LAYER-PROPERTY FLOW PACKAGE The Layer-Property Flow (LPF) Package is an internal flow package like the BCF Package, thus the two should not be used simultaneously. Computes conductance components and the rate of water movement into and out of storage. Computes the conductance components of the finite-difference equation which determine flow between adjacent cells. Calculates flow correction terms that are added to the difference equations when an underlying aquifer becomes partially saturated. Requires the nodes be located at the center of cells like BCF. The differences between LPF and BCF are primarily in the input data that the user specifies. All input data to LPF that define hydraulic properties are independent of cell dimensions. LPF always reads hydraulic conductivities (including vertical) and calculates conductances between nodes. LPF1

LAYER-PROPERTY FLOW PACKAGE Basic Conduction Equations Review Darcy’s law define one-dimensional flow Conductance is defined as, Darcy’s law can be written, LPF1

LAYER-PROPERTY FLOW 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 LPF1

LAYER-PROPERTY FLOW PACKAGE Horizontal Conductance 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) Applying C1C2/(C1-C2) LPF1

LAYER-PROPERTY FLOW PACKAGE 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. Transmissivity is calculated as, TRijk = THICKijk×HKijk and, TCijk = THICKijk×HKijk×HANIijk where, HKijk is the hydraulic conductivity of cell i,j,k in the row direction. HANIijk is the ratio of hydraulic conductivity along the columns to the hydraulic conductivities along the rows. THICKijk is the saturated thickness. Note that the anisotropy can vary from cell-to-cell LPF1

LAYER-PROPERTY FLOW PACKAGE If a layer is confined, the thickness of the cell is, THICKijk= (TOPijk-BOTijk) where, TOPijk is the top elevation of cell i,j,k BOTijk is the bottom elevation of cell i,j,k If a layer is designated as convertible, the saturated thickness calculated, if HNEWijk ≥ TOPijk then THICKijk= (TOPijk-BOTijk) if TOPijk > HNEWijk > BOTijk then THICKijk= (HNEWijk-BOTijk) if HNEWijk ≤ BOTijk then THICKijk= 0 LPF1

LAYER-PROPERTY FLOW PACKAGE At the start of every iteration for solving the flow equation, cell transmissivity values (TR and TC) are recalculated as a product of hydraulic conductivity and saturated thickness, then conductance is recalculated. In LPF, the convertible formulation should be used to represent a layer 1 water table. The top elevation for layer 1 with a water table should be the land surface elevation. LPF1

LAYER-PROPERTY FLOW PACKAGE Alternative Approaches for Calculating Horizontal Conductance When transmissivity varies linearly between nodes, the interblock transmissivity is given by, For an unconfined homogeneous aquifer with a flat bottom, the interblock transmissivity is given by the arithmetic mean, LPF1

LAYER-PROPERTY FLOW PACKAGE For an unconfined aquifer with a flat bottom and with hydraulic conductivity varying linearly between nodes, the interblock transmissivity is given by, Note that the above equation reduces to when the aquifer is homogeneous. LPF1

LAYER-PROPERTY FLOW PACKAGE Vertical Conductance Vertical conductance is given by, where, VKijk is the vertical hydraulic conductivity of cell i,j,k, and THICKijk is the saturated thickness of the cell i,j,k. LPF1

LAYER-PROPERTY FLOW PACKAGE Vertical Conductance Vertical conductance is given by, where, VKCDCB is the vertical hydraulic conductivity of the semi-confining unit between cells i,j,k, and i,j,k+1, and THICKCB is the saturated thickness of the semi-confining unit. LPF1

LAYER-PROPERTY FLOW 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 LPF1

LAYER-PROPERTY FLOW PACKAGE 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) 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, LPF1

LAYER-PROPERTY FLOW 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 maybe set to present iteration because the correction term does not affect the symmetry—it is added only to a diagonal element of equation 1, however, it may not be if diagonal dominance is an issue) LPF1

LAYER-PROPERTY FLOW 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. LPF1

LAYER-PROPERTY FLOW PACKAGE Recall that MODFLOW calculates saturated thickness as if HNEWijk ≥ TOPijk then THICKijk= (TOPijk-BOTijk) if TOPijk > HNEWijk > BOTijk then THICKijk= (HNEWijk-BOTijk) if HNEWijk ≤ BOTijk then THICKijk= 0 The saturated thickness values are then used to calculate transmissivities, TRijk = THICKijk×HKijk and, TCijk = THICKijk×HKijk×HANIijk 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. LPF1

LAYER-PROPERTY FLOW 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 LPF1

LAYER-PROPERTY FLOW PACKAGE Option 1 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. hn dry cell cell n cell i hk hi BOT + THRESH BOT cell k Option 2 dry cell hk BOT + THRESH BOT Cell k LPF1

LAYER-PROPERTY FLOW 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. LPF1

LAYER-PROPERTY FLOW 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. LPF1

LAYER-PROPERTY FLOW 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. LPF1

LAYER-PROPERTY FLOW 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) LPF1

LAYER-PROPERTY FLOW PACKAGE Storage Values If and then, SCA = SSijk(Δrj Δci Δvk) and SCB = SSijk(Δrj Δci Δvk) giving, i,j,k LPF1

LAYER-PROPERTY FLOW PACKAGE Storage Values 2. If and then, SCA = SY(Δrj Δci) and SCB = SY(Δrj Δci) giving, i,j,k LPF1

LAYER-PROPERTY FLOW PACKAGE 3. If and then, SCA = SSijk(Δrj Δci Δvk) and SCB = SY(Δrj Δci) giving, LPF1

LAYER-PROPERTY FLOW PACKAGE 4. If and then, SCA = SY(Δrj Δci) and SCB = SSijk(Δrj Δci Δvk) giving, LPF1

