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Phil Holbrook Ph.D.
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Scientist Force Balanced Petrophysics

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  • Applications of deterministic earth mechanical physics










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

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



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.   RETURN TO TOP

Phil Holbrook Ph.D.

Technical Publications and Technical society presentations

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, 2001b, "Overburden, fracture pressure, and permeability regulate in situ pore pressure profiles" The Oil & Gas Journal December 17, 2001, pp. 37-42. The journal version of this article is copyrighted.

Holbrook, P W, 2000, " Why not solve for Pore pressure using Earth Mechanical Systems?", Overpressure 2000 Compact Disc, 15p, 8 figures.

Holbrook, P W, 2000, " Why not use Earth Physics to solve for Pore Pressure?", Overpressure 2000, extended abstract 4p., April 5-6 London, U.K.

Holbrook, P W, 2000, "How do Poisson’s Ratio and Plasticity relate to Fracture Pressure?", World Oil, March, pp. 91-96. This is a pdf file that is 323 KB in size and will need to be opened with an Adobe Acrobat browser.

Holbrook, P W, 1999, "Pore Pressure Prediction and Detection in Deep Water", the 1999 International AAPG Conference and Exhibition in Birmingham, England, 4p, 3 figures, September 12-15, 1999. Click on Deep Water to hyperlink to this location on this website.

Holbrook, P W, I Goldberg, and B Gurevich, 1999, "Velocity - Porosity - Mineralogy Gassmann Coefficient mixing rules for water saturated sedimentary rocks", 12p., Paper T, 40th 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", 39th 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, 1996, "The Use of Petrophysical Data for Well Planning, Drilling Safety and Efficiency ", paper X in SPWLA 37th Annual Logging Symposium, June 16-19, 1996.

Holbrook, P W, 1995a, "The relationship between Porosity, Mineralogy and Effective Stress in Granular Sedimentary Rocks", paper AA in SPWLA 36th Annual Logging Symposium, June 26-29, 1995.

Holbrook, P W, D A Maggiori, & Rodney Hensley, 1995b, "Real-time Pore Pressure and Fracture Pressure Determination in All Sedimentary Lithologies",pp 215 - 222, SPE Formation Evaluation, December 1995 ( selected for the 1996 edition of the Pore Pressure and Fracture Gradients SPE reprint series)

Holbrook, P W, 1991, " Discussion of Modeling the relationships between sonic velocity, porosity, permeability, and shaliness in sand, shale, and shaley sand", Geophysics.

Holbrook, P W, 1990, "An Accurate Rock Mechanics approach to Pore Pressure/Fracture Gradient Prediction", pp.181-196, MWD Measurement While Drilling Symposium, Feb 26-27, LSU University, Baton Rouge, La. (selected for the SPWLA Reprint Series on MWD).

Holbrook, P W, 1989, "A New Method for Predicting Fracture Propagation Pressure from MWD or wireline log data", pp 475 - 487, SPE Annual Technical Conference, Oct 8-11, 1989 SPE 19566 Drilling.

Holbrook, P W, 1989, "Comments on, How Borehole Ballooning alters Drilling Response", The Oil & Gas Journal, June 12,1989, pp.44-45.

Holbrook, P W, 1989, "Pore Pressure and Fracture Pressure calculated from Rock Mechanics Principles", SPWLA Conference on Formation Evaluation by MWD, Kerrville, TX, Sept 10-14, 1989.

PW Holbrook, and M Hauck, 1987, "Petrophysical-Mechanical Math Model for real-time Wellsite Pore pressure/Fracture Gradient Prediction", SPE 16666, 62nd Annual Technical Conference and Exhibition of the Society of Petroleum Engineers held in Dallas, TX on Sep 27-30

Hauck, Michael L, PW Holbrook, and H. Robertson, 1986, "Quantitative Computer-Based Pore Pressure Determination from MWD data.", Houston Geotech Sept 30, 1986, 13p.

Holbrook, P W, 1985a, " A new method for estimating hydrocarbon saturation in shaley sands", paper GGG in SPWLA 26th Annual Logging Symposium, June 17 – 20 , 1985.

Holbrook, P W, 1985b, " The effect of mud filtrate invasion on the EWR log: a case history", paper NNN in SPWLA 26th Annual Logging Symposium, June 17 – 20 , 1985.

note; Only corporate internal publications were written between 1974 and 1984 for Gulf Science and Technology (now Chevron) and Exxon Production Research (now ExxonMobil).

Holbrook, P W, & E.G. Williams, 1973, " Geologic and Mineralogic Factors Controlling the Properties and Occurrence of Ladle Brick Clays", Special Publication I-73, The Pennsylvania State University, 89p.

Holbrook, P W, 1973, " Geologic and Mineralogic Factors Controlling the Properties and Occurrence of Ladle Brick Clays", Ph.D. Dissertation, The Pennsylvania State University, 318p.   RETURN TO TOP

Phil Holbrook Ph.D.

Holbrook, P W, 1999, "Method and Apparatus for Determining Sedimentary Rock Pore Pressure from Effective Stress Un-loading." U.S. Patent 5,965,810

Holbrook, P W, 1999, "Method and Apparatus for Determining Sedimentary Rock Pore Pressure from Effective Stress Un-loading." U.S. Patent 5,863,752

Holbrook, P W, and Mittal, S, 1994, "System and method for controlling drill bit usage and well plan", U.S. Patent 5,305,836

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

Holbrook, P W, H Robertson, and M Hauck, 1991, "Method for determining pore pressure and horizontal effective stress from overburden and effective vertical stress", U.S. Patent 4,981,037   RETURN TO TOP



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.   RETURN TO TOP


[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.   RETURN TO TOP


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.   RETURN TO TOP

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.   RETURN TO TOP

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.   RETURN TO TOP

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.   RETURN TO TOP


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.   RETURN TO TOP

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.   RETURN TO TOP


Figure 6) Diagrams showing principal stresses in normal fault and strike-slip tectonic settings. [Effective stress/ 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.   RETURN TO TOP

Mechanical equation synthesis for the earth’s sedimentary crust

Unified Matrix

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 & smax) 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.   RETURN TO TOP


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!   RETURN TO TOP

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.   RETURN TO TOP

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.

