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Phil Holbrook Ph.D.
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Scientist Force Balanced Petrophysics

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These two books are a significant advance in physical science that now includes the mechanics
inside of our earth.


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Deterministic Earth
Mechanical Science

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Pore Pressure through Earth Mechanical Systems
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Educational discounts can be obtained from education
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You can get any number of copies at the retail price from Amazon.com

Copies can be found by entering
the unique ISBN's found below.

Deterministic Earth
Mechanical Science
ISBN: 0-9708083-3X

Pore Pressure through Earth Mechanical Systems
ISBN: 0-9708083-2-1



Fellow Earth Scientists,

The earth is principally composed of natural minerals and fluids. Forces are symmetrically generated and borne by the earth's natural constituents.




This is an original constitutive mechanical system that equates [mass-energy] with [stress/strain] in planet earth.


The earth’s Gravity and horizontal tectonic forces are borne by natural mineral and fluid electron clouds. [Mass-energy] is conserved in three overlapping arrangements of physical laws with physical properties. This governing physics is an augmentation of Newtonian mechanics that applies beneath the earth’s surface.

The (physical-mathematical) result is the unified earth mechanical system. The static effective stress theorem applies everywhere. There are plastic and elastic limits to the earth’s scalar mechanics. Plastic and elastic [stress/strain] responses contain the same scalar information.


Five two dimensional planes are inset in a global image. These are real convergent two dimensional planes in the earth’s mechanical system. Each plane is a minimum energy convergence of physical laws and stresses with physical properties (strain). Each plane applies over the depth range of the earth’s sedimentary crust.

(Upper left) Sedimentary rock compaction is governed by the first fundamental in situ [stress/strain] relationship. The colored lines on this plane are power-law (mineral specific) compaction functions. A rock is the scalar weighted average of its mineralogy.

(Upper right) All loading and unloading occurs within the (mineral specific) plastic and elastic limits. The limbs hinge at the peak loading-limb point.

(Lower left) Vectorial stresses are conserved within scalar. They equilibrate at minimum energy. (Center right) The (mineral specific) elastic functions lie along a mineral-fluid lever. Porosity (equivalent plastic strain) is scaled on this lever following the extended elastic equations of Hooke’s law.

(Lower right) The failure envelopes of sedimentary rocks depend upon maximum and minimum principal stresses. The angle of internal friction is a function of scalar average mineralogy. For each tectonic regime the stresses are represented by the scalar and vectorial stresses (shown on the lower left). These tend toward deterministic minimum energy ratios.

The borehole schematic below shows the interdependent scalars and vectors in the normal and strike-slip tectonic regimes.  RETURN TO TOP



Educational and Advisory Services

The Force-balanced companies provide earth mechanical educational and advisory services. The balance of forces in the earth has scores of practical subsurface mechanical engineering applications.

Pore pressure Fracture pressure and overburden are the three primal forces in the earth. Natural minerals and fluids generate and bear the forces in the earth symmetrically.

Two books, Deterministic Earth Mechanical Science, and Pore Pressure through Earth Mechanical Systems were written for scientific and practical purposes. In addition to the exposition of significant new science, the newer book honors the most significant scientific contributors to the present level of the Deterministic Sciences. A small set of physical laws work through the constituents of the earth.

The principles of deterministic science were established by Isaac Newton. He wrote "The Mathematical Principles of Natural Philosophy" also known as (Principia) in 1687. Newton's laws are a set of inter-related postulates that have mathematically descriptive expressions. This set a very high standard for subsequent scientists to follow.

Newton’s laws though discovered in free space also apply in the earth’s confined space. An additional small set of physical laws describe how molecular electron clouds repel each other. Gravity is balanced by molecular electrostatic repulsion in the earth. These laws were first explained by Charles Augustin Coulomb in 1791.

Deterministic Earth Mechanical Science (Phil Holbrook, 2005) is modern physics that applies to the earth's sediments, sedimentary rocks, and SiAlic crust! This science is to the earth’s inner space what Newtonian mechanics is to the earth's surface, and outer space. This book also fills the physical-logical gap between Newtonian and Quantum mechanics. Natural mineral and fluid molecules are the working units of earth mechanics.

Both books synthesize the mechanical laws of Newton, Coulomb, and Hooke. These intuitive men are justly honored by placement on the books' covers. Our earth’s slowly moving continental and oceanic plates are shown as one background that is explained by this synthesis of physical laws. The forces that we measure around an open borehole are shown on the other book's cover. Both mechanical explanations are supported by the same minimalist constitutive mechanical system theory.

Explaining earth and borehole mechanics extends Newton’s gravitational synthesis inward by accounting for the earth's internal forces. Gravity and the Effective Stress Theorem apply simultaneously in the earth's [mass-energy] synthesis. These two books are a significant advancement in physics through their revelation of earthly molecular mechanics.

Earth mechanics are completely scalable from molecular electron clouds which affect borehole walls in basins on drifting continental plates. These two books provide mathematically causal explanations for the earth's subsurface mechanical energy field.

The Venn diagram below illustrates the overlapping characteristics of physical, and inorganic chemical laws with deterministic earth mechanics. These Deterministic Sciences spring from the same <symmetrically balanced> foundation that is illustrated in a central ellipse. The common pattern of the deterministic sciences will be revealed and explained in a newer book but can be taught now.

The dashed lines extending outward symbolize direct algebraic connections to essential physical laws. The whole diagram represents physical realities that have been discovered in nature. The physical and conservation laws that compose the earth's minimum energy equation matrix in Normal Fault Regimes are shown at the bottom of this web page. The other two tectonic regime specific equation matrices are in "Deterministic Earth Mechanical Science."


