7 1_Aquifers_EM

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

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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

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EP - Reservoir Simulation - Aquifers - E.M.

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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

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We(t) and Q(t) plots

EP - Reservoir Simulation - Aquifers - E.M.




EP - Reservoir Simulation - Aquifers - E.M.

Contribution of the different mechanisms

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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

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Bottom Drive

Flow lines are vertical




EP - Reservoir Simulation - Aquifers - E.M.




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Radial Aquifer

Flow lines are radial




Aquifer Drive: Two types of edge aquifers

Linear Aquifer

Flow lines are parallel

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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)

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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

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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

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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

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EP - Reservoir Simulation - Aquifers - E.M.

EP - Reservoir Simulation - Aquifers - E.M.

Aquifers’ Modelling: Bottom Aquifer

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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.

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Analytical Aquifer

Edge Aquifer Modelling: 2 Types of representation

Gridded Aquifer

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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.

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Radial Aquifer Modelling: Use of Analytical Aquifers

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

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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.

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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

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Aquifers’ Representation Radial Aquifer Modelling

EP - Reservoir Simulation - Aquifers - E.M.

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10 km

Radial Aquifer Modelling

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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)

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EP - Reservoir Simulation - Aquifers - E.M.

Radial Aquifer: Pressure Behaviour

Analytical Closed Aquifer

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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

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Steady State Aquifer (Fetkovitch)

This is equivalent to a multiplied pore volume

EP - Reservoir Simulation - Aquifers - E.M.

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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

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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.

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Radial Aquifer: Pressure Behaviour

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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

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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)

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EP - Reservoir Simulation - Aquifers - E.M.

Radial Aquifer: Aquifer geometry (example)

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

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Parameter

A = Contact Area

Data

Voil =

A = θ h ri

(

θhφ V = re2 - ri2 2 aq

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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.

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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

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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

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EP - Reservoir Simulation - Aquifers - E.M.







EP - Reservoir Simulation - Aquifers - E.M.

0,01 0,001 0,01

Reduced water cumulative influx Q(tD)