Deep geothermal: The ‘Moon Landing’ mission in the unconventional energy and minerals space

Deep geothermal from the hot crystalline basement has remained an unsolved frontier for the geothermal industry for the past 30 years. This poses the challenge for developing a new unconventional geomechanics approach to stimulate such reservoirs. While a number of new unconventional brittle techniques are still available to improve stimulation on short time scales, the astonishing richness of failure modes of longer time scales in hot rocks has so far been overlooked. These failure modes represent a series of microscopic processes: brittle microfracturing prevails at low temperatures and fairly high deviatoric stresses, while upon increasing temperature and decreasing applied stress or longer time scales, the failure modes switch to transgranular and intergranular creep fractures. Accordingly, fluids play an active role and create their own pathways through facilitating shear localization by a process of time-dependent dissolution and precipitation creep, rather than being a passive constituent by simply following brittle fractures that are generated inside a shear zone caused by other localization mechanisms. We lay out a new theoretical approach for the design of new strategies to utilize, enhance and maintain the natural permeability in the deeper and hotter domain of geothermal reservoirs. The advantage of the approach is that, rather than engineering an entirely new EGS reservoir, we acknowledge a suite of creep-assisted geological processes that are driven by the current tectonic stress field. Such processes are particularly supported by higher temperatures potentially allowing in the future to target commercially viable combinations of temperatures and flow rates.

A meshless method based on Point Interpolation Method (PIM) for the space fractional diffusion equation

This paper aims to develop a meshless approach based on the Point Interpolation Method (PIM) for numerical simulation of a space fractional diffusion equation. Two fully-discrete schemes for the one-dimensional space fractional diffusion equation are obtained by using the PIM and the strong-forms of the space diffusion equation. Numerical examples with different nodal distributions are studied to validate and investigate the accuracy and efficiency of the newly developed meshless approach.

High order unconditionally stable difference schemes for the Riesz space-fractional telegraph equation

In this paper, a class of unconditionally stable difference schemes based on the Pad´e approximation is presented for the Riesz space-fractional telegraph equation. Firstly, we introduce a new variable to transform the original dfferential equation to an equivalent differential equation system. Then, we apply a second order fractional central difference scheme to discretise the Riesz space-fractional operator. Finally, we use (1, 1), (2, 2) and (3, 3) Pad´e approximations to give a fully discrete difference scheme for the resulting linear system of ordinary differential equations. Matrix analysis is used to show the unconditional stability of the proposed algorithms. Two examples with known exact solutions are chosen to assess the proposed difference schemes. Numerical results demonstrate that these schemes provide accurate and efficient methods for solving a space-fractional hyperbolic equation.

A fast semi-implicit difference method for a nonlinear two-sided space-fractional diffusion equation with variable diffusivity coefficients

In this paper, we derive a new nonlinear two-sided space-fractional diffusion equation with variable coefficients from the fractional Fick’s law. A semi-implicit difference method (SIDM) for this equation is proposed. The stability and convergence of the SIDM are discussed. For the implementation, we develop a fast accurate iterative method for the SIDM by decomposing the dense coefficient matrix into a combination of Toeplitz-like matrices. This fast iterative method significantly reduces the storage requirement of O(n2)O(n2) and computational cost of O(n3)O(n3) down to n and O(nlogn)O(nlogn), where n is the number of grid points. The method retains the same accuracy as the underlying SIDM solved with Gaussian elimination. Finally, some numerical results are shown to verify the accuracy and efficiency of the new method.

A Crank--Nicolson ADI spectral method for a two-dimensional riesz space fractional nonlinear reaction-diffusion equation

In this paper, a new alternating direction implicit Galerkin--Legendre spectral method for the two-dimensional Riesz space fractional nonlinear reaction-diffusion equation is developed. The temporal component is discretized by the Crank--Nicolson method. The detailed implementation of the method is presented. The stability and convergence analysis is strictly proven, which shows that the derived method is stable and convergent of order $2$ in time. An optimal error estimate in space is also obtained by introducing a new orthogonal projector. The present method is extended to solve the fractional FitzHugh--Nagumo model. Numerical results are provided to verify the theoretical analysis.

Numerical analysis for the time distributed-order and Riesz space fractional diffusions on bounded domains

Subdiffusion equations with distributed-order fractional derivatives describe some important physical phenomena. In this paper, we consider the time distributed-order and Riesz space fractional diffusions on bounded domains with Dirichlet boundary conditions. Here, the time derivative is defined as the distributed-order fractional derivative in the Caputo sense, and the space derivative is defined as the Riesz fractional derivative. First, we discretize the integral term in the time distributed-order and Riesz space fractional diffusions using numerical approximation. Then the given equation can be written as a multi-term time–space fractional diffusion. Secondly, we propose an implicit difference method for the multi-term time–space fractional diffusion. Thirdly, using mathematical induction, we prove the implicit difference method is unconditionally stable and convergent. Also, the solvability for our method is discussed. Finally, two numerical examples are given to show that the numerical results are in good agreement with our theoretical analysis.

