Constitutive Modeling of Deformation and Permeability: Relationships between Critical State and Micromechanics
Authors: Crawford B.R. , Yale D.P. 
Published: December 2007

Abstract
The "critical state" concept of elastoplastic constitutive behavior has been utilized extensively to model the stress-strain response of many geomaterials that undergo plastic yielding. This paper uses a critical state model to investigate the deformation and deformation-induced permeability changes observed in laboratory experiments conducted on siliciclastic and carbonate petroleum-reservoir rocks. We find that the shape of the yield surface has significant impact on post-yield fluid flow response. Differences in behavior between the rock types tested can be explained qualitatively through consideration of differences in the micromechanics of deformation. We find that permeability changes with both stress and strain in general follow a similar constitutive model as deformation. However, we observe experimental behavior not fully accounted for by the critical state concept, due to mechanical complexity on the microstructural level.

Introduction
In recent years the coupling of fluid flow and deformation in porous media has culminated in field-scale reservoir studies in which the interdependence of flow and deformation is modeled simultaneously1. While the degree of solid-fluid coupling affects the accuracy of the solution2 all approaches attempt to capture the effect of fluid pressure on solid volume deformation and the corresponding effect of solid deformation on fluid flow, in accordance with Biot's3 self-consistent poroelastic theory. Uncoupled modeling does not account for this interaction that can occur between the fluid and solid portions of the reservoir system. Also, the transmissibility matrix can be a strong function of rock deformation, which leads in turn to another level of coupling4. This second type of fluid flow and rock deformation coupling, termed stress-permeability coupling5 arises when effective stress changes result in pore strains, which modify the connectivity of the reservoir rock.
As Biot's differential equations for coupled fluid-solid interaction assume the permeability of the porous medium to be a constant, stress-permeability coupling is usually implemented in an ad hoc manner. For example, a common approach is to assume that permeability scales with porosity as in the Kozeny-Carman relation, or indeed any state variable that is related to effective stress change and can be measured in the laboratory (fluid pressure, mean effective stress, shear stress, axial strain etc.6).
Common laboratory-applied stress paths used to calibrate permeability as a function of such macroscopic variables include hydrostatic compaction and uniaxial strain compaction, the latter assumed to represent a good approximation of the conditions operative in a reservoir during depletion. However, hydrostatic (isotropic) loading does not truly reflect the anisotropic in situ stress state, and fails to simulate the evolution of deviatoric stresses associated with a producing reservoir. Khan and Teufel7 note that in situ stress measurements in both siliciclastic and carbonate reservoirs indicate reservoir stress paths are always significantly less than the isotropic loading case. Field studies also verify that actual measured reservoir stress paths can deviate substantially from the elastic uniaxial strain model8. Indeed, recent experimental data9,10,11 highlight a marked dependence of permeability on the applied stress path, indicative of deviatoric stress control.
Many coupled simulators modeling reservoir deformation incorporate elastoplastic constitutive models including compressive "caps" within which the material deforms elastically and beyond which the material deforms plastically to "high strain" (many in fact use small strain theory and upodate the geometry). The VISAGE fully coupled, geomechanics/multi-phase flow, finite element code developed by VIPS Ltd.12 allows an elastoplastic "critical state" model of reservoir compaction (yield loci and plastic potentials13) to be implemented in which plastic shearing can continue indefinitely without change in volume or effective stresses.