International Conference «Mathematical and Information Technologies, MIT-2016»

28 August – 5 September 2016

Vrnjacka Banja, Serbia – Budva, Montenegro

## Methods for optimal control of hydraulic fracturing process

### Reporter: Cherny S.

The model of fracture initiation and propagation from the cavity in elastic media caused by viscous fluid pumping is proposed. Based on this model the method for optimal control of fracture initiation and propagation processes is developed. Input parameters of fracture initiation and propagation model are: the surface of the cavity in infinite elastic media; pumping pressure of fluid that causes fracture initiation and propagation (or pumping schedule for fluid of given rheology); elastic media parameters. Output characteristics of the model are: the fracture surface; fracture width distribution; speed of fracture front propagation. Determination of output characteristics using the input parameters is the direct problem of fracture propagation. By solving it one can predict the geometry of forming fracture, volume of oil produced from the fracture, calculate costs of this process, etc. Inverse problem considers finding the input parameters of fracture initiation and propagation model with which the solution of direct problem satisfies the given objective functions of fracture initiation and propagation processes. Optimal control of hydraulic fracturing process consists in the solution of inverse problem. It requires chosing the parameters of rheological laws for fluid, pumping schedule, conditions of fracture initiation (shape of the cavity, its orientation against in-situ stresses of elastic media) that satisfy the needed location of incipient fracture, linearity of fracture propagation trajectory, uniformity of fracture width distribution along the trajectory, no profile twisting along the fracture trajectory, minimal costs for hydraulic fracturing, maximal volume of produced oil, etc. To solve the inverse problem the method of optimization design is used. This method consists choosing the input parameters of direct problem that do provide the best fulfillment of one or several objectives. Strategy of input parameters adjustment is based on the genetic algorithm.

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