MODFLOW is the U.S. Geological Survey modular finite-difference flow model, which is a computer code that solves the groundwater flow equation. The program is used by hydrogeologists to simulate the flow of groundwater through aquifers. The source code is free public domain software, written primarily in Fortran, and can compile and run on Microsoft Windows or Unix-like operating systems. The compiled program is a command line tool that reads text file inputs that define the model grid, boundary conditions, and simulation time frame. Extensions beyond the command line program are several actively developed commercial and non-commercial graphical user interfaces.
Groundwater flow equation The governing partial differential equation for a confined aquifer used in MODFLOW is:
∂ ∂ x [ K x x ∂ h ∂ x ] + ∂ ∂ y [ K y y ∂ h ∂ y ] + ∂ ∂ z [ K z z ∂ h ∂ z ] + W = S S ∂ h ∂ t {\displaystyle {\frac {\partial }{\partial x}}\left[K_{xx}{\frac {\partial h}{\partial x}}\right]+{\frac {\partial }{\partial y}}\left[K_{yy}{\frac {\partial h}{\partial y}}\right]+{\frac {\partial }{\partial z}}\left[K_{zz}{\frac {\partial h}{\partial z}}\right]+W=S_{S}{\frac {\partial h}{\partial t}}}
where
K x x {\displaystyle K_{xx}} , K y y {\displaystyle K_{yy}} and K z z {\displaystyle K_{zz}} are the values of hydraulic conductivity along the x, y, and z coordinate axes (L/T)
h {\displaystyle h} is the potentiometric head (L)
W {\displaystyle W} is a volumetric flux per unit volume representing sources and/or sinks of water, where negative values are extractions, and positive values are injections (T−1)
S S {\displaystyle S_{S}} is the specific storage of the porous material (L−1); and
t {\displaystyle t\,} is time (T)
Finite difference The finite difference form of the partial differential in a discretized aquifer domain (represented using rows, columns and layers) is:
C R i , j − 1 2 , k ( h i , j − 1 , k m − h i , j , k m ) + C R i , j + 1 2 , k ( h i , j + 1 , k m − h i , j , k m ) + C C i − 1 2 , j , k ( h i − 1 , j , k m − h i , j , k m ) + C C i + 1 2 , j , k ( h i + 1 , j , k m − h i , j , k m ) + C V i , j , k − 1 2 ( h i , j , k − 1 m − h i , j , k m ) + C V i , j , k + 1 2 ( h i , j , k + 1 m − h i , j , k m ) + P i , j , k h i , j , k m + Q i , j , k = S S i , j , k ( Δ r j Δ c i Δ v k ) h i , j , k m − h i , j , k m − 1 t m − t m − 1 {\displaystyle {\begin{aligned}&{\mathit {CR}}_{i,j-{\tfrac {1}{2}},k}\left(h_{i,j-1,k}^{m}-h_{i,j,k}^{m}\right)+{\mathit {CR}}_{i,j+{\tfrac {1}{2}},k}\left(h_{i,j+1,k}^{m}-h_{i,j,k}^{m}\right)+\\&{\mathit {CC}}_{i-{\tfrac {1}{2}},j,k}\left(h_{i-1,j,k}^{m}-h_{i,j,k}^{m}\right)+{\mathit {CC}}_{i+{\tfrac {1}{2}},j,k}\left(h_{i+1,j,k}^{m}-h_{i,j,k}^{m}\right)+\\&{\mathit {CV}}_{i,j,k-{\tfrac {1}{2}}}\left(h_{i,j,k-1}^{m}-h_{i,j,k}^{m}\right)+{\mathit {CV}}_{i,j,k+{\tfrac {1}{2}}}\left(h_{i,j,k+1}^{m}-h_{i,j,k}^{m}\right)+\\&P_{i,j,k}\,h_{i,j,k}^{m}+Q_{i,j,k}={\mathit {SS}}_{i,j,k}\left(\Delta r_{j}\Delta c_{i}\Delta v_{k}\right){\frac {h_{i,j,k}^{m}-h_{i,j,k}^{m-1}}{t^{m}-t^{m-1}}}\end{aligned}}}
