Elastoplastic Modelling of Sand Production Using Fracture Energy Regularization Method
Authors: Nouri A. , Kuru E. , Vaziri H. 
Published: March 2010

Abstract
This paper extends the capacity of the current sand production models by eliminating the influence of artificial conditions and numerical mesh on localization and deformation response in the sanding model. Past studies indicate strong size effects when using classical elastoplastic models.
To rectify this deficiency, a fracture energy regularization method is implemented in the numerical model. The model incorporates both the geomechanical aspects (e.g. rock elastoplastic deformation and rock disaggregation), as well as the transport aspects (e.g. the role of seepage on rock deformation and solid release). The model employs a Mohr-Coulomb flow theory of elastoplasticity with friction hardening/cohesion softening. Emphasis is given on calibration procedure and validation of the enriched model through back analysis of triaxial and uniaxial compression tests. Next, the model is used to compare the numerical predictions with laboratory data on sand production. The comparison incorporates the stress and deformation, as well as the sand volume.
The calibration study shows that friction hardening and cohesion softening can satisfactorily reproduce numerically the weak sandstone response to various loading conditions. Further, computation results of strain softening material illustrates that a fracture energy regularization strategy enables the model to exhibit mesh invariance of the energy dissipation.

Introduction
Sand production involves two distinct stages. These are:mechanical degradation of the intact sandstone rock to loose particles by the stress concentration around the wellbore; andthe transport of the loose particles by hydrodynamic forces to the wellbore. An effective sand production model must be adequately equipped with the tools that simulate the phenomena associated with both degradation and seepage forces. One such model is discussed in this paper with an emphasis on modelling of the degradation process and a detailed description of the elastoplastic model calibration.
As it has been discussed in the literature, rock mechanical degradation is related to the development of micro-cracks as failure localizes in narrow bands at post-peak strength. Development of the micro-cracks violates the continuum mechanics assumption leading to spurious influence of the numerical mesh on the formation response(1–7). This mesh dependency is separate from the small numerical error, which should tend to zero with mesh refinement. As a result of the mesh dependency, the numerical model looses its objectivity and needs to be rectified.
Recognition of the deficiencies of the standard continuum theory in the modelling of deformation discontinuity has led to the development of various enrichment methods. De Borst(8) compared the performance of several of these techniques. The common approach in all these methods is the introduction of some sort of length scale that must be built into the constitutive model.
Mesh independence for localization problems can be obtained in a pragmatic fashion by scaling the softening rate in inverse proportion to the element size. This approach was described by Crook et al.(7) and is based on the work of Pietruszczak and Mroz(5) and Bazant and Oh(6). The basic idea in this method is that fracture energy, which is the energy dissipated due to the formation of micro-cracks, must not differ for numerical meshes of various size.