Open-File Report 2006–1104
U.S. GEOLOGICAL SURVEY
Open-File Report 2006–1104
MODFLOW is a computer program that simulates three-dimensional ground-water flow through porous media using the finite-difference method (McDonald and Harbaugh, 1984, 1988). Each process of MODFLOW-2000, the most recent version of MODFLOW, solves an equation with the exception of the Global (GLO) Process. The five processes of MODFLOW-2000 are: Global (GLO) Process, Ground-Water Flow (GWF) Process, Observation (OBS) Process, Sensitivity (SEN) Process, and Parameter-Estimation (PES) Process.
In the GLO Process, input arrays define the physical system including the altitude and spatial discretization. Input data files also describe the model grid geometry, layers, time discretization, and program operation. Data describing the physical properties of the media and the ground-water flow system are input through the various packages of the GWF Process. Finite-difference equations are solved by the GWF Process to compute a hydraulic head solution based on the active processes and packages. Simulated values may be compared to observations in the OBS Process. Various statistics and sensitivities to the observations are calculated in the OBS Process. Upon user initiation, the sensitivity equation for hydraulic heads with respect to each parameter is calculated throughout the model grid by the SEN Process. If both the OBS and SEN Processes are active, observation sensitivities may be calculated. With the PES Process active, parameter estimation may determine parameter values based on optimization of a weighted least-squares objective function that evaluates the fit of simulated values to the observations.
The GLO Process controls overall program flow, opens files, and reads global data such as space and time discretization. The GLO Process has four input files: name, discretization, multiplier, and zone files. Instructions for preparing the name file, and explanations for the variables read in all other files except as noted, are included in the Input Instructions section of Harbaugh and others (2000).
The name file controls the capabilities of MODFLOW-2000 utilized during a model simulation. The name file lists most of the files used by the GLO, OBS, SEN, and PES Processes. MODFLOW capabilities are activated by listing the appropriate file type along with the file name and unit number. A process or file is not active if a “#” sign appears in the first column of the line in the name file (name.txt).
Both space and time information are discretized using information from the Discretization File, specified as file type “DIS”. The length and time units of the model can be specified in the discretization file. In the DVRFS model, these units are meters and days.
Information in the DIS file defines the physical size of the finite-difference grid. The model grid boundary is rectangular. Although the grid can be distorted vertically, model layers cannot pinch out to zero thickness.
The DVRFS model grid is oriented north-south and consists of 194 rows and 160 columns with a constant grid-cell spacing of 1,500 by 1,500 m (fig. 2 and table 2). Each of the 16 model layers generally range in thickness from 50 to 300 m with model thickness generally increasing with depth (Faunt and others, 2004b, table F-1, p. 268). The upper model layers simulate relatively shallow saturated ground-water flow primarily through basin-fill sediments, volcanic rocks, and adjacent mountain ranges. Lower model layers predominantly simulate deep ground-water flow through the regional carbonate-rock aquifer beneath the basin fill and mountain ranges.
Time discretization also is specified in the DIS file. The fundamental component of time discretization is the time step. Time steps are aggregated into stress periods. Time dependent input data can vary every stress period. Individual stress periods within a single simulation may be either transient or steady state. For each stress period, the user specifies the stress period length, the number of time steps, and the multiplier for the length of successive time steps. For transient stress periods, the number of time steps and the time step multiplier can affect the accuracy of the solution. The DVRFS model is structured with an initial steady-state stress period followed by 86 transient stress periods, each one year in duration. Each yearly stress period has two time steps. Simulations with more time steps did not improve model accuracy (Faunt and others, 2004b, p. 266).
The multiplier file, specified as file type “MULT”, defines multiplier arrays for calculation of model-layer characteristics from parameter values. Parameters are defined in section, “Use of Parameters in the Ground-Water Flow (GWF) Process.” The multiplier file may be used to construct new arrays by adding and multiplying existing arrays. For the DVRFS model, the MULT file is used to represent an infiltration array, which is a multiplier for the Recharge Package.
