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

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

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.

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

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)

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)

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

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.

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.

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

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.

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

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

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

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

Percolation Algorithm Phase Mobility Index, γ 𝑝

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

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

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

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

2D-Example: strongly water-wet

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

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

3D-Example: strongly water-wet

3D vs. 2D

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.

Thank You Thank You

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