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Reservoir Simulation - Aquifers

Data review

• Why run a flow simulation ? • Mathematical & Numerical considerations • ECLIPSE Reminder

Introduction

Etienne MOREAU

History matching

Space & Time Discretisation Reservoir description Fluid description Initial State Aquifer Representation Flow description

Production Forecast

• • • • • •

Outline

2

© 2010 - IFP Training

EP - Reservoir Simulation - Introduction - E.M.

]

© 2010 - IFP Training

[

3

Introduction

Aquifers’ Representation Main Concepts

Objective is to know

B N p Bo + Wp B w = Ni[Bo − Bo,i ] + o,i c p + c w,i Sw,i ∆P + We So,i

• Flow term = oil , water productions • Expansion term = oil + water expansion ; pore compaction; water influx

Material Balance Equation

• More or less slows down pressure decline • Drives oil towards producing wells

Principle: Use the energy of the aquifer

EP - Reservoir Simulation - Aquifers - E.M.

• Cumulative water influx We & Pressure support versus time

4

© 2010 - IFP Training

EP - Reservoir Simulation - Aquifers - E.M.

1

0,75

0,5

0,25

0 01/06/1992 30/12/1993

27/09/1998

water injection

Material Balance Equation

26/02/1997

Drive Mechanism - Jafra

Aquifer water influx

Rock Compaction Fluid expansion 30/07/1995

Time (date d/m/y)

Material Balance Equation

5

© 2010 - IFP Training

We(t) and Q(t) plots

EP - Reservoir Simulation - Aquifers - E.M.

EP - Reservoir Simulation - Aquifers - E.M.

Contribution of the different mechanisms

© 2010 - IFP Training

6

We(t) and Q(t) plots

Material Balance Equation

Edge drive

Two Cases Linear or Radial

Flow lines are // to the dip

Aquifer Drive: Three types of aquifers

7

© 2010 - IFP Training

Bottom Drive

Flow lines are vertical

EP - Reservoir Simulation - Aquifers - E.M.

EP - Reservoir Simulation - Aquifers - E.M.

© 2010 - IFP Training

8

Radial Aquifer

Flow lines are radial

Aquifer Drive: Two types of edge aquifers

Linear Aquifer

Flow lines are parallel

© 2010 - IFP Training

9

Aquifer Modelling: Main Concepts

Aquifer Drive: 3 types of aquifers

• Transient behaviour Infinite aquifer • Steady – state behaviour Finite aquifer (open) • Semi steady-state Finite aquifer (closed)

3 types of behaviours

• Can help to better understand aquifer impact on production mechanisms • Can be used to realize sensitivity runs on aquifer parameters

Material Balance Equation

EP - Reservoir Simulation - Aquifers - E.M.

• Bottom Aquifer • Edge Aquifer (Linear) • Edge Aquifer (Radial)

10

© 2010 - IFP Training

EP - Reservoir Simulation - Aquifers - E.M.

11

© 2010 - IFP Training

[

]

Aquifers’ Modelling: Main Concepts

Aquifers’ Modelling

Aquifers’ Representation

Reminder

;

Pinit = Aquifer initial pressure

∆Vaq (t ) = Aquifer expansion

;

Paq (t ) = Aquifer final pressure

C aq = Aquifer compressibility

Vaq (t ) = Aquifer volume

We(t ) = Water Influx

We(t ) = ∆Vaq (t ) = Vaq (t )× C aq × Pinit − Paq (t )

Water influx = Aquifer Expansion

EP - Reservoir Simulation - Aquifers - E.M.

• Vaq increases with time in transient behaviour

12

© 2010 - IFP Training

EP - Reservoir Simulation - Aquifers - E.M.

Aquifers’ Modelling: Main Concepts Aquifer Compressibility = Pore + water Compressibility C aq = C p + C w C p = Pore compressib ility C w = Water compressib ility

Order of magnitudes C w = 0.3 to 0.7 10 −4 vol/vol/b ar C p = 0.3 to 0.7 10 − 4 vol/vol/b ar (consolida ted reservoir) © 2010 - IFP Training

13

Aquifers’ Modelling: Main Concepts

up to 10 − 3 vol/vol/b ar (unconsoli dated reservoir)

Q w, aq

Aquifer flow rate

Reminder

Paq (t) = Aquifer pressure

;

Pres = Reservoir pressure

T ab (t) = Aquifer tr ansmissivi ty µ w = Water visc osity

We (t) = Cumulative water influx vs time

Q w, aq (t) = Aquifer flow rate

d W e (t) T ab (t) = × (Paq (t) − Pres ) (t) = dt µw

EP - Reservoir Simulation - Aquifers - E.M.

