On the use of quarter-point tetrahedral finite elements in linear elastic fracture mechanics

This paper discusses the reproduction of the square root singularity in quarter-point tetrahedral (QPT) finite elements. Numerical results confirm that the stress singularity is modeled accurately in a fully unstructured mesh by using QPTs. A displacement correlation (DC) scheme is proposed in combination with QPTs to compute stress intensity factors (SIF) from arbitrary meshes, yielding an average error of 2–3%. This straightforward method is computationally cheap and easy to implement. The results of an extensive parametric study also suggest the existence of an optimum mesh-dependent distance from the crack front at which the DC method computes the most accurate SIFs

A finite element framework for modeling internal frictional contact in three-dimensional fractured media using unstructured tetrahedral meshes

AbstractThis paper introduces a three-dimensional finite element (FE) formulation to accurately model the linear elastic deformation of fractured media under compressive loading. The presented method applies the classic Augmented Lagrangian(AL)-Uzawa method, to evaluate the growth of multiple interacting and intersecting discrete fractures. The volume and surfaces are discretized by unstructured quadratic triangle-tetrahedral meshes; quarter-point triangles and tetrahedra are placed around fracture tips. Frictional contact between crack faces for high contact precisions is modeled using isoparametric integration point-to-integration point contact discretization, and a gap-based augmentation procedure. Contact forces are updated by interpolating tractions over elements that are adjacent to fracture tips, and have boundaries that are excluded from the contact region. Stress intensity factors are computed numerically using the methods of displacement correlation and disk-shaped domain integral. A novel square-root singular variation of the penalty parameter near the crack front is proposed to accurately model the contact tractions near the crack front. Tractions and compressive stress intensity factors are validated against analytical solutions. Numerical examples of cubes containing one, two, twenty four and seventy interacting and intersecting fractures are presented

A disk-shaped domain integral method for the computation of stress intensity factors using tetrahedral meshes

A novel domain integral approach is introduced for the accurate computation of pointwise J-integral and stress intensity factors (SIFs) of 3D planar cracks using tetrahedral elements. This method is efficient and easy to implement, and does not require a structured mesh around the crack front. The method relies on the construction of virtual disk-shaped integral domains at points along the crack front, and the computation of domain integrals using a series of virtual triangular and line elements. The accuracy of the numerical results computed for through-the-thickness, penny-shaped, and elliptical crack configurations has been validated by using the available analytical formulations. The average error of computed SIFs remains below 1% for fine meshes, and 2–3% for coarse ones. The results of an extensive parametric study suggest that there exists an optimum mesh-dependent domain radius at which the computed SIFs are the most accurate. Furthermore, the results provide evidence that tetrahedral elements are efficient, reliable and robust instruments for accurate linear elastic fracture mechanics calculations

Quantification of fracture interaction using stress intensity factor variation maps

Accurate and flexible models of fracture interaction are sought after in the fields of mechanics and geology. Stress intensity factors (SIFs) quantify the energy concentrated at the fracture tips and are perturbed from their isolated values when two fractures are close to one another. Using a three-dimensional finite element fracture mechanics code to simulate static fractures in tension and compression, interaction effects are examined. SIF perturbations are characterized by introducing three interaction measures: the circumferential and maximum SIF perturbation provide the “magnitude” of the effect of interaction, and the amplification to shielding ratio quantifies the balance between increased and decreased SIFs along the tip. These measures are used to demonstrate the change in interaction with fracture separation and to find the separation at which interaction becomes negligible. Interaction maps are constructed by plotting the values of the interaction measures for a static fracture as a second fracture is moved around it. These maps are presented for several common fracture orientations in tension. They explore interaction by highlighting regions in which growth is more likely to occur and where fractures will grow into nonplanar geometries. Interaction maps can be applied to fracture networks with multiple discontinuities to analyze the effect of geometric variations on fracture interaction

Simultaneous oil recovery and residual gas storage: A pore-level analysis using in-situ X-ray micro-tomography

