Microcomputer Models for the Design of Hydraulic Fractures

Abstract

Abstract This paper describes approximate models which yield credible descriptions of hydraulic fracturing processes and are amenable to calculations on processes and are amenable to calculations on microcomputers. These models provide the basis for both analysis of conventional fracturing treatments and for proposing/designing new procedures suitable to varied reservoir conditions. Satisfying a short-term need for realistic analysis that offer improved fracture designs, the models also admit considerable upgrading as more detail/realism is described by more complex simulators. The two levels of analysis presented are: presented are:Lumped models, in which the spatial variation is represented by integral coefficients. Although simple enough for use on a microcomputer or programmable calculator, the models are detailed programmable calculator, the models are detailed enough to describe geometries varying from long Perkins and Kern-type, through circular shapes, to Perkins and Kern-type, through circular shapes, to high CGD (Christianovich et al.)-type fractures.More complex pseudo-three-dimensional (P3DH) models (which require at least a small micro-computer for proper implementation); these divide the 3D problem into coupled sets of equations governing one-dimensional lateral fluid flow and two-dimensional growth of vertical cross sections. A major feature is a refined mesh that moves with the fracture, rather than one fixed to the reservoir; the result is detailed tracing of evolving geometry, pressure distribution and flow throughout the job and after shut-in. Although not as complex as our full 3D simulations under development, these approximate models contain most important elements governing fracture evolution. They compute such significant design evaluation quantities as length, height, width and pressure variation with specified injection rate (or pressure variation with specified injection rate (or vice versa). To date, the models have produced many significant results that aid in explaining observed pressure profiles and may, in the future, be used at the wellhead for real-time control of fracturing operations. Introduction The modern technology of hydraulic fracturing, as against plentiful ancient and geological examples, has been in development for over three decades (e.g., see ref. 1 for a review). Various models have been developed throughout that period, of both physical analog (e.g., ref. 2) and theoretical/computational kind (e.g., ref. 3); these models have emulated one or more features perceived as important in hydrafrac phenomena, artificial or natural. However, such phenomena, artificial or natural. However, such simulations have usually neglected some vital aspect of the process, so that a host of different models have been formulated/implemented (e.g., refs. 4–11), many of them requiring quite complex/expensive computation. This situation has been remedied somewhat by the increasing sophistication of more recent efforts (e.g., refs. 12,13), but these have retained some degree of incompleteness or complexity: specifically, only some limited examples were worked out, a number of coefficients were introduced without details of their computation and the solution procedures were constrained by their association with procedures were constrained by their association with general, existing reservoir simulators. This paper seeks to amend some of those shortcomings. Firstly, the basic components of hydrafrac are isolated from the reservoir transport processes—which are assumed to be captured by conventional or special techniques (e.g., ref. 14). Then the dominant features are extracted in analytical form, amenable to small (pocket) calculator implementation, which show the role of the various parameters—such as variations of confining stress, rock moduli and frac-fluid rheology. A particular class of fracture geometries is chosen for illustration, motivated by conditions found/assumed to prevail in a great variety of reservoirs (especially low-permeability gas-bearing sandstones); this assumes greater lateral extent than vertical height, due to either natural barriers or reservoir preparation (e.g., ref. 15), but the model can also handle more equi-axedness with some coefficient adjustments. P. 257