This paper presents some derivations and results of lumped numerical models which describe the growth and final geometry of hydraulic fractures. Using generally averaged reservoir 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 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 the general usefulness of these models for fracture design and on -site interpretation of observed versus predicted downhole pressures. predicted downhole pressures.
The literature on hydraulic fracturing modeling is extensive and abounds with different approaches to the various parts of this complex problem. Some of the earliest models problem. Some of the earliest models are universally used in the industry; some of the newer models are increasingly used. In previous papers we presented the basis previous papers we presented the basis for the analytical framework of the levels and techniques used in these numerical models. The three motivations of this present paper are: simplification, unification, and development of comprehensive practical simulators. These are all achieved by means of so-called "simple lumped models," which employ spatial averaging to reduce the problem to manageable form while retaining generality for the incorporation of other more complicated models. The underlying philosophy in developing the lumped models has been to incorporate as many of the significant mechanisms controlling fracture growth in as simple a numerical scheme as possible. As a consequence, the lumped models are possible. As a consequence, the lumped models are comprehensive descriptions, designed to allow the incorporation of results from more general models as reference data bases. The variable fracture geometry capability presented here allows generalized fracture shapes, from equiaxed to either well or poorly contained fractures. Fluid/reservoir effects such as fluid loss, backstress, thermo-rheology, and proppant transport can be accounted for with simplified versions of general 2-D and 3-D models. The simple lumped models presented in this paper include results from several of these models, specifically the P3DH cross-sectional simulations and from simple 3-D models results from more general models can be incorporated as they become available. The result is a fairly simple, efficient numerical routine that captures the essential features of the problem and seems to lead to reasonable predictions of fracture pressures and dimensions in our experiments. An earlier paper presented a description and some basic fracture growth equations; in this paper we present other extensions of the lumped models such as variable fracture geometry and fluid/reservoir interactions.