We imaged sandstone cores at residual gas saturation (Sgr) with synchrotron radiation at a nominal resolution of (9 μm)3. We studied two three-phase flooding sequences: (1) gas injection into a core containing oil and initial water followed by a waterflood (gw process); (2) gas injection into a waterflooded core followed by another waterflood (wgw process). In the gw flood we measured a significantly higher Sgr (=20.6%; Sgr in the wgw flood was 5.3%) and a significantly lower residual oil saturation (Sor; Sor in the gw flood was 21.6% and Sor in the wgw flood was 29.3%). We also studied the size distribution of individual trapped clusters in the pore space. We found an approximately power-law distribution N ∝ s−τ with an exponent τ = 2.02–2.03 for the residual oil clusters and τ = 2.04 for the gas clusters in the gw flood. τ (=2.32) estimated for the gas clusters in the wgw process was significantly different. Furthermore, we calculated the surface area A–volume V relationships for the clusters. Again an approximate power-law relationship was observed, A ∝Vp with p ≈ 0.75. Moreover, in the gw flood sequence we identified oil layers sandwiched between the gas and water phases; we did not identify such oil layers in the wgw flood.These results have several important implications for oil recovery, carbon geo-sequestration and contaminant transport: (a) significantly more oil can be produced and much more gas can be stored using a gw flood; (b) cluster size distributions for residual oil or gas clusters in three-phase flow are similar to those observed in analogue two-phase flow; (c) there is a large cluster surface area available for dissolution of the residual phase into an aqueous phase; however, this surface area is significantly smaller than predicted by percolation theory (p ≈ 1), which implies that CO2 dissolution trapping and contamination of aquifers by hazardous organic solvents is slower than expected because of reduced interfacial contact areas

Evolution of fracture normal stiffness due to pressure dissolution and precipitation

The normal stiffness of a fracture is a key parameter that controls, for example, rock mass deformability, the change in hydraulic transmissivity due to stress changes, and the speed and attenuation of seismic waves that travel across the fracture. Non-linearity of normal stiffness as a function of stress is often attributed to plastic yield at discrete contacts. Similar surface-altering mechanisms occur due to pressure solution and precipitation over larger timescales. These processes partition the fracture surfaces into a flattened contact region, and a rough free surface that bounds the void space. Under low loads, contact occurs exclusively over the flattened part, leading to rapid, exponential stiffening. At higher loads, contact occurs over the rough surface fraction, leading to the conventional linear increase of stiffness with stress. It follows that a relationship exists between the history of in situ temperature and stress state of a rock fracture, and its subsequent deformation behavior

Extracting Information about the Rotator Cuff from Magnetic Resonance Images Using Deterministic and Random Techniques

We consider some methods to extract information about the rotator cuff based on magnetic resonance images; the study aims to define an alternative method of display that might facilitate the detection of partial tears in the supraspinatus tendon. Specifically, we are going to use families of ellipsoidal triangular patches to cover the humerus head near the affected area. These patches are going to be textured and displayed with the information of the magnetic resonance images using the trilinear interpolation technique. For the generation of points to texture each patch, we propose a new method that guarantees the uniform distribution of its points using a random statistical method. Its computational cost, defined as the average computing time to generate a fixed number of points, is significantly lower as compared with deterministic and other standard statistical techniques

Finite element simulations of interactions between multiple hydraulic fractures in a poroelastic rock

A fully coupled three-dimensional finite-element model for hydraulic fracturing in permeable rocks is utilised to investigate the interaction between multiple simultaneous and sequential hydraulic fractures. Fractures are modelled as surface discontinuities within a three-dimensional matrix. This model simultaneously accounts for laminar flow within the fracture, Darcy flow within the rock matrix, poroelastic deformation of the rock, and the propagation of fractures using a linear elastic fracture mechanics framework. The leakoff of fracturing fluid into the surrounding rocks is defined as a function of the pressure gradient at the fracture surface, the fluid viscosity, and the matrix permeability. The coupled equations are solved numerically using the finite element method. Quadratic tetrahedral and triangle elements are used for spatial discretisation of volumes and surfaces, respectively. The model is validated against various analytical solutions for plane-strain and penny-shaped hydraulic fractures. Several cases of simultaneous fracturing of multiple hydraulic fractures are simulated in which the effects of the various parameters (the in situ stresses, the distance between fractures, the permeability of the matrix, the Biot poroelastic coefficient, and the number of the fractures in a group) are investigated. The results show that the stress induced by the opening of the fractures, and the stress induced by the fluid leakoff, each have the effect of locally altering the magnitudes and orientations of the principal stresses, hence altering the propagation direction of the fractures. Opening of a fracture induces excessive compression (also known as the “stress shadow”) that causes adjacent fractures to curve away from each other. This excessive compression competes against the differential in situ stresses, which tend to cause the fracture to grow in the plane normal to the minimum in situ stress. The stress shadow effect is reduced by increasing the distance between fractures, and is increased by increasing the leakoff, which may be due to increased permeability of the rock, or an increase in the Biot coefficient

