1. Hasan Nourdeen Martin Blunt 10 Jan 2017
Impact of Invasion Percolation on Upscaling in Capillary-Controlled Darcy-Scale Flow
Hasan Nourdeen
Martin Blunt
10 Jan 2017

2. Importance of steady-state upscaling approach
Outline
Importance of steady-state upscaling approach
What is Invasion Percolation?
Percolation at Darcy-scale?!
Impact on numerical stability and upscaled properties
Summary and future work

3. Introduction
The modelling of multiphase flow in porous media is a very important topic for many fields.
In the oil industry, in particular, field studies are conducted routinely to optimize the operation of oil and gas fields and to forecast their performance.
Understanding the physics involved in how immiscible fluids flow inside the rock is vital for accurate modelling.

4. Traditional Field-Scale Reservoir Simulation
Rock Properties:
Reservoir Geometry
Porosity, Permeability
Multiphase Flow Properties:
Relative Permeability
Capillary Pressure
Fluid Properties:
Phase Viscosity, Density, …etc.

5. Traditional Field-Scale Reservoir Simulators
Two-phase flow equations (Incompressible fluid):
𝛻. 𝑞 𝑓 +𝜙 𝜕 𝑆 𝑓 𝜕𝑡 = 𝑞 𝑓
𝑞 𝑓 =−𝐊 𝑘 𝑟 𝑓 𝜇 𝑓 𝛻 𝑝 𝑓
𝑆 𝑤 + 𝑆 𝑜 =1
𝑝 𝑐 = 𝑝 𝑜 − 𝑝 𝑤
𝑓=𝑜,𝑤
Conservation of mass
Darcy’s equation (Two-Phase flow)

6. Characterising Multiphase Flow Parameters
Experimental Data(cm-scale):
Special Core Analysis (SCAL):
Pore-Scale Modelling(mm-scale):
Using high-resolution pore-scale images
Two-main measurement methods for rel per (the most important parameter for flow in large-scale)

7. Upscaling Two-Phase Flow
Changes in fluid forces
Heterogeneity
mm scale
cm scale
m scale
Facies-model scale
(Stephen et al., 2001)

8. Two-main Upscaling approaches: Steady-state Unsteady-state
Steady-state Method
Two-main Upscaling approaches:
Steady-state
Unsteady-state
Steady-state is more reliable:
produces “effective properties”
Computationally very expensive when accounting for all fluid forces.
SS is computationally cheaper as compared to dynamic methods and much simple to implement.

9. Two-main Upscaling approaches: Steady-state Unsteady-state
Steady-state Method
Two-main Upscaling approaches:
Steady-state
Unsteady-state
Steady-state is more reliable:
produces “effective properties”
Computationally very expensive when accounting for all fluid forces.
Saturation distribution is estimated from local multiphase flow properties when one force dominates the displacement process.
SS is computationally cheaper as compared to dynamic methods and much simple to implement.

10. Steady-State Method
Indication of rate-dependency at different scale using Capillary Number (Lohne et al., 2006)
Key point in SS: local-saturation distribution in the upscaled region
In Capillary-Limit : 𝑆 𝑤,𝑖 = 𝑆 𝑤,𝑖 ( 𝑃 𝑐,𝑖 ) 𝑖=1:𝑛
In Viscous-Limit : 𝑆 𝑤,𝑖 = 𝑆 𝑤,𝑖 ( 𝑓 𝑤,𝑖 ) 𝑖=1:𝑛
𝑁 𝑐 = 𝛻 𝑝 𝑔 𝑙 𝑝𝑐 ∆ 𝑃 𝑐
𝛻 𝑝 𝑔 global pressure gradient due to viscous flow
𝑙 𝑝𝑐 characteristic length of capillary heterogeneity
∆ 𝑃 𝑐 capillary contrast at VL-conditions
Lohne et al., 2006

11. Capillary-controlled displacement
Saturation distribution is determined from local capillary pressure-saturation relationships.
A phase may fail to form a connected path across a given domain at capillary equilibrium.
Some regions of the domain may produce disconnected clusters that do not contribute to the overall connectivity of the system.

