Reservoir simulation

Transient Coupled Wellbore/Reservoir Model Using a Dynamic IPR Function

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.


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.


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.


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.

The main assumption in these models is that the bottomhole flowing pressure is constant; therefore, it is common practice in the oil industry to use these equations in classical nodal analysis, for example. However, it is equally common to use these IPRs to generate the reservoir response in time-dependent wellbore models, where the bottomhole flowing pressure will vary significantly with time. In this scenario, most IPR models are no longer valid and generate unrealistic reservoir responses to changes in wellbore conditions. Currently, coupling reservoir and wellbore simulations is the only alternative to account for transient effects in both the reservoir and the wellbore, but this can be computationally costly and can have limited application in the lifetime of a well. This paper introduces a new alternative, the dynamic IPR.

Dynamic IPR

IPR models give flow rates for a fixed bottomhole pressure, or vice versa, and are not able to capture reservoir dynamics (i.e., the interactions between Pwf and qo).

This paper postulates that it is possible to represent the dynamic relationship between Pwf and qo by taking into account their time derivatives.

Contrary to the conventional IPR models that rely on algebraic expressions, the dynamic IPR is a linear ordinary-differential equation designed to capture all aspects of the reservoir behavior.

In addition, as opposed to conventional reservoir-simulation systems, the dynamic IPR is not a partial-differential equation (no explicit spatial dependence); hence, it is less expensive computationally than the discretization of the hydraulic diffusivity equation in both time and space. Nevertheless, the information required to determine the parameters of the dynamic IPR still lies within the hydraulic diffusivity.

Case Studies

The complete paper presents four case studies to illustrate the versatility of the dynamic IPR. The idea behind this procedure is to prove that, once the dynamic IPR for a particular reservoir is obtained, it can be used to determine flow-rate and bottomhole-flowing-pressure relationships in any scenario as long as fluid and reservoir properties remain the same. The cases presented are for homogeneous, isotropic, circular-shaped, undersaturated-oil reservoirs with a finite-diameter vertical well in the center.

Cases 1 and 2 correspond to a simple drawdown and a drawdown and buildup, respectively. These two cases involve nonperiodic functions and have simple analytical solutions—at least before the pressure drop reaches the external boundaries of the reservoir. These two cases show that, even though periodic functions were used to determine the coefficients of the dynamic IPR, it can by equally applied to conditions in which nonperiodic dynamics are observed.

Case 3 consists of a casing-heading scenario. Here, the superposition of the sinusoidal solutions at different frequencies yields the analytical solution; however, the input bottomhole flowing pressure first has to be decomposed into a sum of sinusoidal functions. The results of this approach are compared with the results from conventional finite-difference calculations.

Finally, Case 4 couples the dynamic IPR with a simple transient wellbore model to evaluate the effect of the reservoir dynamics in terms of measurable quantities such as production flow rate and wellbore and wellhead pressures. This case study is presented to show that the dynamic IPR is able to characterize the wellbore/reservoir dynamics correctly with minimal computational effort and that the steady-state IPR presents imprecise results.


This work presents classic transient solutions to the hydraulic diffusivity equations that are also relevant to transient simulations of wells. The use of conventional steady-state IPR models is demonstrated to be inadequate in these cases. By implicitly assuming an infinite transmissibility, the steady-state IPR models simply cannot capture the transient behavior of the reservoir. Even though it is a common practice in the industry, the use of a simplified reservoir model in transient well simulations using commercial simulators can have a significant influence on the reliability of the estimated results.

An alternative to the conventional IPRs is the direct coupling of a transient reservoir model with a transient wellbore model. This approach likely is more precise, but it is computationally more expensive and it increases the license cost of the software. This work presents an alternative approach through a dynamic correlation that can capture any transient behavior of the reservoir.

The case studies show that the dynamic IPR is able to estimate the reservoir response reliably in classical cases such as buildup and drawdown, where the maximum relative error was lower than 0.5%. It is also applied in cases of practical relevance to transient simulations such as severe slugging and casing heading, where the amplitude of the flow-rate oscillation predicted by the conventional IPR was ­almost 50% lower than the actual amplitude of oscillation.

This paper also showed that the dynamic IPR can be coupled easily with transient wellbore models and give reliable results, especially when compared with results obtained through the application of conventional steady-state IPR models. A case study shows that the ­casing-heading period increased by 40% when the reservoir dynamics were incorporated into the transient simulation of wells. More important than that, another case showed that neglecting the reservoir dynamics can lead one to conclude that an oscillatory well is actually stable.

Another possible application of the dynamic IPR is to antislugging-control-­system design. A simplified well model is usually used in calculation of the constants of the controller. The dynamic IPR is already generated in a format suitable for control-system applications—the transfer function. On the other hand, there are more-complex situations that were not addressed in this work, such as multiphase flow within the reservoir and the dynamic formation of gas and water cones. At least in principle, a ­generic dynamic IPR could be applied in those cases.

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.