Mathematical Analysis of Exponentially Declining Oil Payments
Authors: Haskett Carl E. 
Published: December 2007

This paper was prepared for the 44th Annual Fall Meeting of the of the Society of Petroleum Engineers of AIME, to be held in Denver, Colo., Sept. 28-Oct. 1, 1969. 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 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
This paper presents the mathematical relationship between interest rate, loan balance, an initial reference payment, total interest and time to pay out for an oil payment-type loan when the stream of payments payment-type loan when the stream of payments follows an exponential decline, or an exponential decline preceded by a constant rate period of payments. Derivation of the period of payments. Derivation of the equations and a graphical presentation of the results are included.

Introduction
For the purchase and development of oil and gas producing properties, it is common to employ loans repaid out of revenue received monthly from oil and gas runs. Such loans provide for allocation of each payment first provide for allocation of each payment first to accumulated interest on the unpaid balance, with the remainder applied to retirement of the principal. To date, there has not been presented in the literature any short method presented in the literature any short method of approximating directly the time to pay out and total interest paid except by calculating these factors based on projected yearly payments, with such parameters as percent of payments, with such parameters as percent of dedication of the oil runs., interest rate demanded by the money market, and the loan principal assumed to provide the basis for the principal assumed to provide the basis for the calculation. Any change in these parameters requires recalculation of pay-out time and total dedication required to repay principal plus interest. The intent of this paper is to plus interest. The intent of this paper is to provide a rapid direct approach through graphs provide a rapid direct approach through graphs or equations resolving the relationship between these parameters, so that several alternatives can be considered quickly, thus allowing selection of appropriate loan principal, percent dedication, etc., before the final percent dedication, etc., before the final detailed analysis is made on the profit projection of the proposed acquisition. projection of the proposed acquisition.

CHARACTER OF THE STREAM OF PAYMENTS
The character of the stream of payments to be applied to repayment of the loan plus interest is basic to the time of pay-out and total interest. If it is presumed that there are two or more properties involved and that the plot of production vs time reveals a constant percentage decline for each of the various leases, then it can be shown that the composite production from these leases will no follow theoretically a constant percentage decline, but will flatten with time from an initially high decline rate to a decline approaching asymptotically the lowest individual decline rate from the group of leases.
Two or three circumstances., however, come into play which allow the presumption that, for all practical purposes, this composite curve of production can indeed be considered a straight production can indeed be considered a straight line on semilog paper, i.e., exponentially declining.