scholarly journals A New Algorithm for Automatic History Matching

1974 ◽  
Vol 14 (06) ◽  
pp. 593-608 ◽  
Author(s):  
W.H. Chen ◽  
G.R. Gavalas ◽  
J.H. Seinfeld ◽  
M.L. Wasserman
Keyword(s):  
Optimal Control ◽  
History Matching ◽  
Computing Time ◽  
Optimization Methods ◽  
Continuous Functions ◽  
Parameter Estimates ◽  
Matching Problem ◽  
Significant Saving ◽  
Sensitivity Coefficients ◽  
Gradient Optimization

Abstract History-matching problems, in which reservoir parameters are to be estimated from well pressure parameters are to be estimated from well pressure data, are formulated as optimal control problems. The necessary conditions for optimality lead naturally to gradient optimization methods for determining the optimal parameter estimates. the key feature of the approach is that reservoir properties are considered as continuous functions properties are considered as continuous functions of position rather than as uniform in a certain number of zones. The optimal control approach is illustrated on a hypothetical reservoir and on an actual Saudi Arabian reservoir, both characterized by single-phase flow. A significant saving in computing time over conventional constant-zone gradient optimization methods is demonstrated. Introduction The process of determining in a mathematical reservoir model unknown parameter valuessuch as permeability and porositythat give the closest permeability and porositythat give the closest fit of measured and calculated pressures is commonly called "history matching." In principle, one would like an automatic routine for history matching, applicable to simulators of varying complexity, one that does not require inordinate amounts of computing time to achieve a set of parameter estimates. In recent years a number of authors have investigated the subject of history matching. All the reported approaches involve dividing the reservoir into a number of zones, in each of which the properties to be estimated are assumed to be uniform. (These zones may, in fact, correspond to the spatial grid employed for the finite-difference solution of the simulator.) Then the history-matching problem becomes that of determining the parameter problem becomes that of determining the parameter values in each of, say, N zones, k1, k2, ..., kN, in such a way that some measure (usually a sum of squares) of the deviation between calculated and observed pressures is minimized. A typical measure of deviation pressures is minimized. A typical measure of deviation is(1) where p obs (j, ti) and p cal (j, ti) are the observed and calculated pressures at the jth well, which is at location j=(xj, yj), j = 1,2,......, M, and where we have n1 measurements at Well 1 at n1 different times, n2 measurements at Well 2 at n2 different times, . . ., and nM measurements at Well M at nM different times. To carry out the minimization of Eq. 1 with respect to the vector k, most methods rely on some type of gradient optimization procedure that requires computation of the gradient of J with respect to each ki, i = 1, 2, . . ., N. The calculation of J/ ki usually requires, in turn, that one obtain the sensitivity coefficients, p cal/ ki, i = 1, 2, . . ., N; i.e., the first partial derivative of pressure with respect to each parameter. The sensitivity coefficients can be computed, in principle, in several ways. 1. Make a simulator base run with all N parameters at their initial values. Then, perturbing each parameter a small amount, make an additional simulator run for each parameter in the system. parameter in the system. SPEJ P. 593

History Matching in Two-Phase Petroleum Reservoirs

1980 ◽  
Vol 20 (06) ◽  
pp. 521-532 ◽  
Author(s):  
A.T. Watson ◽  
J.H. Seinfeld ◽  
G.R. Gavalas ◽  
P.T. Woo
Keyword(s):  
Optimal Control ◽  
Parameter Estimation ◽  
Objective Function ◽  
History Matching ◽  
Computing Time ◽  
Absolute Permeability ◽  
Two Phase ◽  
Automatic History Matching ◽  
Control Approach ◽  
First Order

