Calculation Of Sensitivity Coefficients Research Articles

Kinetic investigation of the effect of the amino acid side chains in the selective acidolysis ofN-acyl-N,α,α-trialkyl glycine amides

Accurate kinetic measurements of the rate constants for the acidolysis of five N-acetyl-N-(4-methoxybenzyl)-alpha,alpha-trialkyl glycine cyclohexyl amides in TFA were performed at 25 degrees C and the reactions monitored by HPLC. In all cases the results were consistent with a first order behaviour with respect to the substrate. No direct correlation was obtained with these data between the rate constant values and structure, but a good correlation coefficient was obtained when a multiple regression analysis was applied by taking advantage of a Taft equation using appropriate polar and steric substituent parameters. In a plot of the values observed for log k against those calculated by this equation all five points fell very close to the line of perfect correlation. The calculated sensitivity coefficients to polar and steric contributions were used to discuss the experimental results and showed that the acidolysis were comparatively less affected by steric effects than expected.

Simultaneous Estimation of Absolute and Relative Permeability by Automatic History Matching of Three-Phase Flow Production Data

Abstract This paper focuses on the simultaneous estimation of the absolute permeability field and relative permeability curves from three-phase flow production data. Irreducible water saturation, critical gas saturation, and residual oil saturations are assumed to be known. The two-phase relative permeability curves for an oil-gas system and the two-phase relative permeability curves for an oil-water system are represented by power law models. The three-phase oil relative permeability curve is calculated from the two sets of two-phase curves using Stone's Model II. The adjoint method is applied to three-phase flow problems to calculate the sensitivity of production data to the absolute permeability field and the parameters defining the relative permeability functions. Using the calculated sensitivity coefficients, absolute permeability, and relative permeability fields are estimated by automatic history matching of production data. Introduction The main objective of this paper is to consider the feasibility of estimating absolute permeability fields and parameters that define relative permeability functions by automatic history matching of production data obtained under multiphase flow conditions. While the topic is not new, to the best of our knowledge, no paper in the petroleum engineering literature has considered this problemunder three-phase flow conditions. It appears that Archer and Wong(1) were the first authors to consider the estimation of relative permeability curves by applying a reservoir simulator to history match laboratory core flood data. They estimated only parameters that define the shape of relative permeability curves for simple empirical relative permeability models and adjusted relative permeabilities by a trial and error method during the history matching. Sigmund and McCaffery(2) were the first to apply nonlinear regression to the problem of history matching laboratory core flood data. They used power law expressions to model relative permeability curves and estimated only the two exponential parameters in these formulas. Kerig and Watson(3) considered a similar problem. They calculated predicted data from the Buckley- Leverett model, used cubic splines to parameterize relative permeability curves and compared relative permeability estimates obtained with such a representation to those obtained using a power law functional form. They showed that, in general, power law models do not contain enough degrees of freedom to represent the truth well, whereas cubic splines with a small number of knots appear to be sufficiently flexible to yield more accurate estimates of true relative permeability curves. In their results, they assume absolute permeability is known. Lee and Seinfeld(4) considered the simultaneous estimation of the absolute permeability field and relative permeabilities for a two-dimensional, two-phase flow oil-water system. They assumed power law relative permeability curves and assumed that the end point values of relative permeabilities were known. Thus, only the two exponents in the power law relative permeability functions were estimated. They modelled the two-dimensional isotropic heterogeneousermeability field using bi-cubic B-splines. In the specific xamples considered, they matched pressure and water cut data at wells producing from an oil reservoir under waterflood. Tikhonov(5) regularization was used to stabilize the nonlinear least squares problem.

