technical publications on Geological Science, Drilling , Reservoirs and Geomechanics  

 Copyright restrictions from some technical journals prevent verbatim reproduction. All relevant technical publications are listed in a personal technical bibliography for full documentation purposes. Hyperlinks to some previously published .pdf files are accessible through this bibliography. Some of these articles are hyper-linked to  Drilling and Deep Water subject bookmarks in this file. These publications reflect the levels of understanding at their publication dates.

Several free publications are reproduced in their entirety on this website. They are below the personal technical bibliography. Much of the recent information on "Deterministic Earth Mechanical Science" is covered in full length technical paper and News essays. However if you wish to actually practice pore pressure prediction, I would strongly recommend some training and/or the purchase of two books that are specifically devoted to the subject of earth mechanics.

Almost all of the technical literature since 1965 have been empirical forced fit to depth methods (P=Pn+DP). Direct correlation of depth to energy or stress is a dimensional mismatch. Science and force-balanced engineering were circumvented by these older (P=Pn+DP)methods.  A physically force-balanced solution to the pore pressure problem was first presented in (Holbrook & Hauck,1987). You can jump start your education by leaving(P=Pn+DP) to history and results are based upon (P=S-s) the effective stress theorem.  These scalars are equated in Nature.

The cited articles in this bibliography involve the [stress/strain] mechanics of the earth's coupled [solid-fluid] mechanical systems. Subsurface science and engineering are coupled as are [mass-energy] and [stress/strain]. A Physical Law Synthesis that explains Earth Mechanics  is a good starting point. This article is at the current (2007) level of earth mechanical science.  The essays in the adjoining News webpage are also reflect current mechanical understanding and a higher level of science. Let the balanced forces and natural energy minimization be with you.

Holbrook, P W, 2005,"Deterministic Earth Mechanical Science" , ISBN: 0-9708083-1-3 , 104 pages. This book explains the earth's macro-mechanics beyond the Normal Fault Regime. It also explains the deterministic relationships between macro-mechanics and fundamental atomic and molecular structures. The book's table of contents, a four paragraph synopsis, and a short professional biography can be seen by clicking on the book title above. A glossary of technical terms is included at  the end of the book. It defines "Deterministic" and most of the other terms that are related.

Holbrook, P W, 2002c, "Overview and Applications of Earth Mechanical Systems", American Society of Civil Engineering, CEWorld virtual conference edited by Walter Marlowe Ph.D. July 1, 2002.   

This was a medium length technical paper that covers the direct mechanical applications. The compactional plastic and elastic end member stress/strain limits are revealed as such in this paper. It is no longer available over the internet, but you may be able to get a copy through Walter Marlowe.

Holbrook, P W, 2002a (Chapter 3 ), "The primary controls over sediment compaction. pp 21-31, 9 figures; in AAPG Memoir 76, Pore Pressure Regimes in Sedimentary Basins and their Prediction.

Holbrook, P W, 2002b (Chapter 14), " Method for determining regional force balanced loading and un-loading pore pressure regimes and applying them in well planning and real-time drilling", pp. 145-157, 9 figures. in AAPG Memoir 76, Pore Pressure Regimes in Sedimentary Basins and their Prediction.

Holbrook, P W, 2001a,"Pore Pressure through Earth Mechanical Systems; The Force Balanced Physics of the Earth’s Sedimentary Crust". ISBN: 0-9708083-0-5, 135p, 33 figures. This book explains how the scalar pore pressure is related to overall natural force balance. The book's table of contents, a four paragraph synopsis, and a short professional biography can be seen by clicking on the book title above. It encompasses the two AAPG articles immediately above in its contents. The molecular scale causal relationships that support this macro-mechanics are in the 2005 book. A glossary of technical terms is included at  the end of each book.

Holbrook, P W, I Goldberg, and B Gurevich, 1999, "Velocity - Porosity - Mineralogy Gassmann Coefficient mixing rules for water saturated sedimentary rocks", 12p., Paper T, 40^th Annual SPWLA International Symposium, May 30 to June 3,1999. This is a pdf file 753 KB in size and will need to be opened with an adobe acrobat browser.

Holbrook, P W, 1999, "A simple closed-form force« balanced solution for Pore pressure, Overburden and the principal Effective stresses in the Earth.", Journal of Marine and Petroleum Geology, Vol. 16, pp. 303-319.

Holbrook, P W, 1998, "Physical explanation of the closed form mineralogic force balanced stress/strain relationships in the Earth’s sedimentary crust." Presented at Overpressures in Petroleum Exploration, the European International Pore Pressure Conference, April 7-8 Pau., France

Holbrook, P W, 1998a, "The primary controls over sediment compaction", in Pressure Regimes in Sedimentary Basins and their Prediction, AADE Industry forum proceedings, September 1 to 4, 1998 Del Lago, Texas. 

Holbrook, P W, 1998b, "The universal fracture gradient/pore pressure force balance upper limit relationship which regulates pore pressure profiles in the subsurface", in Pressure Regimes in Sedimentary Basins and their Prediction, AADE Industry forum proceedings, September 1 to 4, 1998 Del Lago, Texas. 

Holbrook, P W, 1998d, " Method for determining regional force balanced loading and un-loading pore pressure regimes and applying them in well planning and real-time drilling", in Pressure Regimes in Sedimentary Basins and their Prediction, AADE Industry forum proceedings, September 1 - 4, 1998 Del Lago, Texas. 

Goldberg, I, and B. Gurevich, 1998, "Porosity Estimation from P and S sonic log data using a Semi-Empirical Velocity-Porosity-Clay Model", 39^th International SPWLA Symposium at Keystone, Colorado. presented by Phil Holbrook.