Abstract An automatic history-matching algorithm based onan optimal control approach has been formulated forjoint estimation of spatially varying permeability andporosity and coefficients of relative permeabilityfunctions in two-phase reservoirs. The algorithm usespressure and production rate data simultaneously. The performance of the algorithm for thewaterflooding of one- and two-dimensional hypotheticalreservoirs is examined, and properties associatedwith the parameter estimation problem are discussed. Introduction There has been considerable interest in thedevelopment of automatic history-matchingalgorithms. Most of the published work to date onautomatic history matching has been devoted tosingle-phase reservoirs in which the unknownparameters to be estimated are often the reservoirporosity (or storage) and absolute permeability (ortransmissibility). In the single-phase problem, theobjective function usually consists of the deviationsbetween the predicted and measured reservoirpressures at the wells. Parameter estimation, orhistory matching, in multiphase reservoirs isfundamentally more difficult than in single-phasereservoirs. The multiphase equations are nonlinear, and in addition to the porosity and absolutepermeability, the relative permeabilities of each phasemay be unknown and subject to estimation. Measurements of the relative rates of flow of oil, water, and gas at the wells also may be available forthe objective function. The aspect of the reservoir history-matchingproblem that distinguishes it from other parameterestimation problems in science and engineering is thelarge dimensionality of both the system state and theunknown parameters. As a result of this largedimensionality, computational efficiency becomes aprime consideration in the implementation of anautomatic history-matching method. In all parameterestimation methods, a trade-off exists between theamount of computation performed per iteration andthe speed of convergence of the method. Animportant saving in computing time was realized insingle-phase automatic history matching through theintroduction of optimal control theory as a methodfor calculating the gradient of the objective functionwith respect to the unknown parameters. Thistechnique currently is limited to first-order gradientmethods. First-order gradient methods generallyconverge more slowly than those of higher order.Nevertheless, the amount of computation requiredper iteration is significantly less than that requiredfor higher-order optimization methods; thus, first-order methods are attractive for automatic historymatching. The optimal control algorithm forautomatic history matching has been shown toproduce excellent results when applied to field problems. Therefore, the first approach to thedevelopment of a general automatic history-matchingalgorithm for multiphase reservoirs wouldseem to proceed through the development of anoptimal control approach for calculating the gradientof the objective function with respect to theparameters for use in a first-order method. SPEJ P. 521＾

Practical Applications of Optimal-Control Theory to History-Matching Multiphase Simulator Models

1975 ◽  
Vol 15 (04) ◽  
pp. 347-355 ◽  
Author(s):  
M.L. Wasserman ◽  
A.S. Emanuel ◽  
J.H. Seinfeld
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Optimal Control ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
History Matching ◽  
Single Phase ◽  
Continuous Functions ◽  
Partial Differential

Abstract This paper applies material presented by Chen et al. and by Chavent et al to practical reservoir problems. The pressure history-matching algorithm used is initially based on a discretized single-phase reservoir model. Multiphase effects are approximately treated in the single-phase model by multiplying the transmissibility and storage terms by saturation-dependent terms that are obtained from a multiphase simulator run. Thus, all the history matching is performed by a "pseduo" single-phase model. The multiplicative factors for transmissibility and storage are updated when necessary. The matching technique can change any model permeability thickness or porosity thickness value. Three field examples are given. Introduction History matching using optimal-control theory was introduced by two sets of authors. Their contributions were a major breakthrough in attacking the long-standing goal of automatic history matching. This paper extends the work presented by Chen et al. and Chavent et al. Specifically, we focus on three areas.We derive the optimal-control algorithm using a discrete formulation. Our reservoir simulator, which is a set of ordinary differential equations, is adjoined to the function to be minimized. The first variation is taken to yield equations for computing Lagrange multipliers. These Lagrange multipliers are then used for computing a gradient vector. The discrete formulation keeps the adjoint equations consistent with the reservoir simulator.We include the effects of saturation change in history-matching pressures. We do this in a fashion that circumvents the need for developing a full multiphase optimal-control code.We show detailed results of the application of the optimal-control algorithm to three field examples. DERIVATION OF ADJOINT EQUATIONS Most implicit-pressure/explicit-saturation-type, finite-difference reservoir simulators perform two calculation stages for each time step. The first stage involves solving an "expansivity equation" for pressure. The expansivity equation is obtained by summing the material-balance equations for oil, gas, and water flow. Once the pressures are implicitly obtained from the expansivity equation, the phase saturations can be updated using their respective balance equations. A typical expansivity equation is shown in Appendix B, Eq. B-1. When we write the reservoir simulation equations as partial differential equations, we assume that the parameters to be estimated are continuous functions of position. The partial-differential-equation formulation is partial-differential-equation formulation is generally termed a distributed-parameter system. However, upon solving these partial differential equations, the model is discretized so that the partial differential equations are replaced by partial differential equations are replaced by sets of ordinary differential equations, and the parameters that were continuous functions of parameters that were continuous functions of position become specific values. Eq. B-1 is a position become specific values. Eq. B-1 is a set of ordinary differential equations that reflects lumping of parameters. Each cell has three associated parameters: a right-side permeability thickness, a bottom permeability thickness, and a pore volume. pore volume.Once the discretized model is written and we have one or more ordinary differential equations per cell, we can then adjoin these differential equations to the integral to be minimized by using one Lagrange multiplier per differential equation. The ordinary differential equations for the Lagrange multipliers are now derived as part of the necessary conditions for stationariness of the augmented objective function. These ordinary differential equations are termed the adjoint system of equations. A detailed example of the procedure discussed in this paragraph is given in Appendix A. The ordinary-differential-equation formulation of the optimal-control algorithm is more appropriate for use with reservoir simulators than the partial-differential-equation derivation found in partial-differential-equation derivation found in Refs. 1 and 2. SPEJ P. 347

