A New Coupled Fluid Flow/Stress Model for Porous Media Behavior: Numerical Modeling and Experimental Investigation
Authors: Belhaj H.A. , Ryan R.J. , Nouri A.M. , Butt S.D. , Frempong P. , Islam R.M. 
Published: December 2007

Proposal
Reservoir management has been improved dramatically through adaptation of new technologies, enhancement of uncertainty involved and better data management to meet the challenges of today's economic constraints. Yet, these remarkable achievements included little or none about questioning the fundamental concepts of reservoir behavior and the assumptions made at the early days.
In this paper, a comprehensive experimental program has been carried out using a triaxial set-up equipped with on-line acoustic apparatus to study reservoir behavior under depletion process. Fluid flow and reservoir rock properties' behaviors have been investigated as the reservoir is being produced. It has been concluded that the currently used methodology is in great error. Using the diffusivity model based on Darcy's law is one source of this error and the assumption of constant porosity and permeability throughout the life of the reservoir is another source. Therefore, a new fluid model has been proposed to replace the Darcy's model. The new model is derived from the Navier-Stokes equation and is capable of addressing all kinds of fluid flow taking place in reservoirs at low and high flow velocities including both Darcian and non-Darcian regimes1.
On the other hand a new model relates porosity to the mean effective stress changes during a depletion process has been mathematically derived, this model is able to track porosity changes as effective stresses increase during reservoir production. The idea is to couple this model with the fluid flow model introduced in this study so both porosity and permeability can be updated at each time step during simulation of reservoir behavior.
Laboratory results confirm the numerical predictions of the proposed coupled fluid flow/stress model, at the same time the laboratory results are in complete disagreement with Darcy's model predictions. Improvement of reservoir simulators' accuracy is expected through the implementation of the newly proposed model.

Introduction
Traditionally, data about petrophysical properties are essential for reserves' estimates and for the fluid flow characterization of petroleum reservoirs. A great deal of money and effort are expended to determine these properties accurately. Permeability and porosity are among those properties and are by far the most important. Porosity is the key for reserves' estimates while permeability is the main parameter to predict flow rates, design drawdown and therefore wellbore completion. Until recently, the common belief was that, once determined, these properties remain constant throughout the production life of the reservoir. However, numerous research studies2–14 including this one, show that this is not the case.
During the pressure depletion process as production from the reservoir continues, effective stresses within the reservoir increase. The effect of reservoir stresses on porosity and permeability of the reservoir is more severe when porosity and permeability are high (fractured media is an extreme case), although some experimental studies like Hubbert and Willis3, Voight4 and Rosepiler5, showed that this effect is still significant even at low porosity and permeability. When considering percentage losses from the original state of the reservoir, it seems that the same percentage of porosity and permeability reductions was experienced regardless of the original values. It is also understood that stress paths have a large influence on horizontal and vertical permeability as well as on porosity.
The elastic uniaxial strain model is used in reservoir engineering mostly to describe production-induced changes in horizontal stress due to pore pressure decline (pressure depletion). It predicts the total horizontal stress by using overburden stress, reservoir pressure decrease and material mechanical parameters. The principal assumption in this model is that there is no lateral deformation (zero horizontal strain condition) during the depletion process.