Stochastic inverse mapping of hydraulic conductivity and sorption partitioning coefficient fields conditioning on nonreactive and reactive tracer test data

A three‐dimensional geostatistically based iterative inverse method is presented for mapping spatial distributions of the hydraulic conductivity and sorption partitioning coefficient fields by sequential conditioning on both nonreactive and reactive tracer breakthrough data. A streamline‐based semianalytical simulator is adopted to simulate chemical movement in a physically and chemically heterogeneous field and serves as the forward modeling. In this study, both the hydraulic conductivity and sorption coefficient are assumed to be random spatial variables. Within the framework of the streamline‐based simulator an efficient semianalytical method is developed to calculate sensitivity coefficients of the reactive chemical concentration with respect to the changes of conductivity and sorption coefficient. The calculated sensitivities account for spatial correlations between the solute concentration and parameters. The performance of the inverse method is assessed by a synthetic tracer test conducted within an aquifer with distinct spatial features of physical and chemical heterogeneities. The study results indicate that our iterative stochastic inverse method is able to identify and reproduce the large‐scale physical and chemical heterogeneity features. The conditional study on the geostatistical distributions of conductivity and sorption coefficient are expected to significantly reduce prediction uncertainties for solute transport in the medium.

History Matching of Three-Phase Flow Production Data

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.

Inversion of dynamic production data for permeability: fast streamline-based computation of sensitivity coefficients of fractional flow rate

Generation of permeability field in a reservoir model that matchs historical dynamic production data requires an inverse calculation. A gradient method is typically used to solve the inverse minimization problem and requires sensitivity coefficients of reservoir responses, e.g. fractional flow rate or pressure, with respect to the change in the permeability. This paper presents a novel semi-analytical streamline-based method for computing such sensitivity coefficients under the framework of two-phase (oil–water) flow conditions. This method is shown to be significantly faster and generate permeability fields with lower objective function than the traditional perturbation method. The method decomposes the multiple-dimensional full flow problem into multiple 1D problems along streamlines. The sensitivity of fractional flow rate at the production well is directly related to the sensitivity of time-of-flight (TOF) along each individual streamline and the sensitivity of pressure at grid cells along the streamline. The sensitivity of TOF of a streamline can be obtained analytically. The sensitivity of pressure is obtained as part of a fast single phase flow simulation. The proposed method is implemented in a geostatistically based inverse technique, called the sequential self-calibration (SSC) method. Results for fractional flow rate sensitivities are presented and compared with the traditional perturbation method. This new method can be easily extended to compute sensitivity coefficients of saturation (concentration) data.

Sensitivity Coefficients for Three-Phase Flow History Matching

Abstract Changing the value of the porosity or the horizontal or vertical permeability in any grid cell in a reservoir simulator by a small amount often results in a small change in the value of a property predicted by the simulator. The ratio of change in prediction to change in reservoir property is called the sensitivity coefficient. In this paper, we describe the use of the adjoint system of equations to compute the sensitivity of wellbore pressure, water-oil ratio, and gas-oil ratio to changes in gridblock permeability and porosity. Unlike some other methods of computing sensitivity coefficients, this method is applicable for problems with large numbers of model parameters and for problems in which cross-flow and compressibility are significant. Although the adjoint system has previously been used to compute the gradient of an objective function for three-phase flow data with respect to model parameters, the sensitivity coefficients are significantly more powerful as they enable the use of Newton-like methods with quadratic convergence, i.e., estimation of the covariance of model estimates based on the inverse of the Hessian, and they provide insight into the information content of data. We use several three-phase flow examples with solution-gas drive, gas injection, and gravity segregation to illustrate these ideas. Introduction In this work, we develop a computationally efficient adjoint procedure for computing sensitivity coefficients for general threedimensional, three-phase flow problems. Although originally introduced for linear problems by Chen et al.(1) and Chavent et al.,(2) the adjoint method has also been applied to multiphase flow history matching problems; see, for example, Wasserman et al.,(3) Lee and Seinfeld(4), Yang and Watson(5), and Makhlouf et al.(6) These authors applied the adjoint method to compute the gradient of the sum of the squares of the production data mismatch terms and minimized the data mismatch function using steepest descent or conjugate gradient algorithms. The steepest descent and conjugate gradient methods tend to converge slowly, however, so Newton-like methods are advantageous when the number of data is small. For example, Wu et al.(7) applied the adjoint method to calculate sensitivity coefficients of wellbore pressure and WOR with respect to model parameters in two-dimensional, two-phase (oil and water) problems and used a Gauss-Newton algorithm to minimize the data mismatch function. Approximately ten iterations were required to obtain convergence for a problem with 1,250 model parameters. A more common choice for computing sensitivity coefficients for multiphase flow problems is the sensitivity equation(8) or gradient simulator method(9). Tan and Kalogerakis(10) used the sensitivity equations to compute the sensitivity of production data to a relatively few model parameters. Zonation was used in their work as a method for reducing the number of parameters, and correlation between parameters was used to determine which parameters should be combined. They discussed the importance of computation of sensitivities for estimation of covariance of model parameters, and for obtaining the most rapid convergence possible with the Gauss-Newton method. Tan(11) used the sensitivity equation approach to compute sensitivities for a three-dimensional, three-phase flow problem.