• Aquifer Transmissivity decreases with time in transient behaviour

14

© 2010 - IFP Training

EP - Reservoir Simulation - Aquifers - E.M.

EP - Reservoir Simulation - Aquifers - E.M.

Aquifers’ Modelling: Bottom Aquifer

© 2010 - IFP Training

15

Aquifers’ Modelling: Edge Aquifer

10 km

Objective is to know cumulative water influx and pressure support versus time

These results depends on the following parameters: Vaq = Aquifer Volume Ct = Aquifer Compressibility Taq = Transmissivity between reservoir and aquifer There are wo possibilities : 1 - Use of large grid cells 2 - Use of Hurst & van Everdingen functions.

16

© 2010 - IFP Training

EP - Reservoir Simulation - Aquifers - E.M.

Analytical Aquifer

Edge Aquifer Modelling: 2 Types of representation

Gridded Aquifer

EP - Reservoir Simulation - Aquifers - E.M.

© 2010 - IFP Training

17

Radial Aquifer Modelling: Use of large grid cells

Pro’s No special algorithm Aquifer pressure available ----------Con’s Useless calculations Incorrect representation of transient flows Risk of transmissivity overestimation

EP - Reservoir Simulation - Aquifers - E.M.

© 2010 - IFP Training

18

Radial Aquifer Modelling: Use of Analytical Aquifers

Pro’s Good representation of transient flows. Optimization of number of cells. ----------Con’s

© 2010 - IFP Training

19

Aquifer Modelling: Key Points

Need for specific algorithms. Need to handle Hurst & Van Everdingen tabulations.

2 types of representation

− Pore & Water Compressibility

• Aquifer Compressibility

− Contact Area, Aquifer Extension, Net Permeability,

• Aquifer Transmissivity

− Contact Area,, Aquifer Extension, Net Thickness, Net Porosity

• Aquifer Volume

3 Main Parameters

EP - Reservoir Simulation - Aquifers - E.M.

• Bottom Aquifer Gridded Aquifer • Edge Aquifer Use of large cells or analytical functions

EP - Reservoir Simulation - Aquifers - E.M.

© 2010 - IFP Training

20

Pool Radius

Trans.

x

Volume

x

x

Trans.

Compr.

Edge Aquifer

x x x

x

Compr.

Bottom Aquifer Volume

x

x

x

x

Net Thickness x

External Radius

Net Porosity Hor. Permeability

x x

x

Pore Compressibility x

Vert. Permeability

Water Compressibility

EP - Reservoir Simulation - Aquifers - E.M.

21

© 2010 - IFP Training

Aquifers’ Representation Radial Aquifer Modelling

EP - Reservoir Simulation - Aquifers - E.M.

© 2010 - IFP Training

22

10 km

Radial Aquifer Modelling

23

© 2010 - IFP Training

Main Parameters

∂ 2 P 1 ∂P 1 ∂P = + ∂r 2 r ∂r K ∂t

Diffusivity Equation

• Aquifer is infinite and homogeneous • Initial pressure is constant • Reservoir is put into production at a constant flow rate

Main hypotheses

Radial Aquifer: Mathematical model (Infinite Aquifer)

EP - Reservoir Simulation - Aquifers - E.M.

EP - Reservoir Simulation - Aquifers - E.M.

© 2010 - IFP Training

24

EP - Reservoir Simulation - Aquifers - E.M.

Radial Aquifer: Pressure Behaviour

Analytical Closed Aquifer

25

© 2010 - IFP Training

EP - Reservoir Simulation - Aquifers - E.M.

