MODFLOW

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.

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. \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_{xx} , K_{yy} and K_{zz} are the values of hydraulic conductivity along the x, y, and z coordinate axes (L/T) h is the potentiometric head (L) 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} is the specific storage of the porous material (L −1 ); and 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: \begin{align} & \mathit{CR}_{i,j-\tfrac{1}{2},k}\left(h^m_{i,j-1,k}-h^m_{i,j,k}\right) + \mathit{CR}_{i,j+\tfrac{1}{2},k}\left(h^m_{i,j+1,k}-h^m_{i,j,k}\right) + \\ & \mathit{CC}_{i-\tfrac{1}{2},j,k}\left(h^m_{i-1,j,k}-h^m_{i,j,k}\right) + \mathit{CC}_{i+\tfrac{1}{2},j,k}\left(h^m_{i+1,j,k}-h^m_{i,j,k}\right) + \\ & \mathit{CV}_{i,j,k-\tfrac{1}{2}}\left(h^m_{i,j,k-1}-h^m_{i,j,k}\right) + \mathit{CV}_{i,j,k+\tfrac{1}{2}}\left(h^m_{i,j,k+1}-h^m_{i,j,k}\right) + \\ & P_{i,j,k}\,h^m_{i,j,k} + Q_{i,j,k} = \mathit{SS}_{i,j,k}\left(\Delta r_j \Delta c_i \Delta v_k\right) \frac{h^m_{i,j,k}-h^{m-1}_{i,j,k}}{t^m-t^{m-1}} \end{align} where h^m_{i,j,k}\, 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}\, is the sum of coefficients of head from source and sink terms Q_{i,j,k}\, is the sum of constants from source and sink terms, where Q_{i,j,k} is flow out of the groundwater system (such as pumping) and Q_{i,j,k}>0.0\, is flow in (such as injection) \mathit{SS}_{i,j,k}\, is the specific storage \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\, is the time at time step m This equation is formulated into a system of equations to be solved as: \begin{align} &\mathit{CV}_{i,j,k-\tfrac{1}{2}} h^m_{i,j,k-1} + \mathit{CC}_{i-\tfrac{1}{2},j,k} h^m_{i-1,j,k} + \mathit{CR}_{i,j-\tfrac{1}{2},k} h^m_{i,j-1,k} \\ &+ \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^m_{i,j,k} \\ &+ \mathit{CR}_{i,j+\tfrac{1}{2},k} h^m_{i,j+1,k} + \mathit{CC}_{i+\tfrac{1}{2},j,k} h^m_{i+1,j,k} + \mathit{CV}_{i,j,k+\tfrac{1}{2}} h^m_{i,j,k+1} = \mathit{RHS}_{i,j,k} \end{align} where \begin{align} \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^{m-1}_{i,j,k}}{t^m-t^{m-1}} \end{align} or in matrix form as: A\mathbf{h}=\mathbf{q} where A is a matrix of the coefficients of head for all active nodes in the grid \mathbf{h} is a vector of head values at the end of time step m for all nodes in the grid; and \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).

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