A Rock Physics Workflow to Determine Biot's Coefficient

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 −

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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

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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

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

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“Barnett” and “Marble Falls” Intervals

Barnett

Marble Falls

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.

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

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.

Modeled Curves Vs Measured Curves

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▪ 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

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Compressional Velocity Vs Porosity

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▪ The effect of Kerogen

on the modeled

velocity is rather

dramatic

Kdry/KVRH Vs Porosity

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𝐾𝑑𝑟𝑦 from Gassmann 𝐾𝑑𝑟𝑦 from DEM 𝐾𝑑𝑟𝑦 = Ksat

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

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