Least-squares Objective Function Research Articles

Reliability of Horizontal Well Performance On a Field Scale Through Automatic History Matching

Abstract An efficient procedure for the determination of the error bounds (95% confidence intervals) of forecasted horizontal well production rates is presented. In particular, we advocate the use of automatic history matching (based on non-linear regression) to arrive at the best reservoir parameter values (typically porosities and permeabilities) for a given reservoir structure (i.e., for a given grid block representation). If the Gauss-Newton-method is used for the parameter search, the sensitivity coefficients required for the estimation of the standard error of the parameters and the well production rates are computed as part of the parameter search algorithm. Using typical techniques from statistical analysis (and the statistical properties of multivariable non-linear regression estimates), one can readily compute the confidence intervals of the predicted production rates. These confidence intervals around the expected reservoir behaviour incorporate the uncertainty in all reservoir parameters. As a result, a realistic measure of the uncertainty in the model predictions of future reservoir performance is obtained. The proposed procedure is illustrated with a case study. Introduction The final task of a reservoir simulation study is often the computation of future production levels of the history matched reservoir under alternative depletion plans. Moreover, the reservoir engineer often examines the sensitivity of the forecasted performance to different reservoir descriptions in an effort to establish the risks associated with a particular depletion plan. Typically, this is accomplished by varying key reservoir parameters such as porosities and permeabilities and determining how much the performance of a particular well or of the whole reservoir is affected. With the continuous development of powerful workstations and desktop personal computers, a renewed interest in automatic history marching procedures can be observed in the past few years. For the minimization of the resulting least squares (LS) objective function, two different approaches have been widely used: nonlinear regression(1–6) and optimal control methods(7–11). Recently, the simulated annealing(12) method has also proven to be a reliable procedure. In practical applications some form of reguralization(13,14) is always required to overcome the very serious problem of ill-conditioning, particularly as the number of parameters increases to more than ten or so. In our approach to automatic history matching, we minimize the resulting nonlinear LS objective function using an efficient implementation of the Gauss-Newton method(1). The necessary computation of the sensitivity coefficients when he Gauss-Newton method is used enables us to readily determine the uncertainty in the estimated parameter values(2). Furthermore, if we compute the sensitivity coefficients at future times where the performance of the reservoir is forecasted, we can generate 95% confidence intervals of the model predictions. The confidence intervals around the expected reservoir behaviour incorporate the uncertainty in all reservoir parameters and hence this approach provides a more realistic measure of the uncertainty in the model predictions of future reservoir behaviour(15). When several different grid representations of the reservoir are available with a known probability of being correct, the above described approach yields conditional expectations of future production rates which can subsequently be combined to arrive at an overall probability distribution of key variables, like cumulative oil production, WOR, GOR, etc.

The dual regularization approach to seismic velocity inversion

Differential semblance optimization (DSO) is a novel way of approaching a class of inverse problems arising in exploration seismology. The promising feature of the DSO method is that it replaces a non-smooth, highly non-convex cost function (the output least-squares (OLS) objective function) with a smooth cost function that is amenable to standard (local) optimization algorithms. The OLS problem can be written abstractly as a partially linear least-squares problem with linear constraints. The DSO objective function is derived from the associated quadratic penalty function. One way to view the DSO objective function is as a regularization of a function that is dual (in a certain sense) to the OLS objective function. Under suitable assumptions, the DSO method defines a parametrized path of minimizers converging to the desired solution and that, for certain values of the parameter, standard optimization techniques can be used to find a point on the path. The results of the theory are illustrated in the plane wave detection problem, a simple model problem for velocity inversion.

