Mou Seminar

Introduction to Groundwater Modelling

Presentation Outline Groundwater in Hydrologic Cycle Why Groundwater Modeling is needed? Mathematical Models Groundwater Flow Models

Groundwater in Hydrologic Cycle

Types of Terrestrial WaterTypes of Terrestrial Water

Ground waterGround water

SoilSoilMoistureMoisture

SurfaceWater

Unsaturated Zone / Zone of Aeration / Vadose (Soil Water)

Pores Full of Combination of Air and Water

Zone of Saturation (Ground water)

Pores Full Completely with Water

Groundwater

Important source of clean waterMore abundant than SW

Linked to SW systems

Sustains flows in streams

Baseflow

pollution

Groundwater Concerns?

groundwater miningsubsidence

Problems with groundwater

Groundwater overdraft / mining / subsidence

Waterlogging

Seawater intrusion

Groundwater pollution

Why Groundwater Modelling is needed?

Groundwater

• An important component of water resource systems.

• Extracted from aquifers through pumping wells and supplied for domestic use, industry and agriculture.

• With increased withdrawal of groundwater, the quality of groundwater has been continuously deteriorating.

• Water can be injected into aquifers for storage and/or quality control purposes.

GROUND WATER MODELING

WHY MODEL?

•To make predictions about a ground-water system’s response to a stress

•To understand the system

•To design field studies

•Use as a thinking tool

Use of Groundwater models

• Can be used for three general purposes:• To predict or forecast expected artificial

or natural changes in the system. Predictive is more applied to deterministic models since it carries higher degree of certainty, while forecasting is used with probabilistic (stochastic) models.

Use of Groundwater models

• To describe the system in order to analyse various assumptions

• To generate a hypothetical system that will be used to study principles of groundwater flow associated with various general or specific problems.

Processes we might want to model

• Groundwater flow

calculate both heads and flow

• Solute transport – requires information on flow (velocities)

calculate concentrations

TYPES OF MODELS

CONCEPTUAL MODEL QUALITATIVE DESCRIPTION OF SYSTEM "a cartoon of the system in your mind"

MATHEMATICAL MODEL MATHEMATICAL DESCRIPTION OF SYSTEM

SIMPLE - ANALYTICAL (provides a continuous solution over the model domain)

COMPLEX - NUMERICAL (provides a discrete solution - i.e. values are calculated at only a few points)

ANALOG MODEL e.g. ELECTRICAL CURRENT FLOW through a circuit board with resistors to represent hydraulic conductivity and capacitors to represent storage coefficient

PHYSICAL MODEL e.g. SAND TANK which poses scaling problems

Mathematical Models

Mathematical model: simulates ground-water flow and/or solute fate

and transport indirectly by means of a set of governing equations thought to represent the physical processes that occur in the system.

(Anderson and Woessner, 1992)

Components of a Mathematical Model

• Governing Equation

(Darcy’s law + water balance equation) with head (h) as the dependent variable

• Boundary Conditions

• Initial conditions (for transient problems)

R x y Q

yx

z

1. Consider flux (q) through REV2. OUT – IN = - Storage3. Combine with: q = -KK grad h

q

Derivation of the Governing Equation

Numerical Methods All numerical methods involve

representing the flow domain by a limited number of discrete points called nodes.

A set of equations are then derived to relate the nodal values of the dependent variable such that they satisfy the governing PDE, either approximately or exactly.

• Numerical Solutions

Discrete solution of head at selected nodal points. Involves numerical solution of a set of algebraic equations.

Finite difference models (e.g., MODFLOW)

Finite element models (e.g., SUTRA)

Finite difference modelsmay be solved using:

• a computer program (e.g., a FORTRAN program)

• a spreadsheet (e.g., EXCEL)

Groundwater Flow Models

The Two Fundamental Equationsof Ground Water Flow

Basic Form

“First Law of Hydrogeology”

“Second Law of Hydrogeology”

Basic Form

Darcy’s Law:

Average Linear Velocity

Flow Equation: 0dx

hd 2

2

1-D, Steady State

Darcy’s Law

Darcy’s Experiment (1856)

AQxQhQ ,1,

xhAKQ

xhAQ

xQ Q: Volumetric flow rate [L3/T]

h

h

x

Slope = h/x ~ dh/dx

Darcy investigated ground water flow under controlled conditions

hx

h1 h2

h1

h2

x1 x2

K: The proportionality constant is added to form the following equation:

K units [L/T]

: Hydraulic Gradient

A

xhh

A: Cross Sectional Area (Perp. to flow)

Darcy’s Law (cont.)

Other useful forms of Darcy’s LawQA =

QA.n = q

n =

Volumetric Flux

Ave. Linear Velocity

Used for calculating Q given A

Used for calculating average velocity of contaminant transport

Assumptions: Laminar, saturated flow

Introduction to Ground Water Flow Modeling

Predicting heads (and flows) and Approximating parameters

Solutions to the flow equationsMost ground water flow models are solutions of some form of the ground water flow equation

PotentiometricSurface

x

xx

ho

x0

h(x)

x

K q

“e.g., unidirectional, steady-state flow within a confined aquifer

The partial differential equation needs to be solved to calculate head as a function of position and time, i.e., h=f(x,y,z,t)

h(x,y,z,t)?

