GWF BCF6 PACKAGE. Computes the conductance components of the finite-difference equation which determine flow between adjacent cells. Computes the terms that determine the rate of water movement to and from storage. - PowerPoint PPT Presentation
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BCF6 1 GWF BCF6 PACKAGE • Computes the conductance components of the finite-difference equation which determine flow between adjacent cells. • Computes the terms that determine the rate of water movement to and from storage. • Calculates flow correction terms that are added to the difference equations when an underlying aquifer becomes partially saturated. • Requires the node be located in the center of the cell, hence the name Block-Centered Flow.
Transcript
BCF6 1
GWF BCF6 PACKAGE
• Computes the conductance components of the finite-difference equation which determine flow between adjacent cells.
• Computes the terms that determine the rate of water movement to and from storage.
• Calculates flow correction terms that are added to the difference equations when an underlying aquifer becomes partially saturated.
• Requires the node be located in the center of the cell, hence the name Block-Centered Flow.
BCF6 2
GWF BCF6 PACKAGE
Basic Conduction Equations Review
• Darcy’s law define one-dimensional flow
• Conductance is defined as,
• Darcy’s law can be written,
L
hhKAQ 12
L
TW
L
KAC
)( 12 hhCQ
BCF6 3
GWF BCF6 PACKAGE• For a set of conductances
arranged in series, the inverse of the equivalent conductance equals the sum of the inverses of the individual conductances
• When there are only 2 sections, the equivalent conductance reduces to,
Note: the above is call the harmonic mean of the conductances C1 and C2
n
i iCC 1
11
1 2
1 2( )
C CC
C C
BCF6 4
GWF BCF6 PACKAGEHorizontal Conductance under confined
conditions• Conductances are defined between nodes
of adjacent cells rather than within a cell • CR (conductance along rows) and CC
(conductance along columns) are calculated between adjacent horizontal nodes.
• The subscript ½ is used designate conductance between nodes (e.g. CRij+½k
represents the conductance between nodes i,j,k and i,j+1,k)
• Transmissivity is uniform over a cell, but may vary from cell to cell.
• If the transmissivity of both cell is zero, the conductance between the nodes of the two cells is set to zero
Applying C1C2/(C1+C2)
BCF6 5
GWF BCF6 PACKAGEHorizontal Conductance Under Water
Table Conditions• If a model layer is confined, the
horizontal conductance will be constant
• If a layer is unconfined or potentially unconfinined, new values of the horizontal conductance must be calculated as the head fluctuates.
• The transmissivity is calculated (product of hydraulic conductivity and saturated thickness).
• When the head drops below the aquifer bottom, the cell is considered dewatered.
• Transmissivity within a cell in the row direction is calculated,if HNEWijk ≥ TOPijk
then TRijk= (TOPijk-BOTijk)HYRijk
if TOPijk > HNEWijk > BOTijk
then TRijk= (HNEWijk-BOTijk)HYRijk
if HNEWijk ≤ BOTijk
then TRijk= 0HYRijk is the row directed hydraulic
conductivityTOPijk is the top elevation of cell i,j,kBOTijk is the bottom elevation of cell i,j,k
BCF6 6
GWF BCF6 PACKAGEVertical Conductance Formulation
• The vertical interval between two nodes, i,j,k and i,j,k+1, may be considered to contain n geohydrologic layers.
• Let the layers have hydraulic conductivity K,1, K2 … Kn and thickness Δz1, Δz2, … Δzn.
• The map area of the cells around the nodes i,j,k and i,j,k+1 is DELRj×DELCi
• The vertical conductance of an individual geohydrologic layer g is,
g
ijgg z
DELCDELRKC
BCF6 7
GWF BCF6 PACKAGE• The equivalent vertical conductance, Cij+½k, for the vertical interval between i,j,k
and i,j,k+1,
• Rearranging above terms gives,
• The above quantity is called the vertical leakance and is designated Vcontij+½k
n
g g
g
ij
n
g
g
ijg
n
g gkij K
z
DELCDELRz
DELCDELRKCC 111
1111
21
n
g g
gij
kij
K
zDELCDELR
C
1
121
n
g g
gkij
K
zVcont
1
121
BCF6 8
GWF BCF6 PACKAGE• Vcont is the input to the BCF file and is calculated
externally to the model.• Vcont values are read as two dimensional arrays for
each layer.• Each value of Vcontij is the vertical leakance for the
interval between cell i,j,k and cell i,j,k+1 (for the interval between the layer for which the array is read, and the layer below it).
• The Vcont array is not read for the lowermost layer.
