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 hyperlinked
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 forcebalanced engineering were circumvented by
these older (P=Pn+DP)methods.
A physically forcebalanced 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=Ss)
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 [solidfluid] mechanical
systems. Subsurface science and engineering are coupled
as are [massenergy]
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: 0970808313 , 104 pages. This book explains
the earth's macromechanics beyond the Normal Fault
Regime. It also explains the deterministic relationships
between macromechanics 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 2131, 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 unloading
pore pressure regimes and applying them in well planning
and realtime drilling", pp. 145157, 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: 0970808305, 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 macromechanics
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. 3742. 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 56 London,
U.K.
Holbrook, P W, 2000, "How
do Poisson’s Ratio and Plasticity relate to Fracture
Pressure?", World Oil,
March, pp. 9196. 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 1215, 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 closedform force«
balanced solution for Pore pressure, Overburden and
the principal Effective stresses in the Earth.",
Journal of Marine and Petroleum Geology,
Vol. 16, pp. 303319.
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 78 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 unloading pore
pressure regimes and applying them in well planning
and realtime 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 SemiEmpirical VelocityPorosityClay
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.7172
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 910 RuileMalmaisonFrance
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 1619, 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 2629, 1995.
Holbrook, P W, D A Maggiori, & Rodney Hensley, 1995b,
"Realtime 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.181196, MWD Measurement While Drilling Symposium,
Feb 2627, 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 811, 1989 SPE 19566 Drilling.
Holbrook, P W, 1989, "Comments on, How Borehole
Ballooning alters Drilling Response", The
Oil & Gas Journal, June 12,1989, pp.4445.
Holbrook, P W, 1989, "Pore Pressure and Fracture
Pressure calculated from Rock Mechanics Principles",
SPWLA Conference on Formation Evaluation by
MWD, Kerrville, TX, Sept 1014, 1989.
PW Holbrook, and M Hauck, 1987, "PetrophysicalMechanical
Math Model for realtime Wellsite Pore pressure/Fracture
Gradient Prediction", SPE 16666, 62^{nd}
Annual Technical Conference and Exhibition of the Society
of Petroleum Engineers held in Dallas, TX on
Sep 2730
Hauck, Michael L, PW Holbrook, and H. Robertson, 1986,
"Quantitative ComputerBased 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 26^{th} 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 I73, 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.
patents
Holbrook, P W, 1999, "Method and Apparatus for
Determining Sedimentary Rock Pore Pressure from Effective
Stress Unloading." U.S. Patent 5,965,810
Holbrook, P W, 1999, "Method and Apparatus for
Determining Sedimentary Rock Pore Pressure from Effective
Stress Unloading." 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
__________________________________________________
Abstract
Deterministic earth mechanics is a subset of Universal
Physics. The boundary condition is the earth’s
surface. [Massenergy] 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 twophase 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 closedform mechanical system. [Massenergy] 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
Introduction
[Massenergy] conservation is common to the known physics
of the universe. The mechanics in the earth’s
crust unites [massenergy] 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 [massenergy]. 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 Vp^{2}
axis of figure 1. The weighted average of the
minerals that compose the grain matrix resolve to another
end point on the Vp^{2} vs.
Vs^{2} plane. Rock and fluid
moduli are end points of this lever which is linear
on Vp^{2} vs. Vs^{2}
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 graingrain
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 HashinSchtrictman 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
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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 waterwet clay values are close to a “clay
point” as on figure 1. Vp^{2} and Vs^{2}
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 HashinSchtrikman (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 semisolid 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 graingrain contact between 45 and 35% porosity.
At lower porosities their response is quasilinear on
the Vp^{2} vs. Vs^{2} plane.
RETURN
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Plastic
compaction of natural mineralfluid 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 powerlaw 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 nonlinear. 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 subparallel [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 interparticle
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 powerlaw
proportional to clay mineral particle size (Nagaraj
& Murthy, 1983).
Compaction resistance is the sum of inter and intraparticle
repulsions (Holbrook, P.W., 2001). Both depend directly
on Coulomb’s law. Clays have additional interparticle
repulsion. Interparticle repulsion is responsible for
the high (90 to 95%) porosity of freshly deposited fine
clay. Particles with net neutral surface charge have
no net interparticle 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 [massenergy] [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 10^{24} 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 , [massenergy]
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 threedimensional 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 forcedriven 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 sidebyside 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 powerlaw [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 powerlaw 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. Grainmatrix 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 powerlaw linear.
