, the accuracy of the sensitivity coefficients becomes doubtful because of the use of diffusion theory. The purpose of this paper is to derive the generalized perturbation theory using the transport theory, and to evaluate transport effect on the sensitivity coefficients. Sensitivity calculations were performed in 70 energy groups. The calculated sensitivity coefficients were compared with those obtained from direct transport calculations to check the accuracy of the present method, and those obtained from the generalized perturbation calculation based on the diffusion theory to investigate the transport effect on sensitivity coefficients. From the comparison, the difference was remarkably large for the neutronic parameters such as reaction rate distribution, reaction rate ratio, and reactivity worth. For example, the sensitivity coefficient of the 238U capture to 239Pu fission rate ratio in the lower core region of the under-moderated LWR core was large in the energy range 17.6–22.6 eV, and the sensitivity to 238U capture cross section calculated by the transport theory was 20% larger than that by the diffusion theory.

Rockwell hardness testing is a widely used hardness test, and there are many scales depending on the indenter shape and test force for corresponding to various materials. The common Rockwell indenters are the diamond sphero-conical indenter and the tungsten carbide ball indenter. The influence of indenter shape error on Rockwell hardness has been acknowledged, as this shape impacts test results. To accurately estimate this uncertainty, it is essential to understand the sensitivity coefficient, which delineates the relationship between indenter shape and Rockwell hardness. Traditionally, this coefficient has been derived experimentally, but this approach faces challenges that are attributable to the nonuniformity of hardness test blocks and indenter shape measurement errors. This study introduces the estimation of sensitivity coefficients for Rockwell indenters using finite element analysis (FEA). The FEA method offers the advantage of being unaffected by the nonuniformity of test blocks and indenter shape measurement errors. We calculated the sensitivity coefficients by assessing the influence of variations in indenter angle and tip radius for diamond indenters, and indenter diameter for ball indenters, on hardness results through FEA. The sensitivity coefficients obtained via FEA for specific scales aligned with those from previous experimental studies, affirming the validity of FEA-derived coefficients for other scales. This study demonstrates that FEA is a useful tool for estimating the sensitivity coefficients of Rockwell indenters.

MARAVEN S. A., a subsidiary of Petroleos de Venezuela S. A. is operating the majority of Eocene reservoirs in the Maracaibo basin. The CLC, a large volatile oil accumulation of the deep Eocene C, presently undergoing primary depletion, will most likely need additional recovery processes. A multidisciplinary study was therefore carried out within the framework of the Joint Research Agreement between Maraven/Intevep/Shell with the objective of arriving at the best recovery scheme. The preparatory phase of the study involved process simulations which demonstrated that the vaporising gas drive prevailing under gas injection could be modelled sufficiently accurate using the volatile oil version of a black oil simulator. This was extensively checked by simulations in both 1-D and 2-D, using ternary diagram versions of compositional reservoir simulators. The main problem confronting reservoir simulation in this area is, however, how to realistically describe heterogeneities. The wide well spacing in these Eocene sandstone reservoirs precludes meaningful correlations on a well-to-well scale. Therefore, a probabilistic technique was used to create likely reservoir architectures in the form of detailed 2-D cross-section whilst still honoring measured well data. The resulting detailed reservoir models were used to simulate likely performance under water and gas injection. Results indicate that water injection is a very safe process, as both capillary and viscous crossflow from highly permeable into tight layers create a rather even flooding profile. On the other hand, gas injection is very sensitive to internal heterogeneities and vertical permeabilities. Various routes of lumping reservoir properties to arrive at coarser models for large scale field modelling were also attempted. The usefulness has been demonstrated of the concept of dynamic pseudo relative permeabilities/capillary pressures for simplying such complex sandstone reservoir models whilst still maintaining sufficient accuracy.

