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.
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2. GWF BCF6 PACKAGE Darcy’s law define one-dimensional flow
Basic Conduction Equations Review
Darcy’s law define one-dimensional flow
Conductance is defined as,
Darcy’s law can be written,
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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
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4. GWF BCF6 PACKAGE Horizontal 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)
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5. GWF BCF6 PACKAGE Horizontal 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= 0
HYRijk is the row directed hydraulic conductivity
TOPijk is the top elevation of cell i,j,k
BOTijk is the bottom elevation of cell i,j,k
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6. Vertical Conductance Formulation
GWF BCF6 PACKAGE
Vertical 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,
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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
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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.
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9. Case 1-Single Geohydrologic Units
GWF BCF6 PACKAGE
Case 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.
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10. Case 2-Different Geohydrologic Layers
GWF BCF6 PACKAGE
Case 2-Different Geohydrologic Layers
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
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11. GWF BCF6 PACKAGE Case 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.
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12. Vertical Flow Calculations Under Dewatered Conditions
GWF BCF6 PACKAGE
Vertical Flow Calculations Under Dewatered Conditions
The basic flow equation for cell i,j,k is,
This term gives the flow into cell i,j,k through its lower face
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13. GWF BCF6 PACKAGE
Cell 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)
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14. GWF BCF6 PACKAGE Cell 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)
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15. GWF BCF6 PACKAGE Storage 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) confined
SC1ijk =
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.
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16. Storage Term Conversion
GWF BCF6 PACKAGE
Storage 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)
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17. GWF BCF6 PACKAGE i,j,k Storage Values If and then,
SCA = SSijk(Δrj Δci Δvk) and
SCB = SSijk(Δrj Δci Δvk)
giving,
i,j,k
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18. GWF BCF6 PACKAGE i,j,k Storage Values 2. If and then,
SCA = SY(Δrj Δci) and
SCB = SY(Δrj Δci)
giving,
i,j,k
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19. GWF BCF6 PACKAGE SCA = SSijk(Δrj Δci Δvk) and SCB = SY(Δrj Δci)
3. If and then,
SCA = SSijk(Δrj Δci Δvk) and
SCB = SY(Δrj Δci)
giving,
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20. GWF BCF6 PACKAGE 4. If and then, SCA = SY(Δrj Δci) and
SCB = SSijk(Δrj Δci Δvk)
giving,
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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 iterative solution process.
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22. GWF BCF6 PACKAGE Recall that MODFLOW calculates transmissivity as
if HNEWijk ≥ TOPijk
then TRijk= (TOPijk-BOTijk)HYRijk
if TOPijk > HNEWijk > BOTijk
then TRijk= (HNEWijk-BOTijk)HYRijk
if HNEWijk ≤ BOTijk
then TRijk= 0
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.
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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 + THRESH
where,
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.
Option 1 hn
dry cell
cell n
cell i hk hi
BOT + THRESH
BOT
cell k
Option 2
dry cell hk
BOT + THRESH
BOT
Cell k
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) or
h = 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.
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26. GWF BCF6 PACKAGE
There is a non-uniqueness associated with the wetting routine
Threshold
h starting
h final
h final
BOT
h starting
Remains Dry
Remains Wet
The method of wetting and drying cells can cause problems with the convergence of iterative solvers used in MODFLOW.
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27. GWF BCF6 PACKAGE IBCFB—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.
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28. GWF BCF6 PACKAGE
HDRY—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 PACKAGE
IWETIT—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)
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30. GWF BCF6 PACKAGE
Ltype—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 mean
Arithmetic mean
2 Logarithmic mean
3 Arithmetic mean of saturated Thickness and Logarithmic-mean hydraulic conductivity.
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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.
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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.
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33. GWF BCF6 PACKAGE
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.
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34. GWF BCF6 PACKAGE
Sf1—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.
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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
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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.
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37. GWF BCF6 PACKAGE
WETDRY—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
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38. GWF BCF6 PACKAGE
BCF Data Set
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