Holbrook, P W, 1997, "Discussion of A New Simple Method to Estimate Fracture Pressure Gradients", SPE Drilling & Completions, March 1997, pp.71-72

Holbrook, P W, 1996, "A simple closed form force balanced solution for Pore pressure, Overburden and the principal Effective stresses in the Earth.", in COMPACTION and OVERPRESSURE, Current Research 8th conference on Exploration and Production December 9-10 Ruile-Malmaison-France

Holbrook, P W, 1994, "Method for calculating Sedimentary Rock Pore Pressure." U.S. Patent 5,282,384

___________________________________________________________________________________

A Physical Law Synthesis that explains Earth Mechanics

by Phil Holbrook Ph.D.

Abstract

Deterministic earth mechanics is a subset of Universal Physics. The boundary condition is the earth’s surface. [Mass-energy] is explicitly conserved therein. Civil engineering and soil mechanics scientists have been creating and testing small scale constitutive mechanical models for decades.

Deterministic earth mechanics is an earth encompassing field theory. Stresses and fluid pressure are contributing parts of the two-phase continuous field. Existing physical law are correlated directly to the earth's chemically discrete  components ie. (minerals and fluids) of the earth.

Minerals, fluids and gasses are minimum energy forms of matter at earthly pressures and temperatures. Five mineral types account for over 90% of the earth’s sedimentary crust.  Mineral and fluid molecules interact with their nearest neighbor through interactions between their contiguous electron clouds.

Stress ratios in each of three tectonic regimes center about one of three local energy minima. Each energy minimum predicts the overburden relative magnitudes of the principal stresses. Overburden is an accompanying real numbered scaling factor. Each of three equation matrices describes a ratio of vectorial forces that is limited by the same scalar force balance. Each matrix has the same single line definition of strain, gravity, and scalar force balance. Each matrix has a different set of vectorial stress ratios that are symmetrical about the scalar effective stress.

Compaction is a first order scalar [stress/strain] relationship. Plastic compaction depends on the average mineralogy of a sedimentary rock. Elastic [stress/strain] response does as well.

Hooke’s law elastic [stress/strain] limit increases with compaction to the point of zero porosity. Each of the five dominant mineral types has discrete elastic and plastic [stress/strain] coefficients and limits. These have been measured in the laboratory and in the earth.

Overburden, pore fluid pressure, fracture propagation pressure, average mineralogy and porosity are related in a closed-form mechanical system. [Mass-energy] and [stress/strain] are conserved and equated in each equation matrix. Fracture propagation pressure and capillary pressure are the proximal limits of pore fluid pressure. They are components of a comprehensive force balance.

Petrophysicists can make accurate mechanical predictions using their expertise at estimating average mineralogy and porosity. Most often the observed levels of effective stresses and pore fluid pressures are close to that predicted by the local energy minimum. The earth is an accurate two part pressure and strain gauge. One needs its mechanical coefficients to read it. These have been established empirically.

Mechanical predictions can be made deterministically ahead of the bit. The same mechanical model predicts in remote locations so long as the tectonic regime is the same. Reflection seismic signals follow elastic constitutive physics. The most likely pressure and stress predictions are the local energy minimum. The upper mechanical limit is the rock or sediment’s shear strength at in situ confining temperature and pressure.

 Introduction

[Mass-energy] conservation is common to the known physics of the universe. The mechanics in the earth’s crust unites [mass-energy] conservation with the physical properties of natural molecules. The mechanical energy in minerals, fluids, and gasses depends on the configuration of nuclear centered electron clouds. All nuclear centered electron clouds have soft spherical symmetry.

Spherical symmetry and composition control the bulk moduli of minerals, fluids, and gasses. Hooke’s law specifically relates bulk and shear moduli obeying conservation of [mass-energy]. The bulk and shear moduli of the common minerals can be found in handbooks (Carmichael, R.S., 1982). Fluids and gasses have only bulk moduli that are explained by equations of state. D.G. Archer (1992) described the eos for the most abundant earth fluid, Sodium Chloride brine.

 A simple lever rule governs the elastic behavior of natural earth materials (Holbrook, PW, 1999). The weighted average of the fluid and gas bulk moduli in a sedimentary deposit is the intercept point on the Vp² axis of figure 1. The weighted average of the minerals that compose the grain matrix resolve to another end point on the Vp² vs. Vs² plane. Rock and fluid moduli are end points of this lever which is linear on Vp² vs. Vs² plane.

Porosity is scaled on this lever according to which version of Hooke’s law is appropriate. Hooke’s law applies to nonporous isotropic solids with grain-grain contact. The Wood’s equation is a degenerate form of Hook’s law that applies for slurries and clear brines.

Gassmann’s (1951) equations require pore compliance coefficients. The Gassmann and Hashin-Schtrictman symmetrical pore models apply in the absence of systematic fractures. These equations also apply parallel to systemic fractures (joints) that tend to polarize shear waves. These equations do not work across major shear fractures i.e. faults.

 Figure 1. The Hooke’s law plane showing the most common minerals and fluid in the earth’s sedimentary crust (adapted from Kreif et al, 1990). The figure is colorized version of figure 5.2 in Holbrook, P.W, 2001. These relationships are continuous for porosities from 0% to 100% over the entire range of natural sedimentary mixtures.

 Over 90% of the earth’s porous sedimentary crust is composed of mixtures of the four minerals shown. Over 95% of the fluid in pore spaces is Sodium Chloride brine. All water-wet clay values are close to a “clay point” as on figure 1. Vp² and Vs² depend primarily on bulk mineralogy and porosity of a sedimentary rock.  This is a near complete description of rock static elastic properties.

 The Poisson’s ratio of SiAlic sedimentary clays is a virtual constant of 0.29 (Holbrook, 1999). The box in figure 1 shows the range of clay elastic moduli from Hashin-Schtrikman (1963) decomposition. Claystones with very low (10%) porosity still have dynamic elastic properties just above that of grainstone slurry. Claystones fail at very low stresses as well.