A three-dimensional, multiple-well, numerical simulator for simulating single- or two-phase flow of water and oil is developed for fractured reservoirs. The simulator equations are two-phase flow extensions of the single-phase flow equations derived by Warren and Root. The simulator accounts for relative fluid mobilities, gravity force, imbibition, and variation in reservoir properties. The simulator handles uniformly and nonuniformly properties. The simulator handles uniformly and nonuniformly distributed fractures and for no fractures at all. The simulator can be used to simulate the water-oil displacement process and in the transient testing of fractured reservoirs. The simulator was used on the conceptual models of two naturally fractured reservoirs: a quadrant of a five-spot reservoir and a live-well dipping reservoir with water drive. These results show the significance of imbibition in recovering oil from the reservoir rock in reservoirs with an interconnected fracture network. Introduction Numerical reservoir simulators are being used extensively to simulate multiphase, multicomponent flow in "single-porosity" petroleum reservoirs. Such simulators generally cannot be used to petroleum reservoirs. Such simulators generally cannot be used to study flow behavior in the naturally fractured reservoirs that are usually classified as double-porosity systems. In the latter, one porosity is associated with the matrix blocks and the other porosity is associated with the matrix blocks and the other represents that of the fractures and vugs. If fractures provide the main path for fluid flow from the reservoir, then usually the oil from the matrix blocks flows into the fracture space, and the fractures carry the oil to the wellbore. When water comes in contact with the oil zone, water may imbibe into the matrix blocks to displace oil. Combinations of large flow rates, low matrix permeability, and weak imbibition may result in water fingering permeability, and weak imbibition may result in water fingering through the fractures into the wellbore. Once fingering of water occurs, the water-oil ratio may increase to a large value. None of the published theoretical work on multiphase flow in naturally fractured systems has been applied directly to the simulation of a reservoir as a whole. Usually, only a segment of the reservoir was simulated, and the results were extrapolated to the entire reservoir. To simulate a reservoir as a whole, we have developed a mathematical formulation of the flow problem that has been programmed as a three-dimensional, compressible, water-oil reservoir simulator. The simulator equations are two-phase flow extensions of the single-phase flow equations derived by Warren and Root. The theory is based on the assumption of double porosity at each point in a manner that the fractures form a continuum filled by the noncontinuous matrix blocks. In other words, the fractures are the boundaries of the matrix blocks. The flow equations are solved by a finite difference method. A typical finite-difference grid cell usually contains one or several matrix blocks. In this case, all the matrix blocks within the finite-difference grid cell have the same pressure and saturation. Gravity segregation within individual matrix blocks is not calculated, but the over-all gravity segregation from one grid cell to another is accounted for. In many practical problems, this approximation is acceptable. In some situations, a matrix block encloses several finite-difference grid cells. In this case, the gravity segregation within the matrix block is calculated. To include heterogeneity, a redefinition of local porosities and permeabilities provides a method for simulating situations where part of the reservoir is fractured and where part is not fractured. The above description points to the complexity of the situations that one encounters. Therefore, the judicious choice of the number of finite difference grid cells with respect to the number of matrix blocks becomes a critical engineering decision. Later sections will provide insight to alleviate such decisions. SPEJ P. 317

Two-step Monte Carlo sensitivity analysis of alpha- and gamma-eigenvalues with the differential operator sampling method

Perturbation mapping method for sensitivity analysis of three-dimensional cracks near a free surface

The response and operation of a heat flux gage is studied using sensitivity analysis. Sensitivity analysis is the process by which one determines the sensitivity of a model output to changes in the model parameters. This process uses sensitivity coefficients, which are defined as partial derivatives of field variables (e.g. temperature) with respect to model parameters (e.g. thermal properties and boundary conditions). Computing sensitivity coefficients, in addition to the response of a heat flux gage, aids in identifying model parameters that significantly impact the temperature response. A control volume finite element based code is used to implement numerical sensitivity coefficient calculations, allowing general problems to be studied. Sensitivity coefficients are discussed for the well known Gardon gage.

