a rock physics workflow to determine biot's coefficient
TRANSCRIPT
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A Rock Physics Workflow to Determine
Biot's Coefficient for Unconventionals
Mohammad Reza Saberi
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2
Introduction
▪ Introduction
▪ Biot calculation methods
▪ Case study
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▪ Hooke’s Law:
3
Robert Hooke?
(1635-1703)
F = k u
Restoring force
Spring constant (stiffness)
Displacement
Introduction
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▪ Stress effects on an isotropic,homogeneous, linear elastic solid:
Introduction
σ
σ
σσ
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1
0
11
2
ij
ij
ijijij
ijijij
ji
ji
PRPRE
Introduction
σ
σ
σσ
▪ Stress effects on an isotropic,homogeneous, linear elastic solid:
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Introduction
▪ The presence of a freely movingfluid in a porous rock modifies itsmechanical response in twomechanisms:
σ
σ
σσ
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Introduction
▪ The presence of a freely movingfluid in a porous rock modifies itsmechanical response in twomechanisms:
– compression of the rock causes a rise of
pore pressure, if the fluid is prevented
from escaping the pore network.
– an increase of pore pressure induces a
dilation of the rock,
σ
Pp
σ
σσ
𝜎𝑒𝑓𝑓 ≈ 𝜎 − 𝑃𝑝
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▪ First considerations about deformation of porousrocks and soils were done by Terzaghi. He foundtheoretically that there is an effective stress whichcontrols the changes in bulk volume of a sample andinfluences its failure conditions:
▪ The exact form of effective stress is given by Nur &Byerlee (1971) as:
flijceff P
Karl von Terzaghi
(1883-1963)
Introduction
pveff P
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Biot’s coefficient
dry
dry
dry
p
p
KK
KKK
0
0
▪ Effective dry rock pore space stiffness, defined as the ratio of the fractional changein pore volume, vp, to an increment of applied external hydrostatic stress, , atconstant pore pressure:
HK
V
v
dryB
p 1
Drained bulk modulus
Poroelastic expansion factor
▪ (effective-stress coefficient) is a function of stress and is defined as the ratio ofpore-volume change vp to bulk volume change, VB, at constant pore pressure(dry or drained conditions):
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▪ The exact form of effective stress is given by Nur & Byerlee (1971) as:
pveff P
Effective stress
10 ,10
KK
Kdry
▪ α is the “effective-stress coefficient” and is also called the “Biot-Willis coefficient”or simply “Biot coefficient”:
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▪ α=0 Solid rock without pores, and no pore pressure influence (non-porous rock)
▪ α=1 Extremely compliant porous solid with maximum pore pressure influence, i.e.unconsolidated sediments and suspension (fluid with particles in it)
drysat KK
min
11
KKK flsat
𝐾𝑑𝑟𝑦 = 𝐾0
𝐾𝑑𝑟𝑦 = 0
Biot’s coefficient
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▪ Static effective stress coefficient: The traditional method for measuring static Biot’scoefficient is by obtaining a drained triaxial compression measurement underconstant volumetric strain condition:
p
eff
Peff
a
p
a
P
1
(Alam et al. 2012)
▪ In a static case, the strain amplitude is higher than inthe dynamic case and strain contains elastic andplastic components. Therefore, “Biot-Williscoefficient” can be different for dynamic cases.
Calculate Biot’s Coefficient
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▪ Dynamic effective stress coefficient: Using ultrasonic velocities and below equationto calculate dynamic Biot coefficient:
𝛼 = 1 −𝐾𝑑𝑟𝑦
𝐾𝑚𝑖𝑛
𝐾𝑑𝑟𝑦 = ρ𝑑𝑟𝑦𝑉2𝑝 − 𝑑𝑟𝑦 −
4
3𝜌𝑑𝑟𝑦𝑉
2𝑠 − 𝑑𝑟𝑦
Calculate Biot’s Coefficient
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▪ Dynamic effective stress coefficient: Using rock physics models and belowequation to calculate dynamic Biot coefficient:
𝛼 = 1 −𝐾𝑑𝑟𝑦
𝐾𝑚𝑖𝑛
𝐾𝑑𝑟𝑦 = 𝑓(𝑅𝑃𝑀)
Calculate Biot’s Coefficient
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(Mavko and Mukerji, 1995)
0
0~1
1
K
K
K
K sat
Pore Stiffness
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▪ A family of constant k curvescan be drawn on a plot of
Kdry /K0 versus porosity,
▪ This allow us to estimate KØ
trends from rock physicsmeasurements.
