aquifer modelling.pdf

8/10/2019 aquifer modelling.pdf

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AQUIFER MODELLING


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Eclipse 100 User Course Page 324 of 499 08/04/99

Aquifer Modelling Facilities

Figure 108: Aquifer definition

Aquifers can be modelled as numerical, analytical, grid or flux aquifers.

Grid and numerical aquifers are specified in the GRIDsection

Any grid or numerical aquifer NNCs are specified in the GRIDsection

Analytical and flux aquifers are specified in the SOLUTIONsection

Analytical aquifer NNCs are also specified in the SOLUTIONsection.

Different aquifer types may be used in a model but Carter-Tracy and Fetkovich aquifers

cannot be used in the same model.

The number of aquifers and the maximum number of cells to which they are connected is

specified in AQUDIMSin the RUNSPECsection.

Grid Cell Aquifers

Numerical Aquifers

Analytical Aquifers:

Fetkovich

Carter-Tracy

Flux Aquifers


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Aquifer Modelling FacilitiesThere are several ways to specify aquifers in ECLIPSE:

As a grid cell aquifer. To do this: -

Choose cells beneath the OWC to function as an aquifer

Multiply their pore volume as necessary using MULTPV.

Input any extra connections to the oil and/or gas zone with explicit NNCs using

theNNCkeyword.

As a numerical aquifer. To do this: -

Nominate a number of grid cells, using the keyword AQUNUM, to function as an

aquifer

Input the NNCs to the reservoir using AQUCON.

As an analytical aquifer. To do this: -

Create an aquifer using keywords ACUCT (Carter-Tracy aquifer) or AQUFET or

AQUFETP(Fetkovich aquifer)

Join them to the reservoir using the AQUANCONkeyword.

As a flux aquifer. To do this: - Create an aquifer of constant flux per unit area using the AQUFLUXkeyword

Join it to the reservoir grid by NNCs defined in the AQUANCONkeyword.

NOTE that aquifers connected to cells above the OWC will flow into the oil zone. In

numerical aquifers this takes place because the interblock mobility is taken from the

upstream (aquifer) cell, not the downstream cell in which the water relative

permeability may be zero. Analytical aquifer flow is independent of relative

permeability.


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Grid Cell Aquifers

Figure 109: Grid cell aquifer definition

Cells in the water leg of the simulation grid are used as an aquifer

Grid cell aquifers are defined in the GRIDand/or EDITsections.

Pore volume multipliers may be applied and their properties altered in the GRID

and/or EDITsections.

Cell pressure can be reported during the run. The aquifer will behave like a finite aquifer by default.

K=1

BOX

--I1 I2 J1 J2 K1 K2

1 1 2 8 1 1 /

EQUALS

MULTPV 10000 /

/

ENDBOX

J

I

Grid aquifer

cells

Oil zone

Inactive cells


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Grid Cell AquifersAquifers can be incorporated directly into the simulation grid in a number of ways but

the method has a number of limitations.

The simulation grid can be extended artificially below the OWC. This is a valid

approach when modelling aquifers that are small compared to the oil zone. This has the

flexibility that goes with the usage of the entire suite of GRID section keywords to

modify the aquifer properties to match the simulation to the measured aquifer

characteristics. The major disadvantage is that the phase pressures, saturations and

solution ratios are solved in the extra aquifer cells as for any other cell, which may

dramatically increase the run time if the aquifer contains many cells.

In principle aquifers much larger than the oil zone may be defined by multiplying the

pore volumes of the water zone cells. The disadvantages of this approach are

Throughput-related convergence problems are likely to occur if an aquifer cell pore

volume is more than three orders of magnitude greater than the pore volume of any

of its neighbours.

A great deal of time and effort has to be spent in designing a grid to represent the

aquifer.


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Numerical Aquifers

Figure 110: Numerical aquifer definition

Several redundant cells or cells below the OWC are nominated as aquifer cells

Numerical aquifers are defined in the GRIDsection.