© 2010 - IFP Training

26

Analytical Closed Aquifer Instantaneous expansion of a finite volume We = Vaq × C aq × ∆Paq Q w,aq = dWe dt = Vaq × C aq × dPaq dt ---------Vaq = Aquifer Volume C aq = Aquifer Compresibility (Pores + Water) Paq = Aquifer Pressure

This is usually applied to small aquifers with closed boundaries and in equilibrium with the reservoirs. © 2010 - IFP Training

27

Steady State Aquifer (Fetkovitch)

This is equivalent to a multiplied pore volume

EP - Reservoir Simulation - Aquifers - E.M.

EP - Reservoir Simulation - Aquifers - E.M.

© 2010 - IFP Training

28

Steady State Aquifer (Fetkovitch) Constant pressure equal to the initial pressure Q w,aq = dWe dt = Taq × (Paq − Pres ) We = Taq × (Paq − Pres )× t ---------Taq = Aquifer Transmissibility Paq = Aquifer Pressure Pres = Reservoir Pressure

© 2010 - IFP Training

29

Transient aquifer

It is assumed that the aquifer behaves as an infinite aquifer that is to say the pressure at the outer boundary of the aquifer does not change.

EP - Reservoir Simulation - Aquifers - E.M.

EP - Reservoir Simulation - Aquifers - E.M.

© 2010 - IFP Training

30

Radial Aquifer: Pressure Behaviour

© 2010 - IFP Training

31

Transient aquifer

In fact, as diffusivity has a finite value, pressure drop in the reservoir makes aquifer expansion more intense close to the reservoir, and propagating as time passes.

• They mobilize the whole aquifer volume instantaneously. • The whole aquifer volume is supposed to be at the same pressure:

In nearly all applications, the closed or steady-state models discussed before are not adequate in describing the water influx.

EP - Reservoir Simulation - Aquifers - E.M.

In single phase flow and for slightly compressible fluid the diffusivity equation is :

∂ 2 p 1 ∂p φ µ c ∂p 2 + r ∂r = k ∂t ∂r

32

© 2010 - IFP Training

EP - Reservoir Simulation - Aquifers - E.M.

AQUIFER

Permeability = k Porosity = φ Thickness = h Compressibility = c

Transient Radial Aquifer: Hurst & Van Everdingen

External radius = re Internal radius = ri Aperture = θ

re θ rp

Water influx = We (re/ri, θ, k.h, φ.h.c)

33

© 2010 - IFP Training

EP - Reservoir Simulation - Aquifers - E.M.

Radial Aquifer: Aquifer geometry (example)

Aperture = 40° Inner radius = 2 000 m Outer radius = 20 000 m

© 2010 - IFP Training

34

Parameter

A = Contact Area

Data

Voil =

A = θ h ri

(

θhφ V = re2 - ri2 2 aq

35

© 2010 - IFP Training

i

V r = 2 oil Aφ

A2 θ= 2 h Voil

Relationship

Hurst & Van Everdingen

)

Hurst & Van Everdingen

θ = Aperture

Voil = Oil Volume

Relationship

ri = inner radius

Vaq = Aquifer Volume

Data

θhφ 2 r 2 i

re = outer radius

Parameter

= Oil Volume

A = Contact Area

= inner radius

e

V V Vaq = Aquifer Volume r = 2 oil 1 + aq Aφ Voil

Voil

θ = Aperture

EP - Reservoir Simulation - Aquifers - E.M.

ri

re = outer radius

EP - Reservoir Simulation - Aquifers - E.M.

© 2010 - IFP Training

36

1000

100

10

1

0,1

Re/Ri

20 10 6 4 3 2

1000 10000

320

280

240

200

160 01/02/92 31/05/95

Production Simulation - Jafra

30/09/93 Time (date d/m/y)

27/01/97

27/09/98

Radial Aquifer : Influence of aquifer size (example)

Radial Aquifer - Jafra

0,1

1 10 100 Reduced time tD

37

© 2010 - IFP Training

Identify the following points:

Three cases have been simulated with the same production history

Radial Aquifer : Influence of aquifer perm. (exercise)

Tank Pressure (bar)

• At what time field is shut–in • What case correspond to no aquifer, low perm. or high perm. aquifer

38

© 2010 - IFP Training

EP - Reservoir Simulation - Aquifers - E.M.

EP - Reservoir Simulation - Aquifers - E.M.