Explicit calculation of smoothed sensitivity coefficients for linear problems

AbstractA technique of explicit calculation of sensitivity coefficients based on the approximation of the retrieved function by a linear combination of trial functions of compact support is presented. The method is applicable to steady state and transient linear inverse problems where unknown distributions of boundary fluxes, temperatures, initial conditions or source terms are retrieved. The sensitivity coefficients are obtained by solving a sequence of boundary value problems with boundary conditions and source term being homogeneous except for one term. This inhomogeneous term is taken as subsequent trial functions. Depending on the type of the retrieved function, it may appear on boundary conditions (Dirichlet or Neumann), initial conditions or the source term. Commercial software and analytic techniques can be used to solve this sequence of boundary value problems producing the required sensitivity coefficients. The choice of the approximating functions guarantees a filtration of the high frequency errors. Several numerical examples are included where the sensitivity coefficients are used to retrieve the unknown values of boundary fluxes in transient state and volumetric sources. Analytic, boundary‐element and finite‐element techniques are employed in the study. Copyright © 2003 John Wiley & Sons, Ltd.

Calculating measurement uncertainty for complex-valued quantities

A software technique is described that provides support for uncertaintycalculations for complex-valued measurements. The technique is based on classicalmethods of multivariate statistics and propagation of variance as well asautomatic differentiation, which is an established computational method. Itfacilitates propagation of covariance data by automating the calculation ofsensitivity coefficients. The technique introduces a simple abstraction torepresent measurement data and defines arithmetic operations and standardfunctions that manipulate this abstraction. The method is not difficult toimplement and is easy to use. An example source code in C++ is included.

Numerical Calculation of Sensitivity Coefficients to a Permeability Field

AbstractThe calculating formula of sensitivity coefficients to a permeability field are derived from initial boundary value problems of fluid equations in porous media by the Green's reciprocity principle. These coefficients of a discretized form is obtained with the help of the numerical integration and the finite difference method. The sensitivity coefficients of wellbore pressure to permeability parameters are determined by the Peaceman's equation and calculated by the numerical calculation method of 3‐D transient anisotropic fluid equation in porous media and verified with direct method. The computing results by this method are in excellent agreement with those obtained by the direct method.

Uncertainty Propagation Methodology and Nuclear Data Sensitivity Analysis in 3D Heterogeneous Problems