Stability and convergence of a new finite volume method for a two-sided space-fractional diffusion equation

In this paper, we consider a two-sided space-fractional diffusion equation with variable coefficients on a finite domain. Firstly, based on the nodal basis functions, we present a new fractional finite volume method for the two-sided space-fractional diffusion equation and derive the implicit scheme and solve it in matrix form. Secondly, we prove the stability and convergence of the implicit fractional finite volume method and conclude that the method is unconditionally stable and convergent. Finally, some numerical examples are given to show the effectiveness of the new numerical method, and the results are in excellent agreement with theoretical analysis.

A Novel High Order Space-Time Spectral Method for the Time Fractional Fokker--Planck Equation

The fractional Fokker-Planck equation is an important physical model for simulating anomalous diffusions with external forces. Because of the non-local property of the fractional derivative an interesting problem is to explore high accuracy numerical methods for fractional differential equations. In this paper, a space-time spectral method is presented for the numerical solution of the time fractional Fokker-Planck initial-boundary value problem. The proposed method employs the Jacobi polynomials for the temporal discretization and Fourier-like basis functions for the spatial discretization. Due to the diagonalizable trait of the Fourier-like basis functions, this leads to a reduced representation of the inner product in the Galerkin analysis. We prove that the time fractional Fokker-Planck equation attains the same approximation order as the time fractional diffusion equation developed in [23] by using the present method. That indicates an exponential decay may be achieved if the exact solution is sufficiently smooth. Finally, some numerical results are given to demonstrate the high order accuracy and efficiency of the new numerical scheme. The results show that the errors of the numerical solutions obtained by the space-time spectral method decay exponentially.

Fractional models in space for diffusive processes in heterogeneous media with applications in cell motility and electrical signal propagation

This work addresses fundamental issues in the mathematical modelling of the diffusive motion of particles in biological and physiological settings. New mathematical results are proved and implemented in computer models for the colonisation of the embryonic gut by neural cells and the propagation of electrical waves in the heart, offering new insights into the relationships between structure and function. In particular, the thesis focuses on the use of non-local differential operators of non-integer order to capture the main features of diffusion processes occurring in complex spatial structures characterised by high levels of heterogeneity.

Verification and Operation of Adaptive Materials in Space

Piezoelectric polymers based on polyvinylidene fluoride (PVDF) are of interest as smart materials for novel space-based telescope applications. Dimensional adjustments of adaptive thin polymer films are achieved via controlled charge deposition. Predicting their long-term performance requires a detailed understanding of the piezoelectric property changes that develop during space environmental exposure. The overall materials performance is governed by a combination of chemical and physical degradation processes occurring in low Earth orbit as established by our past laboratory-based materials performance experiments (see report SAND 2005-6846). Molecular changes are primarily induced via radiative damage, and physical damage from temperature and atomic oxygen exposure is evident as depoling, loss of orientation and surface erosion. The current project extension has allowed us to design and fabricate small experimental units to be exposed to low Earth orbit environments as part of the Materials International Space Station Experiments program. The space exposure of these piezoelectric polymers will verify the observed trends and their degradation pathways, and provide feedback on using piezoelectric polymer films in space. This will be the first time that PVDF-based adaptive polymer films will be operated and exposed to combined atomic oxygen, solar UV and temperature variations in an actual space environment. The experiments are designed to be fully autonomous, involving cyclic application of excitation voltages, sensitive film position sensors and remote data logging. This mission will provide critically needed feedback on the long-term performance and degradation of such materials, and ultimately the feasibility of large adaptive and low weight optical systems utilizing these polymers in space.

Public space design of knowledge and innovation spaces: Learnings from Kelvin Grove Urban Village, Brisbane

The era of knowledge-based urban development has led to an unprecedented increase in mobility of people and the subsequent growth in new typologies of agglomerated enclaves of knowledge such as knowledge and innovation spaces. Within this context, a new role has been assigned to contemporary public spaces to attract and retain the mobile knowledge workforce by creating a sense of place. This paper investigates place making in the globalized knowledge economy, which develops a sense of permanence spatio-temporally to knowledge workers displaying a set of particular characteristics and simultaneously is process-dependent getting developed by the internal and external flows and contributing substantially in the development of the broader context it stands in relation with. The paper reviews the literature and highlights observations from Kelvin Grove Urban Village, located in Australia’s new world city Brisbane, to understand the application of urban design as a vehicle to create and sustain place making in knowledge and innovation spaces. This research seeks to analyze the modified permeable typology of public spaces that makes knowledge and innovation spaces more viable and adaptive as per the changing needs of the contemporary globalized knowledge society.