Peter P. Valko

Optimal Transformations for Multiple Regression: Application to Permeability Estimation From Well Logs

Conventional multiple regression for permeability estimation from well logs requires a functional relationship to be presumed. Due to the inexact nature of the relationship between petrophysical variables, it is not always possible to identify the underlying functional form between dependent and independent variables in advance. When large variations in petrological properties are exhibited, parametric regression often fails or leads to unstable and erroneous results, especially for multivariate cases. In this paper we describe a nonparametric approach for estimating optimal transformations of petrophysical data to obtain the maximum correlation between observed variables. The approach does not require a priori assumptions of a functional form and the optimal transformations are derived solely based on the data set. An iterative procedure involving the alternating conditional expectation (ACE) forms the basis of our approach. The power of ACE is illustrated using synthetic as well as field examples. The results clearly demonstrate improved permeability estimation by ACE compared to conventional parametric regression methods.

Pipe viscometry of foams

This paper describes a method for extracting useful information from small‐scale pipe viscometer measurements of foam rheology. The rheology of a foamed polymer solution at a given temperature, pressure, and quality was determined in pipes of five diameters. The flow curves showed a marked dependence on the diameter of the pipe. The concept of apparent slip could be used to explain the phenomenon. The classical slip correction of Mooney was not applicable, but the method developed by Jastrzebski (based on the previous work of Oldroyd) provided a consistent means of apparent slip correction. The geometric interpretation of the two slip correction methods revealed the possible reason for the difference of their performance. The slip corrected measurements were interpreted in the framework of the volume equalization principle.

An extended Marquardt-type procedure for fitting error-in-variables models

Abstract The paper presents a simple derivation of the method of fitting nonlinear algebraic models where all variables are subject to error and improves the numerical efficiency of the algorithm. Including a known procedure for equilibrating balance equations and factorizing the weighting matrix, the classical Gauss-Marquardt method of estimating parameters in nonlinear models is shown to handle also the error-in-variables model, thereby extending the efficiency and robustness of Marquardts compromise to this slightly more involved case.

A DIRECT-INDIRECT PROCEDURE FOR ESTIMATION OF KINETIC PARAMETERS

Abstract The direct integral method (a generalization of the Himmelblau-Jones- Bischoff technique) and the conventional least squares approach are compared in the solution of kinetic estimation problems. Some examples demonstrate that the former method may be preferable in terms of mean square error of the estimates. Recent results on biased estimators are used to explain this fact. A three-stage estimation procedure is presented which preserves the advantages of its stages, thereby combining robustness and numerical efficiency of the direct method with unbiasedness of the indirect estimates.

The Method of Distributed Volumetric Sources for Calculating the Transient and Pseudosteady-State Productivity of Complex Well-Fracture Configurations

The method of Distributed Volumetric Sources (DVS) is developed to solve problems of transient and pseudo steady state fluid flow. The basic building block of the method comprises calculation of the analytical response of a rectilinear reservoir with closed outer boundaries to an instantaneous volumetric source, also shaped as a rectilinear body. The solution also provides the well-testing derivative of the response to a continuous source in analytical form. This can be integrated over time to provide the pressure response to a continuous source. For production engineering applications, we cast the results into a transient/pseudo-steady productivity index form. The new method is validated through comparison to results of some of the well known well-testing solutions for simple configurations such as vertical wells with full and partial penetration, horizontal wells with unform flux and infinite conductivity, and fractured wells with uniform flux, finite or infinte conductivity. The results show a very good agreement with the existing models. The main advantage of the new solution is its applicability over the more complex fracture/well configurations, some of which is studied in the paper. The new method has shown to provide a fast, robust, and reliable way to pressure transient analysis, and well performance prediction whenever complex well/fracture configuration is considered.

The Optimization Of The Productivity Index And The Fracture Geometry Of A Stimulated Well With Fracture Face And Choke Skins

