Panos Papanastasiou

THE CRACK TIP REGION IN HYDRAULIC FRACTURING

We present analytical tip region solutions for fracture width and pressure when a power law fluid drives a plane strain fracture in an impermeable linear elastic solid. Our main result is an intermediate asymptotic solution in which the tip region stress is dominated by a singularity which is particular to the hydraulic fracturing problem. Moreover this singularity is weaker than the inverse square root singularity of linear elastic fracture mechanics. We also show how the solution for a semi-infinite crack may be exploited to obtain a useful approximation for the finite case.

A finite element displacement formulation for gradient elastoplasticity

We present a second gradient elastoplastic model for strain-softening materials based entirely on a finite element displacement formulation. The stress increment is related to both the strain increment and its Laplacian. The displacement field is the only field needed to be discretized using a C-1 continuity element. The required higher-order boundary conditions arise naturally from the displacement field. The model is developed to regularize the ill-posedness caused by strain-softening material behaviour. The gradient terms in the constitutive equations introduce an extra material parameter with dimensions of length allowing robust modelling of the post-peak material behaviour leading to localization of deformation. Mesh insensitivity is demonstrated by modelling localization of deformation in biaxial tests. It is shown that both the thickness and inclination of the shear-band zone are insensitive to the mesh directionality and refinement and agree with the expected theoretical and experimental values.

Hydro-mechanical aspects of the sand production problem

This paper examines the hydro-mechanical aspect of the sand production problem and sets the basic frame of the corresponding mathematical modelling. Accordingly, piping and surface erosion effects are studied on the basis of mass balance and particle transport considerations as well as Darcys law. The results show that surface erosion is accompanied by high changes of porosity and permeability close to the free surface. Quantities which can be measured in experiment, like the amount of produced solids or fluid discharge, can be used in an inverse way to determine the constitutive parameters of the problem.

Modelling of localisation and scale effect in thick-walled cylinders with gradient elastoplasticity

We model the progressive localisation of deformation which causes failure around thick-walled cylinders under external radial pressure. The study is based on a second-gradient elastoplastic model developed to regularise the ill-posedness caused by material strain-softening behaviour. The stress increment is related to both the strain increment and its Laplacian. The gradient terms introduce an internal length scale to the material allowing robust modelling of its post-peak behaviour. The numerical implementation is based on a C-1 finite element displacement formulation. Mesh insensitivity in terms of load-displacement and failure mechanism is demonstrated. The internal length in the constitutive equations enables modelling of the scale effect in thick-walled cylinders, according to which the load required to induce failure appears to be much larger for small holes than for large holes.

The influence of plasticity in hydraulic fracturing

This paper examines the influence of plasticity in hydraulic fracturing. Fluid flow in the fracture is modelled by lubrication theory. Rock deformation is modelled by the Mohr–Coulomb flow theory of plasticity and the propagation criterion is based on the softening behaviour of rocks. The coupled, nonlinear problem is solved by a combined finite difference–finite element scheme. The results show that plastic yielding near the tip of a propagating fracture provides an effective shielding, resulting in an increase of the rock effective fracture toughness by more than an order of magnitude. Higher pressure is needed to propogate an elasto-plastic fracture than an elastic fracture, and the created elasto-plastic fracture is shorter and wider than the elastic fracture of the same volume.

The effective fracture toughness in hydraulic fracturing

This paper examines the effective fracture toughness approach which is used in hydraulic fracturing in order to explain the high net-pressures that are often observed in field operations. The effective fracture toughness is calculated using a fully deterministic elasto-plastic hydraulic fracturing model. Rock is modelled by Mohr–Coulomb flow theory of plasticity for cohesive-frictional dilatant material. Fluid flow is modelled by lubrication theory. A cohesive crack model which takes into account the softening behaviour of rocks is employed as the propagation criterion. The fully coupled model is solved numerically by the finite element method and the effective fracture toughness is calculated using the path independent J-integral. The results show that plastic yielding near the tip of a propagating fracture provides an effective shielding, resulting in an increase in the rock effective fracture toughness by more than an order of magnitude. It is demonstrated that an elastic model based on the concept of effective fracture toughness matches the results of plasticity quite well. The effective fracture toughness increases with formation yielding, which is influenced by the deviator of the in-situ stresses, the rock strength, the elastic modulus and the pumping parameters. Tables of effective fracture toughness for a representative set of physical parameters are presented.