Penalized weighted least-squares image reconstruction for positron emission tomography

Presents an image reconstruction method for positron-emission tomography (PET) based on a penalized, weighted least-squares (PWLS) objective. For PET measurements that are precorrected for accidental coincidences, the author argues statistically that a least-squares objective function is as appropriate, if not more so, than the popular Poisson likelihood objective. The author proposes a simple data-based method for determining the weights that accounts for attenuation and detector efficiency. A nonnegative successive over-relaxation (+SOR) algorithm converges rapidly to the global minimum of the PWLS objective. Quantitative simulation results demonstrate that the bias/variance tradeoff of the PWLS+SOR method is comparable to the maximum-likelihood expectation-maximization (ML-EM) method (but with fewer iterations), and is improved relative to the conventional filtered backprojection (FBP) method. Qualitative results suggest that the streak artifacts common to the FBP method are nearly eliminated by the PWLS+SOR method, and indicate that the proposed method for weighting the measurements is a significant factor in the improvement over FBP.

Elastic constants of orthotropic composite materials using plate resonance frequencies, classical lamination theory and an optimized three-mode rayleigh formulation

This paper shows how the four independent elastic constants (longitudinal and transverse Young's moduli, in-plane shear modulus and major Poisson's ratio) of an orthotropic material may be extracted from the modal resonance data of a freely-supported rectangular thin plate made from the material, using the classical lamination theory and an optimized three-mode Rayleigh formulation with a suitably formed least-squares objective function. Results are obtained for orthotropy ratios of unity (aluminum), about three (unidirectional E-glass/vinylester) and about 13 (graphite/epoxy), and plate aspect ratios of unity and about two. The results suggest that the method proposed has potential for rapid characterization of elastic properties of advanced composites.

Modelling of a fixed-bed water-gas shift reactor

This paper describes the development and verification of a non-linear, steady-state model of a laboratory-scale water-gas shift reactor. The objective of the work was to develop a model of the laboratory reactor for use in simulation and model-based control strategies. Heat transfer parameters were determined from the experiments without reaction. Parameter estimates for the pre-exponential factor, the activation energy and the coefficient for heat transfer between the catalyst bed and reactor wall were determined from experiments with reaction. Non-linear optimization with a least-squares objective function was used to determine the parameter values. With six fitted parameters, the simplified reactor model accurately predicted the performance of the laboratory reactor system. A second-order reversible rate expression successfully modelled the water-gas shift reaction over a cobalt-molybdenum catalyst.

A New Algorithm for Estimating Relative Permeabilities From Displacement Experiments

Summary The process of estimating relative permeability curves from two-phase displacement experiments is considered. A parameter estimation approach overcomes significant limitations of the classic calculation procedure. In this approach, functional representations are chosen for the relative permeability curves. Adjustable coefficients (or parameters) in the functional representations are then chosen to minimize a least-squares objective function. Previous applications of this approach have used exponential functional representations with a single adjustable coefficient (the exponent) for each curve. However, significant errors in the relative permeability estimates may be encountered when this function is used. It has been determined, in principle, that cubic spline functional representations may be used to obtain accurate estimates of relative permeability curves. The process of obtaining the least-squares estimates for the parameters is more difficult, however, because a significantly greater number of parameters are to be determined. No algorithm has been reported for this purpose. We present an algorithm for estimating relative permeability curves that uses cubic spline functional representations. A unique and essential feature of the algorithm is that it incorporates inequality constraints that ensure that physically realistic relative permeability curves are maintained throughout the iterative minimization process. The performance of the algorithm is demonstrated with data from hypothetical and laboratory coreflood displacement experiments.

Parameter estimation and sensitivity analysis of a nonlinearly elastic static lung model.

A model for the static pressure-volume behavior of the lung parenchyma based on a pseudo-elastic strain energy function was tested. Values of the model parameters and their variances were estimated by an optimal least-squares fit of the model-predicted pressures to the corresponding data from excised, saline-filled dog lungs. Although the model fit data from twelve lungs very well, the coefficients of variation for parameter values differed greatly. To analyze the sensitivity of the model output to its parameters, we examined an approximate Hessian, H, of the least-squares objective function. Based on the determinant and condition number of H, we were able to set formal criteria for choosing the most reliable estimates of parameter values and their variances. This in turn allowed us to specify a normal range of parameter values for these dog lungs. Thus the model not only describes static pressure-volume data, but also uses the data to estimate parameters from a fundamental constitutive equation. The optimal parameter estimation and sensitivity analysis developed here can be widely applied to other physiologic systems.

