1. Modelling Rate Effects in Imbibition
Pore-Scale Network Modelling
Modelling Rate Effects in Imbibition by
Nasiru Idowu
Supvr. Prof. Martin Blunt

2. Outline Introduction Motivation Pore-Scale Models Results Conclusion
Displacement processes
Displacement forces
Current: quasi-static and dynamic models
New: Time-dependent model
Results
Conclusion
Future Work

3. Introduction What is Pore-Scale Network Modelling?
3 mm
A technique for understanding and predicting a wide range of macroscopic multiphase transport properties using geologically realistic networks

4. Introduction Network elements (pores and throat) will be defined with
properties such as:
Radius
Volume
Clay volume
Length
Shape factor, G = A/P2
Connection number for pores
x, y, z positions for pores
Pore1 and pore2 for throats

5. Motivation
To incorporate a time-dependent model into the existing 2-phase code and study the effects of capillary number (different rate) on imbibition displacement patterns
To reproduce a Buckley-Leverett profile from pore scale model by combining the dynamic model with a long thin network
To study field-scale processes driven by gravity with drainage and imbibition events occurring at the same time

6. Motivation
1-Sor
1-Sor
water
oil
water
oil
Swc
Swc
distance
distance
Ideal displacement
Non-ideal displacement
Evolution of a front: capillary forces dominate at the pore scale while viscous forces dominate globally

7. Pore-Scale Models: Displacement processes
Drainage / oil flooding
Displacement of wetting phase by non-wetting phase, e.g. migration of oil from source rocks to reservoir
This can only take place through piston-like displacement where centre of an element can only be filled if it has an adjacent element containing oil

8. Pore-Scale Models: Displacement processes
Imbibition / waterflooding
Displacement of non-wetting phase by wetting phase, e.g. waterflooding of oil reservoir to increase oil recovery
Displacement can take place through:
piston-like displacement
Pore-body filling
Snap-off : will only occur if there is no adjacent element whose centre is filled with water

9. Pore-Scale Models: Displacement forces
Capillary pressure:
Circular elements:
Polygonal elements:
Viscous pressure drop:
Viscous pressure drop in water:
Viscous pressure drop in oil:
Gravitational forces:
In x-direction
In y-direction
In z-direction A L
Pin
Pout
Capillary number, Ncap is the ratio of viscous to capillary forces:

10. Current: quasi-static and dynamic models
Ncap > 10-6
Both viscous and capillary forces influence displacement
Explicit computation of the pressure field required
Computationally expensive
Applicable to only small network size
Quasi-static
Ncap
Applicable to slow flow
Capillary forces dominate
Displacement from highest Pc
to lowest Pc (for imbibition)
Computationally efficient
Perturbative
Assumes a fixed conductance for wetting layers
Uses the viscous pressure drop across wetting layers and local capillary forces to influence displacement
Retains computational efficiency of the static model
Why dynamic/perturbative?
Quasi-static displacement is not valid for
Fracture flow where flow rate may be very high
Displacements with low interfacial tension e.g. near-miscible gas injection
Near well-bore flows

11. New: Time-dependent model
Drawbacks of current dynamic and perturbative models
Fill invaded (snap-off) elements completely whether there is adequate fluid to support the filling or not
Duration of flow is not taken into consideration
Prevent swelling of wetting fluid in layers & corners by assuming fixed conductance
Fully dynamic models are only applicable to small network size with < 5,000 pores
Time-dependent model
Introduces partial filling of elements whenever there is insufficient fluid within the specified time step
Updates the conductance of wetting layers at specified saturation intervals
Uses the pertubative approach and computationally efficient
Applicable to large network size with around 200,000 pores

12. New: Time-dependent model
Algorithm
Definition:
Qw = desired water flow rate
Vw = QwΔt; total vol. of water injected at the specified time step Δt
vwe = qweΔt; water vol. that can enter invaded element at flow rate qwe at the same time step Δt
vo = initial vol. of oil in the invaded element
vw = initial vol. of water in the invaded element
vt = vw + vo (total vol. of the invaded element)
Complete filling:
if Vw vwe & vwe vo; then set Vw = Vw - vo & vw= vt
Partial filling:
if vwe < vo & Vw vwe; then set Vw = Vw - vwe & vw= vw+ vwe
Last filling:
If Vw < vwe or Vw < vo; then set Vw = 0 & vw= vw+ Vw

13. New: Time-dependent model
Computation of pressure field
From Darcy’s law:
Imposing mass conservation at every pore
ΣQp, ij = (a)
where j runs over all the throats connected to pore i. Qp, ij is the flow rate between pore i and pore j and is defined as
(b)
A linear set of equations can be defined from (a) and (b) that can be solved in terms of pore pressures using the pressure solver
Pressure scaling factor:
For (water into light oil) A L
Pin
Pout
For M > 1 (water into heavy oil) po pw
Pc1
Pc2 po
Pc3
pressure
Psort can be viewed as the inlet pressure necessary to fill an element & we fill the element with the smallest value of Psort
Psort = ∆Pwi + ∆Poi -Pci
Inlet
Distance along model
outlet

14. Results for water into light oil
Network: 30,000 pores with
59,560 throat
Ncap = 3.0E-8
∆t = 400secs
Sw = 0.24
Water viscosity = 1cp
Interfacial tension = 30mN/m

15. Results for water into light oil
Ncap = 3.0E-6
∆t = 4secs
Sw = 0.24
Water viscosity = 1cp
Interfacial tension = 30mN/m

16. Results for water into light oil
Ncap = 3.0E-5
∆t = 0.4secs
Sw = 0.24
Water viscosity = 1cp
Interfacial tension = 30mN/m

17. Results for water into light oil
Ncap = 3.0E-4
∆t = 0.04secs
Sw = 0.24
Water viscosity = 1cp
Interfacial tension = 30mN/m

18. Results for viscosity ratio of 1.0
Ncap = 3.0E-8
∆t = 400secs
Sw = 0.24
Water viscosity = 1cp
Oil viscosity = 1cp
Interfacial tension = 30mN/m

19. Results for viscosity ratio of 10.0
Ncap = 3.0E-8
∆t = 400secs
Sw = 0.24
Water viscosity = 1cp
Oil viscosity = 10cp
Interfacial tension = 30mN/m

20. Conclusions
We have developed a time-dependent model that allows partial filling and prevent complete filling of invaded elements when there is insufficient wetting layer flow within the specified time step
The new model allows swelling of wetting phase in layers and corners and does not assume fixed conductivity for wetting layers
For water into light oil, we have been able to reproduce Hughes and Blunt model results and generate Sw vs distance plots for different Ncap values

21. Future work
Resolve challenges associated with higher rates / viscosity ratios displacements and reproduce a Buckley-Leverett profile from pore scale model
To study field scale processes driven by gravity where drainage and imbibition displacements take place simultaneously

22. Thank you