Conceptual Model A descriptive representation of a groundwater system that incorporates an interpretation of the geological & hydrological conditions. Generally includes information about the water budget.

a set of equations that describes the physical and/or chemical processes occurring in a system. Mathematical Model Governing equation Boundary conditions Initial conditions for transient simulations

R  x  y Q yy xx zz 1.Consider flux (q) through REV 2.OUT – IN = -  Storage K 3.Combine with: q = -K grad h q Derivation of the Governing Equation

div q = 0 q = - K grad h Steady state mass balance eqn. Darcy’s law grad h qequipotential line grad hq IsotropicAnisotropic K x = K z Kx  KzKx  Kz z x

x z x’ z’ globallocal  K xx K xy K xz K yx K yy K yz K zx K zy K zz K’ x K’ y K’ z bedding planes

qxqyqzqxqyqz = - q = - K grad h K xx K xy K xz K yx K yy K yz K zx K zy K zz

2D confined: 2D unconfined w/ Dupuit assumptions: Storage coefficient (S) is either storativity or specific yield. S = S s b & T = K b General 3D equation Hetergeneous, anisotropic, transient, sink/source term

Types of Boundary Conditions 1.Specified head (including constant head) 2.Specified flow (including no flow) 3.Head-dependent flow

From conceptual model to mathematical model…

Toth Problem Laplace Equation 2D, steady state h = c x + z o Cross section through an unconfined aquifer. Water table forms the upper boundary condition

Governing Eqn. for TopoDrive 2D, steady-state, heterogeneous, anisotropic

b h ocean groundwater divide “Confined” Island Recharge Problem R x = 0x = Lx = - L We can treat this system as a “confined” aquifer if we assume that T= Kb. datum Areal view Water table is the solution. Poisson’s Eqn. 2D horizontal flow through an unconfined aquifer where T=Kb.

b h ocean groundwater divide R x = 0x = Lx = - L datum Unconfined version of the Island Recharge Problem Water table is the solution. (Pumping can be accommodated by appropriate definition of the source/sink term.) 2D horizontal flow through an unconfined aquifer under the Dupuit assumptions.

Vertical cross section through an unconfined aquifer with the water table as the upper boundary. 2D horizontal flow in a confined aquifer; solution is h(x,y), i.e., the potentiometric surface. 2D horizontal flow in an unconfined aquifer where v= h 2. Solution is h(x,y), i.e., the water table. All three governing equations are the LaPlace Eqn.

t = 0 t > 0 BC: h (0, t) = 16 m; t > 0 h (L, t) = 11 m; t > 0 datum 0 L = 100 m x IC: h (x, 0) = 16 m; 0 < x < L (represents static steady state) Reservoir Problem 1D transient flow through a confined aquifer. confining bed

Solution techniques… Analytical solutions Numerical solutions finite difference (FD) methods finite element (FE) methods Analytic element methods (AEM)

Toth Problem z x Analytical Solution Numerical Solution h(x,z) = z o + cs/2 – 4cs/  2  … h i,j = (h i+1,j + h i-1,j + h i,j+1 + h i,j-1 )/4 z x continuous solutiondiscrete solution (eqn. 2.1 in W&A) h = c x + z o Mathematical model

h = ft Toth Problem mesh vs block centered grids another view  x =  y = a = 20 ft 200 ft Grid Design

Three options for solving the set of algebraic equations that result from applying the method of FD or FE: Iteration Direct solution by matrix inversion A combination of iteration and matrix solution Note: The explicit solution for the transient flow equation is another solution technique, but in practice is never used.

Examples of Iteration methods include: Gauss-Seidel Iteration Successive Over-Relaxation (SOR)

Let  x=  y=a

Gauss-Seidel Formula for 2D Laplace Equation General SOR Formula Relaxation factor = 1 Gauss-Seidel < 1 under-relaxation >1 over-relaxation, typically between 1 and 2

Gauss-Seidel Formula for 2D Poisson Equation SOR Formula Relaxation factor = 1 Gauss-Seidel < 1 under-relaxation >1 over-relaxation (Eqn. 3.7 W&A)

m m+1 m+2 m+3 solution (Initial guesses) Iteration for a steady state problem. Iteration levels

n n+1 n+2 n+3 Steady state tt tt tt Initial conditions (steady state) Transient Problems require time steps. Time levels

Explicit Approximation Implicit Approximation

where  = 1 for fully implicit  = 0.5 for Crank-Nicolson  = 0 for explicit In general:

Explicit solutions do not require iteration but are unstable with large time steps. We can derive the stability criterion by writing the explicit approx. in a form that looks like the SOR iteration formula and setting the terms in the position occupied by omega equal to 1. For the 1D governing equation used in the reservoir problem, the stability criterion is: < < or Note that critical  t value is directly dependent on grid spacing,  x.

Implicit solutions require iteration or direct solution by matrix inversion or a combination of iteration and matrix inversion.

tt Iteration planes n n+1 m+2 m+1 m+3 Solution of an Implicit transient FD equation by iteration

Boundary conditions always affect a steady state solution. Initial conditions should be selected to represent a steady state configuration of heads. Modeling rules/guidelines A water balance should always be included in the simulation. In general, the accuracy of the numerical solution improves with smaller grid spacing and smaller time step

At steady state:  h /  t = 0 and  V /  t = 0

Note relatively large change in storage Implicit Solution  t = 5 Early time

timeinflow Inflow+  storage At late time  storage is small