FCM: The fuzzy c-means clustering algorithm

This paper transmits a FORTRAN-IV coding of the fuzzy c-means (FCM) clustering program. The FCM program is applicable to a wide variety of geostatistical data analysis problems. This program generates fuzzy partitions and prototypes for any set of numerical data. These partitions are useful for corroborating known substructures or suggesting substructure in unexplored data. The clustering criterion used to aggregate subsets is a generalized least-squares objective function. Features of this program include a choice of three norms (Euclidean, Diagonal, or Mahalonobis), an adjustable weighting factor that essentially controls sensitivity to noise, acceptance of variable numbers of clusters, and outputs that include several measures of cluster validity.

Lung washout during spontaneous breathing: Parameter estimation with a time-varying model

We use a time-varying model to analyze the multibreath washout of nitrogen from lungs of human subjects breathing spontaneously. Based on standard pulmonary function evaluation, the 39 subjects tested are classified in the following categories: nonsmoking normal, “smoking” normal, asthma, diffuse interstitial lung disease, and chronic obstructive lung disease. The degree of ventilation inhomogeneity among these subjects was indicated by two independent parameters of the model, which were estimated by nonlinear optimization. A Gauss-Newton algorithm with a Marquardt modification was applied to a least-squares objective function. Constraints were included in a penalty function. Values of the model parameters from repeated washouts of the same subjects showed wide variability. Also, model parameters did not appear to provide any better distinction among clinical groups than did a moment ratio whose computation is much less expensive and more reliable.

A review of some mathematical models used in hydrology, with observations on their calibration and use

Terms are defined that are frequently used (and misused) where mathematical models are applied to solve hydrological problems, and some of the many available models are classified into four main groups. It is suggested that models with parameters estimated by computing the minimum of a least-squares objective function represent an application of well-known non-linear regression theory to situations in which the assumptions commonly made in this theory are seldom valid. The correction required is not the use of objective functions other than those based on sums of squares, but the use of more realistic assumptions concerning the stochastic structure of the model residuals. Interdependence between model parameters necessitates extensive exploration of the sum-of-squares surface in the neighbourhood of its minimum even when regression assumptions are valid: this is particularly true where the model is to be used to examine the likely effects of a proposed physical change to the catchment, since the complexity of such a change will not generally be represented by a change in one (or even some) of the model's parameters, regardless of the remainder.

A Further Note on Pulse-Test Interpretation

The development of pulse testing, a special type of well interference testing, has resulted in interpretation techniques that require the use of graphical methods. These methods make use of such variables as "time lag" and "pressure response", obtained visually from plots of well pressure vs time. Such methods require that the pulse-rate schedule be a symmetric, rectangular step-function of constant frequency and amplitude. In actual field practice it may be impossible to maintain a pulse-rate schedule of this type. Furthermore, even if such a schedule could be adhered to, the resulting pressure signal in the responding well might possess a high degree of random noise. Noise caused, perhaps, by a low-resolution pressure instrument can obscure the test results so that time lag and pressure response cannot be determined accurately. A more general approach that overcomes these difficulties is that of automatically history-matching observed responding well pressures with pressures computed from a mathematical model. A convenient method is to minimize the least-squares objective function (1) Any obvious long-term reservoir pressure trends must be removed from the measured pressures in order to arrive at po. An analysis of the residuals, (pt - po), computed at the minimum of psi, will determine if pressure trends have been successfully removed. Since pt and its first and second derivatives can be determined analytically from the exponential integral solution to the diffusion equation, a useful minimization technique is the Newton-Raphson method. This method is used to solve the following equations for transmissibility, T, and storage, s: (2) and (3) This interpretation technique was programmed in FORTRAN for the IBM 360/50. The Newton-Raphson method was found to be quite satisfactory and most problems took less than 10 seconds for adequate convergence. P. 1143ˆ