Electrostatically negative surfaces cause clay mineral to form semi-solid gels with porosities as high as 95% and as low as 10%. In claystones acoustic disturbances must pass through fluid electron clouds to reach the next solid electron cloud. SiAlic claystones plot along the Poisson’s ratio 0.29 line as long as porosities are lower than 10%.

Calcite, dolomite and quartz granular slurries reach solid grain-grain contact between 45 and 35% porosity. At lower porosities their response is quasi-linear on the Vp² vs. Vs² plane.

 Plastic compaction of natural mineral-fluid mixtures

The presence of electrostatically negative surfaces also has a profound effect on plastic compaction of the grain matrix. Figure 2 shows the stress/strain relationships of the five most common sedimentary minerals. Compactional strain is directly proportional to solidity in agreement with the law of solid mass conservation (figure 2 right). Solidity is the mathematical complement of porosity that has a distinct upper limit.

Figure 2). The First Fundamental in situ [Stress/Strain] Relationship and the solid mass conserved definition of strain. Effective stress (solid partitioned energy) is power-law proportional solidity (solid mass conserved strain) taken from figure 9.1 in Holbrook, P.W., 2005.

Each of the regression lines on figure 2 covers a wide porosity range. The lines terminate where the [stress/strain] response becomes non-linear. These are near surface effects at effective stresses below 100 psi.

Anhydrite and Halite are late stage evaporites from sodium chloride rich brines. Their porosity reduction is complete at less than 1000 psi. Thereafter salt behavior becomes entirely plastic forming natural salt pillows, ridges, and domes.

Most mineral grains including Quartz, Calcite, Halite, and Anhydrite have electrostatically neutral grain surfaces. These minerals have sub-parallel [stress/strain] relationships as shown on figure 2. The solidity intercept of a compaction function on figure 2 is principally related to the mixed mineral rock’s hardness.

Clay minerals have a distinctly different static [stress/strain] relationship. Coulomb’s law causes inter-particle repulsion. The broad surfaces of clay minerals have negatively charged oxygen anions at their particle surfaces. The balancing (+3 or +4) cations are sandwiched by the (-2) oxygen anions. Thus clay minerals have a net negative surface charge. The repulsive surface charge is power-law proportional to clay mineral particle size (Nagaraj & Murthy, 1983).

Compaction resistance is the sum of inter- and intra-particle repulsions (Holbrook, P.W., 2001). Both depend directly on Coulomb’s law. Clays have additional inter-particle repulsion. Inter-particle repulsion is responsible for the high (90 to 95%) porosity of freshly deposited fine clay. Particles with net neutral surface charge have no net inter-particle repulsion. Grains of quartz and calcite immediately come in direct contact forming a granular solid at near zero effective stress.

 Differential energy minimization in the earth’s sedimentary crust

Newton’s gravitational law and Coulomb’s law operate simultaneously inside the earth. Both are inverse square laws that affect atoms and ions. The interaction of these two laws explains the earth’s general sedimentary crustal [mass-energy] [stress/stain] hypothesis.  That hypothesis is a dimensionally correct summation of accepted physical laws.

 Energy is conserved in each of these physical laws and in the synthesized law hypothesis. The mechanical energy associated with gasses and fluids is pore pressure. The energy associated with solids is effective stress. The effective stress theorem provides that the sum of solid and fluid energy is conserved. Energy conservation is typical of physical laws.

 The principal stresses in the earth’s crust are naturally orthogonal. With the exception of local density anomalies, one principal stress is the vertical mass attraction toward the earth’s core. The earth’s mass is 6 x 10²⁴ kilograms and its center of mass is about 6370 km depth. Natural sediments, sedimentary rocks and fluids, the mantle and the core are gravitationally stratified.

 Energy is conserved in the earth according to the effective stress theorem (S=P+s).  Vectorial effective stress is conserved within volumetric. Net horizontal mass attraction is very close to zero. Coulomb’s law, Newton’s law , [mass-energy] conservation and minimization work in concert to control both average and vectorial stresses.

 The relative magnitudes of the three principal stresses define earth’s three tectonic regimes. Effective stress is three-dimensional and unequal in the sedimentary crust. Differential stress is continually minimized in the earth’s sedimentary crust. Coulomb’s law and Newtonian gravitation tend to balance each other. Positive integer stress ratios result when the earth’s mechanical system is in balance.

 This is a spherical solid case of symmetry in natural physical laws. Newtonian gravitation and Coulomb’s laws are symmetrical about the integer two. Many other physical laws are as well. The earth’s minimum energy states are physically symmetrical about the integer two <2> as well.

 Atoms, ions and minerals have geologic time to adjust to the earth’s environment. Deformation of an ionic cloud in a mineral increases mechanical energy.  Energy is reduced when a stressed ion migrates to a lower stress space. “Pressure solution” is the common name for this ionic migration process. Mass movement also reduces energy when grains are rotated or fractured as faults move. Both ionic and mechanical rearrangements of matter tend to minimize differential stress.

 Consequently high energy does not persist. For every displacive force that increases energy there is an equal and opposite restoring force that reduces it. The energy minimization process is slower and asymmetrical within solids. It is force-driven toward a minimum and has geologic time in which to equilibrate.

 Energy minimization in the earth’s three tectonic regimes

Stress conservation and natural orthogonal stress orientation lead to integer ratio energy minima. In normal fault regime regions, the maximum principal stress is vertical. Conservation of energy dictates that if vertical stress is above average, the sum of the two horizontal stresses is equally below the average. The lowest possible positive even number ratio is ((4:1:1)/3=2) for (v:H:h) vectors. Normal Fault tectonic regimes tend strongly toward this ratio. The absolute minimum differential stress about this ratio is 2:1.

Figure 3. Stress ratios related to in situ strain in Normal Fault Regime Basins. The collection of empirical curves on the left was taken from (Pilkington, P.E., 1978). The empirical strain data on the right was taken from (Bryant et al, 1980).

