Modelling Rate Effects in Imbibition

by

Nasiru Idowu

Supvr. Prof. Martin Blunt

Pore-Scale Network Modelling

Outline• Introduction• Motivation • Pore-Scale Models

– Displacement processes– Displacement forces– Current: quasi-static and dynamic models– New: Time-dependent model

• Results• Conclusion• Future Work

Introduction

A technique for understanding and predicting a wide range of macroscopic multiphase transport properties using geologically realistic networks

3 mm

What is Pore-Scale Network Modelling?

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Sw

Kr PSM

Expt

Introduction

Network elements (pores andthroat) 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

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

Motivation

distance distance

Swc Swc

1-Sor 1-Sor

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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/

oro

wrw

k

kM

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oro

wrw

k

kM

water oil water oil

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

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

• 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

Pore-Scale Models: Displacement forces

rPc

cos2

Pin Poutghow

PL

AKkQ rp

p

A

L

),,()21(cos

GFr

GP dc

/qNcap

Capillary number, Ncap is the ratio of viscous to capillary forces:

Current: quasi-static and dynamic models

Quasi-static•Ncap 10-6

•Applicable to slow flow•Capillary forces dominate •Displacement from highest Pcto lowest Pc (for imbibition)•Computationally efficient

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

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

New: Time-dependent modelDrawbacks 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

New: Time-dependent modelAlgorithm 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

New: Time-dependent modelComputation 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:

pw

po

Distance along modelInlet outlet

pressure

po

Psort = ∆Pwi + ∆Poi -Pci

For (water into light oil)

For M > 1 (water into heavy oil)

)(,ji

ij

ijpp PP

L

gQ

oldett

new PQ

QP arg

Psort can be viewed as the inlet pressure necessary to fill an element & we fill the element with the smallest value of Psort

Pin Pout

L

AP

L

AKkQ

p

rpp

Pc1 Pc2 Pc3

1M

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/mSw vs Distance

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distance(fraction)

Sw Ncap = 3.0E-8

Swc

Results for water into light oil

Ncap = 3.0E-6

∆t = 4secs

Sw = 0.24

Water viscosity = 1cp

Interfacial tension = 30mN/m

Sw vs Distance

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distance(fraction)

Sw

3.0E-8

3.0E-6

Swc

Results for water into light oil

Ncap = 3.0E-5

∆t = 0.4secs

Sw = 0.24

Water viscosity = 1cp

Interfacial tension = 30mN/m

Sw vs Distance

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distance(fraction)

Sw

3.0E-8

3.0E-6

3.0E-5

Swc

Results for water into light oil

Ncap = 3.0E-4

∆t = 0.04secs

Sw = 0.24

Water viscosity = 1cp

Interfacial tension = 30mN/m

Sw vs Distance

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distance(fraction)

Sw

3.0E-8

3.0E-6

3.0E-5

3.0E-4

Swc

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/mSw vs Distance

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0.00 0.20 0.40 0.60 0.80 1.00 1.20

distance(fraction)

Sw

3.0E-8

Swc

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/mSw vs Distance

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distance(fraction)

Sw

3.0E-8

Swc

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

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

Thank you