1. Title
Petrophysical Analysis of Fluid Substitution in Gas Bearing Reservoirs to Define Velocity Profiles – Application of Gassmann and Krief Models
Digital Formation, Inc.
November 2003

2. Contents Benefits Introduction Gassmann Equation in Shaley Formation
Wyllie Time Series Equation
Linking Gassmann to Wyllie
Adding a gas term to Wyllie Equation
Krief Equation
Examples
Conclusions

3. Benefits – Seismic
Reliable compressional and shear curves even if no acoustic data exists.
Quantify velocity slowing due to presence of gas.
Full spectrum of fluid substitution analysis.
Reliable mechanical properties, Vp/Vs ratios.
Reliable synthetics.
Does not involve neural network or empirical correlations.

4. Benefits – Petrophysics
Verifies consistency of petrophysical model.
Ability to create reconstructed porosity logs using deterministic approaches.

5. Benefits – Engineering
Reliable mechanical property profiles for drilling and stimulation design.
Does not rely on empirical correlations, or neural network curve generation, for mechanical properties.

6. Introduction
A critical link between petrophysics and seismic interpretation is the influence of fluid content on acoustic and density properties.
Presented are two techniques which rigorously solve compressional and shear acoustic responses in the entire range of rock types, and assuming different fluid contents.

7. Gassmann Equation in Shaley Formation – I
The Gassmann equation accounts for the slowing of acoustic compressional energy in the formation in the presence of gas.
There is no standard petrophysical analysis that accounts for the Gassmann response and incorporates the effect in acoustic equations (e.g. Wyllie Time-Series).
Terms in the Gassmann equation:
M = Elastic modulus of the porous fluid filled rock
Merf = Elastic modulus of the empty rock frame
Berf = Bulk modulus of the empty rock frame
Bsolid = Bulk modulus of the rock matrix and shale
Bfl = Bulk modulus of the fluid in pores and in clay porosity
FT = Total Porosity
rB = Bulk density of the rock fluid and shale combination
Vp = Compressional wave velocity

8. Gassmann Equation in Shaley Formation – II
In shaley formation, adjustments need to be made to several of the Gassmann equation terms, including porosity and bulk modulus of the solid components.
This allows a rigorous solution to Gassmann through the full range of shaley formations.
Estimates of shear acoustic response are made using a Krief model analogy.

9. Wyllie Time Series Equation
In the approach presented here, we have solved the Gassmann equation in petrophysical terms, and defined a gas term for the Wyllie Time-Series equation that rigorously accounts for gas.
Original Time-Series equation:
Matrix
Contribution
Fluid
Contribution
Dt = Travel time = 1/V
Dtma = Travel time in matrix
Dtfl = Travel time in fluid

10. Linking Gassmann to Wyllie
Calculate Dt values from Gassmann using fluid substitution
Liquid filled i.e. Gas saturation Sg=0
Gas filled assuming remote (far from wellbore) gas Sg
Gas filled assuming a constant Sg of 80%
From Dt values, calculate effective fluid travel times (Dtfl)
Knowing mix of water and gas, determine effective travel time of gas (Dtgas)
Relate Dt values to gas saturation, bulk volume gas

11. Gassmann Sg vs. Ratio of Dtgas to Dtwet
Color coding refers to porosity bins

12. Gassmann Bulk Volume Gas vs. Ratio Dtgas to Dtwet
Color coding refers to porosity bins

13. Gassmann Bulk Volume Gas vs. DtgasC2
Hyperbola = C3 C1

14. Adding a Gas Term to Wyllie Equation
Contribution
Matrix
Contribution
Water
Contribution
Gas term involves C1, C2 and C3 (constants)
Equation reduces to traditional Wyllie equation when Sg=0
If gas is present, but has not been determined from other logs, the acoustic cannot be used to determine reliable porosity values.

15. Krief Equation – Part I
Krief has developed a model that is analogous to Gassmann, but also extends interpretations into the shear realm. We have similarly adapted these equations to petrophysics.
Vp = Compressional wave velocity
VS = Shear wave velocity
rB = Bulk density of the rock fluids and matrix and shale
m = Shear modulus
K = Elastic modulus of the shaley porous fluid filled rock
KS = Elastic modulus of the shaley formation
Kf = Elastic modulus of the fluid in pores
bb = Biot compressibility constant
Mb = Biot coefficient
FT = Total Porosity

16. Krief Equation – Part II
The Krief analysis gives significantly different results from Gassmann, in fast velocity systems (less change in velocity in the presence of gas as compared with Gassmann).
In slow velocity systems (high porosity, unconsolidated rocks), the two models give closely comparable results.

17. Examples
In all of these examples, the pseudo acoustic logs are derived from a reservoir model of porosity, matrix, clay and fluids.
There is no information from existing acoustic logs in these calculations.
On all plots, porosity scale is 0 to 40%, increasing right to left.
Slow Rocks
Gassmann DTP
Krief DTP
Krief DTP & DTS
Fast Rocks
Carbonates
Gassmann DTP & DTS

18. Slow Rocks – Gassmann DTP
Compressional shows
significant
slowing due
to gas

19. Slow Rocks – Krief DTP Compressional shows significant slowing due
to gas

20. Slow Rocks – Krief DTP & DTS
Compressional shows
very good comparison
Ratio and Shear
shows fair to
good comparison

21. Fast Rocks – Gassmann DTP
Actual compressional
meanders between
wet and remote
Noticeable
slowing due to gas

22. Fast Rocks – Krief DTP Actual compressional superimposes
on both wet and
remote
Negligible
slowing due to gas

23. Carbonates – Gassmann DTP
Compressional
shows slight slowing due to gas

24. negligible slowing due to gas
Carbonates – Krief DTP
Compressional shows
negligible slowing due to gas

25. Fast Rocks – Gassmann DTP/DTS
Good comparison with actual Shear
Ratio shows
slight slowing
due to gas

26. Fast Rocks – Krief DTP/DTS
Good comparison with actual Shear
Ratio shows
negligible
slowing due to gas

27. Conclusions – Part I
Pseudo acoustic logs (both compressional and shear) can be created using any combination of water, oil and gas, using either Gassmann’s or Krief’s equations for clean and the full range of shaley formations.
Comparison with actual acoustic log will show whether or not the acoustic log “sees” gas or not – gives information on invasion profile.
Pseudo acoustic logs can be created even if no source acoustic log is available.
Data from either model can be incorporated into the Wyllie Time Series equation to rigorously account for gas.

28. Conclusions – Part II
Interpretation yields better input to create synthetic seismograms and for rock mechanical properties.
Methodology allows for detailed comparisons among well log response, drilling information, mud logs, well test data and seismic.
In fast velocity rocks and in the presence of gas, the Krief model predicts less slowing effect than Gassmann.
In slow velocity gas-bearing rocks, both models give closely comparable results.
The techniques have been applied successfully to both clastic and carbonate reservoirs throughout North America.