The zone file, specified as file type “ZONE,” defines arrays of different zones. Zone arrays may specify cells in a layer variable associated with a specific parameter. Parameters may be composed of either one or many zones. The DVRFS model uses many zone arrays to represent areas of similar porous media properties for a given hydrogeologic unit.
The GWF Process includes all aspects of solving the ground-water flow equation, including formulation of the finite-difference equations, data input, solving the resulting simultaneous equations, and model output. The GWF Process uses the finite-difference method in which the ground-water flow system is divided into a grid of cells. At the center of each cell, or node, the finite-difference method is used to define a set of simultaneous equations, which are solved for head. For steady-state stress periods, the storage term in the ground-water flow equation is set to zero.
With MODFLOW-2000, parameters may be used to define many of the numerous data values that must be specified for each model cell. A parameter is a single value that is used to determine data values for multiple cells. Parameter definition includes a parameter name and value, and the cells for which the input values are calculated using that parameter. The most common and direct approach for determining data values from parameters is to set the data value equal to the parameter value for multiple cells. In a more complex approach, data values are assigned for multiple cells with additive contributions from multiple parameters (Harbaugh and others, 2000, p. 14-19). The DVRFS model does not use additive parameters, but in some cases, such as the recharge parameters, the data value at a cell is equal to a parameter value times a value in a multiplication array. Cell-by-cell model input used in MODFLOW 2000 calculations is a combination of parameter values specified in the package in which the parameters are defined and parameter values specified in the SEN Process, and possibly values specified in the zone and/or multiplier arrays.
Harbaugh and others (2000) document the 15 packages included in the initial release of MODFLOW-2000. The DVRFS model includes 2 packages developed after the initial release of MODFLOW-2000: the internal flow package, HUF2, (Anderman and Hill, 2000, 2003) and the MNW1 Package (Halford and Hanson, 2002). The DVRFS model uses the Output Control Option and 8 GWF packages:
The Basic Package, specified as file type “BAS6”, defines the initial conditions and some of the boundary conditions of the model. Initial conditions consist of starting heads for each cell in all model layers. Boundary conditions are specified using the array IBOUND. The value of the IBOUND array is defined at each cell to indicate that (1) the cell is active and head in the cell will be calculated (for a variable-head cell, value = 1); (2) the cell is active and head should not change from a user-specified value (for a constant- or specified-head cell, value = –1); or (3) the cell is inactive and water cannot flow through the cell (for a no-flow cell, value = 0).
The Output Control Option, specified as file type “OC”, is used in conjunction with flags in other packages to write head, drawdown, and budget information into separate files for specified time steps and stress periods. The OC Option overwrites the default for writing heads and the water budget to the listing file.
In the DVRFS model, head, drawdown, and budget information is printed and saved for the initial steady-state stress period and for stress periods 44 and 86 of the transient simulation. Stress period 44 is a mid-simulation stress period and stress period 86 (representing 1997) is the next-to-last stress period. Head and drawdown information is written to data files HEADSOUT.txt (unit number 46) and DRAWDOUT.txt (unit number 47).
The Hydrogeologic-Unit Flow (HUF2) Package, specified as file type “HUF2”, is an internal-flow package as are the Layer-Property Flow Package (Harbaugh and others, 2000) and the Block-Centered Flow Package (McDonald and Harbaugh, 1988). Internal-flow packages calculate hydraulic conductance between cell centers. The HUF2 Package translates the hydrogeologic framework into flow model layers by optionally defining the geometry of the hydrogeologic units (HGUs) independently of flow model layers. In the DVRFS model, geometries of the HGUs are complex because of considerable structural deformation, which includes folding and faulting; model layers in the DVRFS model generally do not coincide with HGUs.
The HUF2 Package calculates effective hydraulic properties for all model layers based on hydraulic properties of the HGUs. The HUF2 Package determines which HGUs apply to a model layer for each flow model grid cell, and calculates model-layer horizontal hydraulic conductance using arithmetic averaging, vertical hydraulic conductance using harmonic averaging, and specific storage using arithmetic averaging (Anderman and Hill, 2000, p. 7-10).