0,01 0,001 0,01

Reduced water cumulative influx Q(tD)

View more...
Data review

• Why run a flow simulation ? • Mathematical & Numerical considerations • ECLIPSE Reminder

Introduction

Etienne MOREAU

History matching

Space & Time Discretisation Reservoir description Fluid description Initial State Aquifer Representation Flow description

Production Forecast

• • • • • •

Outline

2

© 2010 - IFP Training

EP - Reservoir Simulation - Introduction - E.M.

]

© 2010 - IFP Training

[

3

Introduction

Aquifers’ Representation Main Concepts

Objective is to know

B N p Bo + Wp B w = Ni[Bo − Bo,i ] + o,i c p + c w,i Sw,i ∆P + We So,i

• Flow term = oil , water productions • Expansion term = oil + water expansion ; pore compaction; water influx

Material Balance Equation

• More or less slows down pressure decline • Drives oil towards producing wells

Principle: Use the energy of the aquifer

EP - Reservoir Simulation - Aquifers - E.M.

• Cumulative water influx We & Pressure support versus time

4

© 2010 - IFP Training

EP - Reservoir Simulation - Aquifers - E.M.

1

0,75

0,5

0,25

0 01/06/1992 30/12/1993

27/09/1998

water injection

Material Balance Equation

26/02/1997

Drive Mechanism - Jafra

Aquifer water influx

Rock Compaction Fluid expansion 30/07/1995

Time (date d/m/y)

Material Balance Equation

5

© 2010 - IFP Training

We(t) and Q(t) plots

EP - Reservoir Simulation - Aquifers - E.M.

EP - Reservoir Simulation - Aquifers - E.M.

Contribution of the different mechanisms

© 2010 - IFP Training

6

We(t) and Q(t) plots

Material Balance Equation

Edge drive

Two Cases Linear or Radial

Flow lines are // to the dip

Aquifer Drive: Three types of aquifers

7

© 2010 - IFP Training

Bottom Drive

Flow lines are vertical

EP - Reservoir Simulation - Aquifers - E.M.

EP - Reservoir Simulation - Aquifers - E.M.

© 2010 - IFP Training

8

Radial Aquifer

Flow lines are radial

Aquifer Drive: Two types of edge aquifers

Linear Aquifer

Flow lines are parallel

© 2010 - IFP Training

9

Aquifer Modelling: Main Concepts

Aquifer Drive: 3 types of aquifers

• Transient behaviour Infinite aquifer • Steady – state behaviour Finite aquifer (open) • Semi steady-state Finite aquifer (closed)

3 types of behaviours

• Can help to better understand aquifer impact on production mechanisms • Can be used to realize sensitivity runs on aquifer parameters

Material Balance Equation

EP - Reservoir Simulation - Aquifers - E.M.

• Bottom Aquifer • Edge Aquifer (Linear) • Edge Aquifer (Radial)

10

© 2010 - IFP Training

EP - Reservoir Simulation - Aquifers - E.M.

11

© 2010 - IFP Training

[

]

Aquifers’ Modelling: Main Concepts

Aquifers’ Modelling

Aquifers’ Representation

Reminder

;

Pinit = Aquifer initial pressure

∆Vaq (t ) = Aquifer expansion

;

Paq (t ) = Aquifer final pressure

C aq = Aquifer compressibility

Vaq (t ) = Aquifer volume

We(t ) = Water Influx

We(t ) = ∆Vaq (t ) = Vaq (t )× C aq × Pinit − Paq (t )

Water influx = Aquifer Expansion

EP - Reservoir Simulation - Aquifers - E.M.

• Vaq increases with time in transient behaviour

12

© 2010 - IFP Training

EP - Reservoir Simulation - Aquifers - E.M.

Aquifers’ Modelling: Main Concepts Aquifer Compressibility = Pore + water Compressibility C aq = C p + C w C p = Pore compressib ility C w = Water compressib ility

Order of magnitudes C w = 0.3 to 0.7 10 −4 vol/vol/b ar C p = 0.3 to 0.7 10 − 4 vol/vol/b ar (consolida ted reservoir) © 2010 - IFP Training

13

Aquifers’ Modelling: Main Concepts

up to 10 − 3 vol/vol/b ar (unconsoli dated reservoir)

Q w, aq

Aquifer flow rate

Reminder

Paq (t) = Aquifer pressure

;

Pres = Reservoir pressure

T ab (t) = Aquifer tr ansmissivi ty µ w = Water visc osity

We (t) = Cumulative water influx vs time

Q w, aq (t) = Aquifer flow rate

d W e (t) T ab (t) = × (Paq (t) − Pres ) (t) = dt µw

EP - Reservoir Simulation - Aquifers - E.M.