This paper presents a 3D uncertainty propagation methodology and its application to the case of a small heterogeneous reactor system (“slab” reactor benchmark). Key neutron parameters (keff, reactivity worth, local power, …) and their corresponding cross-section sensitivities are derived by using the French calculation route APOLLO2 (2D transport lattice code), CRONOS2 (3D diffusion code) and TRIPOLI4 (3D Monte-Carlo reference calculations) with consistent JEF2.2 cross-section libraries (punctual or CEA93 multigroup cross-sections) and adapted perturbation methods (the Heuristically-based Generalized Perturbation Theory implemented in the framework of the CRONOS2 diffusion method or the correlation techniques used in Monte-Carlo simulations). The investigation of the slab system underlined notable differences between the 2D/3D computed sensitivity coefficients and consequently a priori uncertainties (when sensitivity coefficients are combined with covariance matrices the discrepancies rise up to 20% due to thermal and fast flux variations). In addition, the induced local power effect of nuclear data perturbations (JEF-2.2 vs. Leal-Derrien-Wright-Larson 235U evaluation) had been be correctly estimated with the standard 3D CRONOS2 depletion calculations. For industrial applications (PWR neutron parameters optimization problems, R&D studies dealing with the design of future fission reactors, …), the same calculation route could be advantageously applied to infer the target accuracies (knowing the required safety criteria) of future nuclear data evaluation (JEFF-3 data library for instance).

Sensitivity coefficients of single-phase flow in low-permeability heterogeneous reservoirs

Theoretical equations for computing sensitivity coefficients of wellbore pressures to estimate the reservoir parameters in low-permeability reservoirs conditioning to non-Darcy flow data at low velocity were obtained. It is shown by a lot of numerical calculations that the wellbore pressures are much more sensitive to permeability very near the well than to permeability a few gridblocks away from the well. When an initial pressure gradient existent sensitivity coefficients in the region are closer to the active well than to the observation well. Sensitivity coefficients of observation well at the line between the active well and the observation well are influenced greatly by the initial pressure gradient.

Drift design of steel‐frame shear‐wall systems for tall buildings

AbstractDrift design methods based on resizing algorithms are presented to control lateral displacements of steel‐frame shear‐wall systems for tall buildings. Three algorithms for resizing of structural members of the steel‐frame shear‐wall systems are derived by formulating the drift design process into an optimization problem that minimizes lateral displacement of the system without changing the weight of a structure. During the drift design process, cost‐effective displacement participation factors obtained by the energy method are used to determine the amount of material to be modified instead of calculating sensitivity coefficients. The overall structural design model with the drift design method for the steel‐frame shear‐wall systems is proposed and applied to the structural design of three examples. As demonstrated in the examples, the lateral displacement and interstorey drift of a frame shear‐wall system can be effectively designed by the drift design method without the time‐consuming trial‐and‐error process. Copyright © 2002 John Wiley & Sons, Ltd.

Residual Ozone Measurement: Indigo Sensitivity Coefficient Adjustment

The determination of residual ozone using indigotrisulfonate (Standard Methods 4500–O3B) assumes a molar absorptivity of–20,000 M -1cm -1 and a constant sensitivity coefficient of 0.42 L mg-1cm-1. Data are presented showing large molar absorptivity differences and subsequent differences in the calculated sensitivity coefficients for various sources of indigo. The variation in molar absorptivity (up to 15%) emphasizes the need for measuring the sensitivity coefficient on a regular basis. A revised simple calculation that compensates for decomposing indigo reagents is proposed. A modified equation that can be used with the current Standard Method procedure is given. However, for the most accurate results, it is recommended that the dry indigo reagent be periodically calibrated.

Construction of geostatistical aquifer models integrating dynamic flow and tracer data using inverse technique