Within a DVRFS model simulation, aquifer transmissivity was not changed as water levels declined because changes in the saturated thickness were small. This condition is documented and specified as the “confined” option in the HUF2 Package. Storage values were assigned to the top model layer, independent of HGU (Anderman and Hill, 2003, p. 4) using the parameter type SYTP. The storage coefficient of the top model layer is not multiplied by model layer thickness.
With the HUF2 Package, horizontal hydraulic conductivity can be simulated as decreasing with depth exponentially using a variable (KDEP) that represents the depth decay coefficient (Anderman and Hill, 2003 p. 16-18). Simulations with depth decay result in relatively more permeable values near the land surface and less permeable values at depth within a single HGU or in a portion (or zone) of an HGU. Values of the depth decay coefficients used in the DVRFS model result in substantially reduced hydraulic conductivity at depth for most HGUs (Faunt and others, 2004b, fig. F-35, p. 322).
Many fault zones contain a core of low permeability gouge, the locus of fault displacement, which acts as a barrier to ground-water flow. The capability to simulate thin, vertical, and low-permeability geologic features is part of the Horizontal-Flow Barrier Package (Hsieh and Freckleton, 1993), specified as file type “HFB6”. The HFB6 Package simulates a ground-water barrier, which is located on the boundary between two adjacent finite-difference cells in the same layer, by adjusting the hydraulic conductance between adjacent cells.
In the DVRFS model, oftentimes juxtaposition of HGUs effectively represented a flow barrier because hydraulic properties differed greatly between HGUs. Consequently, the barrier to flow did not need to be represented with a hydraulic barrier in the HFB6 Package. In some cases, important regional faults that influence ground-water flow were incorporated into the affected HGUs. For example, thrust faults can create a stratigraphic repetition of HGUs that can create sharp contrasts in hydraulic conductivity (Faunt and others, 2004b, p. 181). Structures interpreted as thin, vertical, low-permeability features were represented as 1 m wide (Faunt and others, 2004b, p. 269) using the HFB6 Package. HFB6 parameters with little influence on the simulation of head and discharge were removed from the model during calibration. Consequently, there are nine HFB6 parameters in the DVRFS model that represent eight important regional faults (Faunt and others, 2004b table F-15, p. 324 and fig. F-5, p. 273).
The Recharge Package, specified as file type “RCH”, simulates areally distributed recharge to the ground-water system from precipitation that percolates through the unsaturated zone to the ground-water system. Distributed recharge for the DVRFS model is based on a spatially distributed hydrologic model of net infiltration (Hevesi and others, 2003). The net infiltration model simulates a daily water balance in the unsaturated zone to a maximum depth of 6 m, the depth at which the seasonal effects of evapotranspiration become insignificant (San Juan and others, 2004, p. 115-118). Net infiltration is a reasonable approximation of ground-water recharge because most of the net infiltration and surface runoff that originates as precipitation eventually moves downwards through the unsaturated zone to recharge the ground-water system.
Net-infiltration results (Hevesi and others, 2003) were modified to create a recharge input for the ground-water system that balanced system discharges, the transmissive properties of the less permeable rock units at the water table, and the variable depth to the water table. Average annual net-infiltration was estimated with simulated net-infiltration from 1950 to 1999 (Hevesi and others, 2003). Based on the average annual net-infiltration rate and the relative permeability of rocks in the upper five flow model layers of the DVRFS model, a recharge input for the DVRFS model was created using nine zones and five recharge parameters. The nine zones were created by calculating the recharge rate and the relative permeability of rocks in the upper five flow model layers. In order to fit the recharge input to the observed discharge, recharge was adjusted using a recharge multiplication array and recharge parameters based on the nine zones (Faunt and others, 2004b, fig. F-35, p. 325 and table F-16, p. 326). The values of the five calibrated recharge parameters produced a total volume of recharge to the ground-water system that is not significantly different from the total volume of net-infiltration; for the entire model domain, 92 percent of the estimated net-infiltration was simulated as recharge.