• Aquifer Transmissivity decreases with time in transient behaviour

14

© 2010 - IFP Training

EP - Reservoir Simulation - Aquifers - E.M.

EP - Reservoir Simulation - Aquifers - E.M.

Aquifers’ Modelling: Bottom Aquifer

© 2010 - IFP Training

15

Aquifers’ Modelling: Edge Aquifer

10 km

Objective is to know cumulative water influx and pressure support versus time

These results depends on the following parameters: Vaq = Aquifer Volume Ct = Aquifer Compressibility Taq = Transmissivity between reservoir and aquifer There are wo possibilities : 1 - Use of large grid cells 2 - Use of Hurst & van Everdingen functions.

16

© 2010 - IFP Training

EP - Reservoir Simulation - Aquifers - E.M.

Analytical Aquifer

Edge Aquifer Modelling: 2 Types of representation

Gridded Aquifer

EP - Reservoir Simulation - Aquifers - E.M.

© 2010 - IFP Training

17

Radial Aquifer Modelling: Use of large grid cells

Pro’s No special algorithm Aquifer pressure available ----------Con’s Useless calculations Incorrect representation of transient flows Risk of transmissivity overestimation

EP - Reservoir Simulation - Aquifers - E.M.

© 2010 - IFP Training

18

Radial Aquifer Modelling: Use of Analytical Aquifers

Pro’s Good representation of transient flows. Optimization of number of cells. ----------Con’s

© 2010 - IFP Training

19

Aquifer Modelling: Key Points

Need for specific algorithms. Need to handle Hurst & Van Everdingen tabulations.

2 types of representation

− Pore & Water Compressibility

• Aquifer Compressibility

− Contact Area, Aquifer Extension, Net Permeability,

• Aquifer Transmissivity

− Contact Area,, Aquifer Extension, Net Thickness, Net Porosity

• Aquifer Volume

3 Main Parameters

EP - Reservoir Simulation - Aquifers - E.M.

• Bottom Aquifer Gridded Aquifer • Edge Aquifer Use of large cells or analytical functions

EP - Reservoir Simulation - Aquifers - E.M.

© 2010 - IFP Training

20

Pool Radius

Trans.

x

Volume

x

x

Trans.

Compr.

Edge Aquifer

x x x

x

Compr.

Bottom Aquifer Volume

x

x

x

x

Net Thickness x

External Radius

Net Porosity Hor. Permeability

x x

x

Pore Compressibility x

Vert. Permeability

Water Compressibility

EP - Reservoir Simulation - Aquifers - E.M.

21

© 2010 - IFP Training

Aquifers’ Representation Radial Aquifer Modelling

EP - Reservoir Simulation - Aquifers - E.M.

© 2010 - IFP Training

22

10 km

Radial Aquifer Modelling

23

© 2010 - IFP Training

Main Parameters

∂ 2 P 1 ∂P 1 ∂P = + ∂r 2 r ∂r K ∂t

Diffusivity Equation

• Aquifer is infinite and homogeneous • Initial pressure is constant • Reservoir is put into production at a constant flow rate

Main hypotheses

Radial Aquifer: Mathematical model (Infinite Aquifer)

EP - Reservoir Simulation - Aquifers - E.M.

EP - Reservoir Simulation - Aquifers - E.M.

© 2010 - IFP Training

24

EP - Reservoir Simulation - Aquifers - E.M.

Radial Aquifer: Pressure Behaviour

Analytical Closed Aquifer

25

© 2010 - IFP Training

EP - Reservoir Simulation - Aquifers - E.M.

