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＾

A Comparison Study on Algorithms in Reservoir Automatic History Matching

2012 ◽  
Vol 518-523 ◽  
pp. 4376-4379
Author(s):  
Bao Yi Jiang ◽  
Zhi Ping Li
Keyword(s):  
Objective Function ◽  
Reservoir Simulation ◽  
History Matching ◽  
Optimization Algorithms ◽  
Reservoir Engineering ◽  
Comparison Study ◽  
Automatic History Matching ◽  
Numerical Reservoir Simulation ◽  
Computational Capability ◽  
Matching Procedure

With the increase in computational capability, numerical reservoir simulation has become an essential tool for reservoir engineering. To minimize the objective function involved in the history matching procedure, we need to apply the optimization algorithms. This paper is based on the optimization algorithms used in automatic history matching.

Identification of Geological Shapes in Reservoir Engineering By History Matching Production Data

1999 ◽  
Vol 2 (05) ◽  
pp. 470-477 ◽  
Author(s):  
Daniel Rahon ◽  
Paul Francis Edoa ◽  
Mohamed Masmoudi
Keyword(s):  
Inverse Problem ◽  
Objective Function ◽  
History Matching ◽  
Geometric Parameters ◽  
Production Data ◽  
Phase Flow ◽  
State Equations ◽  
Two Phase ◽  
Discrete Equations ◽  
Phase Water

Summary This paper discusses a method which helps identify the geometry of geological features in an oil reservoir by history matching of production data. Following an initial study on single-phase flow and applied to well tests (Rahon, D., Edoa, P. F., and Masmoudi, M.: "Inversion of Geological Shapes in Reservoir Engineering Using Well Tests and History Matching of Production Data," paper SPE 38656 presented at the 1997 SPE Annual Technical Conference and Exhibition, San Antonio, Texas, 5–8 October.), the research presented here was conducted in a multiphase flow context. This method provides information on the limits of a reservoir being explored, the position and size of faults, and the thickness and dimensions of channels. The approach consists in matching numerical flow simulation results with production measurements. This is achieved by modifying the geometry of the geological model. The identification of geometric parameters is based on the solution of an inverse problem and boils down to minimizing an objective function integrating the production data. The minimization algorithm is rendered very efficient by calculating the gradients of the objective function with respect to perturbations of these geometric parameters. This leads to a better characterization of the shape, the dimension, and the position of sedimentary bodies. Several examples are presented in this paper, in particular, an application of the method in a two-phase water/oil case. Introduction A number of semiautomatic history matching techniques have been developed in recent years to assist the reservoir engineer in his reservoir characterization task. These techniques are generally based on the resolution of an inverse problem by the minimization of an objective function and require the use of a numerical simulator. The matching parameters of the inverse problem comprise two types of properties: petrophysical/porosity and permeability and geometric position, shape, and size of the sedimentary bodies present in the reservoir. To be efficient, minimization algorithms require the calculation of simulated production gradients with respect to matching parameters. Such gradients are usually calculated by deriving discrete state equations solved in the numerical simulator1–5 or by using a so-called adjoint-state method.6,7 Therefore, most of these gradient-based methods only allow the identification of petrophysical parameters which appear explicitly in the discrete equations of state. The case of geometric parameters is much more complex, as the gradients of the objective function with respect to these parameters cannot be determined directly from the flow equation. Recent works8–10 have handled this problem by defining geological objects using mathematical functions to describe porosity or permeability fields. But, generalizing these solutions to complex geological models remains difficult. The method proposed in this paper is well suited to complex geometries and heterogeneous environments. The history matching parameters are the geometric elements that describe the geological objects generated, for example, with a geomodeling tool. A complete description of the method with the calculation of the sensitivities was presented in Ref. 11, within the particular framework of single-phase flow adapted to well-test interpretations. In this paper we will introduce an extension of the method to multiphase equations in order to match production data. Several examples are presented, illustrating the efficiency of this technique in a two-phase context. Description of the Method The objective is to develop an automatic or semiautomatic history matching method which allows identification of geometric parameters that describe geological shapes using a numerical simulator. To be efficient, the optimization process requires the calculation of objective function gradients with respect to the parameters. With usual fluid flow simulators using a regular grid or corner point geometry, the conventional methods for calculating well response gradients on discrete equations are not readily usable when dealing with geometric parameters. These geometric parameters do not appear explicitly in the model equations. With these kinds of structured models the solution is to determine the expression of the sensitivities of the objective function in the continuous problem using mathematical theory and then to calculate a discrete set of gradients. Sensitivity Calculation. Here, we present a sensitivity calculation to the displacement of a geological body in a two-phase water/oil flow context. State Equations. Let ? be a two- or three-dimensional spatial field, with a boundary ? and let ]0,T[ be the time interval covering the pressure history. We assume that the capillary pressure is negligible. The pressure p and the water saturation S corresponding to a two-phase flow in the domain ? are governed by the following equations: ∂ ϕ ( p ) S ∂ t − ∇ . ( k k r o ( S ) μ o ∇ ( p + ρ o g z ) ) = q o ρ o , ∂ ϕ ( p ) S ∂ t − ∇ . ( k k r w ( S ) μ w ∇ ( p + ρ w g z ) ) = q w ρ w , ( x , y , z ) ∈ Ω , t ∈ ] 0 , T [ , ( 1 ) with a no-flux boundary condition on ? and an initial equilibrium condition