Simulation of pumpage from wells with the Well (WEL1) Package (McDonald and Harbaugh, 1988) is limited to withdrawal at specified rates from individual cells. However, the Multi-Node Well (MNW1) Package (Halford and Hanson, 2002) simulates pumpage from wells with screens that span multiple layers, partially penetrating wells, and variations in intraborehole flow. In the DVRFS region, wells typically are completed with screens that span multiple aquifers and thus multiple layers in the flow model. The vast majority of pumping occurs from model layers 1 through 4 (Faunt and others, 2004b, fig. F-8, p. 278). The MNW1 Package uses hydraulic conductivity and thickness to portion well pumpage between model layers. The DVRFS water budget (Faunt and others, 2004b, table F-18, p. 331) includes a small component of simulated well pumpage inflows due to simulated irrigation-return flows and well injection. A minor contribution of the simulated inflows is from intraborehole flow between pumping nodes connecting model layers within a single well in the MNW1 Package.
Return flows were simulated because some return flow of pumpage by infiltration of excess irrigation, lawn water, or septic tank wastewater was anticipated. The magnitude and timing of return flow depends on the method (domestic, irrigation, or mining, public supply, and commercial) by which water is returned to the flow system (San Juan and others, 2004, p. 114-115). At each withdrawal point, return flow was estimated to be 20 percent of the estimated annual pumpage (Moreo and others, 2003) lagged by 7 years. All computed return flows were assumed to return to the water table at the location of the pumped well.
The Drain Package, specified as file type “DRN”, removes water from an aquifer through a head-dependent boundary (McDonald and Harbaugh, 1988). For a finite-difference cell specified as a drain cell, ground water is simulated as discharging when the simulated head in the cell rises above a specified drain altitude. When the head declines below the cell altitude, the drain is turned off. Simulated drain discharge is calculated as the drain conductance multiplied by the difference in altitude between the simulated head and the drain.
In the DVRFS model, evapotranspiration was simulated with the drain package for numerical stability. The altitude of the drain cells approximates the extinction depth for evapotranspiration (Faunt and others, 2004b, p. 271-278). Drain altitudes were set equal to 10 m below the lowest land-surface altitudes for each group of cells where natural discharge is observed (Faunt and others, 2004b, fig. F-7, p. 275 and table F-4, p. 276-277). Areas of natural discharge include wet playas, wetlands with free-standing water or surface flow, narrow drainages lined with riparian vegetation, and broad areas of phreatophytic shrubs and grass. For drain cells representing springs, the drain at that location is connected to the topmost occurrence of the lower carbonate-rock aquifer, located between model layers 1 and 10.
Data for flows emanating from springs and seeps were derived from spring-flow measurements (San Juan and others, 2004, p. 103-110). Evapotranspiration rates were measured at Ash Meadows, Death Valley, and Oasis Valley as a proxy for discharge because most ground water discharging from springs and seeps is evaporated or transpired locally and is accounted for in estimates of evapotranspiration. Discharge simulated through the drain cells is compared to measured values in the section, “Observation (OBS) Process.”
At cells defined as constant head, MODFLOW-2000 calculates flow into and out of the cell as needed to maintain the head. In the DVRFS model, lateral boundary flows are represented using the Constant-Head Boundary Package, specified as file type “CHD”. CHD cells were specified in boundary cells that are at or below the regional potentiometric surface (Bedinger and Harrill, 2004a, 2004b) to allow flow across the boundary (Harrill and Bedinger, 2004). In the DVRFS model, the model boundary is not affected by simulated pumpage, and heads in the constant-head boundary cells do not change over time (Faunt and others, 2004b, table F-2, p. 270). Flows simulated through the constant-head boundaries are compared against reasonable estimates of flow in the section, “Observation (OBS) Process.”
