Modelling Rate Effects in Imbibition

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

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

3. 3 mm Introduction What is Pore-Scale Network Modelling? 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 • Dynamic • 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 10-6 • 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 Vwvwe & vwevo; then set Vw = Vw - vo & vw= vt • Partial filling: • if vwe < vo & Vwvwe; 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 = 0 (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 Pc3 po 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