scholarly journals Optimal Control through Leadership of the Cucker and Smale Flocking Model with Time Delays

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-14
Author(s):  
Adsadang Himakalasa ◽  
Suttida Wongkaew
Keyword(s):  
Optimal Control ◽  
Time Delay ◽  
Optimal Control Problem ◽  
Time Delays ◽  
Control Function ◽  
Discretization Method ◽  
Control Approach ◽  
First Order ◽  
Nonlinear Conjugate Gradient Method ◽  
Nonlinear Conjugate Gradient

The Cucker and Smale model is a well-known flocking model that describes the emergence of flocks based on alignment. The first part focuses on investigating this model, including the effect of time delay and the presence of a leader. Furthermore, the control function is inserted into the dynamics of a leader to drive a group of agents to target. In the second part of this work, leadership-based optimal control is investigated. Moreover, the existence of the first-order optimality conditions for a delayed optimal control problem is discussed. Furthermore, the Runge–Kutta discretization method and the nonlinear conjugate gradient method are employed to solve the discrete optimality system. Finally, the capacity of the proposed control approach to drive a group of agents to reach the desired places or track the trajectory is demonstrated by numerical experiment results.

Automatic History Matching in a Bayesian Framework, Example Applications

2005 ◽  
Vol 8 (03) ◽  
pp. 214-223 ◽  
Author(s):  
Fengjun Zhang ◽  
Jan-Arild Skjervheim ◽  
Albert C. Reynolds ◽  
Dean S. Oliver
Keyword(s):  
Objective Function ◽  
History Matching ◽  
Maximum Likelihood Method ◽  
Likelihood Method ◽  
Production Data ◽  
A Posteriori ◽  
Automatic History Matching ◽  
Flow Problems ◽  
Static Data ◽  
Randomized Maximum Likelihood