where
h i , j , k m {\displaystyle h_{i,j,k}^{m}\,} is the hydraulic head at cell i,j,k at time step m CV, CR and CC are the hydraulic conductances, or branch conductances between node i,j,k and a neighboring node
P i , j , k {\displaystyle P_{i,j,k}\,} is the sum of coefficients of head from source and sink terms
Q i , j , k {\displaystyle Q_{i,j,k}\,} is the sum of constants from source and sink terms, where Q i , j , k < 0.0 {\displaystyle Q_{i,j,k}<0.0\,} is flow out of the groundwater system (such as pumping) and Q i , j , k > 0.0 {\displaystyle Q_{i,j,k}>0.0\,} is flow in (such as injection)
S S i , j , k {\displaystyle {\mathit {SS}}_{i,j,k}\,} is the specific storage
Δ r j Δ c i Δ v k {\displaystyle \Delta r_{j}\Delta c_{i}\Delta v_{k}\,} are the dimensions of cell i,j,k, which, when multiplied, represent the volume of the cell; and
t m {\displaystyle t^{m}\,} is the time at time step m This equation is formulated into a system of equations to be solved as:
C V i , j , k − 1 2 h i , j , k − 1 m + C C i − 1 2 , j , k h i − 1 , j , k m + C R i , j − 1 2 , k h i , j − 1 , k m + ( − C V i , j , k − 1 2 − C C i − 1 2 , j , k − C R i , j − 1 2 , k − C R i , j + 1 2 , k − C C i + 1 2 , j , k − C V i , j , k + 1 2 + H C O F i , j , k ) h i , j , k m + C R i , j + 1 2 , k h i , j + 1 , k m + C C i + 1 2 , j , k h i + 1 , j , k m + C V i , j , k + 1 2 h i , j , k + 1 m = R H S i , j , k {\displaystyle {\begin{aligned}&{\mathit {CV}}_{i,j,k-{\tfrac {1}{2}}}h_{i,j,k-1}^{m}+{\mathit {CC}}_{i-{\tfrac {1}{2}},j,k}h_{i-1,j,k}^{m}+{\mathit {CR}}_{i,j-{\tfrac {1}{2}},k}h_{i,j-1,k}^{m}\\&+\left(-{\mathit {CV}}_{i,j,k-{\tfrac {1}{2}}}-{\mathit {CC}}_{i-{\tfrac {1}{2}},j,k}-{\mathit {CR}}_{i,j-{\tfrac {1}{2}},k}-{\mathit {CR}}_{i,j+{\tfrac {1}{2}},k}-{\mathit {CC}}_{i+{\tfrac {1}{2}},j,k}-{\mathit {CV}}_{i,j,k+{\tfrac {1}{2}}}+{\mathit {HCOF}}_{i,j,k}\right)h_{i,j,k}^{m}\\&+{\mathit {CR}}_{i,j+{\tfrac {1}{2}},k}h_{i,j+1,k}^{m}+{\mathit {CC}}_{i+{\tfrac {1}{2}},j,k}h_{i+1,j,k}^{m}+{\mathit {CV}}_{i,j,k+{\tfrac {1}{2}}}h_{i,j,k+1}^{m}={\mathit {RHS}}_{i,j,k}\end{aligned}}}
where
H C O F i , j , k = P i , j , k − S S i , j , k Δ r j Δ c i Δ k t m − t m − 1 R H S i , j , k = − Q i , j , k − S S i , j , k Δ r j Δ c i Δ v k h i , j , k m − 1 t m − t m − 1 {\displaystyle {\begin{aligned}{\mathit {HCOF}}_{i,j,k}&=P_{i,j,k}-{\frac {{\mathit {SS}}_{i,j,k}\Delta r_{j}\Delta c_{i}\Delta _{k}}{t^{m}-t^{m-1}}}\\{\mathit {RHS}}_{i,j,k}&=-Q_{i,j,k}-{\mathit {SS}}_{i,j,k}\Delta r_{j}\Delta c_{i}\Delta v_{k}{\frac {h_{i,j,k}^{m-1}}{t^{m}-t^{m-1}}}\end{aligned}}}
or in matrix form as:
A h = q {\displaystyle A\mathbf {h} =\mathbf {q} }
where
A is a matrix of the coefficients of head for all active nodes in the grid
h {\displaystyle \mathbf {h} } is a vector of head values at the end of time step m for all nodes in the grid; and
q {\displaystyle \mathbf {q} } is a vector of the constant terms, RHS, for all nodes of the grid.
