Non-Linear Dynamic Programming of Hydraulic Fracturing Models

American Institute of Mining, Metallurgical, and Petroleum Engineers, Inc. This paper was prepared for the 43rd Annual Fall Meeting of the Society of Petroleum Engineers of AIME, to be held in Houston, Tex., Sept. 29-Oct. 2, 1968. 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 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. 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. Non-Linear programing techniques are applied to hydraulic fracturing models resulting in the design of an optimum fracture treatment. The objective function is chosen as the discounted present worth of the well and is optimized with respect to eight decision variables. The region of feasibility is restricted by both upper and lower bounds on each decision variable and by constraints placed on the interactions of the various placed on the interactions of the various variables within the models. The fracture process is considered to be a single stage process, using deterministic models. These models predict the fracture width and length, the propping agent distribution, the stimulation ratio and the production decline. The models consider only packed vertical fractures and uniform undamaged reservoir formations. The optimization method is based on the principle of steepest ascent. From an principle of steepest ascent. From an arbitrarily chosen initial set of decision variables one "climbs" a multidimensional hill, the axes of which represent all the decision variables and the objective function. The summit represents the maximum value of the objective function. The computer program for this calculation automatically adjusts step size, increasing or decreasing the size depending on the ease of progress. Criterion for the attainment of an optimum is the dropping of the step size to a pre-established lower limit. pre-established lower limit. The possibility of applying venture analysis to fracture treatment is also discussed. This involves the assignment of a probability distribution to uncertain parameters in the model. This allows the parameters in the model. This allows the estimate of a probability curve showing the probability of achieving various values of probability of achieving various values of the objective function. Introduction Scientific Decision Making Operations research has been defined by McCurdy as "the rational application of quantitative methods to problems of planning and decision making." The planning and decision making." The tremendous improvement in computer technology in the past two decades has completely revolutionized this type of decision making. It has provided "the capabilities of generating and interpreting vast quantities of data and information."