Summary The Bayesian framework allows one to integrate production and static data into an a posteriori probability density function (pdf) for reservoir variables(model parameters). The problem of generating realizations of the reservoir variables for the assessment of uncertainty in reservoir description or predicted reservoir performance then becomes a problem of sampling this a posteriori pdf to obtain a suite of realizations. Generation of a realization by the randomized-maximum-likelihood method requires the minimization of an objective function that includes production-data misfit terms and a model misfit term that arises from a prior model constructed from static data. Minimization of this objective function with an optimization algorithm is equivalent to the automatic history matching of production data, with a prior model constructed from static data providing regularization. Because of the computational cost of computing sensitivity coefficients and the need to solve matrix problems involving the covariance matrix for the prior model, this approach has not been applied to problems in which the number of data and the number of reservoir-model parameters are both large and the forward problem is solved by a conventional finite-difference simulator. In this work, we illustrate that computational efficiency problems can be overcome by using a scaled limited-memory Broyden-Fletcher-Goldfarb-Shanno (LBFGS) algorithm to minimize the objective function and by using approximate computational stencils to approximate the multiplication of a vector by the prior covariance matrix or its inverse. Implementation of the LBFGS method requires only the gradient of the objective function, which can be obtained from a single solution of the adjoint problem; individual sensitivity coefficients are not needed. We apply the overall process to two examples. The first is a true field example in which a realization of log permeabilities at26,019 gridblocks is generated by the automatic history matching of pressure data, and the second is a pseudo field example that provides a very rough approximation to a North Sea reservoir in which a realization of log permeabilities at 9,750 gridblocks is computed by the automatic history matching of gas/oil ratio (GOR) and pressure data. Introduction The Bayes theorem provides a general framework for updating a pdf as new data or information on the model becomes available. The Bayesian setting offers a distinct advantage. If one can generate a suite of realizations that represent a correct sampling of the a posteriori pdf, then the suite of samples provides an assessment of the uncertainty in reservoir variables. Moreover, by predicting future reservoir performance under proposed operating conditions for each realization, one can characterize the uncertainty in future performance predictions by constructing statistics for the set of outcomes. Liu and Oliver have recently presented a comparison of methods for sampling the a posteriori pdf. Their results indicate that the randomized-maximum-likelihood method is adequate for evaluating uncertainty with a relatively limited number of samples. In this work, we consider the case in which a prior geostatistical model constructed from static data is available and is represented by a multivariate Gaussian pdf. Then, the a posteriori pdf conditional to production data is such that calculation of the maximum a posteriori estimate or generation of a realization by the randomized-maximum-likelihood method is equivalent to the minimization of an appropriate objective function. History-matching problems of interest to us involve a few thousand to tens of thousands of reservoir variables and a few hundred to a few thousand production data. Thus, an optimization algorithm suitable for large-scale problems is needed. Our belief is that nongradient-based algorithms such as simulated annealing and the genetic algorithm are not competitive with gradient-based algorithms in terms of computational efficiency. Classical gradient-based algorithms such as the Gauss-Newton and Levenberg-Marquardt typically converge fairly quickly and have been applied successfully to automatic history matching for both single-phase- and multiphase-flow problems. No multiphase-flow example considered in these papers involved more than 1,500reservoir variables. For single-phase-flow problems, He et al. and Reynolds et al. have generated realizations of models involving up to 12,500 reservoir variables by automatic history matching of pressure data. However, they used a procedure based on their generalization of the method of Carter et al. to calculate sensitivity coefficients; this method assumes that the partial-differential equation solved by reservoir simulation is linear and does not apply for multiphase-flow problems.

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: Incompressible Flow

1977 ◽  
Vol 17 (06) ◽  
pp. 398-406 ◽  
Author(s):  
Bruno van den Bosch ◽  
John H. Seinfeld
Keyword(s):  
Relative Permeability ◽  
Incompressible Flow ◽  
History Matching ◽  
Single Phase ◽  
Absolute Permeability ◽  
Petroleum Reservoir ◽  
Reservoir Properties ◽  
Two Phase ◽  
Pressure Behavior ◽  
Three Phase