The average principal
stress (orange)
is powerlaw proportional to strain
(brown).
This corresponds to[massenergy] conservation as it
occurs in the earth. The (horizontal/vertical) stress
ratio is exactly equal to strain
(1.0f)
as shown on figure 4. This is the minimumenergystate
for the normal fault tectonic regime. RETURN
TO TOP Differential
energy minimization in the strikeslip tectonic regimes
Gravity is the intermediate principal stress in strikeslip
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
strikeslip 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 26foot 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. Minifrac 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 minifrac date but the interpretive
line is closely related to claystone minima on this
plot.
Minifrac 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.
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Figure
6) Diagrams showing principal stresses in normal fault
and strikeslip 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 [massenergy] 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 10^{39} stronger than gravitational.
Earth gravitational forces are strong but don’t
approach the strength of electrostatic forces.
The forcebalanced [stress/strain] relationship has
closed upon itself. This relationship is like the other
[massenergy] relationships are verified physical laws.
This equation synthesis is composed only of physical
laws. [Massenergy] 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
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.) solidmassconserved strain and 2.) mineralfluid
partitioning. The common partitioning coefficient is
the backbone of the [massenergy] [stress/strain] synthesis.
The brown arrow on figure 7 highlights mathematical
simultaneity. Solid mass conservation is the basis for
the simultaneous solution of physical and conservation
laws.
Newton’s gravitational law is the second equation.
It is partitioned according to discrete mineral and
density coefficients. This is the mass attraction between
each atom and the earth’s center of mass. The
mineral and fluid density coefficients were determined
empirically and can be found in physical properties
handbooks (Carmichael, R.S., 1982).
Fluids and gas density coefficients were empirically
derived and are explained by equations of state (Archer,
1990). Thus Newton’s gravitational law is directly
correlated to the physical properties of the mineral
and fluid phases in the earth’s sedimentary crust.
The third equation expresses scalar energy conservation
of the minerals in a sedimentary rock. The mineral coefficients
(a & s_{max})
were derived empirically from the natural compactional
relationships shown on figure 2. Each single mineral
compactional [stress/strain] relationship is a powerlaw
function.
A whole rock powerlaw function is an average of its
single mineral powerlaw 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 minimizeddifferentialstress
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 [massenergy] 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 strikeslip 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 strikeslip 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 physicalmathematical <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
Conclusions
The mechanics of the earth’s sedimentary crust
is explained through a synthesis of universal physical
laws. The synthesis predicts additional mathematical
symmetry which has been empirically verified. It appears
a minimum differential stress condition dominates in
the earths subsurface as it does the rest of the universe.
Natural forces drive the earths two phase mechanical
system from higher to lower energy. This is just like
previously known mechanical systems.
Newton’s law and Coulomb’s law describe
symmetrically conserved energy. Hooke’s law describes
[massenergy] conservation in terms of elastic [stress/strain]
definitions. Solid mass conservation bounds compactional
grainmatrix 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 interparticle repulsion into
both elastic and grainmatrix compaction is also a significant
contributing step. Inter and intraparticle repulsion
is total solid energy conservation. Coulomb’s
law explains clay’s role in elastic and plastic
endmember mechanical systems. It simultaneously explains
nonclay 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. [Massenergy]
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 forcedfit 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.
[Massenergy] 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 [massenergy] 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. [Massenergy]
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 codependant
[massenergy] conserved relationships. Reservoir compaction
falls within the elastic and grainmatrix 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 [massenergy]
conserved physics.
Oilfield earth predictive needs have historically been
filled by forcedfit 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. 793829.
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, p122.
Goldberg, I. & B. Gurevich, 1998, "Porosity
Estimation from P and S sonic log data using a semiempirical
velocityporosityclay model", SPWLA 39^{th}
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.
127140.
Hewitt, P.G.,2002, “Conceptual Physics”,
732p. ISBN 0130542652, PrenticeHall, Needham, Massachusetts.
Holbrook, P W, 1996, "The Use of Petrophysical
Data for Well Planning, Drilling Safety and Efficiency
", paper X in SPWLA 37^{th} Annual Logging
Symposium, June 1619, 1996.
Holbrook, P W, 1995a, "The relationship between
Porosity, Mineralogy and Effective Stress in Granular
Sedimentary Rocks", paper AA in SPWLA 36^{th}
Annual Logging Symposium, June 2629, 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 40^{th} 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. 9196.
