Wellbore stress change due to drawdown and depletion: An analytical model and its application
Authors: Schutjens Peter Maarten , Knowles David Wayne , Dunayevsky Victor , Rambow Frederick H.K. , Love Keith , Fehler Daniel , Khodaverdian Mohamad , Frazer Simon Andrew , Leeftink Gerrit 
Published: February 2016

Abstract
An analytical model is presented to describe total stress change at a vertical wellbore due to drawdown and depletion. The model is applied to two reservoirs with different fluid-flow properties: Reservoir A has a porosity of about 20% and permeability of up to 30 milliDarcy. Reservoir B has a porosity of about 30% and a permeability of a few Darcy. Mechanical properties, including shear-failure criteria, were available from deformation experiments on reservoir core.
Low-permeability reservoir A requires high drawdown values of at least 34 MPa to achieve economic production rates. The maximum of the total stress then lies in the horizontal plane, oriented tangentially to the wellbore wall. The minimum stress also acts in the horizontal plane, but is oriented orthogonal to the wellbore wall. An elastic-brittle shear failure analysis suggests that a drawdown of 34 MPa will be accompanied by wellbore shear failure. This may start after 15 MPa of drawdown, or perhaps even at drawdown values as low as 5 MPa. Adverse effects of wellbore shear failure include casing or liner damage and a high risk of sand production. A benign effect could be enhanced production due to a fracture-induced increase in near-wellbore permeability.
The modeled drawdown in high-permeability reservoir B is about 2 MPa, which is relatively low compared to its depletion (tens of MPa). The total stress is maximal along the wellbore axis (in the vertical direction), and is minimal in the direction orthogonal to the wellbore wall. Our model predicts that shear failure limits in reservoir B will be reached after 20 to 30 MPa of depletion. With first massive-sand production observed after 40 MPa depletion, our model is too conservative. However, massive sand production is typically preceded by inelastic (permanent) deformation of the rock around the perforations by grain and grain-contact-cement breakage, enabling grain sliding and grain rotation. Our model could perhaps be used to predict the start of this mechanism. Field data such as rock acoustic emissions are needed to verify this.

Introduction
Production-induced reduction of reservoir fluid pressure is controlled by structural and stratigraphic inhomogeneities at all scales. From a geomechanical perspective, and simplifying the geology, one can distinguish pore pressure reduction close to the wellbore due to the pressure difference in well and reservoir (drawdown) and pore pressure reduction on a field scale (depletion). Both drawdown and depletion are accompanied by changes in total stress in reservoir and surrounding rock1–4. This implies that the change in local effective stress is not equal to the change in fluid pressure, and differences of several percent to tens of percent could occur. Knowing such stress changes for drawdown and depletion helps to predict and control sand production and casing deformation5, and allows making a pre-production estimate of compaction-induced permeability change6 and minimum total principal stress change affecting the fracture gradient4,7. Knowing total stress change during depletion is also important to predict reservoir-compaction-induced subsidence8 and seismic timeshifts and acoustic impedance changes9,10.
Here we present an analytical model to predict the total stress change around a vertical well due to drawdown and depletion. The model is applied to two sandstone reservoirs of similar size but with different fluid-flow characteristics: Reservoir A has a porosity of about 20% and permeability of up to 30 milliDarcy. Reservoir B has a porosity of about 30% and permeability of a few Darcy. Taking drawdown patterns from fluid-flow models (not described here), we calculate wellbore stress changes as a function of drawdown and depletion. We then study how these affect well integrity and sand production, based on an elastic-brittle analysis with wellbore shear failure.
Wellbore geomechanics can be done with empirical, numerical and analytical models. Empirical models require extensive production data, and are based on correlations rather than on physical mechanisms11. Numerical models are easy to construct these days, but are quite often hard to interpret, and can show cell-size/shape effects and numerical instabilities. Analytical geomechanical models like the one we present here have their limitation in the simplification of geology and rock properties. Yet they are suitable to screen for specific geomechanical effects and for comparative analysis between "extreme cases", paving the way for detailed numerical modeling and for collection of field data to test them (see Kuvshinov12 for a detailed discussion).