Radiogenic Granite as an Energy Resource Leigh Farrar and Mark Holland Leigh Farrar and Mark Holland - TIG Nov 2010 1

Radiogenic Granite as an Energy Resource

Leigh Farrar and Mark Holland

Leigh Farrar and Mark Holland - TIG Nov 20101

Discrete geophysical anomalies

The elusive implications that emerge from appreciation of the discrete anomalies can be captured by employing an analytic approach that induces geometric frames of reference which lead to algebraic formulation and computational efficiency.


Mathematical Expression

Mathematical reasoning has a big role to play, being instrumental to every part of the sequence from exploration to exploitation


The Granite

The granite is considered with regard to

discrete anomolies

statistical mechanics

physical properties

infrastructure



Regional Cells and Continental Facets

Crations

Orogens

Platforms

Basins

Can be regimented as regular network of subdivisible continental divisions



7



Isosahedral Tectonic Frame of Reference

Compabitible with plate disposition

Symmetric about Antartica

Blatant historic implication

Elusive 4D implication






Continental Facets

Continental facets can be counted and tagged in terms that allude to

Climatic

Geographic

Cultural

Lifeform Significance


Geographic

Shield

Plains

Mountains

Coastal



Plains

Subdivision into Regional Cells

Tectonic stress state

Disposition of granite within

Nature of insulating cover




The Granite

As a discrete geophysical anomoly

As a confocal mathematical enclose

Its state of evolution

As a mathematical entity beyond the current geophysical methodology





Heat Modelling in the Exploration and Exploitation of Radiogenic Granites

Understanding the Difficulties in Heat Modelling

What are the most important aspects to model?

Model Steps using Breakthroughs in Mathematics and Computer Science

Close Relationship: Geophysics and Mathematics

Heat Equations or Families of Equations

Symmetries

Exact Solutions

Numerical Analysis Specific Areas of Improvement

Conclusions


Understanding the Difficulties in Heat Modelling

Finite Elements and Traditional Methods in detecting and assessing radiogenic granites:

too slow

spectral methods and other traditional efficiency gains are not directly applicable

need a lot of heat, specific heat and other rock property measurements

proxy data techniques do not confront the core issues of problem formulation

lack explanatary power

A broader base of general mathematics and algorithmic techniques can reveal new approaches


What are the most important aspects to model?

Comparison of heat creation versus heat dissipation

Temperature dependence

Long term temperature behavior – steady or unsteady state

Emphasis on the identification of “hot spots”

Successive computer simulations are a hard road towards the identification of “blow up” conditions

But information is in the equations themselves which could help us anticipate “hot spot” phenomena

Let’s look there first!


Steps using Breakthroughs in Mathematics and Computer Science Focus first on getting the mathematical analysis right prior to crunching

out the numbers

Take advantage of symbolic computation (use of computer applications such as Mathematica, Maple and others to process expressions with thousands of terms with ease)

Find all the symmetries – classical, nonclassical and discrete

Heat models are described by Partial Differential Equations (PDEs). Mathematical analysis has shown the “symmetries” of PDEs tell us many things about the behavior of the model. The “symmetries” provide a “road map” and are the key to understanding everything that the model does or can do and how it relates, can be transformed into simpler models or combined with other models.

With various breakthroughs in mathematical analysis and with the use of symbolic computation it is possible to determine the symmetries for a given PDE or system of PDEs “almost always”


Steps using Breakthroughs in Mathematics and Computer Science

Find all the solutions: Painlevé Test (Algorithm)

Interpret the structure of solutions – far reaching implications

Identify all physical connections with solutions and symmetries

Evaluate solutions in numerical/algebraic/geometric/physical property form exploiting knowledge gained about symmetries and solutions in a variety of ways mathematically and physically


The Heat Equation or Families of Equations


The classic or “point” symmetries are determined by techniques invented by Sophus Lie (1842-99). The algebra is expedited with the use of an algorithm that uses symbolic computation. The result is a classification of various forms of f(u), tables of the corresponding symmetry parameters, symmetry reductions and “hidden” symmetry of the reduced form. In short a “roadmap” containing the various structural details for the entire family (1) of heat equations. An example will be give below when we obtain a formulation of particular interest. (Clarkson and Mansfield (2008).





Numerical Analysis: Specific Areas of Improvement

Depending on the form of the heat model, numerical discretisation potentially on a reduced PDE can be much faster to evaluate

For some situations notably steady or unsteady state solutions, direct , non-iterative methods can be used

Using identified symmetries, the performance of various numerical algorithms can be predetermined and dynamic algorithm switching or concurrency techniques can be used

Fitting data – optimal coordinate systems and geometric/physical property formulation can be determined using knowledge of symmetries


Close Relationship: Geophysics and Mathematics

Heat models can be combined with other geophysical models at inception

Mathematical Symmetry directly relates to Physics Conservation Laws - Noether

Long term temperatures are determined by Symmetry

Boundary and initial conditions use the same techniques of symmetry and solution analysis

hot geothermal domains are described as “blow up” conditions in PDEs

Numerical analysis – a variety of benefits. Often fundamental questions have already been answered by this stage. The specifics of what happens after discretisation vary depending on heat model formulation details but are substantial


Conclusions

Breakthroughs in Mathematical analysis and symbolic computations have been outlined and outcomes of a particular heat model presented.

Close ongoing connections exist between mathematical analysis and the understanding of geophysical phenomena.

The algorithms can be tailored as a system for modelling geothermal phenomena.

A far reaching range of improvements in numerical analysis, adaptability and physical interpretation is available from this approach.


Documents

Radiogenic Granite as an Energy Resource Leigh Farrar and Mark Holland Leigh Farrar and Mark Holland - TIG Nov 2010 1