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Introduction to Mathematical Geoenergy

ABSTRACT

In this introductory chapter, we relate the geophysics of the Earth and its atmosphere and of the influences of the sun and the moon and cast that into a geoenergy analysis. Geoenergy is energy derived from geological and geophysical processes and categorized according to its originating source. The sources are compartmentalized according to whether they are radiation-based (such as from sunlight via the photo-electric effect), gravitational (such as from the moon or terrain), geothermal (such as from volcanic sources), kinetic (from the rotation of the Earth and Coriolis forces), or chemical/nuclear (such as from fossil fuel and ion-based batteries). We use these models to project fossil fuel production and provide analysis tools for renewable technologies.

Our objective is to apply what we know about the geophysics of the Earth and its atmosphere and of the influences of the Sun and the Moon and cast that into a geoenergy analysis. As we define it, geoenergy is energy derived from geological and geophysical processes and categorized according to its originating source. Perhaps most convenient is to compartmentalize the sources according to whether they are radiation based (such as from sunlight via the photoelectric effect), gravitational (such as from the Moon or terrain), geothermal (such as from volcanic sources), kinetic (from the rotation of the Earth and Coriolis forces), or chemical/nuclear (such as from fossil fuel and ion-based batteries).

As the acquisition and use of energy is in essence an active process, geoenergy analysis becomes (1) a study of differentiating between deterministic and stochastic processes and (2) of applying physics or heuristics to come up with adequate models to aid in understanding and to perhaps improve the efficient use of our resources either statistically or with confidence based on sound physical models.

It really is not difficult to understand the first distinction (1), as the Sun rising in the morning and falling in the evening is an example of a deterministic process, while predicting cloud cover during the day is a stochastic process. This of course has impacts for predicting efficiencies in solar energy collection, as we know exactly when the Sun will be at its zenith in any geographic location (a deterministic process; see figure), yet we do not know if there will be significant cloud cover at any specific time (a stochastic process).

The second distinction (2), between physics and heuristics, is based on how well we scientifically understand a phenomenon. This becomes apparent when one realizes that many estimates of remaining fossil fuel reserves are heuristics (i.e., educated guesses), based many times only on historical trends. In neglecting a mathematical physics treatment, however, we unfortunately remain uncertain on projections as we cannot account for how the heuristic may fail. In general, we will have more confidence in a scientifically based physics model.

These distinctions can be combined to create four different basic categories.

For example, stochastic physics would be represented by a detailed weather model which would include differential equations describing atmospheric flow and solved on a supercomputer. Different outcomes based on varying initial conditions would generate a statistical spread to be used in regional weather forecasting.

Stochastic heuristics typically apply to a situation that may be too complicated or detailed in scope, resulting in a model that may simply estimate a mean value and possibly a variance for some quantity. This would include our current best guess at predicting future oil production, which has typically applied the so-called Hubbert curve. But this may not be the best possible guess and explains why we have better and more physically oriented models as we will further detail.

On the deterministic side, a good example of a physics application is the theory of tides and tidal analysis. These have high precision and are routinely used for predicting tides down to the minute.

On the other hand, a deterministic heuristic is rare to come across. It is a behavior that appears very predictable yet one for which we lack a good physical model. For example, countering the easily predictable sunset and sunrise, which we physically understand, we have only a partial understanding with respect to solar sunspots. Sunspots appear to have an 11-year cycle, making them somewhat deterministic, yet we do not fully understand the mechanism. Thus, a heuristic is applied to the sunspot cycle describing an 11-year cycle.

1.1. NONRENEWABLE GEOENERGY

The comprehensive framework we will describe has aspects of probability-based forecasting (Limited by the psychology of collective human actions). The salient reason for using probabilistic-based models results from reasoning in the face of uncertainty. We never have had and probably never will have perfect and complete data to accurately analyze, much less predict, our current situation. Lacking this, imperfect probabilistic approaches serve us very well in our understanding of the fundamentals of oil depletion.

Concerning oil (defined as crude plus condensate) depletion, we know that three things will happen in sequence:

1. Oil output will peak.
2. Oil output will decline.
3. Extraction and use of oil will become counterproductive in terms of energy efficiency and the impact on the environment. This will occur for all sources of oil (such as shale oil, extra heavy oil, etc.).

The dates of these events remain unknown, but we have historical data and stochastic models to help guide us in understanding future energy resource availability.

1.2. RENEWABLE GEOENERGY

To understand how to harness renewable geoenergy, we need to model natural phenomena so that it becomes more predictable. In other contexts, we do that already. For example, for ocean tides, we create tidal tables that allow us to plan typical coastal activities. If we can do the same with related geophysical and climate phenomena, the benefits would be enormous.

We start with knowledge of the external energy sources, focusing on solar and gravitational, and find patterns that allow us to model these natural phenomena as both deterministic and stochastic processes. As of today, not any single one of these processes can take the place of fossil fuels in terms of efficiency, but taken together they may make a dent.

To that end, the scope of the analysis will include models of wind, climate cycles, solar energy conversion, battery technology, etc. The main idea in creating such models is that renewable energy is closely linked to efficiency, and the more we can wring out of these sources, the less the impact we will see during our energy transformation away from nonrenewable fuels to a renewable paradigm.

So, the main themes are to create deterministic and stochastic models of natural phenomena according to gathered empirical data using physics and heuristics where appropriate. The emphasis on mathematical physics is stressed because that has the potential for further insight. In several cases, we will show how machine learning models have uncovered patterns in the data leading directly to the applied physics mathematical models.

Models of the physical environment play an important role in supporting planning, analysis, and engineering. Fundamental principles of thermodynamics and statistical physics can be applied to create compact parameterized models capable of statistically capturing the patterns exhibited in a wide range of environmental contexts. Such models will allow more efficient and systematic assessment of the strengths and weaknesses of potential approaches to harnessing energy or efficiently working with the environment. Further, the models play an important role in computer simulations which can produce better designs of complex systems more quickly, affordably, and reliably.

In terms of renewable energy, models of the weather and climate are vital for planning, optimizing, and taking advantage of energy resources. Every aspect of the climate is important. For example, knowing the long-term climate forecast for the occurrence of El Niños will allow us to plan for hotter than average temperature extremes in certain parts of the world or to plan for droughts or floods. These climate behaviors are examples of geophysical fluid dynamics models (Vallis, 2016) where the distinction between stochastic and deterministic (and deterministically chaotic) causes is under intense research (Caprara & Vulpiani, 2016), and we will describe how we may be able to simplify the models.

From a computational perspective, there has been a steady increase of the use of machine learning to identify deterministic patterns (Jones, 2017; Karpatne & Kumar, 2017; Steinbach et al., 2002). For example, the quasi-biennial oscillation (QBO) behavior of stratospheric winds has long been speculated to be forced by the cyclic lunar tidal potential. A matching lunar pattern was discovered via a symbolic regression machine learning experiment and then verified by aliasing a strong seasonal (yearly) signal onto an empirical model of the lunar tidal potential...