Solver Packages, part of the GWF Process, are used to arrive at the solution of the finite-difference equations at each time-step in a stress period. Many different solvers have been written for MODFLOW (Hill, 1990; Harbaugh, 1995; Harbaugh and others, 2000). The DVRFS model is archived with the Preconditioned Conjugate-Gradient (PCG2) (Hill, 1990) solver, specified with file type “PCG2”. Two additional solvers, Link to Algebraic Multigrid (LMG) solver (Mehl and Hill, 2001) and the Geometric Multigrid (GMG) solver (Wilson and Naff, 2004), both recently developed, typically are 2 to 25 times faster than the PCG (Wilson and Naff, 2004) solver. When used with the DVRFS model, the LMG solver was faster than the GMG solver for steady-state simulations but slower for transient simulations. Due to licensing restrictions, the distribution of the LMG solver is limited (URL: http://water.usgs.gov/nrp/gwsoftware/modflow2000/modflow2000.html, accessed December 22, 2005). For steady-state simulations, the GMG solver is often much faster than the PCG2 solver.
The OBS Process generates simulated (or model-calculated) values for comparison with measured, or observed, values. Additionally, observation sensitivities are calculated for use in sensitivity analysis and for regression during parameter calibration (Hill and others, 2000). By defining an observation value and a weight associated with the observation, either as a standard deviation or a coefficient of variation (standard deviation divided by the mean), observed values can be compared to simulated values to evaluate model fit in the context of expected observation accuracy. Statistics calculated to quantify this comparison include a weighted least-squares objective function (Hill, 1998). In addition, when the OBS Process is active, output files are produced for comparison of simulated and observed values (Hill and others, 2000, p. 23).
The OBS Process requires an input file identified by file type “OBS”. The tasks of the Observation Process are as follows:
Observation inputs for the DVFRS model include: (1) head observation file (specified as file type HOB) for hydraulic heads and changes in hydraulic head over time, (2) drain observation file (specified as file type DROB) for natural flows to or from surface-water bodies and diffuse discharge areas represented with the DRN Package, and (3) flow observations at constant-head boundaries file (specified as file type CHOB) for flow to or from a set of constant-head finite-difference cells along parts of the model boundary as specified in the CHD Package.
Hydraulic head or head change observations are specified in the Head Observation Package, specified as file type “HOB”. For each observation, the head observation and observation weight are defined by layer(s), row, column, and time of the observation. Head observations are located as a row and column offset from the node at the center of the finite-difference cell (Hill and others, 2000, p. 31-40). In the DVRFS model, head observations for wells with water-level measurements over multiple years were determined to be either affected or not affected by pumping. Head observations affected by pumping were simulated as a head change, which is calculated as the difference between the observation of interest and a reference observation (Hill and others, 2000, p. 33–34). The reference observation is the measurement prior to any pumping effect or the first measurement available.
The open intervals of the wells and boreholes determined which model layers were associated with head and head-change observations. Most head and head-change observations (82 percent) are from wells completed in the shallow part of the flow system (no deeper than model layer 5) and none are deeper than model layer 14 (Faunt and others, 2004b, fig. F-8, p. 278). For wells open to more than one model layer, simulated heads are a weighted average calculated by MODFLOW-2000 with user-defined weights (Hill and others, 2000, p. 34–36). In the DVRFS model, the standard deviation of a head observation is based on five potential errors: (1) well-altitude error, (2) well-location error, (3) nonsimulated transient error, (4) measurement-accuracy error, and (5) model-discretization error (San Juan and others, p. 122-132). Well-altitude and well-locations errors vary with the type of instrument or method used to specify the well location. Nonsimulated transient errors result from uncertainty in the magnitude of water-level response caused by stresses not simulated in the flow model, which typically are seasonal and long-term climate changes. A separate method was used to calculate the standard deviation of head-change observations because differences between simulated and observed head changes are expected to be dominated by inaccurate pumpage estimates (Faunt and others, 2004b, p. 282). Standard deviations for head-change observations less than 1 m were assigned a value of 1 m to avoid very small errors that could cause numerical instability during calibration. For head change observations greater than 1 m, a function was developed that produced larger standard deviations for larger head differences but also tempered the range in standard deviation between large and small head-change observations (Faunt and others, 2004b, p. 282, eq. 4).
