Analytical modeling of multi-fractured horizontal well performance can be overly complex and cumbersome due to the use of Laplace space and pseudo-time. State-of-the-art analytical solutions are typically developed using Laplace space, which is not easily understood and often requires numerical inversion. Pseudo-time (for gas reservoirs) is iterative and has demonstrated issues with accuracy, particularly in stress-sensitive reservoirs. In this work, we present a new, practical semi-analytical model which provides a direct time domain solution using the succession of steady-states method without the need for pseudo-time.
The well-reservoir description used in our approach is based on the enhanced fracture region model introduced by Stalgarova and Mattar (2012). It is a composite system consisting of a stimulated (inner) region connected to a fracture and adjacent to an unstimulated (outer) region. Wattenbarger's bounded linear flow solution (1998) is utilized to account for linear flow contributions from both regions, in succession. The proportion of flow through time from the inner and outer regions is determined using a transient productivity index during transient flow and using material balance during boundary-dominated flow.
The accuracy of the model is evaluated by comparing its results to an equivalent numerical model, encompassing a wide variety of tight oil and gas descriptions with varying reservoir properties and operating conditions. For all cases studied, this model achieves either the same consistency with numerical simulation results, in comparison to its Laplace space counterpart. This new model is also more accurate for gas reservoirs with pressure-dependent rock and fluid properties. The associated nonlinearities are handled using a modified productivity index during transient flow, which is a function of the average reservoir pressure within the region of influence. This average pressure is traditionally calculated using a material balance performed over the area of investigation through an iterative procedure. In order to avoid such a procedure, an explicit relationship has been developed which correlates the average reservoir pressure to the initial reservoir pressure and the bottomhole flowing pressure. This new technique provides a simple engineering workflow and an alternative to numerical simulation for modeling complex fracture networks.
To the authors knowledge, this is the first analytical model which does not require an iterative approach to obtain its solution during transient flow. Since it is solved in the time domain (unlike models in Laplace space), it can be more easily implemented in a spreadsheet application. It is also more accurate, requires less calculation effort and can be extended to accommodate additional complexities, such as multiphase flow.