12. Capillary-controlled displacement
We need to identify and remove these isolated clusters: important to the global connectivity of the system and the stability of numerical solvers

13. Invasion Percolation Water can only access open cells Invaded Closed
{𝒌,𝝓, 𝒌 𝒓𝒐 , 𝒌 𝒓𝒘 , 𝒑 𝒄 , 𝑳 𝒙,𝒚,𝒛 }
mm-cm

14. Invasion Percolation 𝜉 𝑜𝑝𝑛 = 𝑁 𝑜𝑝𝑒𝑛 𝑁 𝜉 𝑖𝑛𝑣 = 𝑁 𝑖𝑛𝑣𝑎𝑑𝑒𝑑 𝑁
Invaded
Closed
Open
{𝒌,𝝓, 𝒌 𝒓𝒐 , 𝒌 𝒓𝒘 , 𝒑 𝒄 , 𝑳 𝒙,𝒚,𝒛 }
conventional upscaling process might not be accurate since identification and removal of these isolated clusters are extremely important to the global connectivity of the system and the stability of numerical solvers
Fraction of open cells
Fraction of invaded cells
𝜉 𝑜𝑝𝑛 = 𝑁 𝑜𝑝𝑒𝑛 𝑁
𝜉 𝑖𝑛𝑣 = 𝑁 𝑖𝑛𝑣𝑎𝑑𝑒𝑑 𝑁

15. Percolation Theory 𝜉 𝑜𝑝𝑛 = 𝑁 𝑜𝑝𝑒𝑛 𝑁 𝜉 𝑐 Percolation threshold;
Invaded
Closed
Open
{𝒌,𝝓, 𝒌 𝒓𝒐 , 𝒌 𝒓𝒘 , 𝒑 𝒄 , 𝑳 𝒙,𝒚,𝒛 }
Probability or fraction of a site (or a cell) to be open
𝜉 𝑜𝑝𝑛 = 𝑁 𝑜𝑝𝑒𝑛 𝑁
𝜉 𝑐 Percolation threshold;
depends on network type and dimensionality;
universal for random media as 𝑁 →∞
The fraction of invaded cells or connected cells that first spans the whole space
𝜉 𝑜𝑝𝑛 ≥𝜉 𝑐 , 𝑘 𝑟 >0
𝜉 𝑜𝑝𝑛 <𝜉 𝑐 , 𝑘 𝑟 =0

16. Percolation Algorithm
Phase Mobility Index, γ 𝑝

17. Percolation Algorithm
Phase Connectivity Index, 𝜓 𝑝
𝜓 𝑝 0 → 𝜓 𝑝 ∞
B is the adjacency matrix of a network

18. 2D-Example: strongly water-wet
𝑝 𝑐 ( 𝑆 𝑤 )=2 𝜎 𝑜𝑤 cos 𝜃 𝐽 𝑆 𝑤 𝐾 𝜙 −1

19. 2D-Example: strongly water-wet
𝜓 𝑝 ∞
1000-by-1000
𝜉 𝑐 =0.59
10 m

20. 2D-Example: strongly water-wet
𝜉 𝑐 =0.59

21. 2D-Example: strongly water-wet

22. 3D-Example: strongly water-wet
𝜓 𝑝 ∞
250x250x250
𝜉 𝑐 ≈0.31
2.5 m

23. 3D-Example: strongly water-wet
𝜉 𝑐 ≈0.31

24. 3D-Example: strongly water-wet

25. 3D vs. 2D

26. Summary and Future Work
In a heterogeneous capillary-controlled environment, a phase may fail to form a connected path across a given domain at capillary equilibrium.
If a continuous saturation path exists, some regions of the domain may produce disconnected clusters that do not contribute to the overall connectivity of the system.
conventional upscaling process might not be accurate since identification and removal of these isolated clusters are extremely important to the global connectivity of the system and the stability of numerical solvers.
We presented a comprehensive investigation using random permeability fields, for strongly water-wet media.
Important information is revealed about the average connectivity of the phases and the trapping of saturations due to capillary forces.
The next step is add viscous forces.

27. Thank You
Thank You

28. Multistage Upscaling Approach
Multistage Approach:
Starting from the sub-micron scale – important for rocks with significant micro-porosity
Incorporate pore-, core-, intermediate-, and large-scale heterogeneities, as required.
Including the right balance of forces at each stage.
Represent physics and geology.
Efficient Algorithm to determine the steady-state saturation distribution