LAYER-PROPERTY FLOW PACKAGE ILPFB—is a flag and a unit number If ILPFCB > 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. ILPFCB = 0, cell-by-cell flow terms will not be written. ILPFCB < 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. LPF1

LAYER-PROPERTY FLOW 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. NPLPF—is the number of LPF parameters. LAYTYP—is a layer type flag indicating whether a layer is confined or convertible. Use as many records as needed to enter a value for each layer. LAYTYP=0, the layer is simulated as confined. There is no conversion and no modification of saturated thickness. LAYTYP≠0, the layer is simulated as convertible. LPF1

LAYER-PROPERTY FLOW PACKAGE LAYAVG—contains a flag for each layer that defines the method of calculating interblock transmissivity. Use as many records as needed to enter a value for each layer. LAYAVG=0, use the harmonic mean. LAYAVG=1, use the logarithmic mean. LAYAVG=2, use arithmetic-mean of saturated thickness and the logarithmic-mean hydraulic conductivity LPF1

LAYER-PROPERTY FLOW PACKAGE CHANI—is a horizontal anisotropy flag that indicates whether horizontal anisotropy is a constant for a layer, or whether it can vary at each cell in a layer. CHANI≤0, the horizontal anisotropy can vary at each cell in the layer. A separate layer-data variable of horizontal anisotropy values (HANI) will be read in the input data. If any HANI parameters are used, then CHANI≤0 for all layers. CHANI>0, the horizontal anisotropy is constant for all cells in the layer and the HANI are not read. The hydraulic conductivity along the columns is the product of HK and CHANI. Set CHANI=1.0 for an isotropic layer. LPF1

LAYER-PROPERTY FLOW PACKAGE LAYVKA—is a layer type flag indicating whether variable VKA is the vertical hydraulic conductivity or the ratio of horizontal to vertical hydraulic conductivity. Use as many records as needed to enter a value for each layer. LAYVKA=0, indicates VKA is the vertical hydraulic conductivity. LAYVKA≠0, indicates VKA is the ratio of horizontal to vertical hydraulic conductivity, where the horizontal hydraulic conductivity is specified as HK in the input data (item 10). LPF1

LAYER-PROPERTY FLOW PACKAGE LAYWET—is a layer type flag indicating if wetting is active. Use as many records as needed to enter a value for each layer. LAYWET=0, indicates wetting is inactive for the layer. LAYWET≠0, indicates wetting is active for the layer. 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. LPF1

LAYER-PROPERTY FLOW 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) LPF1

LAYER-PROPERTY FLOW PACKAGE PARNAM—is the name of a parameter. This name can consist of 1 to 10 characters, and is not case specific. PARTYP—is parameter type to be defined. For the LPF Package, the allowed parameter types are: HK—horizontal hydraulic conductivity, HANI—horizontal anisotropy (CHANI≤0), VK—vertical hydraulic conductivity (LAYVKA=0), VANI—vertical anisotropy (LAYVKA≠0), SS—specific storage, SY—specific yield, VKCB—hydraulic conductivity of a quazi-three-dimensional confining layer LPF1

LAYER-PROPERTY FLOW PACKAGE Parval—is the parameter value. NCLU—is the number of clusters required to define a parameter. Each repetition of Item 4 is a cluster. There is usually only one cluster used to define a EVT parameter, but it is acceptable to have more. LPF1

LAYER-PROPERTY FLOW PACKAGE Layer—is the layer number to which a cluster definition applies. Mltarr—is the name of the multiplier array to be used to define cell values that are determined by parameters. The name NONE means that there is no multiplier array, and the cell value will be set to Parval. LPF1

LAYER-PROPERTY FLOW PACKAGE Zonarr—is the name of the zone array to be used to define cell values that are associated with a parameters. The name ALL means that there is no zone array, and all cells in the layer are associated with the parameter. LPF1

LAYER-PROPERTY FLOW PACKAGE IZ—is up to ten zone numbers (specified by spaces) the define the cells that are associated with a parameter. These values are not used if Zonarr is specified as ALL. Values can be negative, but not zero. The end of line, a zero value, or a non-numeric entry terminates the list of values. LPF1

LAYER-PROPERTY FLOW PACKAGE The following variables are either read by U2DREL, or they are defined through parameters. If a variable is defined through parameters, the variable itself is not read, but a single print code is read that determines the printing format. If any parameters of a give type are used, parameters must be used to define corresponding variables for all layers of the model. LPF1

LAYER-PROPERTY FLOW PACKAGE HK—is the hydraulic conductivity along the rows. HK is multiplied by HANI to obtain hydraulic conductivity along the columns. HANI—is the ratio of the hydraulic conductivity along the column to the hydraulic conductivity along the row. Read only if CHANI≤0. The hydraulic conductivity along the columns is the product of HK and HANI. LPF1

LAYER-PROPERTY FLOW PACKAGE VKA—is either vertical hydraulic conductivity or the ratio of horizontal to vertical hydraulic conductivity depending on the value of LAYVKA LAYVKA=0, VKA is the vertical hydraulic conductivity. LAYVKA≠0, VKA is the ratio of horizontal to vertical hydraulic conductivity. For this case, HK is divided by VKA to obtain vertical hydraulic conductivity, and values of VKA typically are greater than or equal to 1. LPF1

LAYER-PROPERTY FLOW PACKAGE SS—is the specific storage. Read only for a transient stress period. SY—is the specific yield. Read only for a transient stress period and if a layer is convertible. VKCB—is the vertical conductance of a quazi-three dimensional confining bed below a layer. VKCB can not be specified for the bottom layer LPF1

LAYER-PROPERTY FLOW 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 LAYTYP≠0 and LAYWET≠0 LPF1

LAYER-PROPERTY FLOW PACKAGE LPF1