Coupled wellbore erosion and stability analysis

This paper extends earlier work on sand erosion and presents an attempt to couple sand erosion to mechanical damage of rock around a wellbore. Porosity which evolves in time and space as surface erosion progresses, is chosen as the coupling parameter. Both rock elasticity and strength (cohesion) are assumed to depend on porosity in such a way that the material becomes weaker with increasing porosity. The mathematical model, consists of erosion equations, mixture flow equations and stress equilibrium equations, is solved numerically by Galerkin finite element method. Numerical results suggest that erosion, resulting in sand production, is high close to the free surface. Erosion is accompained by changes in porosity and a significant permeability increase. Erosion in the vicinity of the wellbore induces alterations in the mechanical behaviour of the medium. Weakening of rock stiffness leads to severe alteration of both effective stresses and pore pressure near the cavity. Since cohesion decreases with increasing porosity, one can also identify the time instant at which rock mechanical failure starts.

ELASTOPLASTIC ANALYSIS OF CYLINDRICAL CAVITY PROBLEMS IN GEOMATERIALS

A large-strain elastoplastic analysis is presented for a cylindrical cavity embedded in an infinite medium under uniform radial pressure. The investigation employs invariant, non-associated deformation-type theories for Mohr-Coulomb (M-C) and Drucker-Prager (D-P) solids, accounting for arbitrary hardening, with the equivalent stress as the independent variable. The M-C model results in a single first-order differential equation, whereas for the D-P solid an algebraic constraint supplements the governing differential equation. Material parameters and response characteristics were determined by calibrating the models with data from triaxial compression tests on Castlegate sandstone and on Jurassic shale. A comparison is presented between predictions obtained from the two models and experimental data from hollow cylinder tests under external loading. A sensitivity of the results to material parameters, like friction and dilation angles, is provided for the case of a cavity subjected to internal pressure in terms of limit pressure predictions. In all cases it has been found that the results of the D-P inner cone model are in close agreement with those obtained from the M-C model.

Sand Erosion in Axial Flow Conditions

A mathematical model of sand erosion in axial flow conditions is presented. The basic mass balance equations and sand erosion constitutive equation were given in Vardoulakis et al. (1996). As opposed to reference Vardoulakis et al. (1996), we consider here the extreme case where convection is null and hydrodynamic dispersion dominates. In addition, Brinkmans extension of Darcys law is adopted to account for a smooth transition between channel flow and Darcian flow. The set of governing PDEs is presented in dimensionless form and is solved numerically. In concordance with the basic constitutive equation for erosion kinetics, the analysis shows that erosion progresses in time as a ‘front’ of high transport concentration. This result is justified by the highly non-linear character of the erosion source term which dominates in the diffusion-like governing equation.

Elastoplastic finite element analysis of inclined wellbores

An effective finite element model which can be used for elastoplastic analysis of inclined wellbores is presented. The same discretised model can be used for any wellbore inclination and azimuth by varying only the applied initial stress field or boundary conditions. The model is applied for calculating the optimum drilling mud-pressure assuming an elastoplastic rock behaviour and different failure criteria. The rock material parameters were derived from triaxial compression tests. The results show that the difference in mud- pressure predictions between elasticity and plasticity depends strongly on the employed failure criteria. The difference between elasticity and plasticity becomes more pronounced with a criterion which allows the material near the borehole wall to reach a critical plastic strain determined from the calibration tests. In such a case differences from 45% to 80% were encountered. Calibration factors given by the ratio of plasticity predictions over the elasticity predictions have been derived and can be used with analytical models for quick field wellbore stability computations.