This paper provides a number of comprehensive algebraic formulae, with readily determinable coefficients, which can be used to predict the extent and width of fractures produced by fluid injection at specified rates or borehole pressures, when these are reasonably well-behaved functions of time. The fracture geometries described include as special cases the models currently in use for industrial design of hydraulic fractures, but extensions to allow variable fracture height are readily achieved. The formulae serve both to simplify the implementation of conventional models and to allow development of more realistic simulations which contain the rather idealised concepts of those models in their rightful place as components of a more general three-dimensional place as components of a more general three-dimensional description. One such pseudo-3-D model is described in its simplest form; it allows physically credible tracing of length, height and width distributions under conditions of slow vertical spreading which a desirable stimulation treatment would achieve. All of the models admit quite general reservoir properties and frac-fluid behavior. A few popular applications are used to illustrate their power and simplicity.


Over the past two decades, a considerable amount of effort has been expended on the development of models intended to describe the effects of a hydraulic fracturing treatment and to aid in the design of pumping sequences aimed at optimisation of the return pumping sequences aimed at optimisation of the return on considerable investments in equipment, labour, and materials employed in a typical field operation. Various analyses and numerical routines have emerged from this activity (e.g. 1–8), but it seems fair to say that few of the authors would claim a satisfactory level of realism for their simulation capabilities, except perhaps in cases of unusually favourable circumstances in the reservoir being fractured. This state of affairs can be readily explained by the exceptionally difficult combined character of the phenomena which must be represented. A renewed effort phenomena which must be represented. A renewed effort has been underway over the past few years (e.g. 9–16) to obtain more realistic descriptions of the hydrofrac process; many insights have resulted from this activity, process; many insights have resulted from this activity, but a worthwhile fully three-dimensional simulator will require a few more years of concentrated endeavor. Numerous models may appear in the meantime, superficially embodying a 3-D capability; they will certainly be lacking many of the complex features which recent work (e.g. 9–10) has shown to be so essential for a physically realistic representation of the process involved in even the simplest reservoir geometries.

In the absence of such an acceptable comprehensive simulation capability, it appears necessary to have at least some approximate means of determining in a credible way what the general features will be for a fracture produced by fluid injection through a borehole, particularly as to effective length, width, and height. While it is true that these quantities are not yet measured accurately in the field, it is certainly possible to make sufficiently good deductions from reservoir data in order to establish what the overall character of hydrofrac evolution will be and thus eliminate some of the more ridiculous models. Indeed, it is also possible to achieve scaled laboratory versions of increasingly complicated reservoir structures and the theoretical predictions should at least agree with the fracture growth observed in these. Although the geometry assumed in almost any model, no matter how simple, can actually be generated in the laboratory, one must recognise and account for the complicated shapes which develop when test conditions approximate those found in the various geological circumstances where oil and gas are present.

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