BCF6 9
GWF BCF6 PACKAGECase 1-Single Geohydrologic Units
If i,j,k and i,j,k+1 both fall with in a single hydrogeologic unit, having a vertical hydraulic conductivity Kzij that is uniform within the two cell region, then
where Δzk+½ is the vertical distance between nodes and is the sum of Δvk/2 and Δvk+1/2 and Δv represents the layer thickness.
If i,j,k and i,j,k+1 fall at the midpoint of two adjacent hydrogeologic units, each having its own vertical hydraulic conductivity that is uniform within its cell region, then
where Δvk is the thickness of layer k,
Δvk11 is the thickness of layer k+1,
Kijk is the hydraulic conductivity of cell i,j,k
Kijk+1 is the hydraulic conductivity of cell i,j,k+1
1
122
121
ijk
k
ijk
kijk
Kz
v
Kz
vVcont
BCF6 11
GWF BCF6 PACKAGECase 3-Quazi 3D
Two aquifers separated by a confining unit,
Where Δzu, Δzc, ΔzL are the thickness of the upper, confined, and lower units, respectively, and Kzu, Kzc, KzL are the their hydraulic conductivities.If Kzc is much smaller than Kzu and KzL then the above equation reduces to
This formulation is called the quasi-3D approach.
Lc
c
u
ijk
Kz
z
Kz
z
Kz
zVcont
Lu22
21
1
c
cijk z
KzVcont
2
1
BCF6 12
GWF BCF6 PACKAGE
Vertical Flow Calculations Under Dewatered Conditions• The basic flow equation for cell i,j,k is,
)1()(
)()(
)()(
)()(
11
11
11
21
21
21
21
21
21
t
hvcrSS
WhP
hhCVhhCV
hhCChhCC
hhCRhhCR
ijkkijijk
ijkijkijk
ijkijkijkijkijkijk
ijkjkijkiijkjkijki
ijkkijkijijkkijkij
This term gives the flow into cell i,j,k through its lower face
BCF6 13
GWF BCF6 PACKAGECell ijk+1 Dewatered
• Assume cell i,j,k and the confining unit are fully saturated, then head at upper surface of confining layer is hijk.
• Below the confining zone is unsaturated, and the head at the base of the confining unit is atmospheric, and the head there is
hijk+1 = Topijk+1
and the flow through the confining bed is given by,
instead of the term from equation 1,
To avoid asymmetry, we leave equation 2 in equation 1 and add,
to the RHS of the flow equation. (notice the iteration parameter is set at the previous iteration, n-1)
)( 11,1
21
ijknm
ijkijkcn TophCVq
)2()( 121
21
mijk
mijkijkijk hhCVq
)( 121
21
mijkijkijkijk hTopCVq
BCF6 14
GWF BCF6 PACKAGECell ijk Dewatered
• A correction must also be applied for the dewatered cell itself.
• Let the dewatered cell be i,j,k an consider flow into i,j,k from overlying cell i,j,k-1.
• The computed flow into i,j,k from the cell above is,
whereas the actual flow into the cell is,
This time we add,
to the RHS of the flow equation. (notice the iteration parameter is set to present iteration because the correction term does not affect the symmetry—it is added only to a diagonal element of equation 1)
)( 121
21
mijk
mijkijkijk hhCVq
)( 121
21 ijk
mijkijkijk TophCVq
)(21
mijkijkijkc hTopCVq
BCF6 15
GWF BCF6 PACKAGEStorage Formulation
• There are two types of layers that are considered,– Layer whose storage values
remain constant– Layer whose storage properties
may convert from confined to unconfined or vice-versa.
• If a layer’s storage values remain constant, the rate of accumulation of water in the cell, ΔV/Δt, is given by,
where
SSijk(Δrj Δci Δvk) confinedSC1ijk =
SY(Δrj Δci ) unconfined
• The SC1ijk is called the primary storage capacity for cell i,j,k.
Note: The primary storage capacity is adequate if the water level in the cell remains either above the top of the cell or below the top of the cell through out the simulation.