Efficient sensitivity analysis method for chaotic dynamical systems

Summary Adjoint equations for 3D, three-phase flow problems are developed and implemented to calculate the sensitivity of production data to permeability fields and well skin factors. Typically, the development of adjoint equations is rather tedious, but the formulation given here makes the development and numerical solution of adjoint equations straightforward. The procedure for calculating sensitivity coefficients is coupled with a fully implicit finite-difference simulator to develop an automatic history-matching procedure. Even if one does not wish to fully automate history matching, sensitivity coefficients are useful for understanding the physics that control 3D multiphase flow and the influence of local reservoir properties on production data. Procedures to estimate the value of particular types of data [pressure, producing gas/oil ratio (GOR), and water/oil ratio (WOR)] for reducing the uncertainty in estimates of reservoir properties are applied and discussed.

Sensitivity coefficients are calculated, for the simple thickness modes of quartz plates, due to perturbations of the elastic, piezoelectric, and dielectric material values. These are given in tabular form for various cuts of practical and analytic interest. Also shown are a number of sensitivity coefficient surfaces using the two plate orientational angles as variables.

To materialize actual designs in computer-aided design, we presented a method to determine structural dimensions in order to make all design requirements accomplish their respective objective values, using the attainability and simultaneous equations with several unknowns. However, most of the designs have no theoretical design formula or sensitivity coefficients. This paper presents an approximation method to identify sensitivity coefficients by trial production evaluations of the designs. The coefficient matrix is specified between k th revised values of the attainabilities and k th revised values of the design variables, and made into a square or a triangular matrix by neglecting insignificant variables. The matrix is solved iteratively until all coefficients converge. This paper also presents a multiobjective satisfactory design using the identified coefficients to prove the effectiveness of the identification, comparing the identified coefficients with the theoretical ones.

The standardized FAO56 Penman-Monteith model, which has been the most reasonable method in both humid and arid climatic conditions, provides reference evapotranspiration (ETo) estimates for planning and efficient use of agricultural water resources. And sensitivity analysis is important in understanding the relative importance of climatic variables to the variation of reference evapotranspiration. In this study, a non-dimensional relative sensitivity coefficient was employed to predict responses of ETo to perturbations of four climatic variables in the Ejina oasis northwest China. A 20-year historical dataset of daily air temperature, wind speed, relative humidity and daily sunshine duration in the Ejina oasis was used in the analysis. Results have shown that daily sensitivity coefficients exhibited large fluctuations during the growing season, and shortwave radiation was the most sensitive variable in general for the Ejina oasis, followed by air temperature, wind speed and relative humidity. According to this study, the response of ETo can be preferably predicted under perturbation of air temperature, wind speed, relative humidity and shortwave radiation by their sensitivity coefficients.

Dependence of sensitivity coefficient of a nude-type Bayard–Alpert gauge on the diameter of an envelope

In naturally fractured reservoirs, the fracture intensity (density) control the fracture permeability and the dimensions of matrix blocks, and hence govern the reservoir production performance and oil recovery process. In the last decade, great progress has been made in the characterization of natural fracture system of naturally fracture reservoirs. The statistical fracture intensity model can be constructed by integrating the direct local data, such as core and image log data, and field scale data derived from seismic attributes, fault-related-strain-field, structural information (curvature), etc. The static fracture model must be converted into flow simulation model for dynamic simulation, and must be adjusted to honor the production history to perform production performance prediction and optimal development planning.This paper discusses the development of a novel method of conditioning the fracture intensity model to the production data. This is achieved by making the sensitivity coefficients available of production data with respect to fracture intensity through the building of correlations between fracture intensity and fracture permeability and fracture/matrix coupling factor.Gradient simulator and adjoint methods are used to calculate the sensitivity coefficients of production data to the parameters of fracture and matrix blocks. Both two methods usually involve solving of a great number of matrix equations. Sparse LU decomposition is used to solve the matrix equations to improve the computational efficiency of sensitivity coefficient. To the best of our knowledge, the particular implementation of the sparse LU direct methods in automatic history matching is new to the petroleum engineering literature.The proposed method and developed software is validated by applying to a giant Middle East carbonate naturally fracture reservoir.

Spatiotemporal interactions between soil moisture and water availability across the Yellow River Basin, China