Transient Coupled Wellbore/Reservoir Model Using a Dynamic IPR Function
Authors: Wilson Adam 
Published: July 2017

This article, written by Special Publications Editor Adam Wilson, contains highlights of paper SPE 181691, “A Transient Coupled Wellbore/Reservoir Model Using a Dynamic IPR Function,” by A. Posenato Garcia, The University of Texas at Austin; P. Cavalcanti de Sousa, Texas A&M University; and P.J. Waltrich, SPE, Louisiana State University, prepared for the 2017 SPE Reservoir Simulation Conference, Montgomery, Texas, USA, 20–22 February. The paper has not been peer reviewed.
Conventional inflow-performance-relationship (IPR) models are used in coupled wellbore/reservoir transient simulations, even if bottomhole-pressure conditions are assumed to be constant on the derivation of such IPR models. The dynamic IPR model proposed in this paper not only captures the relevant reservoir dynamics from the well perspective but also is computationally more efficient than discretized models using hundreds of gridblocks to simulate the near-wellbore region.
Introduction
Traditionally, well deliverability is obtained by combining IPR and vertical-flow-performance (VFP) curves. This method was first discussed in 1954 and provides snapshots of the average- bottomhole-flowing-pressure (Pwf) and average-oil-flow-rate (qo) relationships at given times in the life of the well. It fails, however, in accurately portraying the transient behavior of these variables. Transient relationships are important in the design and analysis of pressure-transient tests, design of production tubing and artificial-lift systems, reservoir management, and estimating flow rates from multiple producing zones.
Capturing the real behavior of these variables requires coupling the reservoir model and wellbore model. General-use coupled models rely on running reservoir simulations (accounting for near-wellbore effects) and using the output of that run as the input to the wellbore model, which, in turn, will retrofit the reservoir simulator for a new run. This can be costly and time consuming and it is not always successful or accurate. On the other hand, coupled models developed for special cases, while still relying upon simulations, are faster and provide reliable results but have no general application.
This paper introduces a new technique of coupling wellbore and reservoir models where the simulation of the reservoir response to changes in the bottomhole flowing pressure is obtained not by numerical methods but rather by solving the diffusivity equation using the Fourier transform. This mathematical tool generates time-dependent equations—the dynamic IPR—which are able to provide the reservoir response to any pressure variation, regardless of how fast or slow this change might be.
Methods
Conventional IPR models are algebraic equations correlating the bottomhole flowing pressure and the flow rate through the production-zone completion.
Because most of these models rely on correlations and experimental-data fit, it is common practice in the industry to perform well tests periodically to correct the IPR models according to the most-updated information on the fluids and reservoir parameters.
The focus of this work is on homogeneous, isotropic, circular-shaped, undersaturated oil reservoirs with a finite-diameter vertical well in the center. Thus, the authors considered IPRs that correspond to these same conditions. These IPRs are defined for three different flowing times: transient, pseudosteady state, and steady state. Please see the complete paper for the relevant equations.