Application of the Variational Fracture Model to Hydraulic Fracturing in Poroelastic Media

Hydraulic fracturing has persisted through the use of simple numerical models to describe fracture geometry and propagation. Field tests provide evidence of interaction and merging of multiple fractures, complex fracture geometry and propagation paths. These complicated behaviors suggest that the simple models are incapable of serving as predictive tools for treatment designs. In addition, other commonly used models are designed without considering poroelastic effects even though a propagating hydraulic fracture induces deformation of the surrounding porous media. A rigorous hydraulic fracturing model capable of reproducing realistic fracture behaviors should couple rock deformation, fracture propagation and fluid flow in the both the fracture and reservoir. In this dissertation, a fully coupled hydraulic fracturing simulator is developed by coupling reservoir-fracture flow models with a mechanical model for reservoir deformation. Reservoir-fracture deformation is modeled using the variational fracture model which provides a unified framework for simultaneous description of fracture deformation and propagation, and reservoir deformation. Its numerical implementation is based on a phase-field regularized model. This approach avoids the need for explicit knowledge of fracture location and permits the use of a single computational domain for fracture and reservoir representation. The first part of this work involves verification of the variational fracture model by solving the classical problem of fracture propagation in impermeable reservoirs due to injection of an inviscid fluid. Thereafter, the developed reservoir-fracture model is coupled to the mechanical model. Iterative solution of the variational fracture model and the coupled flow model provides a simplified framework for simultaneous modeling of rock deformation and fluid flow during hydraulic fracturing. Since the phase field technique for fracture representation removes the limitation of knowing a priori, fracture direction, the numerical solutions provide a means of evaluating the role of reservoir and fluid properties on fracture geometry and propagation paths. First, the proposed approach is validated for simple idealized scenarios for which closed form solutions exist in the literature. Further simulations highlight the role of fluid viscosity and reservoir properties on fracture length, fracture width and fluid pressure. Numerical results show stress shadowing effect on multiple hydraulic fracture propagation. Finally, the effect of in situ stress on fracture propagation direction is reproduced while the role of varying reservoir mechanical properties on fracture height growth is investigated

Shape Optimization for Drag Minimization Using the Navier-Stokes Equation

Fluid drag is a force that opposes relative motion between fluid layers or between solids and surrounding fluids. For a stationary solid in a moving fluid, it is the amount of force necessary to keep the object stationary in the moving fluid. In addition to fluid and flow conditions, pressure drag on a solid object is dependent on the size and shape of the object. The aim of this project is to compute the shape of a stationary 2D object of size 3.5 m2 that minimizes drag for different Reynolds numbers. We solve the problem in the context of shape optimization, making use of shape sensitivity analysis. The state variables are fluid pressure and velocity modeled by the Navier-Stokes equation with cost function given by the fluid drag which depends on the state variables. The geometric constraint is removed by constructing a Lagrangian function. Subsequent application of shape sensitivity analysis on the Lagrangian generates the shape derivative and gradient. Our optimization routine uses a variational form of the sequential quadratic programming (SQP) method with the Hessian replaced by a variational form for the shape gradient. The numerical implementation is done in Python while the open source finite element package, FEniCS, is used to solve all the partial differential equations. Remeshing of the computational domain to improve mesh quality is carried out with the open source 2D mesh generator, Triangle. Final shapes for low Reynolds numbers resemble an american football while shapes for moderate to high Reynolds numbers are more streamlined in the tail end of the object than at the front

Pore-scale lattice Boltzmann simulations of inertial flows in realistic porous media: a first principle analysis of the Forchheimer relationship

With recent advances in the capabilities of high performance computing (HPC) platforms and the relatively simple representation of complex geometries of porous media, lattice Boltzmann method (LBM) has gained popularity as a means of solving fluid flow and transport problems. In this work, LBM was used to obtain flow parameters of porous media, study the behavior of these parameters at varying flow conditions and quantify the effect of roughness on the parameters by relating the volume averaged flow simulation results to Darcy and Forchheimer equations respectively. To validate the method, flow was simulated on regular and random sphere arrays in cubic domains, for which a number of analytical solutions are available. Permeability and non-Darcy coefficients obtained from the simulation compared well with Kozeny and Ergun estimates while deviation from the observed constant permeability and tortuosity values occurred aroundRe≈1-10. By defining roughness as hemispherical protrusions on the smooth spheres in the regular array, it was observed from flow streamlines obtained at different roughness heights that the average length of the flow paths increased with increasing roughness height. As such, the medium tortuosity and non-Darcy coefficient increased while the permeability decreased as height of the roughness increased. Applying the method to a 3D computed tomography image of Castlegate sandstone, the calculated macroscopic permeability and beta factor components were in good agreement with reported experimental values. In addition, LBM beta factors were compared with a number of empirical models for non-Darcy coefficient estimation and were found to be of the same order of magnitude as most of the correlations, although estimates of the models showed wide variation in values. Resolution of the original sample was increased by infilling with more voxels and simulation in the new domain showed better flow field resolution and higher simulated flow regimes compared to those of the original sample, without significant change in the flow parameters obtained. Using the Reynolds number based on the Forchheimer coefficient, the range of transition from Darcy to non-Darcy regime was within the values reported by Ruth and Ma (1993) and Zeng and Grigg (2006).