Holbrook, P W, 2001, “Pore Pressure through Earth
Mechanical Systems”. Force Balanced Publications,
ISBN 0970808305 138p, 33 figures.
Holbrook, P W, 2005, “Deterministic Earth Mechanical
Science”. Force Balanced Publications, ISBN 0970808313
106p, 22 figures.
Katahara, K. W., K.W. Lynch, and R.G. Keck, 1995, “A
SemiEmpirical Model for InSitu Stress Distribution
for a StrikeSlip Regime: The Long Beach Unit, California”,
SPE 29602, pp 581592
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. 355369.
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, pp138148.
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 49212 in Seismic Stratigraphyapplications
to hydrocarbon exploration, edited by Charles E. Payton,
ISBN: 0891813020 RETURN
TO TOP
Pore
Pressure Prediction and Detection in Deep Water;
RETURN
TO TOP
General force
balanced stress/strain physics in the Earth
Pore pressure (P_{p}) is the fluid loadbearing
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 physicalmathematical
expression for porous granular solids that compose the
Earth’s sedimentary crust. In this rigorous physical
expression; the fluid scalar pore pressure (P_{P})
is the difference between the two solid element scalars
average confining load (S_{ave}) and average
effective stress (s _{Ave}) that is (P_{P}
= S_{ave}  s _{ave}).
By coincidence, the Terzaghi (1923) mixed scalarvector
relationship (P_{p} = S_{v}  s _{v})
works in most Normal Fault Regime basin deep water settings.
Terzaghi found empirically that in shallow marine sediments
the scalar pore pressure (P_{p}) was related
to the axial load vector, overburden (S_{v}).
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 solidfluid 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 loadinglimb stress/strain relationships are global
in nature dependent principally upon mineralogic composition
(Holbrook, 1995a). Minerals are the discrete solid loadbearing
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.
Nonclay 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
nonlinear 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 GrainMatrix 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 (S_{ave})
and average effective stress (s _{ave}) are
completely three axis (xyz) confined force balanced
scalars.
The Effective Stress Theorem is the fifth equation in
the Newtonian force« balanced closed formulation.
The scalar Pore pressure (P_{p}) is calculated
as the difference between the two load scalars [(S _{ave})(s
_{ave})]. Fracture propagation pressure (P_{F}
= P_{p}+ s _{h} =S _{h} ) is
thereafter calculated in the sixth equation using pore
pressure (Pp) calculated using the Effective Stress
theorem. All the load terms (S, s _{h} &
P_{p}) to the left of the (=) signs denoted
by the sloping dashed line on figure 1 are a Newtonian
closed form force balance! The Effective Stress theorem
is the missing element in the "Terzaghi" relationship
that makes it work.
All the earth strain terms are on the right side of
the (=) signs denoted by the color background change
on figure 1. Absolute volumetric in situ strain (1.0
 f ) is in each of the individually force balanced
stress/strain equations. The descending arrow on figure
1 indicates the algebraic linkage of these equations
to petrophysically measurable strain. The remaining
strain terms (r , s _{max} & a ) are mineral
and fluid coefficients that are compositionally linked
to each other. The equal (=) signs denoted by the sloping
dashed line on figure 1 mathematically relate force
balanced loads to absolute in situ strain in the earth!
Force« balance and direct measurable strain linkage
are unique to this closedform Newtonian formulation.
This physically representative mathematical formulation
leads to simplicity and accuracy of calibration, prediction
and detection of pore pressure.
RETURN TO TOP
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 (S_{v}); 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 closedform 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 offshelf 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
offshelf settings are low, gradual and uniform.
Initial overburden (S_{v}) is also highly predictable,
as it is mostly seawater. Sediments deposited in cold
deep water typically have low carbonate content. Cold
water stratigraphic sequences are dominantly a simple
binary quartz grainstone  claystone mineralogic continuum.
A normalized gammaray log is a fairly good quartzclay
mineralogic estimator these settings. Favorable geologic
conditions for the petrophysical calculation of overburden
(S_{v}) and average effective stress (s _{ave})
are present.
Both (S_{v} & s _{ave}) are dependent
only on strain, mineral, and fluid coefficients in the
Newtonian closed formulation. Effective stress ratio
(s h/s v) is also directly related to (1.f ) strain.