CALCULATING THE BOUNDARIES OF THE AIRCRAFT EXIT ZONE TO A SET POINT

2021 ◽  
pp. 3-11
Author(s):  
O. N. Korsun ◽  
A. V. Stulovsky ◽  
S. V. Nikolaev
Keyword(s):  
Mathematical Model ◽  
Optimal Control ◽  
Direct Method ◽  
Vertical Plane ◽  
Control Object ◽  
Aircraft Design ◽  
Continuous Functions ◽  
Cubic Splines ◽  
Parameters Estimation ◽  
Parameter Estimates

The article considers the method of calculating the boundaries of the exit zone of the aircraft to a given point based on the optimal control method. To find the optimal control, it is proposed to use a direct method based on parameterization of the desired control signals using third-order Hermitian splines. The choice of Hermitian cubic splines was motivated by the fact that these splines and their first order derivatives are smooth and continuous functions, on the one hand, and, on the other, do not require the additional solution of algebraic equations to meet the specific conditions in spline nodes which is obligatory for classic cubic splines. Spline parameters estimation is achieved through solution of the unconditional multiparametric optimization problem. The target functional includes the squares of mismatches between the desired output signals and the object model output signals. In this paper the parameter estimates are obtained using the widely known numerical optimization algorithm – the particle swarm method. The paper considers the aircraft motion in the vertical plane, for which a mathematical model of the control object is formed and the target functional is formulated. The proposed solution is advisable to apply when calculating the optimal trajectories and flight profiles of aircraft when planning their functioning for the designed purpose. The developed method allows solving a number of tasks in the process of modern aircraft design and flight tests. The application of the proposed method, the required structure of the mathematical model of the object and the features of the formation of the minimized functional are shown in a specific example.

Confidence Limits on the Parameters and Predictions of Slightly Compressible, Single-Phase Reservoirs

1977 ◽  
Vol 17 (01) ◽  
pp. 42-56 ◽  
Author(s):  
A.H. Dogru ◽  
T.N. Dixon ◽  
T.F. Edgar
Keyword(s):  
Measurement Errors ◽  
History Matching ◽  
Optimization Procedure ◽  
Parameter Estimates ◽  
Well Testing ◽  
Model Errors ◽  
Well Test ◽  
Matching Problem ◽  
Matching Process ◽  
The Future

Abstract Methods of nonlinear regression theory were applied to the reservoir history-matching problem to determine the effect of erroneous problem to determine the effect of erroneous parameter estimates obtained from well testing parameter estimates obtained from well testing on the future prediction of reservoir pressures. Two examples were studied: well testing in a radial one-dimensional slightly compressible reservoir and in an undersaturated, two-dimensional, heterogeneous oil field. The reservoir parameters of permeability, porosity, external radius, and pore volume were considered, and the effects of pore volume were considered, and the effects of measurement error, test time, and flow rate on the confidence limits were computed. Introduction The operation of a reservoir simulator requires accurate estimates of the reservoir properties. However, the simulation parameters, such as permeability, porosity, and reservoir geometry, are permeability, porosity, and reservoir geometry, are usually unknown unless coring and physical property analysis have been undertaken. Because of the cost of these procedures, it is more desirable to use the pressures measured at the well during a well test pressures measured at the well during a well test and indirectly compute the important parameters of the system. By using history matching of the test data to obtain the system parameters, the future pressure behavior of the reservoir can be predicted pressure behavior of the reservoir can be predictedSeveral studies on history matching have indicated that the welltest approach for determining the reservoir parameters often suffers from incorrect and nonunique parameter estimates. The factors that affect the parameter estimation can be classified as model errors, observability, measurement errors or noise, history time, test procedure, and optimization procedure. Model errors arise from the inaccuracy of the model and the numerical integration. For example, a reservoir simulator is only a reasonable approximation for flow through porous media. Solution of a model equation by numerical means also introduces roundoff and discretization errors. Observability of the system plays an important role in estimating the reservoir parameters. Depending on the location of the well and the number of data points, it may not be possible to determine uniquely all reservoir parameters from the measurements made at that well. Observability is strictly a function of the reservoir model used. At a given well, pressure measurements may only reflect the values of the parameters in specific zones of the reservoir. If a specific zone away from the well does not affect the measured pressure, then the system is not observable at that particular location. A rigorous definition of observability can be found in other papers. Measurement errors in the pressures and flow rates are another source of unrealistic parameter estimates. Longer history times always give more information about the reservoir as long as the system remains in a dynamic state. The nature of the system input (well flow rate) also affects the accuracy of the estimates and predictions. The final source of incorrect parameter estimates arises because the history-matching problem, posed mathematically, is usually a nonlinear programming problem that must be solved computationally. Such problem that must be solved computationally. Such a problem yields multiple extrema that often can lead to a relative minimum (rather than a global minimum) in the numerical search for the smallest matching error. Also, the magnitude of the objective function can be quite insensitive to the parameters selected, thus causing the optimization procedure to terminate prematurely. The above factors control the history-matching process; with actual data, it is usually impossible process; with actual data, it is usually impossible to identify the exact contributions of each factor to the errors in the parameter estimates. Since a certain amount of error will be introduced into the estimated parameters from the history-matching process, it is parameters from the history-matching process, it is useful to study the magnitude of this error resulting from various sources under controlled simulation conditions. Also, it is important to determine how the errors in the parameters are reflected in the future predictions of the pressures. SPEJ P. 42

