1. Effective and Relative Permeabilities 1
Laboratory Measurement of Relative Permeability - Capillary End Effect -Unsteady State Method

2. Capillary End Effect During immiscible displacement
In the bulk of the core plug
Pc= f (Swet)
At the outflow face
Pc= 0  Swet=1
There must be a gradient of saturation from the the bulk of the core to the outflow face
This saturation gradient is the “Capillary End Effect”

3. Capillary End Effect Comparison for low flow rate
Theoretical gradient (dashed line)
Experimental data (circles)
Saturation gradient extends over half of the length of the core plug

4. Capillary End Effect Comparison for higher flow rate
Theoretical gradient (dashed line)
Experimental data (circles)
At higher flow rate, saturation gradient extends over only 1/5 of the length of the core plug

5. Capillary End Effect
Eliminating errors due to end effect in measurement of relative permeability functions
Measure saturation far enough away from outflow face (e.g. Penn State Method)
Use high flow rates to make error in measured saturation negligible

6. Rel. Perm. - Unsteady State
Unsteady State Method for relative permeability provides
Relative permeability ratio (kr,nonwet/kr,wet) as a function of wetting phase saturation (Swet)
Irreducible wetting phase saturation (drainage)
Residual nonwetting phase saturation (imbibition)

7. Rel. Perm. - Unsteady State
Saturation in Core Plug
Production Rates
Imbibition Relative Permeability Ratio Function
Stage 1: Preparation for drainage
core saturated with wetting phase
Stage 2: Irreducible wetting phase (drainage)
inject non-wetting phase until steady state, measure saturation
no wetting phase will be produced after steady state

8. Rel. Perm. - Unsteady State
Saturation in Core Plug
Production Rates
Imbibition Relative Permeability Ratio Function
Stages 3-6: Inject wetting phase
Stage 3 (A) Wetting phase has not yet reached outflow face
only nonwetting phase produced at outflow face
Stage 4 (B) Wetting phase just reaches outflow face, called breakthrough
wetting phase will be produced at outflow face

9. Rel. Perm. - Unsteady State
Saturation in Core Plug
Production Rates
Imbibition Relative Permeability Ratio Function
Stages 3-6: Inject wetting phase
Stage 5 (C) As injection of wetting phase continues, production of nonwetting phase decreases (unsteady state)
Important to take many data points during this decrease
cummulative nonwetting phase produced
production rate for both phases

10. Rel. Perm. - Unsteady State
Saturation in Core Plug
Production Rates
Imbibition Relative Permeability Ratio Function
Stages 3-6: Inject wetting phase
Stage 6 (D) Eventually, no more nonwetting phase is produced, allowing residual nonwetting phase saturation to be determined

11. Rel. Perm. - Unsteady State
Analysis Procedure
Assumptions
immiscible displacement
incompressible fluids
linear, 1-D flow
capillary pressure neglected
Determination of average saturation
cumulative nonwetting phase production
Determination of relative permeability ratio from fractional flow
fwet = qwet/qtotal ; where qtotal = qwet + qnonwet
production rate measured at outflow face

12. Rel. Perm. - Unsteady State
Relative Permeability Ratio usually plotted semi-log
hysteresis due to saturation process (drainage, imbibition)