Given these equation redundancies, and the compositionally
bound mineral and fluid physical constants, the Newtonian
closed formulation is mathematically simple. Given geologic
uniformity; pore pressure determination from petrophysical
data is further simplified in offshelf 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
offshelf 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
finetune the topoflog initial overburden (S_{v})
constant and the local water conductivity (C_{w})
profile. Both types of borehole pressure measurements
can be simultaneously calibrated to Newtonian closed
form force balance using linked pore pressure/fracture
pressure method. The calibration to Newtonian force
balance physical approach is the most accurate for both
pore pressure and fracture pressure predictiondetection
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 2629, 1995.
Holbrook, P W, D A Maggiori, & Rodney Hensley, 1995b,
"Realtime 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 1619, 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 78 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. 107122.
RETURN
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___________________________________________________________
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 1619, 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 textbookmanual.
CONTACT
the author for information. RETURN
TO TOP
Abstract
The two open borehole
fluid pressure limits, pore
pressure (P_{p})
and fracture gradient
(P_{f})
are represented by linked stress/strain
relationships which are best obtained directly from
in situ
petrophysical data. The key to in
situ petrophysical
determination of these stress
 fluid pressure
relationships is that rock solidity
(1.0  f
) is an absolute measure of granular
matrix strain.
A new rock mechanics
system has been developed from and
for use with downhole petrophysical data related to
in situborehole
fluid pressure measurements. Using
gravitational force balance (s
_{v} = S_{v}
 P_{p}), two new
in situ
mineralogic stress/strain
(1.0  f )
relationships were derived directly from subsurface
measurements of porosity on granular sedimentary
rocks. These in
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 ( P_{f}
= s _{h}
+ P_{p}) is therefore
also linked to in situ
compactional
strain
(1.0  f )
and average sedimentary
rock mineralogy.
This new compactional
strain
fracture pressure relationship
has been shown to be very accurate (» 4% SD) in
5 separate statistical studies. Accuracy is consistently
high for shales, sandstones and limestones over the
entire effective stress
range of drilling interest. The compactional
strain
relationship is similar to but more accurate than the
empirical fracture
pressuredepth 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 pressuredepth
relationships of previous
authors.
All the pressure
and stress
parameters ( P_{p}, P_{f}
, S_{v}, s
_{v}, s _{h} ) are related
to bulk volumetric strain
(1.0  f )
in the above linked equations.
The four linked equations constitute the force
balanced in
situ Rock
Mechanics System. With this system continuous
logs calculated from in situ petrophysical measurements
can be calibrated to all relative and absolute borehole
fluid pressure measurements and leakoff tests simultaneously
for an entire well. The single well calibration is regional
if the overburden (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
Introduction
Petrophysicists and log analysts are becoming
increasingly involved in the technical decision making
related to well planning and realtime 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 reprecipitated
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
(S_{v}) & (S_{h})
are applied, and pore
fluid pressure (P_{p})
is measured. Effective
stresses (s
_{v}) & (s
_{h}) are calculated by force
balance diffence. Two relative length
strains
and a bulk volume strain
are also measured. RETURN
TO TOP
Using
the in situ
rock mechanics system
on the right; the vertical load (S_{v})
is calculated from an integrated bulk density log. Pore
fluid pressure (P_{p})
is measured directly inside a borehole from wireline
RFT's. Relative mud weight borehole
fluid pressures ( P_{b}
/ P_{p}
) opposite moderately permeable formations provide an
upper pore pressure
(P_{p})
limit in an open borehole if no well flow is observed
during drilling. The effective
stresses (s_{v})
& (s_{h})
are calculated by force
balance difference from Overburden
(S_{v}),
leakoff tests (P_{f}),
and pore pressure (P_{p}).
Solidity
(1.0  f )
is the in situ measure
of bulk volumetric strain.
This parametric measurement shortfall is not a handicap
in NFR »
biaxial
basins because both the effective
stresses (s_{v})
& (s_{h})
are closely related to in situ
bulk volumetric strain.
Laboratory based systems
use the measured final/initial ratio of a length or
volume as a measure of strain.
Using the laboratory approach the relative final/initial
strain
always depends on the initial unloaded state of the
rock sample being tested. The results of each laboratory
experiment depend on the initial porosity,
composition and extent of stress
relief microfractures in the sample. Each laboratory
rock sample has its own unloading
reloading stress
path. The general conclusion that can be reached from
interpreting laboratory experiments is that the reloading
 unloading limbs are different
but are usually close to linear
elastic.
