Non-linear dynamic programming of hydraulic fracturing models

Abstract

Non-Linear programming 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 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 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. 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 estimate of a probability curve showing the probability of achieving various values of the objective function.