Goldberg, I. & B. Gurevich, 1998, "Porosity Estimation from P and S sonic log data using a semi-empirical velocity-porosity-clay model", SPWLA 39th Annual Logging Symposium, paper QQ.

Hashin, Z., & S. Shtrikman, 1963, "A variational approach to the theory of the elastic behavior of multiphase materials", J. Mech. Phys. Solids, vol. 11, pp. 127-140.

Hewitt, P.G.,2002, “Conceptual Physics”, 732p. ISBN 0-13-054265-2, Prentice-Hall, Needham, Massachusetts.

Holbrook, P W, 1996, "The Use of Petrophysical Data for Well Planning, Drilling Safety and Efficiency ", paper X in SPWLA 37th Annual Logging Symposium, June 16-19, 1996.

Holbrook, P W, 1995a, "The relationship between Porosity, Mineralogy and Effective Stress in Granular Sedimentary Rocks", paper AA in SPWLA 36th Annual Logging Symposium, June 26-29, 1995.

Holbrook, P.W. I.Goldberg & B. Gurevich, 1999b, “Velocity - Porosity - Mineralogy Gassmann coefficient mixing relationships for water saturated sedimentary rocks”. , paper T in SPWLA 40th Annual Logging Symposium, May –30 – June 3, 1999.

Holbrook, P W, 2000, “ How do Poisson’s Ratio and Plasticity relate to Fracture Pressure?”, World Oil , March, pp. 91-96.

Holbrook, P W, 2001, “Pore Pressure through Earth Mechanical Systems”. Force Balanced Publications, ISBN 0-9708083-0-5 138p, 33 figures.

Holbrook, P W, 2005, “Deterministic Earth Mechanical Science”. Force Balanced Publications, ISBN 0-9708083-1-3 106p, 22 figures.

Katahara, K. W., K.W. Lynch, and R.G. Keck, 1995, “A Semi-Empirical Model for In-Situ Stress Distribution for a Strike-Slip Regime: The Long Beach Unit, California”, SPE 29602, pp 581-592

Krief, M., Garat, J., Stellingwerf, J., and Ventre, J., 1990, "A petrophysical interpretation using the velocities of P and S waves (full waveform sonic), The Log Analyst, vol. 31, pp. 355-369.

Nagaraj, T.S., and Murthy, B.R.S., 1983, “Rationalization of Skempton’s compressibility equation,” Geophysique, Vol. 33, #4, pp 433 - 443.

Newton, Isaac, 1687, "The Mathematical Principles of Natural Philosophy", his famous latin titled (Philosphiae Naturalis Principia Mathematica)

Pilkington, P E, 1978, "Fracture Gradient Estimates in Tertiary Basins", Petr. Eng. International, May 1978, pp138-148.

P.R. Vail, R.M. Mitchum Jr., R.G. Todd, J.M. Widmier, S. Thompson, J.B. Sangree, J.N. Bubb, W.G.Hatlelid, 1977, pp 49-212 in Seismic Stratigraphy-applications to hydrocarbon exploration, edited by Charles E. Payton, ISBN: 0-89181-302-0    RETURN TO TOP

Pore Pressure Prediction and Detection in Deep Water;



General force balanced stress/strain physics in the Earth

Pore pressure (Pp) 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 (PP) is the difference between the two solid element scalars average confining load (Save) and average effective stress (s Ave) that is (PP = Save - s ave).

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

Forces are balanced in deep water settings, as they are everywhere else in the subsurface. The entire load is both generated and borne by the Earth’s matter. The Earth solid-fluid mechanical system is described using a closed form force balanced formulation that is an extension of Newtonian physics (Holbrook, 1998). Fluids are the pore pressure transmission system; and solids are the pore fluid pressure regulation and measurement system.   RETURN TO TOP

General mineralogic stress/strain loading for granular solids

The loading-limb stress/strain relationships are global in nature dependent principally upon mineralogic composition (Holbrook, 1995a). Minerals are the discrete solid load-bearing elements of the Earth. Each mineral has a stress/strain coefficient that has been measured in situ under Effective Stress Theorem force balance boundary conditions. These in situ static equilibrium mineralogic coefficients are related to the average bond strength of that mineral's crystalline lattice. The reversible thermal and elastic stress/strain properties of minerals were measured in laboratories decades ago (Carmichael, 1982).

Solidity (1. - f ) is a direct scalar measure of volumetric in situ strain for porous granular solids. Rocks pass into a purely elastic stress/strain regime at the zero porosity limit. During natural loading through burial compaction, sedimentary grains are brought closer together and contact area between grains is increased. The solid element load is borne at these grain contacts and through the mineral lattice to the neighboring grains. Under increasing loads elastic energy is accumulated in the mineral lattice in proportion to strain. Elastic strain is a miniscule fraction of total in situ strain that is dominantly plastic.