A side-by-side comparison of stress ratio to solid mass conserved strain on figure 3 shows considerable similarity. Stress ratio and strain could be even closer than they appear. Examination of the supporting data in Pilkington’s (1978) paper reveals a very poor fit to the curved depth functions. Early leakoff tests were notoriously unreliable. Also the early data was taken when offshore drilling was in less than 400 feet water depth. See figures 6a in Holbrook 1996 that is reproduced further below.

The plot of the single power-law [stress/strain] function shows an excellent fit to all of Bryant’s (1980) data. The compactional [stress/strain] and (h/v) stress ratio is a power-law type of [mass–energy] relationship. The data fit to this mechanically sensible relationship is excellent.

The custom of plotting energy vs. depth is dimensionally incorrect. However many researchers have seized upon the energy vs. depth correlation as an obvious observable shortcut. Unfortunately the shortcut is data and region specific. The wide variance in Pilkington’s  empirical depth function data (figure 6a) reflects how far off a depth function can be.

The largest difference in stress and pressure  occurs in the first 4000 feet below the mud line. Any drilling risks taken at less than 4000 feet could result in the loss of a short relatively inexpensive hole. The application of physically rational [stress/strain] relationships at shallower depths can save much greater trouble costs that could occur later.

Figure 4. The First and Second in situ [Stress/Strain] relationships in Normal Fault Regime Basins (taken and colorized from figure 4.2 in Holbrook P.W., 2001).

Vectorial and volumetric effective stresses are conserved energy on figure 4.  When minimized they are directly proportional to volumetric strain. Grain-matrix strain depends directly on solid mass conservation. Thus mass and energy are conserved in Normal Fault Regime Basins as shown on figure 4.  All the minimum energy [stress/strain] equations are power-law linear.

The average principal stress (orange) is power-law proportional to strain (brown). This corresponds to[mass-energy] conservation as it occurs in the earth. The (horizontal/vertical) stress ratio is exactly equal to strain (1.0-f) as shown on figure 4.  This is the minimum-energy-state for the normal fault tectonic regime.

Differential energy minimization in the strike-slip tectonic regimes

Gravity is the intermediate principal stress in strike-slip tectonic regimes. Orthogonal energy conservation dictates that one horizontal stress must be lower and one higher by an equal amount. This is in fact the case and the three principal stresses are close to their minimum energy state.

Figure 5.  Quantitative vectorial stress relationships in a strike-slip tectonic regime. Reproduced from Holbrook (2005) figure 4.4. which was adapted from Katahara et al (1995).

Keith Katahara (1995) measured minimum, vertical, and maximum stresses in the Long Beach unit in California. The three lines shown on figure 5 are in exact 1:2:3 ratios assuming a 26-foot water table. Such exactitude is unusual if the laws of nature are not directly involved.

Boxed red “x” indicate leakoff tests where the upper hole section was protected by casing. These show excellent agreement with the maximum horizontal stress that is plotted. Mini-frac measurements are represented by red “x” symbols. The overburden data are indicated by black “+” signs. The minimum principal stress line for claystones was copied from Katahara et al’s (1995) plot. They noted considerable variance in all mini-frac date but the interpretive line is closely related to claystone minima on this plot.

 Mini-frac tests are difficult to perform in sandstones owing to their much higher permeability. The unusual readings could easily be due to the sandstone permeabilities. Flow around packers or into sandstones can be mistaken for flow into fractures. One cannot put much faith in this data. Simple stress conservation based on the maximum and intermediate curves would put the minimum stress curve exactly where Keith Katahara put it.

 Figure 6)  Diagrams showing principal stresses in normal fault and strike-slip tectonic settings. [Effective tress/ strain (Solidity)] plots that correspond to these two tectonic regimes are immediately below. The color convention of all plots is vertical (purple), major horizontal (black), minor horizontal (green), Average volumetric scalar (orange); improved from Holbrook, 2005 figure 12.3.

 Figure 6 summarizes the stress ratio data for these very different tectonic regimes. Vectorial stresses are conserved within volumetric in both cases. Energy and mass are conserved in both cases. Although not shown on this figure, fluid energy (pore pressure) is also accounted for under general [mass-energy] conservation.

 The total strain intercept varies with the mineralogic composition of the grain matrix. A mineralogically weighted average from figure 2 is used on figure 6. Thus the stress/strain properties of the granular solid rock are directly related to its constituent minerals in both tectonic regimes.

 Coulomb’s law is insensitive to the direction of stress because electron cloud symmetry is fundamentally spherical. If an individual electron path is forced away from its nucleus in one place, it will be closer in another. Electrostatic energy balance also applies to the electron clouds of neighboring nuclei. Electrostatic forces are 10³⁹ stronger than gravitational. Earth gravitational forces are strong but don’t approach the strength of electrostatic forces.

 The force-balanced [stress/strain] relationship has closed upon itself. This relationship is like the other [mass-energy] relationships are verified physical laws. This equation synthesis is composed only of physical laws. [Mass-energy] conservation of the synthesis is inherited from the parent physical laws.

The importance of this synthesis is that it has been verified empirically under some very different conditions. For example figure 6 explains the observations of figures 3, 4, and 5. The purely elastic relationships of figure 1 were compositionally related to the purely plastic relationships of figure 2. The mechanical response of all matter fits somewhere between these two mechanical limits. All solid and fluid matter is composed of atoms and ions with spherically symmetrical electron clouds.

The attributes of figures 1 through 6 can all be directly related to the energy that is stored as compression of electron clouds. This force is transmitted from ion to neighboring ion. This is how the force applied at one edge of a continental plate is transmitted to the other. A simple set of equations, all of which are physical or conservation laws, describes this.

Mechanical equation synthesis for the earth’s sedimentary crust

Figure 7) Mechanical equations that define the static force balance of for Normal Fault Regime basins in the earth’s sedimentary crust. Energy is background shaded tan, and mass properties are background shaded light blue (taken from Holbrook, P.W., 2005, figure 13.1).

