a rock physics workflow to determine biot's coefficient
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A Rock Physics Workflow to Determine
Biot's Coefficient for Unconventionals
Mohammad Reza Saberi
2
Introduction
▪ Introduction
▪ Biot calculation methods
▪ Case study
▪ Hooke’s Law:
3
Robert Hooke?
(1635-1703)
F = k u
Restoring force
Spring constant (stiffness)
Displacement
Introduction
▪ Stress effects on an isotropic,homogeneous, linear elastic solid:
Introduction
σ
σ
σσ
1
0
11
2
ij
ij
ijijij
ijijij
ji
ji
PRPRE
Introduction
σ
σ
σσ
▪ Stress effects on an isotropic,homogeneous, linear elastic solid:
Introduction
▪ The presence of a freely movingfluid in a porous rock modifies itsmechanical response in twomechanisms:
σ
σ
σσ
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
▪ 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
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):
▪ 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”:
▪ 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
▪ 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
▪ 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
▪ 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
(Mavko and Mukerji, 1995)
0
0~1
1
K
K
K
K sat
Pore Stiffness
▪ 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
▪ 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
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
18
Barnett Wells with Solid Log Data
19
▪ 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
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
Ksat and Gsat Crossplots
21
“Barnett” and “Marble Falls” Intervals
Barnett
Marble Falls
Vclay Determination
22
▪ 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.
Stochastic Model for Barnett
23
▪ 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
Rock Physics Modeling Workflow
24
▪ 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.
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).
Quality Control Check on Modeled Curves
26
Compressional Velocity Vs Porosity
27
▪ The effect of Kerogen
on the modeled
velocity is rather
dramatic
Kdry/KVRH Vs Porosity
28
𝐾𝑑𝑟𝑦 from Gassmann 𝐾𝑑𝑟𝑦 from DEM 𝐾𝑑𝑟𝑦 = Ksat
Biot’s Coefficient
29
Biots Coefficient
▪ Red curve is
calculated using
inverse Gassmann
on the measured
logs.
▪ Blue is calculated
assuming
Kdry =Ksat
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.
30
Thank you
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