Also during natural loading, grain contact area is increased irreversibly following a plastic stress/strain relationship. The limit of plastic compaction is when all fluid filled porosity is gone and the rock is totally solid. Solidity (1.0 - porosity) is an absolute measure of volumetric in situ strain. Holbrook (1998) provides a physical explanation of the linked force balanced stress/strain relationships in the Earth.   RETURN TO TOP

Deep Water in situ Petrophysical Newtonian Force Balanced method to calculate Pore Pressure

Figure 1 shows three flowpaths to calculate water filled porosity from different petrophysical sensors. One minus porosity is absolute in situ strain. Each flowpath depends upon using the appropriate in situ density, conductivity, and elastic coefficients of the minerals and fluid that compose a sedimentary rock. Non-clay mineral coefficients are essentially constant at normal geothermal gradients. The coefficients for clay minerals and Sodium Chloride brines vary with prevailing PV/T and Salinity conditions. The grain density of average sedimentary clay minerals increases gradually with the regional geothermal gradient.

The resisitivity and sonic sensor flowpaths contain non-linear pore volume and shape coefficients that must be taken into account in order to calculate porosity. If these sensor porosity transforms are executed properly and there is no significant borehole wall damage, all three sensor paths should indicate the same in situ True Rock Porosity for water filed rocks. True Rock Porosity is central to many reservoir petrophysical considerations and is the central force balance strain consideration in figure 1.   RETURN TO TOP


Figure 1 Linkage of petrophysical sensor readings through mineralogically sensitive porosity transforms to the Newtonian closed form force balanced stress/strain relationship in deep water NFR » biaxial basins. The Extended Elastic and Grain-Matrix Compactional Mechanical Systems link on common definitions of mineralogy and porosity. These definitions represent conservation laws in both mechanical systems domains.

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 (Save) 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 (Pp) is calculated as the difference between the two load scalars [(S ave)-(s ave)]. Fracture propagation pressure (PF = Pp+ s h =S h ) is thereafter calculated in the sixth equation using pore pressure (Pp) calculated using the Effective Stress theorem. All the load terms (S, s h & Pp) 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.  

Deep Water basin Geologic correspondences that simplify Force Balanced Pore Pressure Prediction

There are only three free parameters required to calculate pore pressure in the Gulf of Mexico deep water setting; Initial overburden (Sv); a formation water conductivity vs. depth profile; and a dry clay mineral grain density profile. All of the rest of the parameters that are needed are mineralogic constants that are part of the closed-form mechanical systems model.

Seafloor water temperatures are near 4.0 degrees Celsius for water depths below 2000 feet. These waters are essentially normal saline with nearly constant electrical conductivity. Deep water off-shelf depositional environments have been cold water marine for millions of years. These sediments have low thermal conductivity, have been buried with cold seawater and heated very gradually from below. Water conductivity vs. depth profiles in the deep water off-shelf settings are low, gradual and uniform.

Initial overburden (Sv) 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 (Sv) and average effective stress (s ave) are present.

Both (Sv & 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.

Figure 2 shows the average formation water conductivity in the Gulf of Mexico 4000 feet below the mudline. Figure 3 is the conductivity increase 8000 feet below the mudline. The water conductivity data was derived from force balanced petrophysical logs at each of the well locations shown on the maps.

The water conductivity contour lines on each map are generally parallel to strike in the Gulf of Mexico basin. The conductivity gradients are more gradual in the colder off-shelf sediments. Decreasing geothermal gradients in the seaward direction are the dominant cause for the mapped patterns. Referring back to figure 1, the physical quantities needed to calculate pore pressure anywhere along a well depth profile can be calculated from mineral and fluid physical constants and these water conductivity profiles.

Deep Water Force Balanced Pore Pressure/ Fracture Pressure Prediction Method Conclusions
Newtonian closed form force balanced pore pressure prediction in deep water settings is mathematically simple and straightforward. The linking coefficients required for in situ strain calibrations are mineral and fluid physical properties. Two variable calibration parameters are needed, initial overburden and a regional water conductivity profile. These are 1.) physically representative, 2.) Can be determined through measurement; 3.) Are continuous between locations that; 4.) Can be estimated through regional mapping.

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 (Sv) constant and the local water conductivity (Cw) 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.  RETURN TO TOP


References cited

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

Holbrook, P W, 1995a, "The relationship between Porosity, Mineralogy and Effective Stress in Granular Sedimentary Rocks", paper AA in SPWLA 36th Annual Logging Symposium, June 26-29, 1995.

Holbrook, P W, D A Maggiori, & Rodney Hensley, 1995b, "Real-time Pore Pressure and Fracture Pressure Determination in All Sedimentary Lithologies", pp 215 - 222, SPE Formation Evaluation, December 1995

Holbrook, P W, 1996, "The Use of Petrophysical Data for Well Planning, Drilling Safety and Efficiency ", paper X in SPWLA 37th Annual Logging Symposium, June 16-19, 1996.

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.

Terzaghi, K. Van, 1923, "Die Berchnung der Durchassigkeitziffer des Tones aus dem Verlauf der Hydrodynamischen Spannungscheinungen", Sitzunzsber Akad Wiss. Wein Math Naturwiss, K1.ABTS 2a, pp. 107-122.   RETURN TO TOP



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

Phil Holbrook, Force Balanced Petrophysics, Houston,Texas

Adapted from Holbrook, P W, 1996, "The Use of Petrophysical Data for Well Planning, Drilling Safety and Efficiency ", paper X in SPWLA 37th Annual Logging Symposium, June 16-19, 1996 with permission.