scholarly journals Formulating the history matching problem with consistent error statistics

2021 ◽  
Author(s):  
Geir Evensen
Keyword(s):  
Simulation Model ◽  
Measurement Errors ◽  
History Matching ◽  
Critical Role ◽  
Bayes Theorem ◽  
Conditional Probability Density ◽  
Model Parameters ◽  
Rate Data ◽  
Matching Problem ◽  
Reservoir Prediction

AbstractIt is common to formulate the history-matching problem using Bayes’ theorem. From Bayes’, the conditional probability density function (pdf) of the uncertain model parameters is proportional to the prior pdf of the model parameters, multiplied by the likelihood of the measurements. The static model parameters are random variables characterizing the reservoir model while the observations include, e.g., historical rates of oil, gas, and water produced from the wells. The reservoir prediction model is assumed perfect, and there are no errors besides those in the static parameters. However, this formulation is flawed. The historical rate data only approximately represent the real production of the reservoir and contain errors. History-matching methods usually take these errors into account in the conditioning but neglect them when forcing the simulation model by the observed rates during the historical integration. Thus, the model prediction depends on some of the same data used in the conditioning. The paper presents a formulation of Bayes’ theorem that considers the data dependency of the simulation model. In the new formulation, one must update both the poorly known model parameters and the rate-data errors. The result is an improved posterior ensemble of prediction models that better cover the observations with more substantial and realistic uncertainty. The implementation accounts correctly for correlated measurement errors and demonstrates the critical role of these correlations in reducing the update’s magnitude. The paper also shows the consistency of the subspace inversion scheme by Evensen (Ocean Dyn. 54, 539–560 2004) in the case with correlated measurement errors and demonstrates its accuracy when using a “larger” ensemble of perturbations to represent the measurement error covariance matrix.

scholarly journals Recent Design Optimization Methods for Energy-Efficient Electric Motors and Derived Requirements for a New Improved Method—Part 1

Proceedings ◽  
2018 ◽  
Vol 2 (22) ◽  
pp. 1400
Author(s):  
Johannes Schmelcher ◽  
Max Kleine Büning ◽  
Kai Kreisköther ◽  
Dieter Gerling ◽  
Achim Kampker
Keyword(s):  
Rare Earth ◽  
Design Optimization ◽  
Energy Efficient ◽  
Degrees Of Freedom ◽  
High Efficiency ◽  
Optimization Problems ◽  
Computing Time ◽  
Optimization Methods ◽  
Electric Motors ◽  
Electric Cars

Energy-efficient electric motors are gathering an increased attention since they are used in electric cars or to reduce operational costs, for instance. Due to their high efficiency, permanent-magnet synchronous motors are used progressively more. However, the need to use rare-earth magnets for such high-efficiency motors is problematic not only in regard to the cost but also in socio-political and environmental aspects. Therefore, an increasing effort has to be put in finding the best design possible. The goals to achieve are, among others, to reduce the amount of rare-earth magnet material but also to increase the efficiency. In the first part of this multipart paper, characteristics of optimization problems in engineering and general methods to solve them are presented. In part two, different approaches to the design optimization problem of electric motors are highlighted. The last part will evaluate the different categories of optimization methods with respect to the criteria: degrees of freedom, computing time and the required user experience. As will be seen, there is a conflict of objectives regarding the criteria mentioned above. Requirements, which a new optimization method has to fulfil in order to solve the conflict of objectives will be presented in this last paper.