The cell properties are modified by the AQUNUMkeyword

Cells are attached to the oil zone by NNCs defined in the AQUCONkeyword

The number of numerical aquifers and NNC is defined in AQUDIMSin RUNSPEC

K=1

GRIDAQUNUM--1 2 3 4 5 6 7 8 9 10 11 12--Aquifer I J K Area Length K Depth Initial PVT SAT--Id pressure table table 1 8 9 1 1E2 1E2 1 / 1 9 9 1 1E4 1E3 1 / 1 10 9 1 1E6 1E4 1 //AQUCON--1 2 3 4 5 6 7 8 9 10 11--Aquifer I1 I2 J1 J2 K1 K2 Face Trans Trans Connection--Id mult option option 1 1 1 2 8 1 1 I- //

J

I

Numericalaquifer cells

Oil zone

Inactive cells

NNCs to Oil Zone


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Numerical AquifersThe user is free to select a number of cells to function as an aquifer. In Figure 110 cells

(8-10, 9, 1) have been nominated. They are joined to one another and the reservoir in

the order of entry in the AQUNUM keyword and form a single aquifer. The order of

connections is

(10, 9, 1) flows into (9, 9, 1)

(9, 9, 1) flows into (8, 9, 1)

(8, 9, 1) flows into the reservoir.

The AQUNUM keyword automatically sets zero transmissibility multipliers between the

chosen aquifer cells and their neighbours in the grid to prevent unwanted flows into

adjacent portions of the grid. Note that aquifer cells cannot be deactivated by keywords

includingACTNUMand MINPV.

The cell properties including dimensions, depth, porosity, permeability and regions

definitions are unaltered by default. Despite the fact that these and other quantities are

set using AQUNUM, they must still be defined elsewhere with the standard GRID and

REGIONSsection keywords.

The choice of cell dimensions is significant. In Figure 110, cell pore volumes increase

progressively from the oil zone to cell (10, 8, 1) by a factor no greater than 10 3 between

connected cells. This is intended to minimise throughput-related convergence problems.

It is often recommended to place an extra row of cells to act as a buffer between the

aquifer itself and the oil zone for the same reason. This is unnecessary if the aquifer has

been designed to minimise throughput-related convergence problems from the outset.

The initial aquifer pressure is usually defaulted to ensure it is in hydrostatic equilibrium

with the rest of the simulation grid after initialisation. Instability may nevertheless arise.

Consider Figure 111. The OWC is at a depth, which does not coincide with a cell centre

depth. The attached aquifer is joined to the entire lateral faces of several cells. There is a

difference between the OWC depth and the aquifer depth, i.e. a hydrostatic pressure

difference between the aquifer and oil zone. Water will flow into the reservoir from the

aquifer in the absence of injection and production and the reservoir pressure will drop

until equilibrium is reached. Although this is normally a very minor effect since the

height difference is small, users are strongly recommended to design reservoir grids to

avoid this. The effect may become very significant if:

A large number of aquifer cells are connected to the oil zone


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The grid cells are large

The initial aquifer pressure is not defaulted and is significantly different from the

pressure at the OWC.

Figure 111: Model instability from poor aquifer design

The effect is not restricted to numerical aquifers.

The oil zone-aquifer NNCs are defined using AQUCON. Non-neighbour connections must

be enabled and the RUNSPEC NONNC keyword should not be used. The transmissibilitybetween cell (8, 8, 1) and the rest of the reservoir is defined according to the rules

outlined in an earlier section on Cartesian grid transmissibility and is defined as:

gridaq TTT

111+=

EQ. 57

where

aq

aqaqaq

lAkT 2=

OWC

Hydrostatic pressuredifference (

w-

o)gh


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EQ. 58

Where kaq, Aaqand laq are the permeability, cross-sectional area and length, respectively,

of cell (8, 8, 1) and Tgrid is calculated as usual. Transmissibility multipliers may be

applied to these connections in the 9thitem of AQUCON. Item 8 of AQUCONspecifies which

face of cell (8, 8, 1) is joined to the aquifer. The options are I+, I-, J+, J-, K+ and K-

which represent the direction of increasing and decreasing I, J, and K index,

respectively. In Figure 110 the I- face of any cell is at the left and I+ face is at the right.

The connection option in item 11 determines whether aquifers are permitted to connect

to cell faces that are already joined to other active cells. The default is NO. The

alternative is used in hydrogeological modelling to allow aquifers to be joined to the

interiors of simulation grids in simulations of groundwater propagation within fractures

of insignificant size compared to the grid cells.


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Fetkovich Aquifers

Figure 112: Fetkovich aquifer definition

Fetkovich aquifers are defined in the SOLUTIONsection.