The topic headings in the article below are book marked for easy access from here. If you would like to hyperlink to any one of these topics, simply click on that topic. Introduction | Stress/Strain Hysteresis | In Situ Loading Limb Stress Path | In Situ Petrophysical Data Rock Mechanics System | Comparison to Laboratory Rock Mechanics systems| Terzaghi Force Balance | Biaxial Stress Ratio (s h/s v) vs. Strain (1.0 - f ) Relationship | Fracture pressure accuracy criterion | Empirically Adjusted Elastic Theories |Unadjusted In Situ Stress/Strain Theory | In Situ Rock Mechanics Theory | Drilling Applications Significance | Log Calibration Example | The Safe Drilling Window | Conclusions | References Cited | (note the bookmarks and hyperlinks into the glossary of terms are not linked in this document. They are however linked in a CD version of the textbook-manual. CONTACT the author for information.  RETURN TO TOP


The two open borehole fluid pressure limits, pore pressure (Pp) and fracture gradient (Pf) 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 situborehole fluid pressure measurements. Using gravitational force balance (s v = Sv - Pp), 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 situcompactional 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 ( Pf = s h + Pp) 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 ( Pp, Pf , Sv, 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 (Sv) 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.  RETURN TO TOP

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.  RETURN TO TOP

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..  RETURN TO TOP


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.  RETURN TO TOP 


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 force balanced rock mechanics systems. Both force balanced 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 (Sv) & (Sh) are applied, and pore fluid pressure (Pp) 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.  RETURN TO TOP


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

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 (sv) & (sh) 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 rock mechanics system. Rocks vary considerably in porosity and mineralogy, but all rocks compacts to the same final (solidity=1.0) final end point regardless of composition. Consequently general quantitative compactional conclusions relating strain (1.0 - f ) to porosity and effective stress can be drawn using the in situ rock mechanics system(Holbrook, 1995).

The in situ rock mechanics system uses an effective stress / total in situ strain (1.0 - f ) relationship. Total in situ strain includes reversible (elastic) and geologically irreversible (plastic) strain as well as reversible thermal expansion.RETURN TO TOP 


Terzaghi Force Balance Figure 4 shows the equilibrium force balance which occurs in » biaxial Normal Fault Regime sedimentary basins. The maximum effective stress (sv) is vertical resulting from overburden (Sv) in NFR basins. Effective vertical stress (sv) is calculated as the static equilibrium force balance difference between the gravitational overburden (Sv) load minus pore

sv = Sv - Pp (1)

Effective vertical stress (sv) has been related to gravitational compaction at maximum loading. Holbrook (1995) demonstrated a power law linear effective stress (sv) / total in situ strain(1 - f ) relationship for single mineral and mixed mineralogy sedimentary rocks. Panel 3 of figure 4 shows the effective stress compactional relationships for the 5 most common single mineral sedimentary rocks.

The compaction of the 5 common sedimentary minerals and mixtures thereov in NFR » biaxial basins can be expressed with the same effective vertical stress (sv) / total in situ compactional strain(1 - f )relationship;

sv = s max( 1.0 - f )a (2)

The two in situ stress/strain coefficients s max and a are compaction resistance sedimentary rock properties. They are calculated from the weighted average mineralogic composition of the sedimentary rock. This mineralogic loading limb stress/solidity relationship has been extensively tested through pore pressure determination in over 200 wells in NFR » biaxial basins worldwide. Table 1 in (Holbrook, 1995) lists the single mineral compactional constants displayed on the figure. Table 1 also shows mineral hardness and solubility which also affect geologic compaction.  RETURN TO TOP 


Biaxial Stress Ratio (sh/sv) vs. Strain (1.0 - f ) Relationship In NFR » biaxial basins; the minimum principal stress (sh) increases with vertical stress (sv) and depth. Figure 5a taken from Pilkington (1978) shows four average horizontal/vertical effective stress ratio vs. depth curves from four different authors. Minimum horizontal stress (sh) in shales was measured from in situ leakoff test data. Fracture propagation pressure (Pf );

Pf = sh + Pp (3)

is the minimum borehole fluid pressures required to extend a pre-existing tensile fracture which is perpendicular to the minimum principal stress (Sh). Each of the 4 studies shows that effective stress ratio (sh/sv) increases from about 0.3 at the mudline to about 0.9 or greater below 15000 feet.

Figure 5b shows the average strain (1.0 - f ) vs. depth compaction curve for average Gulf Coast slightly silty shales determined by Bryant (1980). The shale grain density increased from 2.64 g/cc at the mudline to 2.69 g/cc at 16000 feet. This grain density and solidity data was used to generate a corresponding overburden vs. depth function. The curve plotted with the Bryant (1980) data is a force balanced power law linear effective stress/in situ strain function. The compactional in situ strain (1.0 - f ) vs. depth relationships on figure 5b also increase from about 0.3 at the mudline to 0.9 or greater below 15000 feet.

The compactional in situ strain (1.0 - f ) vs. depth function (figure 5b) is very much like the leakoff test empirical stress ratio (sh/sv) vs. depth functions (figure 5a). This apparent in situ stress ratio vs. in situ strain relationship could reasonably be causal. The studies shown on figure 5a used different overburden functions some of which were not very realistic. If these curves were properly effective stress normalized, stress ratio (5a) and strain (5b) might vary even more closely than they appear to on this figure. Figure 5 suggests that stress ratio (sh/sv) could be predicted directly from in situ strain (1.0 - f ) rather than depth. Direct stress from strain predictions would be mechanically more satisfying and could also be more accurate.   RETURN TO TOP


Fracture pressure accuracy criterion In NFR » biaxial basins; the increase minimum principal stress (s h) with depth is probably related to vertical stress (s v) through a natural stress/strain relationship. Equation 4 and equation 5 below are two possible mathematical expressions supported by two different stress/strain theories. Each theory will be tested below by comparison with leakoff test measured in situ stress data.