Natural aquifers are heterogeneous, and geostatistical methods are widely used to simulate the heterogeneity of aquifer properties. Due to limited available data, it is essential to integrate as much information as possible to reduce the uncertainty in aquifer models and flow predictions. Traditional geostatistical techniques efficiently consider static hard and soft information, such as core data and seismic data. However, dynamic flow and transport data, such as flow rates, pressure, and tracer breakthrough, are important information that are not easily considered with the traditional techniques. Integrating such dynamic data into a model requires the solution of a difficult inverse problem, since dynamic data and aquifer properties are related to each other through the non-linear flow and transport equations. A recently developed geostatistically based inverse technique, the sequential self-calibration (SSC) method, is introduced to integrate those dynamic data. The SSC method is an iterative inverse technique that is coupled with an optimization procedure. It provides for fast generation of multiple realizations of aquifer property models that jointly match pressure and tracer breakthrough data, yet display the same geostatistical characteristics. This method is flexible, computationally efficient, and robust. The main features of SSC include (1) a master point concept that reduces the number of parameters, (2) a perturbation mechanism based on kriging that accounts for the spatial correlation of the aquifer properties, (3) a fast streamline-based tracer flow simulator for integration of tracer data, and (4) a new semi-analytical streamline-based method for computing sensitivity coefficients of tracer breakthrough. Applications of the SSC method are demonstrated with a synthetic data set. Results show that tracer breakthrough data carry important information on the spatial variation of aquifer permeability in the inter-well areas. As a contrast, pressure data provide information at near well-bore areas only. Integrating pressure and breakthrough data jointly leads to significant improvement in the aquifer heterogeneity representation and a reduction in the uncertainty of aquifer model. The accuracy of flow and transport predictions can be dramatically improved by integrating dynamic data.

Sensitivity Analysis of Nuclide Importance to One-Group Neutron Cross Sections

The importance of nuclides is useful when investigating nuclide characteristics in a given neutron spectrum. However, it is derived using one-group microscopic cross sections, which may contain large errors or uncertainties. The sensitivity coefficient shows the effect of these errors or uncertainties on the importance.The equations for calculating sensitivity coefficients of importance to one-group nuclear constants are derived using the perturbation method. Numerical values are also evaluated for some important cases for fast and thermal reactor systems.Many characteristics of the sensitivity coefficients are derived from the derived equations and numerical results. The matrix of sensitivity coefficients seems diagonally dominant. However, it is not always satisfied in a detailed structure. The detailed structure of the matrix and the characteristics of coefficients are given.By using the obtained sensitivity coefficients, some demonstration calculations have been performed. The effects of error and uncertainty of nuclear data and of the change of one-group cross-section input caused by fuel design changes through the neutron spectrum are investigated. These calculations show that the sensitivity coefficient is useful when evaluating error or uncertainty of nuclide importance caused by the cross-section data error or uncertainty and when checking effectiveness of fuel cell or core design change for improving neutron economy.

Eigenvalue Sensitivity Theory for Resonance-Shielded Cross Sections

A method is presented to compute sensitivity coefficients for the eigenvalue of a critical assembly, including implicit effects associated with changes in resonance-shielded multigroup cross sections. Two alternative approaches, based on a forward and an adjoint solution, respectively, are developed to determine the effect of perturbations on the weight function used in group averaging of resonance cross sections. The forward method uses an automated methodology to compute the flux derivative with respect to various cross-section processing parameters. The adjoint method introduces adjoint equations for a multigroup cross-section functional and presents adjoint slowing-down equations for two common methods of resonance self-shielding. Expressions are presented for sensitivity coefficients of self-shielded group cross sections. These sensitivity coefficients are combined with conventional eigenvalue sensitivity coefficients to obtain a general perturbation expression for the multiplication factor. An example application determines the sensitivity of the critical eigenvalue to hydrogen density changes in a homogeneous sphere containing low-enriched uranium. It is shown that changes in 238U-shielded cross sections caused by perturbations in hydrogen concentrations are important components in the overall eigenvalue sensitivity coefficient, which is predicted well by the developed method.

Kinetic analysis of the chemical processes in the decomposition of gaseous dielectrics by a non-equilibrium plasma - part 2: SF6 and SF6/O2

In this work, a numerical modelling analysis of the gas-phase decomposition of pure SF6 and SF6/O2 mixtures, in the presence of silicon was performed. The relative rate of individual processes, the effect of the parameters uncertainties and the sensitivity coefficients were determined. The results were compared with literature experimental data for the plasma etching of silicon and with previous simulated results to adjust the model parameters. As in the CF4 system, the main etching agent is atomic fluorine and the concentration of the major species depends on the composition of the mixture. The shape of the sensitivity curves follows the general shape of the individual rate curves and the ratio between the calculated sensitivity coefficients is closely related to the contribution of each reaction.