Limitations The water must have a constant density, dynamic viscosity (and consequently temperature) throughout the modelling domain (SEAWAT is a modified version of MODFLOW which is designed for density-dependent groundwater flow and transport)
The principal components of anisotropy of the hydraulic conductivity used in MODFLOW is displayed on the right. This tensor does not allow non-orthogonal anisotropies, as could be expected from flow in fractures. Horizontal anisotropy for an entire layer can be represented by the coefficient "TRPY" (Data Item 3 Page 153).
Versions
"Modular Model" The USGS throughout the 1970s had developed several hundred models, written in different dialects of FORTRAN. At the time, it was common practice to rewrite a new model to fit the need of a new groundwater scenario. The concept for MODFLOW was originally designed in 1981 to provide a common modular groundwater model, which could be compiled on multiple platforms without major (or any) modification, and can read and write common formats. Different aspects of the groundwater system would be handled using the modules, similar to the idea of a "component stereo system". The original name of the code was "The USGS Modular Three-Dimensional Finite-Difference Ground-Water Flow Model", or informally as "The Modular Model". The name MODFLOW was coined several years after the initial code development, which started in 1981. The first version of MODFLOW was published on December 28, 1983, and was coded entirely in FORTRAN 66. The source code for this version is listed in USGS Open File Report 83-875 referred to above.
MODFLOW-88 This version of MODFLOW was rewritten in FORTRAN 77, and was originally released on July 24, 1987. The current version of MODFLOW-88 is 2.6, released on September 20, 1996. MODPATH, was initially developed in 1989 to post-process the steady-state MODFLOW-88 data to determine three-dimensional pathlines of particles. This innovation has been indispensable for the fields of contaminant hydrogeology. It is still used as a post-processor in recent versions of MODFLOW. A separate program, MODFLOWP, was developed in 1992 to estimate various parameters used in MODFLOW. This program was eventually built into MODFLOW-2000.
MODFLOW-96 MODFLOW-96 (version 3.0) was originally released on December 3, 1996, and is a cleaned-up and revised continuation of MODFLOW-88. There are three final releases of MODFLOW-96:
MODFLOW-96 (version 3.3, May 2, 2000) MODFLOW-96h (version 3.3h, July 10, 2000), with HYDMOD package MODFLOWP (version 3.2, Oct 9, 1997), MODFLOW-96 with parameter-estimation Several graphical interfaces were first developed using the MODFLOW-96 code.
MODFLOW-2000 MODFLOW-2000 (version 1.0; version numbering was reset) was released on July 20, 2000, which merged MODFLOWP and HYDMOD codes into the main program and has integrated observation, sensitivity analysis, parameter estimation, and uncertainty evaluation capabilities. Many new packages and enhancements were also included, including new solvers, stream and saturated flow packages. The internal design concepts also changed from previous versions, such that packages, processes and modules are distinct. This version was coded in a mixture of FORTRAN 77, Fortran 90, and one solver was programmed in C. MODFLOW-2000 can also be compiled for parallel computing, which can allow multiple processors to be used to increase model complexity and/or reduce simulation time. The parallelization capability is designed to support the sensitivity analysis, parameter estimation, and uncertainty analysis capabilities of MODFLOW-2000. The final version of MODFLOW-2000 (or MF2K) is version 1.19.01, released on March 25, 2010. There are four related or branched codes based on MODFLOW-2000:
MF2K-GWM or GWM-2000 (version 1.1.4, May 31, 2011, branched from mf2k 1.17.2), with groundwater management capability using optimization MF2K-FMP (version 1.00, May 19, 2006, based on mf2k 1.15.03), with Farm Process MF2K-GWT (version 1.9.8, October 28, 2008, based on MF2K 1.17.02), groundwater flow and solute-transport model SEAWAT (version 4.00.05, October 19, 2012), variable-density flow and transport processes VSF (version 1.01, July 5, 2006), variably saturated flow
MODFLOW-2005 MODFLOW-2005 differs from MODFLOW-2000 in that the sensitivity analysis, parameter estimation, and uncertainty evaluation capabilities are removed. Thus, the support for these capabilities now falls to "clip on" codes that are supported externally to the MODFLOW support effort. In addition, the code was reorganized to support multiple models within one MODFLOW run, as needed for the LGR (Local Grid Refinement) capability. MODFLOW-2005 is written primarily in Fortran 90 and C, with C being used for one solver. The current ver