A statistical study was made comparing the accuracy of fracture pressure prediction theories. Depth predicted fracture pressures from figure 5a are shown on figure 6a. These showed a rather high standard deviation of 1.13 pounds/gallon from the empirically best fit depth curve (Holbrook, 1989).  RETURN TO TOP

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Fracture pressures where (s v) came from equation 2 and minimum horizontal stress (s h) was calculated directly from the effective stress ratio vs. in situ strain ( 1.0 - f ) relationship;

s h / s v = (1.0 - f) (4)

without reference to depth are shown on figure 5b. Figure 6b shows a set of leakoff tests vs. that predicted directly from in situ strain for the dataset in (Holbrook, 1989). These had a much smaller standard deviation of 0.413 ppg. The 0.413 ppg standard deviation was measured from the equation 4 fracture propagation pressure relationship which is an expected minimum; not the best fit average curve as was figure 6a. Without accounting for this statistical handicap, fracture pressure values calculated directly from in situ strain ( 1.0 - f ) are conservatively 3 times more accurate than those predicted from depth (figure 6a). Allowing for the statistical handicap, the direct effective stress ratio vs. in situ strain method is probably about 8 times more accurate than the empirical depth methods.  RETURN TO TOP

The most likely explanation for the greatly improved accuracy of the direct effective stress vs. in situ strain fracture pressure (Pf) method is that equations 1, 2, 3 and 4 are compactional strain and force balance linked. Overburden (Sv), pore pressure (Pp), vertical stress (sv), horizontal stress (sh), in situ compactional strain ( 1.0 - f ), and average sedimentary rock mineralogy are linked in their calculation as they are in the subsurface.

The fracture pressure vs. depth methods lump each of the above variables into a single regional average function of depth. Each real deviation of a physical parameter from the regional depth function contributes to the error of that function. The regional average depth methods clearly demonstrate consistent regional trends but are relatively poor predictors of fracture pressure as their high standard deviation indicates.

Four separate statistical studies with different but similar size datasets have been performed comparing leakoff tests to fracture pressures using the direct effective stress vs. total in situ strain relationship. Each of the post 1989 statistical studies had a standard deviation of about 0.5 ppg. Accuracy has since been consistently high and is now actually better after 7 years of use.

Empirically Adjusted Elastic Theories Fracture pressure is known to be generally lower in porous sandstones in NFR » biaxial basins. Warpinski & Teufel (1989) have demonstrated from in situ minifrac measurements that nonmarine sandstones exert lower horizontal stress than adjacent shales. The lithology related differences in leakoff and minifrac tests have been explained using elastic or poroelastic theory (Anderson et al 1973, Watt & Dvorkin 1994). Under biaxial stress conditions assuming entirely recoverable linear elastic strain; Poisson's ratio (n ) relates maximum vertical stress (sv) to confining horizontal (sh) stress following the relationship;

sh = sv x n /( 1.0 - n ) (5)

Shales and limestones normally have higher Poisson's ratios than porous sandstones. Over short stratigraphic intervals empirically adjusted poroelastic equations can generate a reasonable looking continuous fracture pressure log (Anderson et al 1973). Reasonableness in log appearance is achieved through an empirical scaling term (a ) which is an empirical Biot poroelastic multiplier. The empirical fit is between the poroelastic coefficient (a ) and the total set of in situ leakoff tests. There was no data left out to actually test the poroelastic theory. The authors expressed some concern that the Poisson's ratio used in the poroelastic equation did not match the Poisson's ratio of the in situ sedimentary rocks. The empirical (a ) multiplier must be adjusted frequently to preserve the match between poroelastic theory calculated horizontal stress vs. leakoff test observed.

Eaton (1969) derived an apparent Poisson's ratio (na) depth function to be substituted into equation 5 so that horizontal stresses measured in leakoff tests would match the assumed elastic theory. The leakoff test apparent Poisson's ratio (n a) increases from about 0.2 at the mudline to 0.5 at 20,000 feet. Eaton's (1969) apparent Poisson's ratio depth function inserted into equation 5 yields the Eaton stress ratio (sh/sv) shale vs. depth function shown on figure 5a.  RETURN TO TOP

Eaton's (1969) apparent Poisson's ratio (na ) shows a trend which is the reverse of that expected from laboratory measurements on rocks of decreasing porosity with depth. Poisson's ratio (n ) decreases with porosity in all sedimentary rocks. Representative measured Poisson's ratio (n ) values for mudstones decrease from about 0.45 at the mudline to about 0.3 at 20000 feet. If actual Poisson's ratios (n ) were inserted into equation 5 over the 20000 foot depth range, leakoff tests would be much higher than observed at the mudline, and much lower than observed at 20000 feet.

The elastic assumption correction factor is a multiplier to the value of the Poisson term (n /(1.0-n )) in order to match the leakoff test measures effective horizontal stress (sh). The correction multiplier required ranges from 0.37 at the mudline to 2.33 at 20000 feet. This elastic assumption correction factor is about 3 times larger than the Poisson's ratio ((n /1.0-n )) term which it is intended to correct. The fact that the correction factor decreases the effective horizontal stress (sh) by about 1/3 at the mudline and increases (sh) by more than 2 at 20000 feet should be cause for concern. The empirical correction factor not only adjusts but reverses the expected outcome when using representative shale Poisson's ratios (n ) with known (equation 5) elastic stress/strain theory. In both cases cited above, elastic or poroelastic theory was assumed and not verified by comparison to in situ rock Poisson's ratios (n ).

