1.72, Groundwater Hydrology Prof. Charles Harvey Lecture Packet #9: Numerical Modeling of Groundwater Flow

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1 1.7, Groundwater Hydrology Prof. Carles Harvey Lecture Packet #9: Numerical Modeling of Groundwater Flow Simulation: Te prediction of quantities of interest (dependent variables) based upon an equation or series of equations tat describe system beavior under a set of assumed simplifications. Groundwater Flow Simulation Predict ydraulic eads (1D, D, 3D) For particular conditions confined, unconfined, isotropic, anisotropic, omogeneous, eterogeneous, infinite, finite, steady, transient. Varying levels of complexity: Analystic solutions Teim, Teis, etc. Advantages: exact, simple, ceap, can provide sufficient insigt Analog simulation Pysical models scale models of aquifers Numerical Simulation Numerical Simulation Given a PDE and appropriate ICs and BCs Discretize te system Approximate te PDE corresponding to te discretization Solve te approximated PDE on a computer Commonly finite differences or finite elements for GW flow Advantages: can andle complex geometries, ICs, and C conditions; can be used for nonlinear systems. Beware of te term Model and ow it is used A model sould be used as simple as possible, but not simpler Matematical model a PDE Numerical model a particular tecnique is applied Computer or simulation model a code Finite Differences A numerical metod tat approximates te governing PDE by replacing te derivatives in te equation wit teir respective difference representations. Procedure involves: Grid (or Mes) and Equation Grid a representation of te pysical domain tat enables one to account for te boundaries and internal features 1.7, Groundwater Hydrology Lecture Packet 9 Prof. Carles Harvey Page 1 of 7

2 Equation Difference Approximation of Derivatives S + x y T t D flow equation Approximating te Time Derivative: Backward Difference: i, j i, j,n i, j,n 1 t t t n t n ead n-1 t time t n-1 t n End of were, time step n te current time step index t time step i,j x and y coordinate indices Approximating te Space Derivatives: Consider a D discretization, if we assume tat te grid spacing in te x-direction and y-direction are te same, our discretized grid for an internal node will be: i,j+1 i-1,j i,j i+1,j i-1/ i+1/ i,j-1 y y x x 1.7, Groundwater Hydrology Lecture Packet 9 Prof. Carles Harvey Page of 7

3 In te x-direction: Approximate derivative at location i,j : x Approximate te second spatial derivative at i,j as follows: i + 1/, j i 1/, j x x x x x We can approximate te first spatial derivative at i-1/,j and i+1/,j as follows: i 1/, j i, j i 1, j i + 1 /, j i + 1, j i, j x and x Substitute te above equations to obtain: i, x j + i, i i 1, j j, j i 1, j or i x +, j i 1, j i, j i + 1, j In te y-direction: i, y i, 1 i, + i, + 1 y j j j j Combining Flow Equation Terms: i, j i, j S i, j + x y T t η t For te backwards difference time derivative (note: all left and side values of are for time n-1) i 1, j i, j +i+ 1, j i, j 1 i, j + i, j +1 S i, j, n i, j, n 1 + y T t Linear diff. eq n. Tis eq n is not explicit for at any particular time If x and y are equal tis is te finite difference groundwater flow equation 1.7, Groundwater Hydrology Lecture Packet 9 Prof. Carles Harvey Page 3 of 7

4 i 1, j + i + 1, j + i, j 1 + i, j i, j S i, j, n i, j, n 1 x T t Wat does te finite-difference equation indicate for steady state conditions? S S i, j, n i, j, n 1 0 t t j i + 1, j i, j 1 i, j + 1 i, j x i, 0 i 1, j + i + 1, j + i, j 1 + i, j i, j j i + 1, j i, j 1 i, j + 1 i, j 4 i, i,j+1 i-1,j i,j i+1,j i,j-1 Flow Value of ead at node is te average of te surrounding nodes (for SS isotropic omogeneous case) Consider te 1D steady-state flow equation wit a sink is: d H T w ' 0 or dx d H w ' dx T x H 1 3 H9 Node , Groundwater Hydrology Lecture Packet 9 Prof. Carles Harvey Page 4 of 7

5 Te nodal finite difference equation is: i, j 1 i, j + i, j +1 w ' T Or ( x ) w ' i, j 1 i, j + i, j +1 T Node 1: 1H () Node : Node 3: Node 3 as a forcing term due to te pumping of te well. Tis translates into an initial rigtand side of te equation: x w ' ( 0. 1) 1 ( ) Given 0.1, 1 T H (9) Te system of finite-difference equations consists of 3 equations and 3 unknowns Unknowns Or in coeffiecient matrix form Knowns FD Coeff. Unknown RHS containing Matrix Heads boundary conditions and known pumping 1.7, Groundwater Hydrology Lecture Packet 9 Prof. Carles Harvey Page 5 of 7

6 Or in matrix notation it can be written as A b A is te matrix of difference coefficients H is te vector of unknown eads b is te RHS vector of known quantities A computational linear solve yields te vector : Transient Simulation Using Finite Differences Procedure: Marc Troug Time Start wit initial conditions (tese are known) Solve for eads at end of first time step t; tis give te spatial distribution of ead (a map) after a small time increment. Given known eads at end of first time step solve for eads at te end of te second time step. Wit known value at te end of time step solve for next time step tis is called marcing troug time A n b* were b* b + n-1 Te rigt-and side always contains knowns. Te matrix A is te matrix of finite difference coefficients reflecting te system parameters and discretization. So if you can solve spatial equations for one time step. Ten you can solve it for as many time steps as you like. Te time step must be small wen canges in eads are rapid suc as, wen you start to pump a well, t must be seconds or minutes. It can be increased as canges in ead become smaller. FD Simulation Models: codes tat solve te above system of linear algebraic equations fairly robust. 1.7, Groundwater Hydrology Lecture Packet 9 Prof. Carles Harvey Page 6 of 7

7 Time Stepping S i, n, n i 1 i + i + 1 x n? n-1? T t n Someting else? Explicit i 1 Crank- Nicolson Implicit ead n-1 t T t s x ( i 1 i + i + 1 ) i, n, n i 1 t n-1 t n Explicit: i,n c i-1,n-1 + (1-C) i,n-1 + c i+1,n-1 T t Were: c s x Easy to calculate no linear algebra, but unstable if time-steps are too large. Implicit: c i-1,n - (1+C) i,n + c i+1,n - i,n-1 Results in system of linear equations tat must be solved simultaneously. Stable, but issues of accuracy. c Crank Nicolson: i 1, n (1+ c ) i, n + c i + 1, n ( c 1) i 1, n c ( i 1, n 1 + i, Not muc more numerical expense tan fully implicit, but more accurate Boundary Conditions: 1 n 1 ) Constant Head Don t write equation for constant ead node. Use value of constant ead in equations for neigboring nodes. Flux Boundary replace first derivative term wit constant i 1, j i, j i, j i + 1, j i, j x x x x i, j i +, x 1 j Q T 1.7, Groundwater Hydrology Lecture Packet 9 Prof. Carles Harvey Page 7 of 7