Groundwater Modeling - 1 Groundwater Hydraulics Daene C. McKinney

Models …? Input (Explanatory Variable) Precipitation Soil Characteristics Model (Represents the Phenomena) ET Evaporation Infiltration Output (Results – Response variable) Run off

Models and more models … Hydrologic Simulation Simulation Model Optimization Model Input (Explanatory Variable) Inflow Data Basin Water Allocation Policy Response to the Policy Inflow Data Basin Objectives and Constraints Optimum Policy Precip. & Soil Charact. Mimic Physics of the Basin Runoff Model (Phenomena) Output (Results) Source for Input data of other models Predict Response to given design/policy Identify optimal design/policy

Problem identification Modeling Process Problem identification and description Model verification & sensitivity analysis Model Documentation Model application Model calibration & parameter estimation Model conceptualization development Data Present results 1 Problem identification (1) Important elements to be modeled Relations and interactions between them Degree of accuracy Conceptualization and development (2 – 3) Mathematical description Type of model Numerical method - computer code Grid, boundary & initial conditions Calibration (4) Estimate model parameters Model outputs compared with actual outputs Parameters adjusted until the values agree Verification (4) Independent set of input data used Results compared with measured outputs 2 3 4 5 6 7

Tools to Solve Groundwater Problems Physical and analog methods Some of the first methods used. Analytical methods What we have been discussing so far Difficult for irregular boundaries, different boundary conditions, heterogeneous and anisotropic properties, multiple phases, nonlinearities Numerical methods Transform PDEs governing flow of groundwater into a system of ODEs or algebraic equations for solution

Conceptual Model Descriptive representation of groundwater system incorporating interpretation of geological & hydrological conditions What processes are important to model? What are the boundaries? What parameter values are available? What parameter values must be collected?

What Do We Really Want To Solve? Horizontal flow in a leaky confined aquifer Governing Equations Boundary Conditions Initial conditions Flux Leakage Source/Sink Storage Ground surface Bedrock Confined aquifer Qx K x y z h Head in confined aquifer Confining Layer b

Finite Difference Method Replace derivatives in governing equations with Taylor series approximations Generates set of algebraic equations to solve 1st derivatives

Taylor Series Taylor series expansion of h(x) at a point x+Dx close to x If we truncate the series after the nth term, the error will be

First Derivative - Forward Consider the forward Taylor series expansion of a function h(x) near a point x Solve for 1st derivative

First Derivative - Backward Consider the backward Taylor series expansion of a function f(x) near a point x Solve for 1st derivative

Finite Difference Approximations 1st Derivative (Backward) 1st Derivative (Forward)

Grids and Discretrization Discretization process Grid defined to cover domain Goal - predict values of head at node points of mesh Determine effects of pumping Flow from a river, etc Finite Difference method Popular due to simplicity Attractive for simple geometry i,j i,j+1 i+1,j i-1,j i,j-1 x, i y, j Domain Mesh Node point D x y Grid cell

Three-Dimensional Grids An aquifer system is divided into rectangular blocks by a grid. The grid is organized by rows (i), columns (j), and layers (k), and each block is called a "cell" Types of Layers Confined Unconfined Convertible j, columns i, rows k, layers Layers can be different materials

1-D Confined Aquifer Flow Homogeneous, isotropic, 1-D, confined flow Governing equation Initial Condition Boundary Conditions Ground surface Aquifer x y z hB Confining Layer b hA Dx i = 1 2 3 4 5 6 7 8 9 10 Node Grid Cell

Derivative Approximations Need 2nd derivative WRT x

Derivative Approximations Governing Equation 2nd derivative WRT x Need 1st derivative WRT t Which one to use? Forward Backward

Time Derivative Explicit Implicit Use all the information at the previous time step to compute the value at this time step. Proceed point by point through the domain. Implicit Use information from one point at the previous time step to compute the value at all points of this time step. Solve for all points in domain simultaneously.

Explicit Method Use all the information at the previous time step to compute the value at this time step. Proceed point by point through the domain. Can be unstable for large time steps. FD Approx. Forward

Explicit Method l+1 time level unknown l time level known

1-D Confined Aquifer Flow Initial Condition Boundary Conditions Ground surface Aquifer x y z hB Confining Layer b hA Dx i = 1 2 3 4 5 6 7 8 9 10 Node Grid Cell L Dx = 1 m L = 10 m T=bK = 0.75 m2/d S = 0.02

Explicit Method Consider: r = 0.48 r = 0.52 Dx = 1 m L = 10 m Ground surface Aquifer hB Confining Layer b hA Dx i = 1 2 3 4 5 6 7 8 9 10 Node Grid Cell Consider: r = 0.48 r = 0.52 Dx = 1 m L = 10 m T = 0.75 m2/d S = 0.02

Explicit Results (Dt = 18.5 min; r = 0.48 < 0.5)

Explicit Results (Dt = 20 min; r = 0.52 > 0.5)

What’s Going On Here? At time t = 0  no flow At time t > 0  flow Water released from storage in a cell over time Dt Water flowing out of cell over interval Dt Ground surface Confining Layer hA Aquifer Dx hB b Dx i = 1 2 … i-1 i i+1 … 8 9 10 Grid Cell i r > 0.5 Tme interval is too large Cell doesn’t contain enough water Causes instability

Implicit Method Use information from one point at the previous time step to compute the value at all points of this time step. Solve for all points in domain simultaneously. Inherently stable FD Approx. Backward

Implicit Method l+1 time level unknown l time level known

2-D Steady-State Flow General Equation Homogeneous, isotropic aquifer, no well Equal spacing (average of surrounding cells) Node No. Unknown heads Known heads

2-D Heterogeneous Anisotropic Flow Tx and Ty are transmissivities in the x and y directions

2-D Heterogeneous Anisotropic Flow Harmonic average transmissivity

Transient Problems

MODFLOW USGS supported mathematical model Uses finite-difference method Several versions available MODFLOW 88, 96, 2000, 2005 (water.usgs.gov/nrp/gwsoftware/modflow.html) Graphical user interfaces for MODFLOW: GWV (www.groundwater-vistas.com) GMS (www.ems-i.com) PMWIN (www.ifu.ethz.ch/publications/software/pmwin/index_EN) Each includes MODFLOW code

What Can MODFLOW Simulate? Unconfined and confined aquifers Faults and other barriers Fine-grained confining units and interbeds Confining unit - Ground-water flow and storage changes River – aquifer water exchange Discharge of water from drains and springs Ephemeral stream - aquifer water exchange Reservoir - aquifer water exchange Recharge from precipitation and irrigation Evapotranspiration Withdrawal or recharge wells Seawater intrusion