Unadjusted In Situ Stress/Strain Theory There is an equally plausible in situ hypothesis for » biaxial NFR basins. effective horizontal stress (sh) could also arise from a dominantly plastic loading limb stress/ in situ strain relationship. Figure 5 showed that the depth trends of in situ effective stress ratio (sh/sv) and in situ silty shale strain ( 1.0 - f ) vary closely over the entire depth range. No adjustment was made to Bryant's in situ strain vs. depth function (figure 5b) to match the in situ effective stress ratio (sh/sv) vs. depth functions (figure 5a).

Additionally, no adjustments were made to to match any of the leakoff tests shown on figure 6b which is a different dataset. These most accurate predictions were made directly from petrophysically measured in situ strain ( 1.0 - f ) without respect to depth.  RETURN TO TOP



Figure 7 is a log showing a continuous fracture pressure trace in track 3. This log trace was made using the in situ rock mechanics system with no empirical adjustments. The 17.8 ppg leakoff test shown at 14060 feet in a shale was measured in 4 different wells from a drilling platform at this preferred casing depth. The 16.8 ppg leakoff test in the overlying sandstone at 14020 feet was measured when on one occasion casing got stuck at a shallower than intended depth. Five repeat cement squeezes were followed by five leakoff tests all measuring the same 16.8 ppg value in the sandstone. Repeated borehole fluid pressure measurements suggest that horizontal stress is lower by 1.0 ppg in the sandstone than in the shale.

The continuous log fracture propagation pressure (Pf) trace in figure 7 shows the same 1.0 ppg offset from 14020 feet to 14060 feet. Using similar continuous log examples, the accuracy of unadjusted direct strain fracture pressures is much better than the adjusted elastic and poroelastic relationships (Anderson et al,1973) described in the previous section.

The unadjusted direct effective (h/v) stress ratio vs. strain ( 1.0 - f ) relationship matches leakoff test data very well when viewed as a continuous log (figure 7) where lithology is varying. It matches very well when compared as a shale depth function (figure 5a & b). Ignoring depth and without adjustment it matches even better as a direct in situ stress/stain function (figure 6b & figure 7). Greater accuracy without the need for empirical adjustment favor the effective stress vs. total in situ strain relationship as a predictor of horizontal stress and fracture pressureRETURN TO TOP

In Situ Rock Mechanics Theory
AStatic force balance (equation 1) is a reasonable assumption for an in situ sedimentary rock which has been at it's maximum loading point for thousands to millions of years. After accounting for force balance in NFR » biaxial basins; the compactional loading limb effective vertical stress (sv) vs. volumetric strain ( 1.0 - f ) relationship was found to be a simple power law function of mineralogy (equation 2). The plastic compactional coefficients a and s max used with equation 2 are mineralogic constants. All the effective stress (s ave) load is borne by mineral ionic bonds. Equation 2 with the mineralogically weighted average of a and s max loading limb compaction coefficients states quantitatively that sedimentary rocks composed of harder less soluble minerals compact less for the same effective stress load.

Fracture propagation pressure (equation 3) is the horizontal stress static force balance. In NFR » biaxial basins; the relative in situ effective stress ratio (sh/sv) also varies very closely with in situ volumetric strain ( 1.0 - f ). The very high accuracy of equation 4 at predicting both the average fracture pressure trend of shales (figure 5 and 6) as well as local mineralogically related fracture pressure variability (figure 7) without any empirical adjustment is another important factor that should be accounted for in a general in situ rock mechanics stress/strain theory.

The in situ loading limb compactional relationships lead to two material properties related conclusions. Mineralogy and stress control in situ volumetric strain (equation 2). The loading limb in situ stress/strain relationship is dominantly plastic in all sedimentary rocks regardless of mineralogy (equation 4). These two effective stress relationships are linked to the same in situ strain. Pore pressure and fracture pressure are linked to them by static force balance relationships. These four equations have not been proposed before to explain a general in situ stress/strain relationship. However they are a reasonable unifying hypothesis based on in situ rock properties honoring static force balance with considerable predictive power.

The unifying hypothesis honors both static force balance and the average effective stress theorem (Carroll, 1980). The latter is a grain matrix force balance which states that the grain matrix bears the average effective stress load (s ave) hydrostatically just like the pore fluid (Pp). In » biaxialNFR basins, the relative horizontal load (sh/sv) on the grain matrix is apparently directly proportional to in situ strain ( 1.0 - f ) over the entire depth range for all minerals. At the solidity = 1.0 upper strain limit of equation 4, (sh = sv) which honors the effective stress theorem and is a functional definition of an ideal plastic. The zero lower strain limit of equation 4 is never encountered by granular solids in nature. The closest physical approximation to the limit is a montmorillonite gel with solidity as low as 0.05. This physical lower limit is also an ideal plastic.

At geologic loading rates, the grain matrix can be considered as a plastic solid system within a continuous fluid system. Honoring the effective stress theorem, the plastic solid and the fluid each bears it's portion of the total external load. Mineral hardness plays the role of a viscosity term in the compaction of the plastic solid which is immersed in a partial load bearing fluid. Grain matrix compaction ceases and holds in this fluid immersed plastic solid as long as the mineral ionic bonds can support the remaining average effective stress portion of the total load. Figure 2a & b showed highly irregular quartz and mica grain contacts that appear to have flowed plastically into one another. The hardest most compaction resistant sedimentary mineral quartz is apparently plastic after geologic loading .RETURN TO TOP

Drilling Applications Significance
Greatly improved accuracy of calculated fracture pressure is a major benefit. Much greater detail and accuracy of pore pressure and fracture pressure predictions is possible as these values are calculated from measured in situ petrophysical data at each foot accounting for lithologic variability.