COMPARATIVE ANALYSIS OF NEURO- FUZZY AND SIMPLEX OPTIMIZATION MODEL FOR CONGESTION CONTROL IN ATM NETWORK.

Congestion always occurred when the transmission rate increased the data handling capacity of the network. Congestion normally arises when the network resources are not managed efficiently. Therefore if the source delivers at a speed higher then service rate queue, the queue size will be higher. Also if the queue size is finite, then the packet will observed delay. MATLAB Software was used to carry out simulations to develop Congestion control optimization Scheme for ATM Network with the aims to reducing the congestion of Enugu ATM Network. The results of the research reveal the minimization of congestion application model for Enugu ATM using optimization and Neuro-fuzzy. The result shows that congestion control model with Optimization and Neuro-fuzzy were 0.00003153 and 0.00002098 respectively. The ATM Congestion was reduced by 0.0000105, which is 18.2% decrease after Neuro-fuzzy controller was used. The results show the application of Neuro-fuzzy model which can use to control and minimized the ATM Congestion of Enugu ATM Network. The result shows that when Neuro-fuzzy is applied the congestion and the packet queue length in the buffer will be minimized. Key words: Congestion, MATLAB, Optimization, Neuro-fuzzy, ATM DOI: 10.7176/CTI/10-05 Publication date:July 31st 2020

A variational phase-field model for hydraulic fracturing in porous media

Rigorous coupling of fracture–porous medium fluid flow and topologically complex fracture propagation is of great scientific interest in geotechnical and biomechanical applications. In this paper, we derive a unified fracture–porous medium hydraulic fracturing model, leveraging the inherent ability of the variational phase-field approach to fracture to handle multiple cracks interacting and evolving along complex yet, critically, unspecified paths. The fundamental principle driving the crack evolution is an energetic criterion derived from Griffith\u27s theory. The originality of this approach is that the crack path itself is derived from energy minimization instead of additional branching criterion. The numerical implementation is based on a regularization approach similar to a phase-field model, where the cracks location is represented by a smooth function defined on a fixed mesh. The derived model shows how the smooth fracture field can be used to model fluid flow in a fractured porous medium. We verify the proposed approach in a simple idealized scenario where closed form solutions exist in the literature. We then demonstrate the new method\u27s capabilities in more realistic situations where multiple fractures turn, interact, and in some cases, merge with other fractures

A Variational Approach to the Numerical Simulation of Hydraulic Fracturing

One of the most critical capabilities of realistic hydraulic fracture simulation is the prediction of complex (turning, bifurcating, or merging) fracture paths. In most classical models, complex fracture simulation is difficult due to the need for a priori knowledge of propagation path and initiation points and the complexity associated with stress singularities at fracture tips. In this study, we follow Francfort and Marigo\u27s variational approach to fracture, which we extend to account for hydraulic stimulation. We recast Griffith\u27s criteria into a global minimization principle, while preserving its essence, the concept of energy restitution between surface and bulk terms. More precisely, to any admissible crack geometry and kinematically admissible displacement field, we associate a total energy given as the sum of the elastic and surface energies. In a quasi-static setting, the reservoir state is then given as the solution of a sequence of unilateral minimizations of this total energy with respect to any admissible crack path and displacement field. The strength of this approach is to provide a rigorous and unified framework accounting for new cracks nucleation, existing cracks activation, and full crack path determination (including complex behavior such as crack branching, kinking, and interaction between multiple cracks) without any a priori knowledge or hypothesis. Of course, the lack of a priori hypothesis on cracks geometry is at the cost of numerical complexity. We present a regularized phase field approach where fractures are represented by a smooth function. This approach makes handling large and complex fracture networks very simple yet discrete fracture properties such as crack aperture can be recovered from the phase field. We compare variational fracture simulation results against several analytical solutions and also demonstrate the approach\u27s ability to predict complex fracture systems with example of multiple interacting fractures. Copyright 2012, Society of Petroleum Engineers