Drain observations in the Drain Observation Package are specified as file type “DROB”. At each finite-difference cell specified as a drain cell, discharge observations, observation weight, and time of observation are defined.
In the DVRFS model, ground-water discharge observations did not vary during the transient simulation with the exception of Manse and Bennetts Springs in Pahrump Valley (fig. 1). At this location, changes in discharge have been documented during the model simulation period (fig. 3). For each of these springs, a single observation occurs during the steady-state stress period, and two transient discharge observations occur during the stress periods representing 1960 and 1998. For all other 43 ground-water discharge locations in the DROB Package, observations only occur once during the stress period representing 1997 (San Juan and others, 2004, p. 103-110).
In the DVRFS model, the weight of a ground-water discharge observation is represented as a coefficient of variation based on Monte Carlo analysis (Laczniak and others, 2001; D’Agnese and others, 2002; Faunt and others, 2004b, p. 283). The coefficients of variation for discharge observations range from 0.10 to 0.71 (Faunt and others, 2004b, table F-4, p. 276-277).
At finite-difference cells defined as constant head cells, the model calculates the flow into and out of cells needed to maintain the head. If anything is known about the likely flow rate, it is important to include it to constrain model calibration (Poeter and Hill, 1997). These flows are input through the Constant-Head Flow Observation Package, specified as file type “CHOB”.
In the DVRFS model, locations of the constant-head flow observations coincide with the locations of the 12 constant-head boundary parameters that represent segments of the model boundary. These 12 segments potentially have flow across them based on water budget analyses (Harrill and Bedinger, 2004) (fig. 2). Harrill and Bedinger (2004) estimated the potential for flow through boundary segments of the DVRFS model domain (table A2-9, p. 406-407). Constant-head flow observations are specified only during the first stress period. Observation weights are calculated from specified standard deviations that were based on the method used to estimate flow at the boundary (Harrill and Bedinger, 2004). For boundary-flow estimates based on water-budget analyses (Harrill and Bedinger, 2004), the standard deviation was set to one-half of the estimated flow value which reflects the large uncertainty in these values. In eight of the segments, water budget analyses were not available, so Darcy’s law (with the regional potential gradient) and professional judgment were used to estimate flow across the boundary (Harrill and Bedinger, 2004). In these poorly constrained cases, the standard deviation was set to the estimated flow value rounded down to the nearest 500 m3/d. For poorly constrained boundary-flow estimates, the large standard deviation reflects the large uncertainty.
The Sensitivity Process (Hill and others, 2000), specified as file type “SEN”, calculates the sensitivity of hydraulic heads with respect to parameters. These sensitivities are grid sensitivities. If the OBS Process is active, grid sensitivities are used to calculate sensitivities for the simulated values associated with the observations with respect to a parameter. These sensitivities are called observation sensitivities. Observation sensitivities are used to calculate a number of statistics that can be utilized to: (1) diagnose inadequate data, (2) identify parameters that probably cannot be estimated by regression using the available observations, and (3) evaluate potential new data (Hill and others, 2000). Model output for the sensitivity process includes: (1) composite scaled sensitivities, (2) dimensionless scaled sensitivities, and (3) one-percent scaled sensitivities (Hill, 1998, p. 14-16). Flags in the first data row of the SEN Process, in combination with processes specified in the name file, control whether the SEN Process or PES Process (described in section, “Sensitivity (SEN) Process”) are active in MODFLOW-2000.
In the DVRFS model, parameter values specified in the SEN Process are used to calculate the horizontal-flow barrier characteristic [(m/d)/m], drain conductance [(m/d)/m], hydraulic conductivity (m/d), recharge multiplier (unitless), exponential depth decay coefficient (1/m), specific storage (1/m), and vertical anisotropy (ratio of horizontal to vertical hydraulic conductivity) (unitless) (table 3). All parameters, with the exception of the specific storage parameters, were log-transformed during parameter estimation.