(Russell and Smith, 2007)
0K
Kk
Pore Stiffness
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▪ Static modules are of practical interest ingeomechanical modeling and prediction of theminimum and maximum stresses and reservoirfracturing calculations.
▪ Core samples analysis may not reflect the full extent ofthe elastic properties changes along the well, andneeded to be linked with seismic cube.
▪ Often dynamic parameters are transformed in staticmoduli.
Static and Dynamic moduli
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Case Study Workflow
▪ Examine well log data
▪ Calculate the elastic properties of the rocks
▪ View the elastic properties (Ksat and Gsat)
▪ Determine Vclay
▪ Generate lithological model of the reservoir
▪ Use lithological model to build rock physics model
▪ From rock physics model compute Poisson’s Ratio, Young’s Modulus, Kdry,
and Biot’s Coefficient
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Barnett Wells with Solid Log Data
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▪ The log data for this
study are coming from
Barnett field located in
suburb of Dallas.
▪ The available data
contains three wells
having high-quality
well log data with
detailed petrophysical
interpretation for
reservoir properties
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Calculate Elastic Properties
▪ The proposed workflow
starts with examining
well log data and
calculating the elastic
properties of the rocks
and checking quality of
the saturated bulk and
shear modulus
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Ksat and Gsat Crossplots
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“Barnett” and “Marble Falls” Intervals
Barnett
Marble Falls
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Vclay Determination
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▪ Then, volume of clay is
determined and lithological
model of the reservoir are
generated accordingly.
▪ Clay volume is calculated
by using clay indicators
such as: Gamma Ray, SP,
Resistivity, and Neutron.
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Stochastic Model for Barnett
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▪ The lithological description of
the formation is created using
stochastic methods
▪ This lithological model will be
used to build rock physics
model, and from there
Poisson’s Ratio, Young’s
Modulus, Kdry and Biot’s
Coefficient will be calculated
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Rock Physics Modeling Workflow
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▪ The mineral volumes are
used to compute K0 using
the Voigt-Reuss-Hill
average model.
▪ This is followed,
furthermore, by developing
a rock physics workflow to
determine rock elastic
properties.
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Modeled Curves Vs Measured Curves
25
▪ Elastics are
modelled using the
rock physics
model.
▪ The good match
between measured
and modelled logs,
confirms accuracy
of the inputs into
rock physics model
(interpreted logs).
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Quality Control Check on Modeled Curves
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Compressional Velocity Vs Porosity
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▪ The effect of Kerogen
on the modeled
velocity is rather
dramatic
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Kdry/KVRH Vs Porosity
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𝐾𝑑𝑟𝑦 from Gassmann 𝐾𝑑𝑟𝑦 from DEM 𝐾𝑑𝑟𝑦 = Ksat
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Biot’s Coefficient
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Biots Coefficient
▪ Red curve is
calculated using
inverse Gassmann
on the measured
logs.
▪ Blue is calculated
assuming
Kdry =Ksat
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Conclusion
▪ A solid petrophysical interpretation is required to perform quality rock physics
analysis.
▪ The process is iterative wherein the rock physics results can aid in determining
input parameters for the petrophysical model.
▪ A rock physics model is built using lithology volumes, water saturation, porosity,
pressure, temperature, and fluid characteristics provides a rigorous test of the
petrophysical analysis.
▪ These inputs, furthermore, are used to calculate dynamic Biot’s coefficient.
▪ The assumption of “Kdry=Ksat“ makes calculation easier and faster to calculate
dynamic Biot’s coefficient and it shows less noisy behavior compared with other
methods.
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Thank you