The aquifer properties are defined by the AQFET or AQUFETP keywords

Fetkovich aquifers are attached to the oil zone by NNCs defined in the AQUANCON

keyword

The total number of analytical aquifers and aquifer NNCs is defined in AQUDIMSinRUNSPEC

Fetkovich and Carter-Tracy aquifers cannot be used in the same run

( )aiiaiwwi hhgPPJQ += (

The aquifer inflow is:

From material balance the aquiferpressure response is

( )aawta PPVCW = 00

Integrating these gives

Define Fetkovich Aquifers with

RUNSPEC

AQUDIMS

SOLUTION

AQUFET

--or

AQUFETP

AQUANCON

( )( ) ( )

+=

0

0exp1

wt

wtaiiaiai

VCtJ

VCtJ

hhgPPJQ


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Fetkovich AquifersFetkovich aquifers are based on a pseudo-steady state productivity index and material

balance between aquifer pressure and cumulative influx. The flow is modelled by the

equations in Figure 112 where

the subscripts a and i denote the aquifer and grid cell i, respectively.

Qaiis the inflow rate from aquifer to cell i

Jwis the aquifer productivity index;

iis the area fraction for cell i;

Pais the aquifer pressure at time t

Pi is the cell pressure at time t

is the aquifer water density

hiand hathe cell depth and aquifer datum depth, respectively

Waiis the cumulative influx from aquifer to cell i.

Ctis the total aquifer compressibility

Vw0is the initial aquifer volume

Pa0is the initial aquifer pressure

The aquifer flow in Figure 112 is very similar to the familiar well inflow performance

equation. The relationship of aquifer to reservoir is very similar to the relationship of

reservoir to well. Solution of the radial diffusivity equation in which the well is treated

like a reservoir whilst the reservoir is treated like an aquifer provides results analogous

to the familiar results obtained for wells. The consequence is that, given the same

boundary conditions, the aquifer PI is virtually identical in form to a well PI. Fetkovich

aquifers can effectively represent a wide range of aquifer types from the steady state

infinite aquifer which provides constant pressure support to the pot aquifer, which is

small compared to the reservoir and whose behaviour is determined by the reservoir

influx. If the aquifer has a large time constant, it responds slowly to variations in

reservoir pressure and the behaviour approaches that of a steady state aquifer. If the PI

is large so that the time constant is small, the behaviour approaches that of a pot

aquifer which is close to pressure equilibrium with the reservoir at all times. The topic

is also discussed in the ECLIPSE 100 TECHNICAL APPENDICES.

Fetkovich aquifers can be specified using in two ways

AQUFETis used to specify a single aquifer connected to one reservoir face:AQUFET

--1 2 3 4 5 6


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--Datum Initial Initial rock+ PI PVTW

--depth pressure volume water table

-- @ datum compressibility No.

--7 8 9 10 11 12 13 14

--I1 I2 J1 J2 K1 K2 Face Initial

-- Salt concn

AQUFETPand AQUANCONare used to specify multiple Fetkovich aquifers and/or aquifers

connected to more than one reservoir face.

AQUFETP

--1 2 3 4 5 6

--Id Datum Initial Initial rock+ PI

-- depth pressure volume water

-- @ datum compressibility

--7 8

--PVTW Initial

--table No. Salt concn

AQUANCON

--1 2 3 4 5 6 7 8 9

--Id I1 I2 J1 J2 K1 K2 Face Influx

-- coefficient

--10 11

--Influx Connection

--coefficient option

--multiplier

AQUFETPis followed by up to NANAQUrecords of analytical aquifer data, where NANAQU

is defined in AQUDIMS in the RUNSPEC section. Refer to the section titled Numerical

Aquifers for a discussion on the individual items in each record.

AQUANCONspecifies the connection data for the aquifer(s). The items that are common to

theAQUCON keyword are discussed in the section on Numerical Aquifers. The aquifer

influx coefficient determines the total communication between the aquifer and cells to


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which it is joined. The default for each cell is its face area. The influx coefficient

multiplier may be applied to the influx coefficient of each aquifer-cell connection.


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Carter-Tracy Aquifers

Figure 113: Carter-Tracy aquifer definition

Carter-Tracy aquifers are defined in the SOLUTIONsection.

The aquifer properties are defined by the AQCT keyword

The pressure response is defined by an influence function, which may be entered

with the AQUTABkeyword.

Carter-Tracy aquifers are attached to the oil zone by NNCs defined in the AQUANCONkeyword

The total number of analytical aquifers and aquifer NNCs is defined in AQUDIMSin

RUNSPEC

Fetkovich and Carter-Tracy aquifers cannot be used in the same run

a

twc

kc

rCT

1

20

=

The main parameters governingCarter-Tracy aquifer behaviour arethe time constant T

c, which is t/t

D ,

and the aquifer influx constant .