)(
)(1
1
1
mm
mijk
mijk
ijk tt
hhSC
t
V
BCF6 16
GWF BCF6 PACKAGEStorage Term Conversion
• During any time step, there are four possible storage conditions for each cell– The cell is confined for the entire time step– The cell is unconfined for the entire time step– The cell converts from confined to unconfined– The cell converts from unconfined to confined
• The following expression for the rate of accumulation in storage in cell i,j,k is used,
where Top is the elevation of the top of the model cell, SCA is the storage capacity in effect in the cell at the start of the time step, and SBC is the “current” storage capacity (current iteration)
)(
)()(
1
1
mm
mijkijkijk
mijk
tt
hTopSCATophSCB
t
V
BCF6 17
GWF BCF6 PACKAGEStorage Values
1. If and
then,
SCA = SSijk(Δrj Δci Δvk) and
SCB = SSijk(Δrj Δci Δvk)
giving,
ijkmijk Toph ijk
mijk Toph 1
1mijkh
mijkh
ijkTop
i,j,k
)(
)()(
1
1
mm
mijk
mijk
kijijk tt
hhvcrSS
t
V
BCF6 18
GWF BCF6 PACKAGEStorage Values
2. If and then,
SCA = SY(Δrj Δci) and
SCB = SY(Δrj Δci)
giving,
ijkmijk Toph 1
ijkmijk Toph 1m
ijkh
mijkh
ijkTop
i,j,k
)(
)()(
1
1
mm
mijk
mijk
ij tt
hhcrSY
t
V
BCF6 19
3. If and
then,
SCA = SSijk(Δrj Δci Δvk) and
SCB = SY(Δrj Δci)
giving,
ijkmijk Toph 1
ijkmijk Toph 1m
ijkh
mijkh
ijkTop
GWF BCF6 PACKAGE
)(
))(())((
1
1
mm
mijkijkkijijkijk
mijkij
tt
hTopvcrSSTophcrSY
t
V
BCF6 20
GWF BCF6 PACKAGE
1mijkh
4. If and
then,
SCA = SY(Δrj Δci) and
SCB = SSijk(Δrj Δci Δvk)
giving,
mijkh
ijkTopijkmijk Toph 1
ijkmijk Toph
)(
))(())((
1
1
mm
mijkijkijijk
mijkkijijk
tt
hTopcrSYTophvcrSS
t
V
BCF6 21
GWF BCF6 PACKAGE• As part of the simulation of unconfined aquifers and aquifers that can
convert between confined and unconfined, MODFLOW can change a variable-head cell to a no-flow cell.
• If the saturated thickness becomes zero, MODFLOW converts the cell to no-flow—called “drying” the cell.
• Based on heads in surrounding cells, MODFLOW will attempt to wet cells that are dry.
• The user can specify the cells for which wetting is attempted.• Wetting capability useful for,
– Recovery of water levels when wells are turned off,– Modeling mounds of recharge water from irrigation application, and– Situations where cells incorrectly go dry (convert to no-flow) as part of the
• The transmissivity values are then used to calculate row and column conductances.• Vertical conductance is constant till the cell becomes dry, at which point is changed
to zero.• When a cell becomes dry, IBOUND is set to zero, all conductance to the cell are
set to zero, and head is set to a very large value to serve as a visual indicator.
BCF6 23
GWF BCF6 PACKAGE
• A dry cell is allowed to become wet if the head from the previous iteration in a neighboring cell is greater than or equal to a turn-on threshold,
TURNON = BOT + THRESHwhere,
BOT is the bottom elevation of a dried-out cell,
THRESH is a user-specified constant called the wetting threshold
BCF6 24
GWF BCF6 PACKAGE• There are two options to select
which neighboring are checked to see if the turn-on threshold has been reached,– Check the cell immediately below
the dry cell and the four horizontally adjacent cells, or
– Check only the cell immediately below the dry cell.
• If the neighboring cell is either no-flow or constant-head, then that cell is not checked for TURNON.
BOT
dry cell
hn
BOT + THRESH
hi
Option 1
BOT
BOT + THRESH
dry cell
Cell k
hk
cell k
cell icell n
hk
Option 2
BCF6 25
GWF BCF6 PACKAGE• When a cell is wetted, IBOUND for the cell is set to 1, the vertical conductance for
the cell are set to their origin values, and the head is set to either,h = BOT + WETFCT(hn – BOT)
orh = BOT + WETFCT(THRESH)
where,BOT is the bottom elevation of the dry cell,hn is the head at the neighboring cell that caused the cell to wet,
WETFCT is a user-specified constant called the wetting factor.
Note: The head assigned to a cell might exceed the wetting threshold of a neighboring dry cell, however a neighboring cell cannot become wet in the same iteration.
BCF6 26
GWF BCF6 PACKAGE• There is a non-uniqueness associated with the wetting routine
BOT
Threshold
h starting
h final
Remains Dry
h starting
h final
Remains Wet
The method of wetting and drying cells can cause problems with the convergence of iterative solvers used in MODFLOW.
BCF6 27
GWF BCF6 PACKAGEIBCFB—is a flag and a unit number
If IBCFCB > 0, it is a unit number to which cell-by-cell terms are written when SAVE BUDGET or a non-zero value for ICBCFL is specified in Output Control.IBCFCB = 0, cell-by-cell flow terms will not be written.IBCFCB < 0, cell-by-cell flows for constant-head cells will be written in the LIST FILE when SAVE BUDGET or a non-zero value for ICBCFL is specified in Output Control. Cell-by-cell flow to storage and between adjacent cells will not be written to any file.