In the DVRFS model, the SEN Process specifies the value of all model parameters with the exception of the simulated heads at constant-head boundaries. The SEN Process provides one location to change all parameter values. Because the simulated lateral boundary flows were not calibration targets, constant-head boundary parameters are defined in the CHD and CHOB Packages but are not included in the SEN Process (effectively a parameter value of 1).
Generally, in the DVRFS model, the value of a parameter in the SEN Process is multiplied by additional values (termed factors) in other package(s) to obtain the cell-by-cell model input used in MODFLOW-2000 calculations. For example, in the HFB6 Package, the parameter value in the SEN Process is multiplied by a factor of 1 (table 4). A HFB6 parameter is defined as the hydraulic characteristic, which is the flow barrier hydraulic conductivity divided by the flow barrier width. At cells specified as HFB6 cells, the Factor variable is used to calculate the hydraulic characteristic of the flow barrier as the product of Factor and the parameter value. All hydraulic barriers in the DVRFS model were represented as 1 m wide (Faunt and others, 2004b, p. 269). Consequently, the hydraulic characteristic and the hydraulic conductivity are numerically equivalent, and for all cells specified as HFB6 cells, the value of Factor is 1.
In other cases, such as the DRN Package, the value of the drain parameter in the SEN Process is multiplied by a factor other than 1 at each grid cell. In the DRN Package, the multiplicative factor represents the fractional area of the cell that contributes to natural discharge, evapotranspiration and/or spring flow (table 4). It is convenient to represent parameters in the DRN Package with a factor other than 1 because drain parameters define a conductance that depends on the area of the cell contributing discharge. This area dependence is represented by the value of the multiplicative factor at each drain cell. For each cell specified as a drain cell, the variable “Condfact”, the factor used to calculate the hydraulic conductance of the drain from the parameter value, is the fractional area of the cell that the drain occupies.
In the DVRFS model, recharge is specified by values in the MULT and ZONE files, the RCH Package, and SEN Process. The distributed net infiltration array (Hevesi and others, 2003) input for the DVRFS model is the multiplier array rch1_modmd.asc. For each of the five parameters in the RCH Package, the parameter value, which varies by zone, is multiplied by values in the multiplier array.
Model input for parameter types not listed in table 4 include hydraulic conductivity (HUF2 Package parameter type HK), KDEP, vertical anisotropy (parameter type VANI), and specific storage (parameter type SY and SYTP). For these parameter types, cell-by-cell model values are calculated with the multiplier and zone arrays listed in the HUF2 Package and parameter values specified in the SEN Process (Anderman and Hill, 2000, 2003).
The Parameter-Estimation Process (Hill and others, 2000), specified as file type “PES”, uses a modified Gauss-Newton method to adjust values of user-selected input parameters to minimize the value of the weighted least-squares objective function via iteration. The objective function, the sum of squared weighted residuals, is used to evaluate the fit of simulated to observed values. Residuals are the difference between simulated and observed model values (observed minus simulated). Weighted residuals in the DVRFS model most often reflect a weight equal to the inverse of the standard deviation of the observation squared. Thus, squared weighted residuals for all types of observations can be summed as dimensionless quantities.
Parameter estimation is initiated using parameter values from the SEN Process. Hydraulic heads and sensitivities are calculated for each time step. After all time steps are computed, simulated values are subtracted from observed values to calculate residuals, which are used to compute the least-squares objective function. Convergence criteria specified in the PES Process include maximum allowable values of: (1) the largest fractional change in any of the parameter values, and (2) the change in the weighted least-squares objective function. If either of the two convergence criteria is met, parameter estimation converges. If parameter estimation does not converge and the maximum number of iterations has not been exceeded, updated parameter values are used in subsequent parameter-estimation iterations. When parameter estimation converges or the maximum number of iterations is reached, the PES Process halts.
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