The volume changes of grains (solids) are usually neglected in numerical geomechanical simulations of production from -or injection into -reservoir rocks, assuming that all rock volume changes translate into porosity changes. When grain compressibility is taken into account, it is usually done by modifying the pore pressure changes with the so called Biot-factor (α). Rock failure criteria like fracturing or pore collapse are fully dependent on (Terzaghi's) effective stress and not on total stress or Biot-modified effective stress. This pore pressure modification provides a wrong estimation of plasticity and fracturing and is hence not suitable for these type of analyses. In this paper we propose a different method for incorporating grain compressibility in Finite Element modelling. The effect of stress, pressure and temperature changes on the volume of the grains can be calculated separately and translated into a total volume change and a pore volume change. These effects can be incorporated into the constitutive model of plasticity, which uses standard effective stresses for plastic behaviour. The mismatch of elastic volume changes between total stress and pore pressure changes is corrected for by a volume strain loading, that is equal to pore pressure change (Δp) divided by the Grain Bulk Modulus (K g ). For linear elasticity, the equations reduce to poro-elasticity. This volumetric strain can be directly calculated when the fluid pressure change is known or for undrained materials. The pore volume changes are also computed, since they are relevant for undrained behaviour and for the amount of expelled fluids in the drained situation, which is an input for Reservoir Engineering (fluid flow) computations.This method has been incorporated in Shell's propriety FEM application GEOMEC, based on TNO-Diana FEM. In this paper, we explain the method above fully, starting from poro-elasticity theory.
The literature characterizes cartilaginous tissues as osmoviscoelastic. Understanding the damage and failure of these tissues is essential for designing treatments. To determine tissue strength and local stresses, experimental studies-both clinical and animal-are generally supported by computational studies. Verification methods for computational studies of ionized porous media including cracks are hardly available. This study provides a method for verification and shows its performance. For this purpose, shear loading of a finite crack is addressed analytically and through a commercial finite element code. Impulsive shear loading by two-edge dislocation of a crack was considered in a 2D plane strain model for an ionized porous medium. To derive the analytical solution, the system of equation is decoupled by stress functions. The shear stress distribution at the plane of the crack is derived using Fourier and Laplace transformations. The analytical solution for the shear stress distribution is compared with computer simulations in ABAQUS version 6.4-5. Decoupling of the equations makes it possible to solve some boundary value problems in porous media taking chemical effects into account. The numerical calculations underestimate the shear stress at the crack-tips. Mesh refinement increases accuracy, but is still low in the neighborhood of the crack-tips.
Intervertebral disc tissue consists of a fluid-filled extra-cellular matrix, in which living cells are sparsely dispersed. The mechanical function is highly dependent on the composition of the extra-cellular matrix, which primary consists of collagen fibrils and negatively charged proteoglycans. Due to the fixed charges of the proteoglycans (PG’s), the cation concentration inside the tissue is higher than physiological. This excess of ion particles leads to an osmotic pressure difference, which causes swelling of the tissue [1]. Because the intervertebral disc is gripped between two vertebrae, the swelling is constrained in vivo, resulting in a intradiscal pressure of 0.1 to 0.2 MPa in supine position. It has been shown that the osmotic pressure inside cartilaginous tissues is much higher than would be expected based on its FCD [2]. This is because part of the water in the tissue is absorbed by the collagen fibers. The proteoglycan molecules, because of their large size, are excluded from this intra-fibrillar space. This means that their effective concentrations are much higher in the extra-fibrillar space than if they were distributed uniformly throughout the entire matrix. Hence, the effective fixed charge density is higher than if computed from total tissue water content. A recent study demonstrates that intrafibrillar water increases osmolarity within the annulus fibrosus substantially [3]. On the other hand, Wognum et al. [4] showed by means of a physical and a numerical model of the disc that high osmolarity within the disc has a protective effect against crack propagation within the disc. Hence, the decrease in osmolarity associated with degeneration may be an explanation of (1) the growing number of cracks observed in the degenerating disc as well as (2) the poor correlation between external loading and crack propagation [5]. The purpose of the present study is to test the hypothesis of Wognum et al. [4] through direct observation of the deformation of annulus fibrosus tissue around discontinuities within its collagen network.
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