Characterization of Early-Time Performance of a Well with a Vertical Fracture Producing at a Constant Pressure

Abstract

The flowback performance of multi-fractured horizontal wells completed in an ultra-low permeability (shale) reservoir often includes an anomalous flowrate feature at early times prior to the development of a characteristic reservoir flow regime (i.e., linear or bilinear flow). This flowrate feature is evident in all of the flow phases (gas, oil, water, and total fluid) and is caused by the "unloading" of the well (essentially the "clean-up" behavior following well stimulation). Our concept is that this flowrate feature is caused by decaying skin effects, a changing wellbore storage effect, or a combination of both a decaying skin effects and changing wellbore storage effects. For simplicity — and as a proof-of-concept, this research considers only the case of a vertical well with a single vertical fracture, but the concept and relations developed in this work can be directly extended to the solution for a multi-fractured horizontal well.

As noted, the goal of this research is to develop series of time-dependent skin and wellbore storage models to characterize the early-time flowrate behavior observed in practice, under the constraint of a constant wellbore flowing pressure. Our procedure is to couple case of time-dependent wellbore storage and skin effects with a set of reservoir flow models (i.e., power-law, bilinear and formation linear flow) and by applying the convolution integral to the constant pressure condition, we generate various scenarios of production performance.

Specifically, in this work we provide derivations of the development of each constant pressure solution, where all work is performed Laplace domain and the Stehfest Algorithm is used to numerically invert each case to the real domain. A graphical illustration of the performance of each model is provided and a generalized workflow is presented (we note that this workflow can easily be extended to more complex fracture structures — i.e., the multi-fractured horizontal well case).

Although these models represent different physical phenomena, we observe that all of the proposed models provide some mechanism for representing early-time variations in flowrates. We demonstrate the relevance of these models, which are based on empirical time-dependent models for wellbore storage and skin effects, as proxies that can be used to represent early-time flowrate behavior. In short, we demonstrate that each time-dependent model has unique characteristics which could theoretically allow for characterization of fracture behavior prior to the onset of an undistorted "reservoir" flow regime (i.e., formation linear or bilinear flow).