# A physically consistent solution for describing the transient response of hydraulically fractured and horizontal wells

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## Abstract

Conventional horizontal well transient response models are generally based on the line source approximation of the partially penetrating vertical fracture solution. These models have three major limitations: (i) it is impossible to compute wellbore pressure within the source, (ii) it is difficult to conduct a realistic comparison between horizontal well and vertical fracture transient pressure responses, and (iii) the line source approximation may not be adequate for reservoirs with thin pay zones. This work attempts to overcome these limitations by developing a more flexible analytical solution using the solid bar approximation. A technique that permits the conversion of the pressure response of any horizontal well system into a physically equivalent vertical fracture response is also presented.

A new type curve solution is developed for a hydraulically fractured and horizontal well producing from a solid bar source in an infinite-acting. Analysis of computed horizontal wellbore pressures reveals that error ranging from 5 to 20% depending on the value of dimensionless radius (rwD) was introduced by the line source assumption. The proposed analytical solution reduces to the existing fully/partially penetrating vertical fracture solution developed by Raghavan et al. [1976]. as the aspect ratio aspect ratio (m) approached zero (m < 10-4), and to the horizontal fracture solution developed by Gringarten and Ramey [1974] as m approaches unity. Our horizontal fracture solution yields superior early time (tDxf < 10-3) solution and improved computational efficiency compared to the Gringarten and Ramey's solution, and yields excellent agreement for tDxf > 10-3.

A dimensionless rate function (Beta-function) is introduced to convert the pressure response of a horizontal well into an equivalent vertical fracture response. A step-wise algorithm for the computation of Beta -function is developed. This provides an easier way of representing horizontal wells in numerical reservoir simulation without the rigor of employing complex formulations for the computation of effective well block radius.