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**Modelling Rate Effects in Imbibition**

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

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**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

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**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

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**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

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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

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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

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**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

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**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

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**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:

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**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

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**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

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**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

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**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

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**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

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**Results for water into light oil**

Ncap = 3.0E-6 ∆t = 4secs Sw = 0.24 Water viscosity = 1cp Interfacial tension = 30mN/m

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**Results for water into light oil**

Ncap = 3.0E-5 ∆t = 0.4secs Sw = 0.24 Water viscosity = 1cp Interfacial tension = 30mN/m

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**Results for water into light oil**

Ncap = 3.0E-4 ∆t = 0.04secs Sw = 0.24 Water viscosity = 1cp Interfacial tension = 30mN/m

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**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

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**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

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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

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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

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Thank you

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