This paper presents some derivations and results of lumped numerical models which describe the growth and final geometry of hydraulic fractures. Using generally averaged resevoir and fracture parameters the model employs simple numerical routines to solve coupled non-linear ordinary differential equations for the pressure and dimensions of the fracture. Results of more complex models are included as a data-base in the spatial lumping process; thus the models represent all the essential features of fully 3-D, physically realistic hydraulic fractures. The present model provides a consistent description of fracture growth and includes the effects of mechanisms such as fluid loss, backstress, heat transfer, proppant and variable fracture geometry. Results are presented for a variety of typical field applications and show the general usefulness of these models for fracture design and on-site interpretation of observed versus predicted downhole pressures.

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