All seven equations shown on figure 7 are physical or conservation laws. The first equation simultaneously defines 1.) solid-mass-conserved strain and 2.) mineral-fluid partitioning. The common partitioning coefficient is the backbone of the [mass-energy] [stress/strain] synthesis. The brown arrow on figure 7 highlights mathematical simultaneity. Solid mass conservation is the basis for the simultaneous solution of physical and conservation laws.

 Newton’s gravitational law is the second equation. It is partitioned according to discrete mineral and density coefficients. This is the mass attraction between each atom and the earth’s center of mass. The mineral and fluid density coefficients were determined empirically and can be found in physical properties handbooks (Carmichael, R.S., 1982).

Fluids and gas density coefficients were empirically derived and are explained by equations of state (Archer, 1990). Thus Newton’s gravitational law is directly correlated to the physical properties of the mineral and fluid phases in the earth’s sedimentary crust.

 The third equation expresses scalar energy conservation of the minerals in a sedimentary rock. The mineral coefficients (a & s_max) were derived empirically from the natural compactional relationships shown on figure 2. Each single mineral compactional [stress/strain] relationship is a power-law function.

 A whole rock power-law function is an average of its single mineral power-law functions. This compactional [stress/strain] synthesis is applicable to minerals. Solid energy resides in the compression of the constituent ionic electron clouds.

The first, second, and third physical equations describe global properties of the earth’s sedimentary crustal mechanical system. The equations that follow are necessarily conserved within these scalars. The individual (v:H:h) vectors follow both Newton’s mass attractive and Coulomb’s electrostatic repulsive energy laws. The earth’s mechanical system is at rest when static forces are balanced. Otherwise we have earth movement that can sometimes be catastrophic.

The fourth equation on figure 7 describes the minimized-differential-stress state of normal fault tectonic regimes. Under this condition, (horizontal/vertical) stress ratio is directly proportional to strain. The empirical relationships shown on figures 3 and 4 are consistent with [mass-energy] conservation and differential stress minimization. Both attributes are characteristics of known physical laws (Feynman, R P, 1965).

 The fifth equation on figure 7 is specific to normal fault regimes. It states that vectorial effective stresses are conserved within volumetric. This energy conservation boundary condition applies to the solid phases.

 Differential stress is minimized when (H=h). This condition is a mathematical lower limit. Lacking other information, this is the usual assumption. The spherical symmetry of electron clouds and Coulomb’s law causes this algebraic minimum energy condition.

 The distribution of vectorial stresses is different in strike-slip tectonic regimes. Figure 6 showed the minimized vectorial differential stress ratio of (1:2:3) was empirically very close to the observed (h:v:H) stress ratios. A minimization principle is consistent with known physical laws (Feynman, R P, 1965).

 Discrete power of two integer relationships are characteristics of Newton’s mass and Coulomb’s electrical charge energy laws as well. The inverse square of distance is an integer that defines perfect energy balance. Different stress ratios indicate the potential for movement.

For strike-slip tectonic regimes, the fourth and fifth vectorial equations are replaced. Minimum and maximum horizontal stress differs from the average scalar by the integer one in both tectonic regimes. These discrete integers indicate physical-mathematical <2> symmetry. This is another characteristic of known physical laws (Feynman, R P, 1965).

 The two energy conditions that are of greatest interests to geologic science are the sixth and seventh equations on figure 7. Pore pressure is the fluid mechanical energy in a porous sedimentary rock. It is the scalar relationship of average confining load minus the solid born load of effective stress. The sixth equation is the effective stress theorem.

Fracture propagation pressure is the seventh equation. It is the minimum load that holds fractures closed. It is the sum of the scalar pore pressure with the minimum principal stress vector.

 Fracture propagation pressure is a limit to pore pressure in all tectonic regimes. Fracture propagation pressure can be no greater than overburden in static equilibrium. Single phase fluids are free to move in any direction the subsurface.

 The smallest flaw in a minimally stressed orientation will be opened by single fluid phase pressure. Fluids will escape rapidly with respect to geologic time.  Natural minimum energy conditions for both solids and fluids will quickly be restored.

Conclusions

The mechanics of the earth’s sedimentary crust is explained through a synthesis of universal physical laws. The synthesis predicts additional mathematical symmetry which has been empirically verified. It appears a minimum differential stress condition dominates in the earths subsurface as it does the rest of the universe. Natural forces drive the earths two phase mechanical system from higher to lower energy. This is just like previously known mechanical systems.

 Newton’s law and Coulomb’s law describe symmetrically conserved energy. Hooke’s law describes [mass-energy] conservation in terms of elastic [stress/strain] definitions. Solid mass conservation bounds compactional grain-matrix strain and Hooke’s law.

 The earth’s crustal mechanics overlaps with universal mechanical systems. It has a newly discovered and physically consistent minimization principle. Fluid and solid phase energy is simultaneously conserved and minimized. It is a revolutionary step in the progress of physics and geology.

 The incorporation of clay inter-particle repulsion into both elastic and grain-matrix compaction is also a significant contributing step. Inter- and intra-particle repulsion is total solid energy conservation. Coulomb’s law explains clay’s role in elastic and plastic end-member mechanical systems.  It simultaneously explains non-clay mineral compaction and elasticity.

Niels Bohr’s correspondence principle states that a new physical law must also explain the verified results from existing physical laws. This synthesis is composed only of verified physical laws.

 Andersonian tectonic regime classification specifies rank ordinal stress ratios. This synthesis predicts their magnitudes. Scalar and vectorial stresses occur in the lowest possible positive integer orders. All the new predictions are consistent with an earth internal energy minimization principle. This synthesis passes Niels Bohr’s test and additional empirical tests that are vastly different.