202 rChc t=

The pressure drop at the aquifer

boundary is

)(0 DDa

a tPQ

PP

=

( ) ( )[ ]{ }tPittPibaQ iai +=

and the average flow rate to cell ifrom time t to t+ t is

To define Carter-Tracy aquifers useRUNSPECAQUDIMS

SOLUTIONAQUCTAQUTABAQUANCON

Influence

Function


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Carter-Tracy AquifersCarter-Tracy aquifers use tables of dimensionless time tdversus dimensionless pressure

Pd(td)to determine the amount of influx. The model approximates a fully transient

model. Limiting cases of the Carter-Tracy aquifer model can represent steady state or

pot aquifers. It has the advantage that intermediate behaviour can also be simulated,

i.e. an aquifer which behaves as a steady state aquifer at first but gradually approaches

the behaviour of a pot aquifer. The flow is modelled by the equations in Figure 113,

where

kais the aquifer permeability

is the aquifer porosity

wis aquifer water viscosity

Ctis the total aquifer compressibility

r0is the aquifer inner radius (or reservoir outer radius)

c1, c2are constants

h is aquifer thickness

is the angle subtended by the aquifer boundary to the centre of the reservoir (the

influence angle)

Qais aquifer flow rate

Pa0is the initial aquifer pressure

P is the average water pressure at the aquifer/reservoir boundary

i is the area fraction

tDand PDare dimensionless time and pressure, respectively

a, b are functions of time, , Tc, dimensionless pressure.

The topic is discussed in more detail in the ECLIPSE 100 TECHNICAL APPENDICES.

Carter-Tracy aquifers are specified using AQUCT,AQUTABand AQUANCON.

AQUCT

--1 2 3 4 5 6 7

--Id Datum Initial K rock+ External

-- depth pressure water radius

-- @ datum comp.

--8 9 10 11 12

--Thickness Influence PVTW Influence Initial

-- angle table No. fn table No. salt concn


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The radius is the external radius of the reservoir, or the internal radius of the aquifer.

The influence angle is the angle subtended by the aquifer at the aquifer-reservoir

boundary. Item 11 is a pointer (default value 1) to an influence function defined in

AQUTAB.AQUTABconsists of columns of dimensionless time and dimensionless pressure.

Table number 1 is the default and cannot be altered by the user. It represents a constant

rate terminal aquifer as given by van Everdingen and Hurst.


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This page is intentionally blank


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Flux Aquifers

Figure 114: Flux aquifer definition

Flux aquifers are defined in the SOLUTIONsection.

The aquifer has no properties as such

The flow rate is specified directly by the user. It may be negative, representing flux

out of the reservoir.

As regards theRUNSPEC

section, flux aquifers are treated the same as analyticalaquifers.

Flux aquifers are defined using AQUFLUX

Connections to the reservoir are created using AQUANCON.

Flux aquifers cannot be used with the AQUFETkeyword.

iiaai mAFQ =

A constant flux aquifer has

water flow

To define a flux aquifer use

RUNSPECAQUDIMS

SOLUTION

AQUFLUXAQUANCON

SCHEDULEAQUFLUX


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Flux AquifersThe water flow Qaiinto grid cell i from a flux aquifer is as shown in Figure 114 where

Fais the flux

Aithe area of the connecting cell block, from the cell geometry

miis an aquifer influx multiplier.

The AQUFLUX keyword contains up to NANAQU records of data, each consisting of an

aquifer identification number and the flux. The flux can be modified during the

simulation by re-entering AQUFLUXin the SCHEDULEsection.


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Output Control

Figure 115: Output control

Summary quantities are requested in the normal manner

RPTGRIDcan output numerical aquifer definitions and NNCs.

RPTSCHEDcan output Fetkovich or Carter-Tracy aquifer status

RPTSOLcan output analytic aquifer data and individual connection data

Summary Quantities AAQR, FAQR, FAQT, AAQT, AAQP

Print file Data RPTGRID, RPTSCHED, RPTSOL


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Output Control

The AAQPaquifer pressure summary quantity applies only to Fetkovich aquifers

OtherSUMMARYquantities report instantaneous and cumulative aquifer influxes. The AQUNUM and AQUCON mnemonics in RPTGRID output numerical aquifer

definitions and NNCs, respectively, in tabular form to the PRTfile

The AQUCTor AQUFETor AQUFETPmnemonics in RPTSCHEDoutput status reports on

Fetkovich or Carter-Tracy aquifers in tabular form to the PRTfile.

The AQUFET or AQUFETP or AQUCT or AQUANCON mnemonics of RPTSOL output

analytic aquifer data to the PRTfile in tabular form. If any of these is set to 2 (e.g.

AQUFET=2) then additional data on the aquifer-grid cell connections is written to

thePRTfile.

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