1. Capillary End Effects during Core Flooding – Analytical Solutions and Capillary Numbers accounting for Saturation Functions
Pål Østebø Andersen, Postdoc Lunch & Learn University of Stavanger, 10 May 2017 Co-authors: Dag C. Standnes & Svein M. Skjæveland

2. Outline Context Modelling Application Conclusions
Core flooding experiments
Correction of lab artifacts
Modelling
Application
Conclusions

3. Capillary End Effects Multiphase core flooding experiments
Capillary pressure (CP) often neglected:
simple Buckley-Leverett analysis using relative permeabilities (RP)
CP can have significant impact on interpretation
Pc = 0 at the outlet [Leverett 1941]
Affects transient and steady state behavior.
Accumulation of wetting phase at outlet
Magnitude of end effects can be reduced by increasing factor (Length * viscosity * rate)
Richardson et al., 1952: Measured in situ accumulation of oil (wetting) at outlet during co-injection of gas and oil
Rapoport & Leas, 1953: Measured recovery at water breakthrough of oil-wet cores. Stabilization at high scaling factor (end effects become negligible).

4. Role of capillary pressure
Advantagous to minimize end effects to simplify analysis
Increase rate to check for additional production
Can be impractical:
Very high rates can destroy the core
Important to distinguish between:
Reduction of remaining saturation towards residual saturation (reduce end effects)
Reduction of the residual saturation (desaturation) at very high capillary numbers
Chatzis & Morrow, 1984: Reduction of residual saturation at very high capillary numbers

5. Steady state of water flooding
Oil displaced by water in mixed-wet media
Steady state:
No time derivatives
Oil is immobile
Water is produced with same rate as injected
Saturation and pressure gradients depend on:
Distribution of capillary pressure derivative and water mobility
Not on oil mobility

6. Goal of this work
Derive analytical solutions and explicit formulations for
Saturation profiles
Pressure profiles
Average saturation
Pressure drop
Relative permebility of water
Estimate extent of end effects
Correct for end effects

7. Assumptions Simple saturation function correlations
Capture physical trends from literature
Adapted to experimental measurements on mixed-wet media [Abeysinghe et al. 2012]

8. Saturation profiles Depends on new capillary number N and value of m+n
At higher rates more of the saturation profile approaches Sor
End effects vanish for x>L_E
End effects reach inlet at the critical value N=1
Solution matches Eclipse simulation also when end effects go out of the core (Sor not obtained any locations)

9. Average saturation
Changes behavior whether end effects are limited to within the core or not
Linear with 1/N for
N>1=critical value
Unique slope in this region
Capillary end effects can be scaled by new capillary number N_canew
Cap number represents how far average saturation is from residual value
Higher m (weaker pc) and higher n (stronger advective forces) contribute to reduce water saturation

10. Scaled pressure drop
Also different trends whether end effects have reached inlet
Scaled dP
proportional to 1/N (large N)
goes to 1 (large N)
given by mobility at Seq (very low N)
Higher m (weaker pc) reduce dP, but higher n (stronger advective forces) increase dP
Average saturation and dP depend differently on saturation function parameters

11. Correction of lab measurements
Measure steady state average sw at 3 rates
In linear regime linear interpolation in sw vs 1/rate plot gives correct residual saturation
In nonlinear regime a Langmuir correlation captures that slope becomes linear for low 1/rate
Residual saturation can be estimated

12. Intercept method
At high rates steady state pressure drop is linear with rate
Presented by Gupta and Maloney 2016, used to correct steady state saturation for end effects
Theoretically derived here for water flooding
Gupta and Maloney, 2016: Intercept method to correct saturations

13. Calculation of saturation curves
Assume 3+ steady state measurements: rate, pressure drop and average saturation
Based on linear slope and intercept in high rate region:
Sor
Krw(1-Sor)
m (curvature of Pc)
n (curvature of water rel perm)
Based on critical point where linear region begins (N=1):
J0 (coefficient of Pc)
Only undetermined parameter is Seq
Extrapolate measurements with model
Or spontaneous imbibition test

14. Summary
Explicit analytical solutions for water flooding with end effects at steady state
More general than analytical solutions for strongly wet media by Huang & Honarpour 1998 and more intuitive
Physically meaningful capillary numbers with saturation function parameters
Theoretical derivation of the ‘Intercept Method’
to correct residual saturation for end effects
Procedure to calculate saturation curves from 3 (or more) steady state measurements

15. Future work Extend the work to injection of two phases simultaneously
Relevant for steady state measurement of relative permeability
Perform experimental measurements to validate the model and estimate saturation curves both based on transient and steady state measurements

16. Co-authors: Dag C. Standnes & Svein M. Skjæveland
Acknowledgments
Co-authors: Dag C. Standnes & Svein M. Skjæveland
Sponsors: The Research Council of Norway and The Industry Partners of The National IOR Center of Norway

17. References
Chatzis, I., Morrow, N.R., Correlation of capillary number relationships for sandstone. SPE Journal 24, doi: /10114-PA.
Gupta, R., Maloney, D.R., Intercept method - a novel technique to correct steady-state relative permeability data for capillary end effects. SPE Reservoir Evaluation & Engineering 675 (19), doi: / PA.
Huang, D.D., Honarpour, M.M., Capillary end effects in core flood calculations. Journal of Petroleum Science and Engineering 19, doi: /S (97)
Leverett, M.C., Capillary behavior in porous solids. Trans. AIME 142, doi: / G.
Rapoport, L.A., Leas, W.J., Properties of linear water floods. Journal of Petroleum Technology 5, doi: /213-G.
Richardson, J.G., Kerver, J.K., Hafford, J.A., Osoba, J.S., Laboratory determination of relative permeability. Journal of Petroleum Technology 4, doi: / G.

18. QUESTIONS?