A PDE-constrained optimization formulation for discrete fracture network flows

We investigate a new numerical approach for the computation of the 3D flow in a discrete fracture network that does not require a conforming discretization of partial differential equations on complex 3D systems of planar fractures. The discretization within each fracture is performed independently of the discretization of the other fractures and of their intersections. Independent meshing process within each fracture is a very important issue for practical large scale simulations making easier mesh generation. Some numerical simulations are given to show the viability of the method. The resulting approach can be naturally parallelized for dealing with systems with a huge number of fractures

Dual virtual element method for discrete fractures networks

Discrete fracture networks is a key ingredient in the simulation of physical processes which involve fluid flow in the underground, when the surrounding rock matrix is considered impervious. In this paper we present two different models to compute the pressure field and Darcy velocity in the system. The first allows a normal flow out of a fracture at the intersections, while the second grants also a tangential flow along the intersections. For the numerical discretization, we use the mixed virtual finite element method as it is known to handle grid elements of, almost, any arbitrary shape. The flexibility of the discretization allows us to loosen the requirements on grid construction, and thus significantly simplify the flow discretization compared to traditional discrete fracture network models. A coarsening algorithm, from the algebraic multigrid literature, is also considered to further speed up the computation. The performance of the method is validated by numerical experiments

A Review of Hydraulic Fracturing Simulation

Along with horizontal drilling techniques, multi-stage hydraulic fracturing has improved shale gas production significantly in past decades. In order to understand the mechanism of hydraulic fracturing and improve treatment designs, it is critical to conduct modelling to predict stimulated fractures. In this paper, related physical processes in hydraulic fracturing are firstly discussed and their effects on hydraulic fracturing processes are analysed. Then historical and state of the art numerical models for hydraulic fracturing are reviewed, to highlight the pros and cons of different numerical methods. Next, commercially available software for hydraulic fracturing design are discussed and key features are summarised. Finally, we draw conclusions from the previous discussions in relation to physics, method and applications and provide recommendations for further research

A hybrid mortar virtual element method for discrete fracture network simulations

The most challenging issue in performing underground flow simulations in Discrete Fracture Networks (DFN), is to effectively tackle the geometrical difficulties of the problem. In this work we put forward a new application of the Virtual Element Method combined with the Mortar method for domain decomposition: we exploit the flexibility of the VEM in handling polygonal meshes in order to easily construct meshes conforming to the traces on each fracture, and we resort to the mortar approach in order to ``weakly'' impose continuity of the solution on intersecting fractures. The resulting method replaces the need for matching grids between fractures, so that the meshing process can be performed independently for each fracture. Numerical results show optimal convergence and robustness in handling very complex geometries

On simulations of discrete fracture network flows with an optimization-based extended finite element method

Following the approach introduced in [Berrone,Pieraccini,Scialò,2013], we consider the formulation of the problem of fluid flow in a system of fractures as a PDE constrained optimization problem, with discretization performed using suitable extended finite elements; the method allows independent meshes on each fracture, thus completely circumventing meshing problems usually related to the DFN approach. The application of the method to discrete fracture networks of medium complexity is fully analyzed here, accounting for several issues related to viable and reliable implementations of the method in complex problems

Doctor of Philosophy

dissertationNumerical simulation of the geometrically complex fractured reservoirs has been a major engineering challenge. The deficiencies of continuum models are often addressed using the discrete fracture network (DFN) models which represent the complex fracture geometry explicitly. The primary goal in this dissertation is to explore ways of applying the DFN methodology to solve a variety of multiphase problems in oil reservoir simulation. Three-dimensional, three-phase simulators using the control-volume finiteelement scheme were used. After completing validation and fracture-property sensitivity studies, the limitation of employing the often-used Oda homogenization method was shown followed by the development of a simpler geometric scheme. The important question of oil recovery from basement reservoirs (Type I) composed of fractures of various sizes was examined in detail. Oil recovery and breakthrough behavior of this system comprised of seismic and subseismic features were investigated for different oil distributions, permeability values, levels of heterogeneity and rate. In general having more oil distributed in smaller systems led to lower recovery and quicker breakthrough. Lower permeabilities in the subseismic features also led to lower recovery. The recovery at given pore volume of water injected was rate dependent in all of the scenarios explored, with the lower rate production leading to about 5% higher oil in place recovery. This phenomenon was consistent when viewed from the point of view of gravity number for each displacement. The mechanism of gravity-dominated oil recovery in two-phase applications was explored, and a "critical rate" concept for obtaining higher recoveries in gravity-dominated flow was developed A multiscale upscaling exercise was performed to match the oil recovery performance from a structured fault zone using a single feature with different sets of relative permeability curves. The effectiveness of using DFN simulations for reservoirs containing matrix and fractures (Type II) was shown using two different systems. It was shown that placing wells either in the fault zone or in the matrix can have significant impact on recovery and breakthrough behavior. It was also demonstrated that fracture networks bring apparent anisotropy, and water-flooding from one direction or the other may affect oil recovery. Fractured reservoir simulation is high-performance computing - data and file management, computation, visualization, etc. are integral components of this exercise. A workflow to facilitate creation of fracture networks, gridding and simulation, and visualization was developed. A fully integrated two-dimensional graphical user interface (java-based) was also built