M. Stavropoulou

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.

A closed-form elastic solution for stresses and displacements around tunnels

Abstract A closed-form plane strain solution is presented for stresses and displacements around tunnels based on the complex potential functions and conformal mapping representation. The tunnel is assumed to be driven in a homogeneous, isotropic, linear elastic and pre-stressed geomaterial. Further, the tunnel is considered to be deep enough such that the stress distribution before the excavation is homogeneous. Needless to say that tunnels of semi-circular or “D” cross-section, double-arch cross-section, or tunnels with arched roof and parabolic floor, have a great number of applications in soil/rock underground engineering practice. For the specific type of semi-circular tunnel the distribution of stresses and displacements around the tunnel periphery predicted by the analytical model are compared with those of the FLAC 2D numerical model, as well as, with Kirschs “circular” solution. Finally, a methodology is proposed for the estimation of conformal mapping coefficients for a given cross-sectional shape of the tunnel.

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.

A semi-analytical elastic stress–displacement solution for notched circular openings in rocks

Abstract A semi-analytical plane elasticity solution of the circular hole with diametrically opposite notches in a homogeneous and isotropic geomaterial is presented. This solution is based on: (i) the evaluation of the conformal mapping function of a hole of prescribed shape by an appropriate numerical scheme and (ii) the closed-form solutions of the Kolosov–Muskhelishvili complex potentials. For the particular case of circular notches––which resemble to the circular cavity breakout in rocks––it is demonstrated that numerical results pertaining to boundary stresses and displacements predicted by the finite differences model FLAC 2D , as well as previous analytical results referring to the stress-concentration-factor, are in agreement with analytical results. It is also illustrated that the solution may be easily applied to non-rounded diametrically opposite notch geometries, such as “dog-eared” breakouts by properly selecting the respective conformal mapping function via the methodology presented herein. By employing a stress-mean-value brittle failure criterion that takes into account the stress-gradient effect in the vicinity of the curved surfaces in rock as well as the present semi-analytical solution, it is found that a notched hole, e.g. borehole or tunnel breakout, may exhibit stable propagation. The practical significance of the proposed solution lies in the fact that it can be used as a quick-solver for back-analysis of borehole breakout images obtained in situ via a televiewer for the estimation of the orientation and magnitude of in situ stresses and of strain–stress measurements in laboratory tests.

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.

Rayleigh wave propagation in intact and damaged geomaterials

Abstract An analytical model of an elastically deforming geomaterial with microstructure and damage is assumed to be a material where the second spatial gradients of strain are included in the constitutive equations. Based on this assumption, a linear second gradient (or grade-2) elasticity theory is employed, to investigate the propagation of surface waves in either intact or weathered—–although homogeneous and isotropic at the macroscale—materials with microstructure such as soils, rocks and rock-like materials. First, it is illustrated that in contrast to classical (grade-1) elasticity theory, the proposed higher-order elasticity theory yields dispersive Rayleigh waves, as it is also predicted by the atomic theory of lattices (discrete particle theory), as well as by viscoleasticity theory. Most importantly, it is demonstrated that the theory: (a) is in agreement with in situ non-destructive measurements pertaining to velocity dispersion of Rayleigh waves in monumental stones, and (b) it may be used for back analysis of the test data for the quantitative characterization of degree of surface cohesion or damage of Pendelikon marble of the Parthenon monument of Athens.

Hydrodynamic erosion; A potential mechanism of sand production in weak sandstones

Abstract Fluidized column experiments have been performed, where a 10 cm in diameter and 15 cm high cylindrical sand column was subjected to linear upwards flow of water. The tests were monitored by means of an X-ray CT-scanner. The experiments showed that a change in the flow rate gave rise to a propagating density wave starting upstream at the bottom of the sand body propagating to the top. The experimental observations have been used to calibrate a model which describes erosion processes. Flow rate changes were best described by a temporary change into mildly turbulent flow conditions, which leads to mobilisation of sand particles.

A Theoretical and Experimental Study of Rock Cutting

Two types of numerical models, namely a continuum and a micromechanical discrete element model, are employed for the theoretical study of rock cutting measurements by a new portable rotary microdrilling tool. Our objectives are: (a) to gain insight in the cutting mechanism of granular cohesive-frictional rocks, and (b) to examine the comparability of numerical model predictions with experimental results. In the first type of model, a plane strain continuum calculation is done with a non-hardening, elastic-plastic, Mohr-Coulomb material. In the second type of numerical model, distinct element calculations are done on a simulated plane strain sample of 540 discs. In both models measurements are made of components of traction applied to the cutting tool and compared with measurements taken with the acquisition system of the portable microdrilling tool during specially designed tests on Dionysos marble.

A Method for Inversion of Tunnel Closure Measurements - Application in the Tempi Railway Tunnel

A simple hierarchical methodology is proposed herein for back-analysis of in situ closure measurements in tunnels that is based on in situ convergence measurements, the equivalent plane strain concept, analytical and numerical modeling of continuous elastic and elasto-plastic rocks and dimensional analysis. An example of the application of the proposed methodology is given for the Tempi high-speed railway tunnel in Greece. It is demonstrated that both the in situ lateral-to-vertical stress ratio, rock mass deformation modulus and cohesion are indirectly inferred from the proposed data-inversion analysis, which in turn may be used for future design of tunnels in similar geotechnical conditions. It is also shown that deformation modulus of the rock mass exhibits size effect and stress-dependency, hence the in situ stress ratio depends on the deformation modulus.