Groundwater Modeling 1 Groundwater Hydraulics Daene C Mc

Groundwater Modeling - 1 Groundwater Hydraulics Daene C. Mc. Kinney 1

Models …? Input (Explanatory Variable) Precipitation Model (Represents the Phenomena) Soil Characteristics ET Evaporation Infiltration Output (Results – Response variable) Run off 2

Models and more models … Input (Explanatory Variable) Model (Phenomena) Output (Results) Hydrologic Simulation Model Optimization Model Precip. & Soil Charact. Inflow Data Mimic Physics of the Basin Water Allocation Policy Basin Objectives and Runoff Response to the Policy Optimum Policy Constraints Source for Input Predict Response Identify optimal data of other models to given design/policy 3

Modeling Process • 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 • 1 Model conceptualization 2 Model development 3 Problem identification (1) – Important elements to be modeled – Relations and interactions between them – Degree of accuracy • Problem identification and description Verification (4) – Independent set of input data used – Results compared with measured outputs Data Model calibration & parameter estimation 4 Model verification & sensitivity analysis Model Documentation 5 Model application 6 Present results 7 4

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 5

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? 6

What Do We Really Want To Solve? • Horizontal flow in a leaky confined aquifer Flux Leakage Source/Sink Storage Ground surface Head in confined aquifer • Governing Equations • Boundary Conditions • Initial conditions Confining Layer Qx z y x Bedrock Confined aquiferb K h 7

Finite Difference Method • Finite-difference method – Replace derivatives in governing equations with Taylor series approximations – Generates set of algebraic equations to solve 1 st derivatives 8

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 9

First Derivative - Forward • Consider the forward Taylor series expansion of a function h(x) near a point x • Solve for 1 st derivative 10

First Derivative - Backward • Consider the backward Taylor series expansion of a function f(x) near a point x • Solve for 1 st derivative 11

First Derivative Approximations i 1 st x Derivative (Backward) 1 st x Derivative (Forward) 12

First Derivative Approximations 1 st t Derivative (Backward) 1 st t Derivative (Forward) 13

Second Derivative Approximation 14

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 y, j Mesh Domain i, j+1 Dy i-1, j i+1, j i, j-1 Dx Node point x, i Grid cell 15

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 j, columns – Confined – Unconfined – Convertible i, rows k, layers Layers can be different materials 16

1 -D Confined Aquifer Flow • Homogeneous, isotropic, 1 -D, confined flow • Governing equation Ground surface Confining Layer h. A Aquifer Node Dx h. B • Initial Condition h(x, t = 0) = h 0 b z y x i= 0 • Boundary Conditions h(x = 0, t) = h. A h(x = L, 0) = h. B 1 2 3 4 5 6 7 8 9 10 Grid Cell 17

Derivative Approximations • Governing Equation • LHS - 2 nd derivative WRT x • RHS - 1 st derivative WRT t Which one to use? Forward Backward 18

Time Derivative • Explicit – 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. 19

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. PDE Finite Difference Approx. 20

Explicit Method l+1 time level l time level unknown 21

1 -D Confined Aquifer Flow • Initial Condition Ground surface • Boundary Conditions Confining Layer h. A Aquifer Node Dx h. B b z Dx = 1 m y x i= 0 L = 10 m 1 2 3 4 5 Grid Cell 6 7 8 9 10 L T=b. K = 0. 75 m 2/d S = 0. 02 22

Explicit Method Ground surface Confining Layer h. A Aquifer Node Dx h. B b i= 0 1 2 3 4 5 6 7 8 9 10 Grid Cell Consider: r = 0. 48 r = 0. 52 23

Explicit Results (Dt = 18. 5 min; r = 0. 48 < 0. 5) 24

Explicit Results (Dt = 20 min; r = 0. 52 > 0. 5) 25

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 Ground surface Confining Layer h. A Aquifer Dx • Water flowing out of cell over interval Dt h. B b Dx i= 0 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 26

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 27

Implicit Method l+1 time level unknown l time level known 28

2 -D Steady-State Flow • General Equation Node No. Unknown heads Known heads • Homogeneous, isotropic aquifer, no well • Equal spacing (average of surrounding cells) 29

2 -D Heterogeneous Anisotropic Flow Tx and Ty are transmissivities in the x and y directions 30

2 -D Heterogeneous Anisotropic Flow • Harmonic average transmissivity 31

Transient Problems 32

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 (http: //www. groundwatermodels. com/) – GMS (http: //www. aquaveo. com/software/gms-groundwater-modeling-system-introduction) – PMWIN (www. ifu. ethz. ch/publications/software/pmwin/index_EN) – Each includes MODFLOW code 33

What Can MODFLOW Simulate? 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 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 34