scholarly journals On geometric duality for state-constrained control problems

1979 ◽  
Vol 20 (2) ◽  
pp. 301-312
Author(s):  
T.R. Jefferson ◽  
C.H. Scott
Keyword(s):  
Optimal Control ◽  
Control Problem ◽  
State Constraints ◽  
Strong Duality ◽  
State Constraint ◽  
Continuous Functions ◽  
Control Problems ◽  
Absolutely Continuous Functions ◽  
Pure State Constraints ◽  
Constrained Control Problems

For convex optimal control problems without explicit pure state constraints, the structure of dual problems is now well known. However, when these constraints are present and active, the theory of duality is not highly developed. The major difficulty is that the dual variables are not absolutely continuous functions as a result of singularities when the state trajectory hits a state constraint. In this paper we recognize this difficulty by formulating the dual probram in the space of measurable functions. A strong duality theorem is derived. This pairs a primal, state constrained convex optimal control problem with a dual convex control problem that is unconstrained with respect to state constraints. In this sense, the dual problem is computationally more attractive than the primal.

scholarly journals Efficient Maximum-Likelihood Inference For The Isolation-With-Initial-Migration Model With Potentially Asymmetric Gene Flow

2016 ◽  
Author(s):  
Rui J. Costa ◽  
Hilde Wilkinson-Herbots
Keyword(s):  
Gene Flow ◽  
Maximum Likelihood ◽  
Dna Sequences ◽  
Computing Time ◽  
Likelihood Method ◽  
Ancestral Population ◽  
Parameter Estimates ◽  
Fast Method ◽  
Data Set ◽  
Im Model

AbstractThe isolation-with-migration (IM) model is commonly used to make inferences about gene flow during speciation, using polymorphism data. However, Becquet and Przeworski (2009) report that the parameter estimates obtained by fitting the IM model are very sensitive to the model's assumptions (including the assumption of constant gene flow until the present). This paper is concerned with the isolation-with-initial-migration (IIM) model of Wilkinson-Herbots (2012), which drops precisely this assumption. In the IIM model, one ancestral population divides into two descendant subpopulations, between which there is an initial period of gene flow and a subsequent period of isolation. We derive a very fast method of fitting an extended version of the IIM model, which also allows for asymmetric gene flow and unequal population sizes. This is a maximum-likelihood method, applicable to data on the number of segregating sites between pairs of DNA sequences from a large number of independent loci. In addition to obtaining parameter estimates, our method can also be used to distinguish between alternative models representing different evolutionary scenarios, by means of likelihood ratio tests. We illustrate the procedure on pairs of Drosophila sequences from approximately 30,000 loci. The computing time needed to fit the most complex version of the model to this data set is only a couple of minutes. The R code to fit the IIM model can be found in the supplementary files of this paper.

scholarly journals An Improved Metric Learning Approach for Degraded Face Recognition

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Guofeng Zou ◽  
Yuanyuan Zhang ◽  
Kejun Wang ◽  
Shuming Jiang ◽  
Huisong Wan ◽  
...  
Keyword(s):  
Face Recognition ◽  
Metric Learning ◽  
Computing Time ◽  
Feature Space ◽  
Projection Algorithm ◽  
Learning Approach ◽  
Learning Method ◽  
Matching Problem ◽  
Locality Preserving Projection ◽  
Locality Preserving

To solve the matching problem of the elements in different data collections, an improved coupled metric learning approach is proposed. First, we improved the supervised locality preserving projection algorithm and added the within-class and between-class information of the improved algorithm to coupled metric learning, so a novel coupled metric learning method is proposed. Furthermore, we extended this algorithm to nonlinear space, and the kernel coupled metric learning method based on supervised locality preserving projection is proposed. In kernel coupled metric learning approach, two elements of different collections are mapped to the unified high dimensional feature space by kernel function, and then generalized metric learning is performed in this space. Experiments based on Yale and CAS-PEAL-R1 face databases demonstrate that the proposed kernel coupled approach performs better in low-resolution and fuzzy face recognition and can reduce the computing time; it is an effective metric method.

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