BCF6 28
GWF BCF6 PACKAGEHDRY—is the head that is assigned to
cell that are converted to dry during simulation. HDRY is similar to HNOFLO, it is an indicator.
IWDFLG—is a flag that determines if the wetting capability is active.IWDFLG = 0, the wetting capability is
inactive.IWDFLG ≠ 0, the wetting capability is
active.WETFCT—is a factor that is included in
the calculation of head that is initially establish at a cell when it is converted from dry to wet.
BCF6 29
GWF BCF6 PACKAGEIWETIT—is the iteration interval for
attempting to wet cells. Wetting is attempted every IWETIT iteration (outer iterations if PCG). If IWETIT is 0, it is set to 1.
IHDWET—is a flag that determines which equation is used to define the initial head to cells that become wet:If IHDWET = 0, then h = BOT + WETFCT(hn − BOT)
If IHDWET ≠ 0, then h = BOT + WETFCT(WETDRY)
note: the absolute value of WETDRY is the wetting threshold (THRESH)
BCF6 30
GWF BCF6 PACKAGELtype—contains a combined code for each
layer that specifies both the layer type (LAYCON) and the method of computing interblock conductance.Values are two digit numbers.
The left digit defines method of calculating interblock transmissivity
0 or blank Harmonic mean1 Arithmetic mean2 Logarithmic mean
3 Arithmetic mean of saturated
Thickness and Logarithmic-mean hydraulic conductivity.
BCF6 31
GWF BCF6 PACKAGE
The right digit defines layer type (LAYCON)0 confined—Transmissivity and storativity of the layer are constant the entire simulation.1 unconfined—Transmissivity is calculated from hydraulic conductivity and saturated thickness. Storativity is constant. Type code valid only in first layer.2 confined/unconfined—Transmissivity is constant. Storativity may alternate between confined and unconfined values.3 confined/unconfined—Transmissivity is calculated from hydraulic conductivity and saturated thickness. Storativity may alternate between confined and unconfined values.
BCF6 32
GWF BCF6 PACKAGE
TRPY—is a one-dimensional real variable containing a horizontal anisotropy factor for each layer.It is the ratio of transmissivity or hydraulic conductivity along the column to the comparable values along the row.Set to 1.0 for isotropic values. There is a single value for each layer and is read using U1DREL.
BCF6 33
A subset of the following 2-Dimensional are used to describe each layer.
• The variable needed for each layer depends on LAYCON, whether the simulation has any transient stress periods, and if the wetting capability is active (IWDFLG≠0).
• If a variable is not needed , it must be omitted.
• All variables are read for layer 1 first, then layer 2 and so forth.
GWF BCF6 PACKAGE
BCF6 34
GWF BCF6 PACKAGESf1—is the primary storage. Read only if
there are one or more transient stress periods. For LAYCON=1, Sf1 will always be specific yield.For LAYCON=2 or 3, Sf1 will always be the storage coefficient.For LAYCON=0, Sf1 would normally be the storage coefficient. However, if it is assumed that the drawdowns in an unconfined aquifer always remain small compared to the saturated thickness (transmissivity does not vary with head), Sf1 may be specific yield.
BCF6 35
GWF BCF6 PACKAGE
Tran—is the transmissivity along the rows. Tran is multiplied by TRPY to obtain transmissivity along the columns.Read only for layers where LAYCON is 0 or 2
HY—is the hydraulic conductivity along the rows.HY is multiplied by TRPY to obtain transmissivity along the columns.Read only for layers where LAYCON is 1 or 3
BCF6 36
GWF BCF6 PACKAGE
Vcont—is the vertical hydraulic conductivity divided by the thickness from a layer to a layer below.Vcont is not specified for the bottom layer
Sf2—is the secondary storativity value.Read only for layers where LAYCON=2 or 3, and only if there are one or more transient stress periods.The secondary storativity value is always specific yield.
BCF6 37
GWF BCF6 PACKAGEWETDRY—is a combination of the wetting
threshold and a flag to indicated which neighboring cells can cause a dry cell to become wet.WETDRY<0, only the cell below the dry cell can cause the dry cell to become wet.WETDRY>0, the cell below the dry cell and the four horizontally adjacent cells can cause the dry cell to become wet.WETDRY=0, the cell can not be wetted.The absolute value of WETDRY is the wetting threshold.Read only if LAYCON=1 or 3 and IWDFLG≠0