Einstein determined the uppermost limit of mechanics i.e. the speed of light. The earth mechanical synthesis applies at and near the lowermost speed limit. [Mass-energy] is conserved at both speed limits.

 Minerals and fluids are discrete molecules that compose almost all of the earth. Molecular mechanics fills a gap between Newtonian and Niels Bohr’s quantum mechanics. It uses their physical laws as limits. The laws of physics now have explicit continuity within the earth!

 Applications of deterministic earth mechanical physics

Major oil companies spend billions of dollars each year drilling for and producing hydrocarbons. Pore fluid and fracture pressures are the open borehole pressure limits. These can now be calculated deterministically. Reservoir mechanics are a time and pressure dependent subset of earth mechanics. These limits are deterministic as well.

 Pore pressure and fracture pressures have been estimated by 250+ different forced-fit depth methods. Uncertainties in the choice of methods with method specific coefficients are the major problem. A new set of empirical coefficients must be developed for each local region. Errors during drilling and engineering for earth uncertainties are usually very costly.

 The borehole fluid pressure limits can be calculated simply and accurately from synthesized physical laws. [Mass-energy] is conserved and differential stresses are minimized by the earth. Mass and energy reside in the minerals and fluid phases of the earth. In a given tectonic regime the major questions are how much of each and where are answered.

 Petrophysics has answered mineral and fluid partitioning questions for decades. Mass and energy are the sum of mineral and fluid physical coefficients. Petrophysicist’s measurement skills are adequate to describe this constitutive mechanical system. The [mass-energy] conserved physical model for the earth is the new ingredient.

Reservoir volumetrics are the first issue of the petroleum industry. Recovery factors vary more than three fold depending on mineral and fluid resident energies. [Mass-energy] conserved physics is ideally suited to refine the hydrocarbon recovery factor answer.

Production rate is important in the commercial assessment of a reservoir. Production rate decline depends on fluid pressure and reservoir compaction. These are co-dependant [mass-energy] conserved relationships. Reservoir compaction falls within the elastic and grain-matrix compactional limits of the earth’s mechanical system.

Reservoir pressure decline curves are between the lower and upper mechanical response limits of the earth. A decline curve that intersects these limits and obeys physical laws should have greatly improved predictive power. These limits are both specified by [mass-energy] conserved physics.

Oilfield earth predictive needs have historically been filled by forced-fit empiricism. The same predictions can be made from physical models that depend on physical laws. The connective path through physical laws is the shortest and most direct.

It is the most reliable path known to man. There are fewer steps and the fewest possible empirical coefficients. Whenever possible, dependence on verified physical laws would be preferred by rational thinkers.

References cited

Archer, D.G., 1992, “Thermodynamic properties of NaCl + H2O System II. Thermodynamic properties of NaCl(aq), NaCl.2H2O(cr), and phase equilibria,” by J. Phys. Chem. Ref. Data, Vol. 21, No. 4, pp. 793-829.

Bryant, W, R Bennett, & C Katherman, 1980, "Shear strength, porosity, and permeability of Oceanic sediments", pp 1555 - 1660. in Vol. 7, "The Sea, the Oceanic Lithosphere", C Emiliani editor, John Wiley & Sons.

Carmichael, R.S., 1982, "Handbook of Physical Properties of Rocks", CRC Press.

Feynman, R. P.,1965, “The Character of Physical Law” , MIT Press, ISBN 0 262 56003 8  173 pages.

Gassman, Fritz, 1951, Uber  die Elastizitat Poroser Medien:Vierteljahsschrift der Natorurforschenden Gesellschaft in Zurich, vol. 96, p1-22.

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                     Pore Pressure Prediction and Detection in Deep Water;  

Pore pressure (P_p) is the fluid load-bearing element in the Earth’s subsurface. Deep water settings are particularly amenable to a rigorous force balanced approach for pore pressure detection and prediction. The remaining load borne by a fluid filled rock in the subsurface is the average effective stress (s _ave). The Effective Stress Theorem is the force balanced physical-mathematical expression for porous granular solids that compose the Earth’s sedimentary crust. In this rigorous physical expression; the fluid scalar pore pressure (P_P) is the difference between the two solid element scalars average confining load (S_ave) and average effective stress (s _Ave) that is (P_P = S_ave - s _ave).

By coincidence, the Terzaghi (1923) mixed scalar-vector relationship (P_p = S_v - s _v) works in most Normal Fault Regime basin deep water settings. Terzaghi found empirically that in shallow marine sediments the scalar pore pressure (P_p) was related to the axial load vector, overburden (S_v).

Below the True Rock Porosity midpoint on figure 1 is the Newtonian mathematically closed formulation relating force balance to strain in Normal Fault Regime » biaxial basins. The load elements of the Newtonian closed formulation are on the left side of the individually force balanced equations. On the left, confining loads are denoted with an (S). Force balanced corresponding effective stress loads are denoted with a (s ). Both (S & s ) are subscripted vectors. The "v" subscripts denote vertical gravitational loads and "h" subscripts denote the two corresponding orthogonal horizontal vectorial loads. Average confining load (S_ave) and average effective stress (s _ave) are completely three axis (xyz) confined force balanced scalars.

The Effective Stress Theorem is the fifth equation in the Newtonian force« balanced closed formulation. The scalar Pore pressure (P_p) is calculated as the difference between the two load scalars [(S _ave)-(s _ave)]. Fracture propagation pressure (P_F = P_p+ s _h =S _h ) is thereafter calculated in the sixth equation using pore pressure (P_p) calculated using the Effective Stress theorem. All the load terms (S, s _h & P_p) to the left of the (=) signs denoted by the sloping dashed line on figure 1 are a Newtonian closed form force balance! The Effective Stress theorem is the missing element in the "Terzaghi" relationship that makes it work.

