Estimation of Permeability and Porosity From Well Test Data

Abstract

American Institute of Mining, Metallurgical, and Petroleum Engineers, Inc. This paper was prepared for the 45th Annual California Regional Meeting of the Society of Petroleum Engineers of AIME, to be held in Ventura, Calif., April 2–4, 1975. Permission to copy is restricted to an abstract of not more than 300 words. Illustrations may not be copied. The abstract should contain conspicuous acknowledgment of where and by whom the paper is presented. Publication elsewhere after publication in the JOURNAL OF PETROLEUM TECHNOLOGY or the SOCIETY OF publication in the JOURNAL OF PETROLEUM TECHNOLOGY or the SOCIETY OF PETROLEUM ENGINEERS JOURNAL is usually granted upon request to the Editor PETROLEUM ENGINEERS JOURNAL is usually granted upon request to the Editor of the appropriate journal provided agreement to give proper credit is made. Discussion of this paper is invited. Three copies of any discussion should be sent to the Society of Petroleum Engineers office. Such discussion may be presented at the above meeting and, with the paper, may be considered for publication in one of the two SPE magazines. Abstract For a well that is being produced at rate q, let pi denote the measured wellbore pressure at time ti. Let P(ti,,) denote the pressure at time ti at the wellbore that is pressure at time ti at the wellbore that is calculated from a chosen mathematical model of the reservoir. Here, denotes the parameter for porosity while denotes the parameter for permeability. In order to determine the actual permeability. In order to determine the actual porosity and permeability k, consider the porosity and permeability k, consider the equations pi = P(ti, ,). The implicit function theorem implies that there exist functions gi () such that a = gi() satisfy pi= P(ti, gi(),). In short, given a, the P(ti, gi(),). In short, given a, the equations can be solved for the various that satisfy them. Hence, the functions gi can be generated numerically,. Since the actual and k must satisfy pi = P(ti, , k) (provided the correct model has been chosen), it follows that = gi(k) for all i. This means that if the graphs of = gi() are plotted in the, plane, then the point (,k) must lie on each plane, then the point (,k) must lie on each graph. It is shown via numerical experiments that the graphs are all distinct and intersect in the single point (,k) when no measurement error is present. The method is considered for various cases of measurement error. It leads to a statistical method for determining and k that yields estimates of them which are significantly better than a nonlinear least squares method. Introduction The determination of permeability and porosity from well-test data is an inverse porosity from well-test data is an inverse problem that belongs to the class of problems problem that belongs to the class of problems involving the determination of coefficients of parabolic and elliptic partial differential parabolic and elliptic partial differential equations from overspecified boundary data. Since most models of flow in a porous medium involve a parabolic partial differential equation or system, the initial reservoir pressure distribution and the flow rate at the pressure distribution and the flow rate at the wellbore constitutes a complete set of data needed to determine the bounded solution of the parabolic problem provided that all relevant parabolic problem provided that all relevant parameters such as permeability, porosity, etc., parameters such as permeability, porosity, etc., are specified as data. In this fortuitous situation the wellbore pressure is an additional piece of information that can serve as a check piece of information that can serve as a check on the validity of the model. It was recognized quite early by petroleum engineers that permeability and porosity are rarely known in permeability and porosity are rarely known in advance. Consequently, the wellbore pressure became an extremely important additional piece of information that has been used quite a long time for the determination of the permeability and porosity. Estimation of the permeability and porosity from well data is often attempted by porosity from well data is often attempted by reservoir engineers via history matching. The reservoir is usually divided into zones with one well per zone. The zone is assumed to possess its own independent permeability and porosity. The calculated performance is fitted to the production history by adjusting the parameters production history by adjusting the parameters of each zone. But the estimation of the permeability and porosity via such a method permeability and porosity via such a method presents some difficulties, one of which is the presents some difficulties, one of which is the existence of a high correlation between the parameters of permeability and porosity. parameters of permeability and porosity.