aquifer modelling.pdf
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AQUIFER MODELLING
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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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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.