© 2010 - IFP Training

26

Analytical Closed Aquifer Instantaneous expansion of a finite volume We = Vaq × C aq × ∆Paq Q w,aq = dWe dt = Vaq × C aq × dPaq dt ---------Vaq = Aquifer Volume C aq = Aquifer Compresibility (Pores + Water) Paq = Aquifer Pressure

This is usually applied to small aquifers with closed boundaries and in equilibrium with the reservoirs. © 2010 - IFP Training

27

Steady State Aquifer (Fetkovitch)

This is equivalent to a multiplied pore volume

EP - Reservoir Simulation - Aquifers - E.M.

EP - Reservoir Simulation - Aquifers - E.M.

© 2010 - IFP Training

28

Steady State Aquifer (Fetkovitch) Constant pressure equal to the initial pressure Q w,aq = dWe dt = Taq × (Paq − Pres ) We = Taq × (Paq − Pres )× t ---------Taq = Aquifer Transmissibility Paq = Aquifer Pressure Pres = Reservoir Pressure

© 2010 - IFP Training

29

Transient aquifer

It is assumed that the aquifer behaves as an infinite aquifer that is to say the pressure at the outer boundary of the aquifer does not change.

EP - Reservoir Simulation - Aquifers - E.M.

EP - Reservoir Simulation - Aquifers - E.M.

© 2010 - IFP Training

30

Radial Aquifer: Pressure Behaviour

© 2010 - IFP Training

31

Transient aquifer

In fact, as diffusivity has a finite value, pressure drop in the reservoir makes aquifer expansion more intense close to the reservoir, and propagating as time passes.

• They mobilize the whole aquifer volume instantaneously. • The whole aquifer volume is supposed to be at the same pressure:

In nearly all applications, the closed or steady-state models discussed before are not adequate in describing the water influx.

EP - Reservoir Simulation - Aquifers - E.M.

In single phase flow and for slightly compressible fluid the diffusivity equation is :

∂ 2 p 1 ∂p φ µ c ∂p 2 + r ∂r = k ∂t ∂r

32

© 2010 - IFP Training

EP - Reservoir Simulation - Aquifers - E.M.

AQUIFER

Permeability = k Porosity = φ Thickness = h Compressibility = c

Transient Radial Aquifer: Hurst & Van Everdingen

External radius = re Internal radius = ri Aperture = θ

re θ rp

Water influx = We (re/ri, θ, k.h, φ.h.c)

33

© 2010 - IFP Training

EP - Reservoir Simulation - Aquifers - E.M.

Radial Aquifer: Aquifer geometry (example)

Aperture = 40° Inner radius = 2 000 m Outer radius = 20 000 m

© 2010 - IFP Training

34

Parameter

A = Contact Area

Data

Voil =

A = θ h ri

(

θhφ V = re2 - ri2 2 aq

35

© 2010 - IFP Training

i

V r = 2 oil Aφ

A2 θ= 2 h Voil

Relationship

Hurst & Van Everdingen

)

Hurst & Van Everdingen

θ = Aperture

Voil = Oil Volume

Relationship

ri = inner radius

Vaq = Aquifer Volume

Data

θhφ 2 r 2 i

re = outer radius

Parameter

= Oil Volume

A = Contact Area

= inner radius

e

V V Vaq = Aquifer Volume r = 2 oil 1 + aq Aφ Voil

Voil

θ = Aperture

EP - Reservoir Simulation - Aquifers - E.M.

ri

re = outer radius

EP - Reservoir Simulation - Aquifers - E.M.

© 2010 - IFP Training

36

1000

100

10

1

0,1

Re/Ri

20 10 6 4 3 2

1000 10000

320

280

240

200

160 01/02/92 31/05/95

Production Simulation - Jafra

30/09/93 Time (date d/m/y)

27/01/97

27/09/98

Radial Aquifer : Influence of aquifer size (example)

Radial Aquifer - Jafra

0,1

1 10 100 Reduced time tD

37

© 2010 - IFP Training

Identify the following points:

Three cases have been simulated with the same production history

Radial Aquifer : Influence of aquifer perm. (exercise)

Tank Pressure (bar)

• At what time field is shut–in • What case correspond to no aquifer, low perm. or high perm. aquifer

38

© 2010 - IFP Training

EP - Reservoir Simulation - Aquifers - E.M.

EP - Reservoir Simulation - Aquifers - E.M.

0,01 0,001 0,01

Reduced water cumulative influx Q(tD)

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