All the earth strain terms are on the right side of the (=) signs denoted by the color background change on figure 1. Absolute volumetric in situ strain (1.0 - f ) is in each of the individually force balanced stress/strain equations. The descending arrow on figure 1 indicates the algebraic linkage of these equations to petrophysically measurable strain. The remaining strain terms (r , s _max & a ) are mineral and fluid coefficients that are compositionally linked to each other. The equal (=) signs denoted by the sloping dashed line on figure 1 mathematically relate force balanced loads to absolute in situ strain in the earth! Force« balance and direct measurable strain linkage are unique to this closed-form Newtonian formulation. This physically representative mathematical formulation leads to simplicity and accuracy of calibration, prediction and detection of pore pressure.

Initial overburden (S_v) is also highly predictable, as it is mostly seawater. Sediments deposited in cold deep water typically have low carbonate content. Cold water stratigraphic sequences are dominantly a simple binary quartz grainstone - claystone mineralogic continuum. A normalized gamma-ray log is a fairly good quartz-clay mineralogic estimator these settings. Favorable geologic conditions for the petrophysical calculation of overburden (S_v) and average effective stress (s _ave) are present.

Both (S_v & s _ave) are dependent only on strain, mineral, and fluid coefficients in the Newtonian closed formulation. Effective stress ratio (s _h/s _v) is also directly related to (1.-f ) strain. Given these equation redundancies, and the compositionally bound mineral and fluid physical constants, the Newtonian closed formulation is mathematically simple. Given geologic uniformity; pore pressure determination from petrophysical data is further simplified in off-shelf deep water settings.

Fracture pressure is physically linked to pore pressure through strain and force balance in the Newtonian closed formulation. Because of this, leakoff tests serve as additional calibration points for pore pressure when using the Newtonian force balanced method. The standard deviation of leakoff test measured vs. force balanced calculated fracture pressure is about 0.2 pounds/gallon equivalent mud weight in deep water settings. Leakoff tests are compared to force balanced logs locally to fine-tune the top-of-log initial overburden (S_v) constant and the local water conductivity (C_w) profile. Both types of borehole pressure measurements can be simultaneously calibrated to Newtonian closed form force balance using linked pore pressure/fracture pressure method. The calibration to Newtonian force balance physical approach is the most accurate for both pore pressure and fracture pressure prediction-detection in deep water settings.

The Use of Petrophysical Data for Well Planning,
Drilling Safety and Efficiency

Phil Holbrook, Force Balanced Petrophysics, Houston,Texas

The two open borehole fluid pressure limits, pore pressure (P_p) and fracture gradient (P_f) are represented by linked stress/strain relationships which are best obtained directly from in situ petrophysical data. The key to in situ petrophysical determination of these stress - fluid pressure relationships is that rock solidity (1.0 - f ) is an absolute measure of granular matrix strain.

A new rock mechanics system has been developed from and for use with downhole petrophysical data related to in situ borehole fluid pressure measurements. Using gravitational force balance (s _v = S_v - P_p), two new in situ mineralogic stress/strain (1.0 - f ) relationships were derived directly from subsurface measurements of porosity on granular sedimentary rocks. These in situ compactional relationships vary with average mineral ionic bond strength and are independent of any particular material response law.

Only two compaction coefficients s _max and a , are used to relate vertical stress (s _v) to in situ grain matrix compactional strain (1.0 - f ) in Normal Fault Regime » biaxial basins. The compaction coefficients are weighted average mineralogic constants in a general compactional power law linear stress / in situ strain (1.0 - f ) relationship;

s _v = s _max(1.0 - f )^a

In NFR » biaxial basins; the horizontal/vertical stress ratio (s _h/s _v) increases in direct proportion to in situ compactional strain (1.0 - f ) following the relationship;

s _h / s _v = (1.0 - f )

Fracture propagation pressure ( P_f = s _h + P_p) is therefore also linked to in situ compactional strain (1.0 - f ) and average sedimentary rock mineralogy.

This new compactional strain fracture pressure relationship has been shown to be very accurate (» 4% SD) in 5 separate statistical studies. Accuracy is consistently high for shales, sandstones and limestones over the entire effective stress range of drilling interest. The compactional strain relationship is similar to but more accurate than the empirical fracture pressure-depth relationships of previous authors.

Accuracy is consistently high for shales, sandstones and limestones over the entire effective stress range of drilling interest. The compactional strain relationship is similar to but more accurate than the empirical fracture pressure-depth relationships of previous authors.

All the pressure and stress parameters ( P_p, P_f , S_v, s _v, s _h ) are related to bulk volumetric strain (1.0 - f ) in the above linked equations. The four linked equations constitute the force balanced in situ Rock Mechanics System. With this system continuous logs calculated from in situ petrophysical measurements can be calibrated to all relative and absolute borehole fluid pressure measurements and leakoff tests simultaneously for an entire well. The single well calibration is regional if the overburden (S_v) gradient is regional.

Comparison of these petrophysical force balance calibrated stress and pressure logs to previous well plans often reveals how well plans could be changed to eliminate a casing string. The cost savings from one casing string is usually greater than the total well logging budget.

Introduction

Petrophysicists and log analysts are becoming increasingly involved in the technical decision making related to well planning and real-time drilling operations. The theory and approach herein described involves the direct use of stress/strain relationships related to parameters which are routinely measured from in situ log and borehole fluid pressure data.

Normal Fault Regime basins are a large proportion of the world's sedimentary basins. The maximum principal stress (s _v) is vertical in NFR basins. The two lesser horizontal principal stresses are often approximately equal in NFR basins (ie) the stress field is approximately biaxial. The NFR » biaxial stress boundary condition is like that applied in most laboratory rock mechanics experiments. Also in NFR » biaxial basins most of the sedimentary rocks are at their maximum burial depth and their maximum loading point. NFR » biaxial basins provide the opportunity to determine static equilibrium stress/in situ strain relationships under laboratory equivalent » biaxial boundary conditions.