Caprock integrity and public perception studies of carbon storage in depleted hydrocarbon reservoirs

Capture and subsurface storage of CO2 is widely viewed as being a necessary component of any strategy to minimise and control the continued increase in average global temperatures. Existing oil and gas reservoirs can be re-used for carbon storage, providing a substantial fraction of the vast amounts of subsurface storage space that will be required for the implementation of carbon storage at an industrial scale. Carbon capture and storage (CCS) in depleted reservoirs aims to ensure subsurface containment, both to satisfy safety considerations, and to provide confidence that the containment will continue over the necessary timescales. Other technical issues that need to be addressed include the risk of unintended subsurface events, such as induced seismicity. Minimisation of these risks is key to building confidence in CCS technology, both in relation to financing/liability, and the development and maintenance of public acceptance. These factors may be of particular importance with regard to CCS projects involving depleted hydrocarbon reservoirs, where the mechanical effects of production activities must also be considered. Given the importance of caprock behaviour in this context, several previously published geomechanical caprock studies of depleted hydrocarbon reservoirs are identified and reviewed, comprising experimental and numerical studies of fourteen CCS pilot sites in depleted hydrocarbon reservoirs, in seven countries (Algeria, Australia, Finland, France, Germany, Netherlands, Norway, UK). Particular emphasis is placed on the amount and types of data collected, the mathematical methods and codes used to conduct geomechanical analysis, and the relationship between geomechanical aspects and public perception. Sound geomechanical assessment, acting to help minimise operational and financial/liability risks, and the careful recognition of the impact of public perception are two key factors that can contribute to the development of a successful CCS project in a depleted hydrocarbon reservoir

CONTAIN D11 : integrated final results and conclusions

Carbon capture and storage is a technology capable of reducing CO2 outputs on a large scale; the concept usually requires CO2 to be removed from post-combustion flue gases and sequestered in geological formations. Depleted gas fields constitute “the most important storage type for the UK” and will provide a large and important potential future offshore storage capacity (DECC, 2012). Over the last 4 years, the CONTAIN research project has focussed on the geomechanical behaviour of depleted hydrocarbon fields in response to injection with CO2, combining a modelling and experimental approach with the public perceptions of CCS into three work packages. The project has provided a better understanding of the hydromechanical impacts of depletion on caprocks and the effect of subsequent CO2 injection, in order to assist with the implementation of CCS in this type of reservoir. Work package 1 outlined a phenomenological approach to assessing possible deformation during operation. Focus was placed on rock mechanics and transport experiments on material from the geologies of target formations in the North Sea, providing information that could be incorporated into numerical simulations. Work package 2 expanded this understanding by considering fractured caprock. Numerical modelling was used to study the deformation of an initially intact caprock caused by the depletion of an underlying reservoir during oil extraction. Deformation and flow were geomechanically modelled in three dimensions using a fully coupled poroelastic model, incorporating discrete fractures and faults into the caprock. Work package 3 offered new and valuable insight on future public awareness campaigns aimed at gaining acceptance of CCS. Qualitative expert interviews have been used, a CCS expert survey and a public survey across four countries to gain an understanding of perceptions of CCS risks and benefits, and has allowed for comparison of views on CCS between experts and public. In addition, the work package has explored the impact of different message framings on CCS attitudes. The findings of each work package are summarised in this report, with each work package represented by a report chapter. A synthesis of the findings and discussion of the work as a whole follows