The post-treatment performance of hydraulically fractured wells has been a recurring theme in the petroleum literature, covering the spectrum of understanding the physics of flow to the optimization of design. Optimization itself has taken different hues meaning comprehensive economic, or just the reduction of execution costs, or the maximization of the production or injection rates. Irrespective of the ultimate criterion, the magnitude of the reservoir permeability has been central to the fracture morphology. Long fractures are warranted for lowpermeability reservoirs; wide but short fractures are indicated for high-permeability formations. For a given reservoir of known permeability and dimensions the mass of proppant injected to the pay describes a unique and constant proppant number for which a maximum well productivity index can be achieved at the optimum dimensionless fracture conductivity. The proppant number and the optimum dimensionless fracture conductivity determine exclusively the optimum fracture dimensions. However, damaged hydraulic fracture performance deviates substantially from that of undamaged fractures. Two types of damage are considered, fracture face, often caused by fluid leakoff into the reservoir and choke fracture, which is caused by proppant flow-back, over-displacement or polymer damage. These damages, described by skin effects cause a departure, at times substantial, from the indicated undamaged optimum fracture geometry. In this work, the performance of a fractured well is calculated using a direct boundary element method. The method calculates the dimensionless productivity index and the model allows for the presence of the two different skin effects. The fracture face skin effect was found to have a significant detrimental effect on the dimensionless productivity index, especially for high-permeability reservoirs. The effect of the choke skin was found to be potentially also very detrimental, but less complex to account for, because it can be represented as an apparent reduction to the proppant

Inversion of Noise-Free Laplace Transforms: Towards a Standardized Set of Test Problems

The numerical inversion of Laplace transform arises in many applications of science and engineering whenever ordinary and partial differential equations or integral equations are solved. The increasing number of available numerical methods and computer codes has generated a need for well-documented sets of test problems. Using such sets, algorithm developers can evaluate the relative merits and drawbacks of their suggested new methods, and end-users can make judgments on the applicability of an individual method for a specific problem. Many areas in science and engineering, lead to problems that share three important properties: (i) the image function can be evaluated for real arguments, but not necessarily for complex ones; (ii) the original is known to be infinitely differentiable for times t > 0, (iii) the values of the image function can be obtained with any prescribed accuracy. The published test sets do not properly cover these applications, as many included problems are beyond of the specific class, while the remaining ones fail to address some of the potential difficulties arising in practice. The goal of this paper is to establish a common ground for problem classification, to list the requirements for the above class of problems, and to provide a carefully selected test set by addressing the deficiencies of the ones currently available. The findings of the paper are used to solve a problem of practical importance in modeling underground flow. Accompanying the paper, WEB links are provided to a list of more than 800 relevant publications (going back to 1795), to a Mathematica program to generate and solve the suggested set of test problems, and to a user friendly Java program to solve inversion problems for a restricted class of image functions.

Development and Application of the Multiwell Productivity Index (MPI)

In this article we generalize the concept of the pseudosteady-state productivity index for the case of multiple wells producing from or injecting into a closed rectangular reservoir of constant thickness. The work complements the analytical study by Rodriguez and Cinco-Ley for systems produced at constant flowing pressures. Wells are represented by fully penetrating vertical line sources located arbitrarily in a homogeneous and isotropic reservoir. The multiwell productivity index ~MPI! is a square matrix of dimension n, where n is the number of wells. The MPI provides a simple, reasonably accurate and fast analytical tool to evaluate well performance without dividing the cluster into single-well drainage areas. The MPI approach is used to obtain approximate analytical solutions for constant ~but possibly different! wellbore flowing pressures, and to visualize the resulting pressure field. In addition, the skin factor trace technique is introduced as a tool to monitor a cluster of wells. The MPI technique is illustrated using a synthetic example taken from Ref. 2, as well as two field cases.

Numerical deconvolution using system identification methods

A deconvolution method is presented for use in pharmacokinetic applications involving continuous models and small samples of discrete observations. The method is based on the continuous-time counterpart of discrete-time least squares system identification, well established in control engineering. The same technique, requiring only the solution of a linear regression problem, is used both in system identification and input identification steps. The deconvolution requires no a prioriinformation, since the proposed procedure performs system identification (including optimal selection of model order), selects the form of the input function and calculates its parametric representation and its values at specified time points.

Fluid-Leakoff Delineation in High-Permeability Fracturing

Starting from the original concept proposed by Carter, Howard and Fast, this paper reviews the description of fracturing fluid leakoff in view of modeling flow in porous media. It is shown how various linear leakoff models have been developed and why a new, radial leakoff concept is necessary for high-permeability fracturing, where the injection time is commensurable to the response time of the reservoir. Using Laplace space methods, the new radial leakoff law is calculated and compared to linear leakoff. For comparison purposes a calibration test executed in high-permeability formation is interpreted using several approaches, namely: linear leakoff1bulk leakoff coefficient; filtercake resistance1linear flow in the formation and finally, filtercake resistance1radial flow in the formation.