The in situ rock mechanics system is bounded by force balance. Overburden (Sv) is physical fluid pressure and stress limit in NFR basins which can be calibrated in several ways as will be shown below. The in situ rock mechanics system allows several types of absolute and relative borehole fluid pressure measurements to be used simultaneously with the same calibration effect. The following log example demonstrates most of these features showing how petrophysical measurements can be properly calibrated to all borehole fluid pressure measurements (Pp, Pf, & Pmax = Sv) simultaneously.


Log Calibration Example Figure 8 is a PP/FG log from a deep water well in the Gulf of Mexico. The four traces shown are from left to right 1. Pore pressure gradient ( Pp = Sv - sv), 2. Mud weight (Pb), 3. Fracture Propagation Pressure gradient (Pf = sh + Pp), and 4. Overburden gradient (Sv). The depth scale is feet and the horizontal scale is pounds/gallon equivalent mud weight. Casing was set at 6986 feet slightly above the log interval shown. A leakoff test measured 14.1 ppg EMW in the ten feet of open hole immediately below that casing. There is some missing and unrepresentative petrophysical data down to 7600 feet caused by a severe hole washout. The hole is approximately in gauge below this. If one projects the slope of the fracture Propagation Pressure trace between 7600 feet and 8600 feet up to 6986 feet, the petrophysically predicted equation 3 fracture Propagation Pressure would be 14.1 ppg.

Casing was again set at 10608 feet and a leakoff test made. A peak pressure of 16.77 ppg was measured followed by a pressure bleedoff down to 15.7 ppg where it held. The leakoff pressure stress/strain diagram was like that of the graph inset into figure 8. The inset bar at 10608 shows these two values with respect to the Overburden gradient (Sv) and fracture propagation pressure (Pf ) traces. The in situ petrophysically calculated fracture propagation pressure is 15.7 ppg at that depth and overburden was 16.5 ppg. At this depth measured vs calculated fracture pressure is an exact match and the peak pressure was 1.6% higher than overburden gradient.

Overburden (Sv) is the force balance borehole fluid pressure maximum upper limit in NFR basins (Pmax = Sv ). If a single sub-horizontal bedding plane fracture exists in the short open borehole below casing, borehole fluid would enter that fracture at Overburden. The 1.6% deviation from (Pmax = Sv ) is a very small acceptable deviation of calculated vs. borehole fluid pressure measured Overburden in this case.

Calculated fracture propagation pressure was 15.4 ppg in a sandstone at 10900 feet as shown on figure 8. Later in the drilling of this well, mud weight was raised from 15.0 to 15.4 ppg at 13080 feet. Circulation was lost at 15.4 ppg which cost 3 days rig time to recover. Although not intended, this is a third exact match of petrophysically calculated fracture pressure with measured borehole fluid pressure. All 3 of these (Pf) matches as well as (Pmax = Sv ) were achieved without any adjustments to petrophysical or force balance parameters.

Repeat formation tests are direct measurements of fluid pressure in some reservoir sands from 12830 to 12880 feet. These measurements ranged from 12.59 ppg to 12.67 ppg. The petrophysically calculated pore pressure in that depth range is 12.6 ppg as shown on figure 8. This is an exact match to measured pore pressure (Pp) which is a different force balance variable in the same unadjusted in situ rock mechanics system.

All five borehole fluid pressure measurements mentioned thusfar constitute different matches to different petrophysically calculated values in the same four equation in situ rock mechanics system. The system linkage gives equal weight to each measurement and offers the possibility to use all three exact bore hole fluid pressure measurements (Pp, Pf, & Pmax = Sv) simultaneously in the same calibration. This was done in this example case and could be done in any other well where the in situ borehole fluid pressure measurements are available.

Relative borehole fluid pressure calibration between mud weight (Pb) and calculated pore fluid pressure (Pp) is always available and covers a much broader depth range. For this relative calibration compare the mud weight trace to the pore fluid pressure trace on figure 8. Over the entire log interval, Mud weight (Pb) is at least 0.2 ppg higher than calculated pore fluid pressure (Pp) and the well never flowed. This is a relative (flow vs no flow) pressure calibration. It is an upper limit rather than an exact match calibration. It is a useful and equally valid calibration compared to the five exact calibration points mentioned above. There are at least forty highly permeable intervals in this well which would flow instantly if pore pressure (Pp) were underbalanced by mud weight (Pb). Here again measurement matches observation using the same in situ rock mechanics system with no adjustment.

Connection gas provided pore pressure calibration in the depth range below 12600 feet. In this range mud weight (Pb) was raised in small steps in response to increases in connection gas. Two factors contribute to the driller's success in this depth range. First there were low levels of gas in the formations at this depth range to provide the connection gas signal. The drillers reacted to this connection gas signal in a timely fashion and kept the well slightly overbalanced. A stratigraphic interval with these characteristics is very good for calibrating absolute pore fluid pressure (Pp) from relative borehole fluid pressure (Pb).

There is an unseen but very important aspect to calibrating the in situ rock mechanics system. The same rock porosity is used to calculate effective stresses and overburden. If there is a systematic error in the porosity calculation, it will eventually affect the pressures and stresses below. The porosity at any depth affects (p & Pf) directly. That same porosity value integrated with grain and fluid densities over depth affects calculated (Sv). The in situ rock mechanics stress/strain linkage also applies to petrophysical data over depth.