Abstract The estimation of porosity, absolute permeability, and relative permeability-saturation relations in a two-phase petroleum reservoir is considered The data available for estimation are assumed to be the oil flow rates and the pressures at the wells. A situation in which the reservoir may be represented by incompressible flow of oil and water also is considered. A hypothetical, circular reservoir with a centrally located producing well is studied in detail. In principle, the porosity can be estimated on the basis of saturation behavior, absolute permeability on The basis of pressure behavior, and permeability on The basis of pressure behavior, and coefficients in the relative permeability-saturation relations on the basis of both saturation and pressure behavior. The ability to achieve good pressure behavior. The ability to achieve good estimates was found to depend on the nature of the flow in a given situation. Introduction The estimation of petroleum reservoir properties on the basis of data obtained during production, so-called history matching, has received considerable attention. By and large, the development of theories for history matching and their application have been confined to reservoirs that can be modeled as containing a single phase. (Wasserman et al. considered the estimation of absolute permeability and porosity in a three-phase reservoir permeability and porosity in a three-phase reservoir by the use of pseudo single-phase model.) Since in the single-phase case only a single partial-differential equation is needed to describe partial-differential equation is needed to describe the reservoir, identification techniques can be tested most conveniently on such a system. The customary parameters to be estimated are the rock porosity (or the storage coefficient) and the porosity (or the storage coefficient) and the directional permeabilities (or the transmissibilities), which are not uniform throughout the reservoir but a function of location. The history matching of single-phase reservoirs through the estimation of these functional properties now appears to be understood quite well. Numerical algorithms have been thoroughly studied and tested. The most difficult aspect is the ill-conditioned nature of the problem arising from the large number of unknowns problem arising from the large number of unknowns relative to the available data. A recent study has elucidated the basic structure of single-phase history-matching problems and has shown how the degree of ill-conditioning may be assessed quantitatively. Reservoirs generally contain more than one fluid phase, however, and consequently are described by phase, however, and consequently are described by mathematical models accounting for the multiphase nature of the system. The porosity and absolute permeabilities still must be estimated as in the permeabilities still must be estimated as in the single-phase case. In addition, it may be necessary to estimate the relative permeability-saturation relationships. Ordinarily, relative permeability vs saturation curves are determined through experiments on core samples. Because it may be difficult to reproduce actual reservoir flow conditions in a laboratory core sample, it is desirable to consider the direct estimation of relative permeability-saturation relationships on the basis of permeability-saturation relationships on the basis of reservoir data that ordinarily would be available during the course of production. This paper represents an initial investigation of the complex identification problem in two-phase reservoirs. The major objective problem in two-phase reservoirs. The major objective of this study is to investigate the feasibility of parameter estimation in two-phase reservoirs in parameter estimation in two-phase reservoirs in which the reservoir is described by a two-phase incompressible flow model. In the next section we present basic equations governing two-phase (oil-water) reservoirs. We first define the general history-matching problem for these reservoirs and then consider a hypothetical reservoir, circularly symmetric with a central producing well in which the flow may be taken as producing well in which the flow may be taken as incompressible. The radial flow reservoir represents a situation in which oil is produced from a water drive. We wanted to estimate reservoir properties based on data obtained at the well. Considering the flow as incompressible enables us to draw a direct comparison to the classic incompressible linear-flow case for which the problem of estimating relative permeabilities is well established. Thus, we permeabilities is well established. Thus, we seek to understand fully the incompressible flow case as a prelude to the general problem of history matching in two-phase compressible flow reservoirs. SPEJ P. 398

AN OPTIMAL CONTROL APPROACH TO SEMICONDUCTOR DESIGN

2002 ◽  
Vol 12 (01) ◽  
pp. 89-107 ◽  
Author(s):  
MICHAEL HINZE ◽  
RENÉ PINNAU
Keyword(s):  
Optimal Control ◽  
Diffusion Model ◽  
Minimization Problem ◽  
Lagrange Multipliers ◽  
Design Problem ◽  
Drift Diffusion Model ◽  
Drift Diffusion ◽  
Control Approach ◽  
First Order ◽  
Optimal Control Approach

The design problem for semiconductor devices is studied via an optimal control approach for the standard drift–diffusion model. The solvability of the minimization problem is proved. The first-order optimality system is derived and the existence of Lagrange-multipliers is established. Further, estimates on the sensitivities are given. Numerical results concerning a symmetric n–p-diode are presented.

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