Integrating Dynamic Data Into High-Resolution Reservoir Models Using Streamline-Based Analytic Sensitivity Coefficients

SummaryOne of the outstanding challenges in reservoir characterization is to build high-resolution reservoir models that satisfy static as well as dynamic data. However, integration of dynamic data typically requires the solution of an inverse problem that can be computationally intensive and becomes practically infeasible for fine-scale reservoir models. A critical issue here is computation of sensitivity coefficients, the derivatives of dynamic production history with respect to model parameters such as permeability and porosity.We propose a new analytic technique that has several advantages over existing approaches. First, the method utilizes an extremely efficient three-dimensional multiphase streamline simulator as a forward model. Second, the parameter sensitivities are formulated in terms of one-dimensional integrals of analytic functions along the streamlines. Thus, the computation of sensitivities for all model parameters requires only a single simulation run to construct the velocity field and generate the streamlines. The integration of dynamic data is then performed using a two-step iterative inversion that involves (i) "lining up" the breakthrough times at the producing wells and then (ii) matching the production history. Our approach follows from an analogy between streamlines and ray tracing in seismology. The inverse method is analogous to seismic waveform inversion and thus, allows us to utilize efficient methods from geophysical imaging.We have applied the proposed approach to a highly heterogeneous carbonate reservoir in west Texas. The reservoir model consists of 50,000 cells and includes multiple patterns with 42 wells. Water-cut histories from 27 producing wells are utilized to characterize porosity and permeability distribution in the reservoir, a total of 100,000 parameters.

A direct method of calculating sensitivity coefficients of chemical kinetics

In this paper, a new direct method of calculating the first-order sensitivity coefficients using sparse matrix technology to chemical kinetics is presented. The Gear type procedure is used to integrate a model equation and its coupled auxiliary sensitivity coefficient equations. Because the Jacobian matrix of the model equation is the same as that of the sensitivity coefficient equation with respect to a parameter, it is only necessary to triangularize the matrix related to the Jacobian matrix of the model equation. The FORTRAN subroutines of the model equation, the sensitivity coefficient equations, and their Jacobian analytical expressions are generated automatically from chemical mechanism. This method greatly increases the efficiency of computation by taking advantage of the fact that the auxiliary equations for different sensitivity coefficients are linear and quite similar. Two sets of chemical reactions are used to illustrate this approach: oxidation of formaldehyde in the presence of carbon monoxide and photo-oxidation of dimethyl sulfide. The accuracy and computational efficiency of the new direct method is demonstrated by comparing the results from the new direct method and from the indirect method.

APPLICATION OF SENSITIVITY COEFFICIENTS FOR HEAT CONDUCTION PROBLEMS

In parameter estimation, considerable insight is provided by examining sensitivity coefficients. This article focuses on the use of sensitivity coefficients in connection with estimating thermal properties in the heat conduction equation. It is demonstrated how sensitivity coefficients are used in the design analysis of parameter estimation experiments. A general methodology for computing sensitivity coefficients can be an important design tool in many other regards also. A control-volume, finite-element program is used to implement numerical sensitivity coefficient calculations. In this approach, general problems can be studied. Several example cases are presented to demonstrate the insight gained from sensitivity coefficients. The cases are selected from experimental studies to characterize the thermal properties of carbon carbon composite. However, no experimental data are reported. Sensitivity coefficients show that in an experiment that is not well designed, additional materials in the experimental configuration can have a larger impact on the temperature than the material of interest. Two-dimensional configurations demonstrate that there can be isolated areas of insensitivity and the difficulty of estimating multiple parameters.