A new rock mechanics system which uses in situ petrophysical data and borehole fluid pressure measurements in lieu of laboratory applied external loads is described below. The in situ rock mechanics system is to the loading limb what the present laboratory data based rock mechanics system is to the unloading - reloading limb. The stress paths of the in situ and the laboratory rock mechanics systems intersect at the maximum loading point. The choice of which rock mechanics system will provide best quantitative results depends on which stress path is followed.

Stress/Strain Hysteresis

There is a pronounced hysteresis in the stress/strain relationships of all sediments and sedimentary rocks. Very different stress paths are followed by granular solids during initial loading vs. unloading - reloading. The appropriate stress/strain path must be used when estimating stress from strain. Figure 1 illustrates the primary loading vs. unloading - reloading phenomenon which is the characteristic stress/strain response of sediments and sedimentary rocks.

Consolidation during loading follows upper limit loading limb stress path denoted by solid circle data. Below about 100 kPa effective stress, there is curvature in the plastic loading limb. At higher stresses the loading limb is very close to a power law linear plastic stress/strain relationship. The effective stress/strain slope of the power law linear portion of the plastic loading limb is related to liquid limit (Skempton, 1970) or mineralogy (Holbrook, 1995). Mineralogy is the compositional control over liquid limit. Mineralogy can be estimated indirectly from in situ petrophysical measurements whereas liquid limit can only be measured on a laboratory sample.

Unloading Reloading Stress Path  Unloading begins the moment that effective stress decreases from its previous maximum loading point. Unloading - reloading data are denoted with open circles on figure 1. Starting from the loading limb, the initial unloading expansion of a sediment or sedimentary rock is close to linear elastic. Stress/strain hysteresis loops as shown on figure 1 are sometimes observed below the loading limb. The unloading and reloading stress paths are often linear for more permeable consolidated rocks.

When the laboratory reloading rate of a sediment or sedimentary rock is relatively slow, the reloading stress path will be followed until it intersects the point of departure from the loading limb. When effective stress is increased the loading limb stress/strain path will be followed as shown on figure 1. The lower left hysteresis loop of figure 1 demonstrates the aforementioned loading limb capping relationship.

All rock samples from which we gain our laboratory experimental knowledge are on the unloading - reloading stress path. A major concern with the interpretation of laboratory stress/strain data on more consolidated sedimentary rocks is the opening and closing of stress relief microfractures. This phenomenon also causes slight hysteresis in the unloading - reloading stress paths. With minor exceptions, laboratory rock sample unloading - reloading stress paths are steep, deviating only slightly from linear elastic. If one is attempting to determine if a borehole will fail during short term unloading - reloading; preference should be given to the laboratory based rock mechanics system which measures that dominantly elastic stress/strain relationships.

In Situ Loading Limb Stress Path During geologic loading effective stress increases very slowly. Pressure solution is a slow acting chemical mass transport mechanism which operates during geologic loading. Minerals tend to dissolve at points of high stress and reprecipitate at points of low stress. Matter and stress tend to be redistributed in proportion to the relative bond strength of the constituent minerals. The differential stress on the load bearing mineral lattice and points of grain contact which occurs at laboratory loading rates is minimized through pressure solution over geologic time.

Figure 2a shows extensive sedimentary grain microfractures which have been completely healed through pressure solution. Figure 2b shows how sedimentary grains have compacted through dissolution at grain boundaries and re-precipitated in the intergranular pore space. A mica grain is shown to have penetrated a quartz grain which is over 1000 times harder.

Both of these important in situ compaction and grain fracture healing processes occur only on the loading limb stress path at geologic loading rates. Most of the elastic strain in these two samples has been accommodated through pressure solution into the total strain. Where unsupported grain fractures are filled, or pressure solution has occurred, the strain is unrecoverable (ie) plastic. The new in situ rock mechanics system measures stress and total strain including pressure solution at static equilibrium after geologic time. If one is attempting to determine the state of stress from strain at the moment before a borehole is cut; preference should be given to in situ measured petrophysical data.

In Situ Petrophysical Data Rock Mechanics System

A new force balanced rock mechanics system has been developed from and for use with in situ petrophysical data. Figure 3 shows a side by side comparison between laboratory and in situ measurement based rock mechanics systems. Both rock mechanics systems are biaxial with the two lesser principal stresses being equal. In the laboratory data based system shown on the left; external loads (S_v) & (S_h) are applied, and pore fluid pressure (P_p) is measured. Effective stresses (s _v) & (s _h) are calculated by force balance diffence. Two relative length strains and a bulk volume strain are also measured.

Using the in situ rock mechanics system on the right; the vertical load (S_v) is calculated from an integrated bulk density log. Pore fluid pressure (P_p) is measured directly inside a borehole from wireline RFT's. Relative mud weight borehole fluid pressures ( P_b / P_p ) opposite moderately permeable formations provide an upper pore pressure (P_p) limit in an open borehole if no well flow is observed during drilling. The effective stresses (s _v) & (s _h) are calculated by force balance difference from Overburden (S_v), leakoff tests (P_f), and pore pressure (P_p).

Solidity (1.0 - f ) is the in situ measure of bulk volumetric strain. This parametric measurement shortfall is not a handicap in NFR » biaxial basins because both the effective stresses (s _v) & (s _h) are closely related to in situ bulk volumetric strain.

Laboratory based systems use the measured final/initial ratio of a length or volume as a measure of strain. Using the laboratory approach the relative final/initial strain always depends on the initial unloaded state of the rock sample being tested. The results of each laboratory experiment depend on the initial porosity, composition and extent of stress relief microfractures in the sample. Each laboratory rock sample has its own unloading reloading stress path. The general conclusion that can be reached from interpreting laboratory experiments is that the reloading - unloading limbs are different but are usually close to linear elastic.

Solidity (1.0 - f ) is the present/final volumetric strain ratio which can be directly measured using the in situ data based