Modelling of Transverse Hydraulic Fracturing
Authors: Lietard Olivier , Maniere J. , Norris Mark Robert 
Published: December 2007

Abstract
The expansion of horizontal well technology over the last twenty years has led to the parallel increase of the number of hydraulic fracturing treatments in highly deviated wells.
The non colinearity of the wellbore axis and of the fracture plane has initially induced significant tortuosity effects and premature proppant screenouts. The length of the perforated interval has therefore been reduced to the acceptable minimum.
Although operational problems have been solved this way, the net pressure response while successfully fracturing did not obey any of the existing 2D models (PKN, KGD or Radial). As a consequence, job designs remained impossible and optimum pumping schedules were typically established by trial and error within a given reservoir.
The present paper offers the required equations to correctly design transverse fracturing treatments, or collinear fractures with short perforation intervals. As compared to previous models, it only replaces the line source and linear flow assumptions (a consequence of colinearity and long perforated intervals) by the point source and radial flow ones. The new model therefore combines both radial fracture and flow geometries, instead of the conventional Radial model where linear flow (orthogonal to the wellbore axis) is assumed.
The theoretical diagnostic (Nolte) plot is in perfect agreement with actual job responses, i.e. a positive, however extremely low slope (almost constant net pressure all along the pad stage). Such a behaviour cannot be interpreted with the responses of the previous models (steep slopes, either positive - PKN - or negative - KGD and Radial).
An actual, well documented treatment in the North Sea validates the proposed model for a short point source hydraulic fracturing from a horizontal wellbore.

Introduction
Hydraulic fracturing of deep enough and highly deviated wells is known to potentially propagate transverse fractures, the wellbore axis not being contained in the fracture plane. In order to prevent the initiation of several parallel fractures and to mitigate the risk of premature proppant screenout (as a result of low rate into, hence narrow width of, each fracture), these wells are cased, cemented and perforated over a very short, high shot density cluster. A unique transverse fracture of radial geometry is propagated from the central cluster, which can be considered as a source point.
To design such fracturing jobs and to interpret pressure responses while pumping and after shut-in, it seemed obvious at first sight that the so-called Radial model was the best suited to the case, owing to the fracture geometry. Over the past twenty years, however, it became even more obvious that none of the available fracturing models were matching with the observed pressure responses. While pumping such jobs, the net pressure was high and almost stable throughout the pad stage, instead of being high and increasing (PKN) or low and decreasing (KGD or Radial).
The reason why current available models are not adapted to the case of point source hydraulic fracturing is that they all assume linear flow, a legacy of the past of fracturing vertical wells, either open hole or across long perforated intervals. In such cases indeed, the flow source is assumed to be a line, of either fixed (PKN) or increasing (KGD and Radial) length, with orthogonal flow along the fracture.
A new model is herein proposed, with both the fracture geometry and flow being radial. Actually not so new, since Sneddon and Elliot had started to derive it as early as 1946.1 At this time however nobody could imagine that the era of horizontal drilling would eventually come, so their difficult mathematical approach of the source point "true" radial model was abandoned. Its present resurrection therefore stems from the simple recognition that most horizontal well hydraulic fracturing treatments are nowadays performed in wells oriented to propagate transverse fractures or collinear fractures from short perforation clusters.
The mathematical developments of the new LMN model (from the names of the authors, as usual in the industry) follow the methodology of Gulrajani and Nolte.2 They are, however, more rigourous in the sense that they are based on equations rather than on proportionalities. In addition, they are not based on the static elliptical profile of the so-called penny-shaped crack in an infinite elastic body.3