For those that are interested in a litho logically general and mechanically sensible way to estimate pore pressure click on, "Why learn about the earth's two-phase mechanics?" The most recent thoughts and scientific information can be found in four essays by clicking on the News tab of this website.

Book Order Information: Either "Pore Pressure through Earth Mechanical Systems" or "Deterministic Earth Mechanical Science" can be ordered by phone, through book distributors or over the internet. A book synopsis and table of contents can be seen by clicking on either of the book titles that are highlighted blue.

Each book includes a glossary of technical terms. Years of work were involved in study and preparation. The books are modestly priced for maximum educational benefit.

Hardbound copies are much more durable and only a little bit more expensive. You can order them by their unique ISBN's that are shown with the book covers shown below.



"Deterministic Earth Mechanical Science" . . . . $60.00

Order Now


"Pore Pressure through Earth Mechanical Systems" . . . . $70.00

Order Now

Educational discounts for student and teachers can be obtained from phil by e-mail



You can get any number of copies at the retail price from

Copies can be found by entering the unique ISBN's found below.

Deterministic Earth Mechanical Science
ISBN: 0-9708083-3X

Pore Pressure through Earth Mechanical Systems
ISBN: 0-9708083-2-1



A Brief Summary of Earth Mechanical Science

The mechanics of the earth has both plastic and elastic limits. These limits are deterministically related as shown below. Earth [stress/strain] reactions are usually near one of the two most recent limits. These limbs hinge at the peak loading limb point. The slight differences from perfect linearity have additional physical explanations such as [stress/strain] time lags.

Both plastic and elastic limits are scalar mineral and strain properties. In both cases, strain is a force balanced distance relationship that is governed by a mineral's molecular electron clouds.

Home 4

Well planning requires earth load and fluid pressure information that depend directly on the earth's plastic compact ional mechanical system. The elastic and Grain matrix compact ional [stress/strain] limits are physically coupled as indicated by the panel immediately above. For an overview of earth constitutive mechanical systems limits, see earth mechanical systems limits.

This is a quantum step in understanding the relationship between rock properties and their deterministic causes. Earth mechanics and scalar pore pressures are co-dependant in accordance with [mass-energy] conservation and energy minimization principles!

Drilling, setting casing, reservoir evaluation and compaction during hydrocarbon production are all carried out within the earth's mechanical field. The inter-dependencies within the earth's mechanical field can be used to engineering advantage.

Geomechanics, rock mechanics, and soil mechanics are parts of the same sediment-to-rock physical continuum. The energies in the earth's mechanical systems are transmitted through their contiguous molecular mineral and fluid electron clouds.



The Constitutive Mechanical System Equation Matrix for
the Normal Fault Regime

[Mass-energy] equivalence is expressed as a mathematical equal sign (=) in Newton’s gravitational equation. Newton’s laws of motion apply to macroscopically distinct objects that are separated by space. The earth is primarily composed of natural minerals and fluids.
The earth’s internal space is compositionally complex.

This natural variability must be taken into account by any physical explanation.

[Mass-energy] is equated with [stress/strain] in planet earth. The equation matrix below describes how the earth’s forces are related to each other and to its mass.

Earth Gravity is continuous with Universal Gravity. They share the small set of physical laws that appear in the equation matrix below. The equation matrix shown is one of three overlapping matrices that describe the earth’s three tectonic regimes.

The earth and universe are minimum energy seeking mechanical systems. The energy terms are background shaded tan on this equation matrix. Mass terms are background shaded light blue.

Equation matrix rows are dimensionally correct physical laws or conservation laws. Equation matrix columns are solid and fluid mass conserved scalars. The scalar [mass] terms equate in various ways to the [effective stress scalar] on the left (tan) side of the equation matrix.

Earth Gravity and the two mutually orthogonal horizontal vectors average to the [effective stress scalar]. This is the vector-scalar case of energy conservation. The three cases for the earth’s three tectonic regimes also involve vector-scalar energy difference minimization.

An absolute minimum occurs where one of the three vectors is equal to the [effective stress scalar]. There is one vectorial minimum for each of the earth’s three tectonic regimes.

Under these conditions the deterministic minimum vector has a minimum scalar magnitude. Minimum fracture propagation pressure is always equal to the minimum principal stress plus the scalar pore fluid pressure.

The earth’s mechanical system is balanced in this way. Fluids in the subsurface tend to move toward locations of lower fluid pressure. The minimum energy equation for each tectonic regime is a mathematically closed form. As with other natural mechanical systems it will tend to equilibrate toward minimum energy. These equation matrices are predictors of earth and fluid movement.

The earth’s mechanical system is continuous but it is not smooth. There are at least five linear minimum energy grooves in the earth’s mechanical field. The field’s dimensionality drops from three to two at the minima.

These five two dimensional planes are shown on the website banner. The minimum energy planes are explained in various places throughout the website which was developed progressively over about ten years.

Unified Matrix

Deterministic earth mechanics is what engineers are seeking to reproduce through empirical forced-fits to depth functions. We already know that these approximations are a dimensional mismatch. These bear no resemblance to the actual mechanics of the earth or any other mechanical system.

Engineering forced-fits commonly bring their own special semantic confusions. People will argue about these and never get to the actual subject.

This Constitutive mechanical system is an unambiguous natural basis for subsurface geology and engineering. The equation matrix shown above links previously accepted physical and conservation laws to physical properties.

Energy minimization and differential energy minimization are already parts of accepted physics. To my pleasant surprise they also explained a lot about the major questions that we were asking about the earth.

Minimum energy mechanics are just that. A person that can perform simple mathematical operations can produce deterministically predictive answers. The human energy expended in learning the inter-related laws is well worth the effort for everyone concerned.

Training is recommended for geologists and engineers that are or might be directly involved in earth mechanics. Good judgment in application comes from knowledge and experience.


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