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

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

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).

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

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

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

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

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)

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

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

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

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

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

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

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

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

References Chatzis, I., Morrow, N.R., 1984. Correlation of capillary number relationships for sandstone. SPE Journal 24, 555-562. doi:10.2118/10114-PA. Gupta, R., Maloney, D.R., 2016. Intercept method - a novel technique to correct steady-state relative permeability data for capillary end effects. SPE Reservoir Evaluation & Engineering 675 (19), 316-330. doi:10.2118/171797-PA. Huang, D.D., Honarpour, M.M., 1998. Capillary end effects in core flood calculations. Journal of Petroleum Science and Engineering 19, 103-117. doi:10.1016/S0920-4105(97)00040-5. Leverett, M.C., 1941. Capillary behavior in porous solids. Trans. AIME 142, 152-169. doi:10.2118/941152-G. Rapoport, L.A., Leas, W.J., 1953. Properties of linear water floods. Journal of Petroleum Technology 5, 139-148. doi:10.2118/213-G. Richardson, J.G., Kerver, J.K., Hafford, J.A., Osoba, J.S., 1952. Laboratory determination of relative permeability. Journal of Petroleum Technology 4, 187-196. doi:10.2118/952187-G.

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