Porosity evaluation and interpretation are best handled by a petrophysicist who is familiar with the formations and the nuances of log interpretation. The other important petrophysical parameter to enter the in situ rock mechanics system is shale volume. This too can involve petrophysical judgment and depend on outside information.RETURN TO TOP

The Safe Drilling Window The safe drilling window is the range of allowable mud weights which a driller can confidently use in a given open hole interval. The lower limit is the pore pressure profile plus a reasonable level of uncertainty. The upper limit is the fracture propagation pressure minus a level of uncertainty associated with that measurement. The standard deviation of observed vs. predicted values is a measure of that uncertainty.


Figure 9 The inset panel to the right of the well planning log demonstrating the effect of greater accuracy on calculated pore pressure and fracture pressure. Greater statistical accuracy on pore pressure and fracture pressure widens the safe drilling window. The solid distribution curves are from the statistically more accurate force balanced method for calculating pore pressure and fracture pressure. The dashed distribution curves show the less accurate depth function methods.

Given the same average expected profile for pore pressure and fracture propagation pressure, a lower standard deviation on either measurement widens the safe drilling window by that difference. A statistically more accurate method will have lower uncertainty and a wider safe drilling window. For example if the 1.13 ppg standard deviation of fracture pressure calculated from empirical depth functions (figure 5a) were replaced by the 0.413 ppg standard deviation of fracture pressure using only in situ strain (figure 5b) the safe drilling window would be on average 0.72 ppg wider. Looking at the scales on figure 8 and figure 9, this ~0.7 ppg widening of the safe drilling window would have a large impact on well planning casing design. Frequently a widening such as this could add 400 to 1000 feet to the length of an individual casing run.

If a casing depth is set strategically, for example in the ramp at the bottom of figure 9, one casing run can often be eliminated by taking advantage of the increased fracture pressure that is a consequence of the ramp. This could be termed geologically informed well planning.

A new rock mechanics system that is shown above has been developed from and for use with downhole petrophysical data. It is a set of seven linked equations applicable to » biaxial Normal Fault Regime basins. These force, load, and mass conservation equations operate simultaneously in the earth . The relationship between in situ porosity, mineralogy and effective vertical stress (sv) was documented in (Holbrook, 1995). The correspondence between the same parameters and effective horizontal stress (sh) is documented in this paper. A general theory which simultaneously explains both the observed mineralogic in situ stress/strain relationships is also presented herein. This theory is more thoroughly developed in a quantum step in the history of natural science.

The in situ rock mechanics system generates force balance linked continuous log traces of pore pressure (Pp), fracture propagation pressure (Pf), and Overburden (Sv) from wireline or MWD petrophysical data. These can be calibrated separately or simultaneously to three corresponding borehole fluid pressure measurements; leakoff tests and lost circulation zones.

Fracture pressures calculated within the new in situ rock mechanics system are several times more accurate than older depth trend methods. Greater accuracy and force balance linkage to in situ petrophysical data provide a more accurate and consequently wider safe drilling window for well planning applications. Once petrophysical calibration is accomplished on an offset well; the same rock and fluid properties constants applied with MWD petrophysical data provide real-time pore pressure and fracture gradient continuous logs from near bit in situ petrophysical data to avoid drilling trouble. RETURN TO TOP

References Cited
Anderson, R A, D S Ingram & A M Zanier, 1973, "Determining Fracture pressure Gradients from Well Logs", JPT (Nov 1973) p1259

Bjorkum, P A, 1996, "How important is pressure in causing dissolution of Quartz in Sandstones?", pp 147-154, Journal of Sedimentary Research, vol 66 #1.

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.

Carroll, M M, 1980, "Compaction of Dry or Fluid-filled Porous Materials", Journal of Engineering Mechanics Devision, Proceedings of the American Society of Civil Engineers, Vol. 106, No EM5, Oct 1980 pp969 - 990.

Eaton, B A, 1969, "Fracture Gradient Prediction and its Application in Oilfield Operations", JPT (Oct. 1969) pp 1353- 1360

Holbrook, P W, 1989, "A new method for predicting Fracture Propagation Pressure from MWD or wireline Log Data", SPE 19566 Drilling pp475 - 487.

Holbrook, P W, 1995, "The relationship between Porosity, Mineralogy and Effective Stress in Granular Sedimentary Rocks", paper AA in SPWLA 36th Annual Logging Symposium, June 26-29, 1995.

Holbrook, P W, D A Maggiori, & Rodney Hensley, 1995, "Real-time Pore Pressure and Fracture Pressure Determination in All Sedimentary Lithologies",pp 215 - 222, SPE Formation Evaluation, December 1995

Milliken, K L, 1994, "The widespread occurrence of healed microfractures in siliciclastic rocks: Evidence from scanned cathodoluminescence imaging", pp 825-832. in Nelson & Laubach (eds) Rock Mechanics Models and Measurement Challenges from Industry; Proceedings of the 1st North American Rock Mechanics Symposium.

Pilkington, P E, 1978, "Fracture Gradient Estimates in Tertiary Basins", Petr. Eng. International, May 1978, pp138-148.

Skempton, A. W., 1970, "The consolidation of clays by gravitational compaction", Quarterly Journal of the Geologic Society of London; vol 125, pp 373-411, 22 figures.

Taylor, E & J Leonard, 1990, "Sediment consolidation and permeability at the Barbados forearc", in Moore, J C, and Mascle A, et al, 1990, Proceedings of the Ocean Drilling Program, Scientific Results, Vol. 110

Walls, J D & J Dvorkin, 1994, "Measured and Calculated Horizontal Stresses in the Travis Peak Formation", SPE Formation Evaluation (Dec. 1994) pp 259-263.

Warpinski & Teufel L 1989, "In situstress in low permeability non-marine rocks", JPT (April p405) RETURN TO TOP