Solidity
(1.0  f )
is the present/final volumetric strain
ratio which can be directly measured using the in
situ data based 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 (s_{v})
is vertical resulting from overburden
(S_{v})
in NFR basins. Effective vertical stress (s_{v})
is calculated as the static equilibrium force
balance difference between the gravitational
overburden
(S_{v})
load minus pore
s_{v}
= S_{v}
 P_{p}
(1)
Effective vertical stress
(s_{v})
has been related to gravitational compaction at maximum
loading. Holbrook (1995) demonstrated a power
law linear
effective stress (s_{v})
/ 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 (s_{v})
/ total in situ compactional
strain(1
 f )relationship;
s_{v}
= 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 (s_{h}/s_{v})
vs. Strain (1.0
 f ) Relationship
In NFR »
biaxial
basins; the minimum
principal
stress (s_{h})
increases with vertical
stress (s_{v})
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 (s_{h})
in shales was measured from in situ
leakoff test
data. Fracture propagation
pressure (P_{f}
);
P_{f}
= s_{h}
+ P_{p}
(3)
is the minimum borehole
fluid pressures required to extend
a preexisting tensile fracture which is perpendicular
to the minimum
principal
stress (S_{h}).
Each of the 4 studies shows that effective
stress ratio (s_{h}/s_{v})
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 (s_{h}/s_{v})
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 (s_{h}/s_{v})
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
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 (P_{f})
method is that equations 1, 2, 3 and 4 are compactional
strain
and force balance
linked. Overburden
(S_{v}),
pore pressure
(P_{p}),
vertical stress
(s_{v}),
horizontal stress
(s_{h}),
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 (s_{v})
to confining horizontal
(s_{h})
stress
following the relationship;
s_{h}
= s_{v}
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
(s_{h}/s_{v})
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 /(1.0n )) in order
to match the leakoff
test measures
effective horizontal stress
(s_{h}).
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 /1.0n )) term which it is intended to correct.
The fact that the correction factor decreases the effective
horizontal stress (s_{h})
by about 1/3 at the mudline and increases (s_{h})
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 (s_{h})
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 (s_{h}/s_{v})
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 (s_{h}/s_{v})
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 (P_{f})
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
pressure. RETURN
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 (s_{v})
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
(s_{h}/s_{v})
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 (P_{p}).
In »
biaxialNFR
basins, the relative horizontal
load (s_{h}/s_{v})
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, (s_{h}
= s_{v})
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
(S_{v})
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 (P_{p},
P_{f},
& P_{max}
= S_{v})
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 ( P_{p}
= S_{v}
 s_{v}),
2. Mud weight (Pb), 3. Fracture
Propagation Pressure gradient (P_{f}
= s_{h}
+ P_{p}),
and 4. Overburden gradient
(S_{v}).
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 (S_{v})
and fracture propagation
pressure (P_{f}
) 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
(S_{v})
is the force balance
borehole fluid pressure maximum
upper limit in NFR
basins (P_{max}
= S_{v}
). If a single subhorizontal bedding plane fracture
exists in the short open borehole below casing, borehole
fluid would enter that fracture at Overburden.
The 1.6% deviation from (P_{max}
= S_{v}
) 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 (P_{f})
matches as well as (P_{max}
= S_{v}
) 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
(P_{p})
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 (P_{p},
P_{f}, & P_{max}
= S_{v})
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 (P_{b})
and calculated pore fluid pressure (P_{p})
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 (P_{b})
is at least 0.2 ppg higher than calculated pore fluid
pressure (P_{p})
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 (P_{p})
were underbalanced by mud weight (P_{b}).
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 (P_{b})
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 (P_{p})
from relative borehole
fluid pressure (P_{b}).
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 & P_{f})
directly. That same porosity value integrated with grain
and fluid densities over depth affects calculated (S_{v}).
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.
Conclusions
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 (s_{v})
was documented in (Holbrook, 1995). The correspondence
between the same parameters and effective
horizontal stress (s_{h})
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 (P_{p}),
fracture
propagation pressure
(P_{f}),
and Overburden
(S_{v})
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 realtime pore
pressure and fracture
gradient continuous logs from near bit
in
situ petrophysical data to
avoid drilling trouble. RETURN
TO TOP
References
Cited
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