Darcy’s Law Integrated

Flow Modeling (cont.)

Analytical models (a.k.a., closed form models) The previous model is an example of an analytical model

is a solution to the 1-D Laplace equation

i.e., the second derivative of h(x) is zero

With this analytical model, head can be calculated at any position (x) Analytical solutions to the 3-D transient flow equation would give head at any position and at any time, i.e., the continuous function h(x,y,z,t)Examples of analytical models:

1-D solutions to steady state and transient flow equationsThiem Equation: Steady state flow to a well in a confined aquiferThe Theis Equation: Transient flow to a well in a confined aquiferSlug test solutions: Transient response of head within a well to a

pressure pulse


Common Analytical Models Thiem Equation: steady state flow to a well within a confined aquifer

Analytic solution to the radial (1-D), steady-state, homogeneous K flow equation

Gives head as a function of radial distance

Theis Equation: Transient flow to a well within a confined aquifer Analytic solution of radial, transient, homogeneous K flow equation Gives head as a function of radial distance and time


Forward Modeling: Prediction

Models can be used to predict h(x,y,z,t) if the parameters are known, K, T, Ss, S, n, b…

Heads are used to predict flow rates,velocity distributions, flow paths, travel times. For example:

Velocities for average contaminant transport Capture zones for ground water contaminant plume capture Travel time zones for wellhead protection

Velocity distributions and flow paths are then used in contaminant transport modeling

1-D, SS Thiem Theis


Inverse Modeling: Aquifer Characterization Use of forward modeling requires estimates of aquifer

parameters Simple models can be solved for these parameters

e.g., 1-D Steady State:

This inverse model can be used to “characterize” K This estimate of K can then be used in a forward model to

predict what will happen when other variables are changed

ho

h1

Clay

b

x

ho h1

Q Q


Inverse Modeling: Aquifer Characterization The Thiem Equation can also be solved for K

Pump Test: This inverse model allows measurement of K using a steady state pump test

A pumping well is pumped at a constant rate of Q until heads come to steady state, i.e.,

The steady-state heads, h1 and h2, are measured in two observation wells at different radial distances from the pumping well r1 and r2

The values are “plugged into” the inverse model to calculate K (a bulk measure of K over the area stressed by pumping)


Inverse Modeling: Aquifer Characterization Indirect solution of flow models

More complex analytical flow models cannot be solved for the parameters

Curve Matching or Iteration

This calls for curve matching or iteration in order to calculate the aquifer parameters

Advantages over steady state solution• gives storage parameters S (or Ss) as well as T (or K)• Pump test does not have to be continued to steady state• Modifications allow the calculation of many other parameters

e.g., Specific yield, aquitard leakage, anisotropy…


Limitations of Analytical Models Closed form models are well suited to the

characterization of bulk parameters However, the flexibility of forward modeling

is limited due to simplifying assumptions: Homogeneity, Isotropy, simple geometry,

simple initial conditions… Geology is inherently complex:

Heterogeneous, anisotropic, complex geometry, complex conditions…

This complexity calls for a more powerful solution to the flow equation Numerical modeling

Numerical Modeling in a NutshellA solution of flow equation is approximated on a

discrete grid (or mesh) of points, cells or elements

Within this discretized domain: 1)Aquifer parameters can be set at each cell

within the grid2)Complex aquifer geometry can be modeled3)Complex boundary conditions can be

accounted forRequires detailed knowledge of 1), 2) , and 3)As compared to analytical modeling, numerical modeling is:

Well suited to prediction but More difficult to use for aquifer characterization


The parameters and variables are specified over the boundary of the domain (region) being modeled

An Introduction to Finite Difference Modeling

Approximate Solutions to the Flow Equation

Partial derivatives of head represent the change in head with respect to a coordinate direction (or time) at a point.e.g., thoryh

h

y

hy

h1

h2

y1 y2

yh

yh

These derivatives can be approximated as the change in head (h) over a finite distance in the coordinate direction (y) that traverses the point

i.e., The component of the hydraulic gradient in the y direction can be approximated by the finite difference h/y

The Finite Difference Approximation of Derivatives

Finite Difference Modeling (cont.)

Approximation of the second derivative The second derivative of head with respect to x represents the change

of the first derivative with respect to x The second derivative can be approximated using two finite differences

centered around x2

This is known as a central difference

h

xx

ha

ho

xo xbxa

xhb

ha-ho

ho-hb

x

xhh

yh

oa

xhh

yh

bo


Finite Difference Approximation of 1-D, Steady State Flow Equation


Physical basis for finite difference approximation

y

z

h

x

ha

ho

xo xbxa

xhb

ha-ho

ha-hb

x

xhh

xh

oa

xhh

xh

bo

xhhKzy

qzyQ ii

oaoa

xhhKzy

qzyQ oo

boob

x

Kab: average K of cell and K of cell to the left; Kab: average K of cell and K of cell to the left

2

2boob

aooa

KKKKKK

KaKaKo


Discretization of the Domain Divide the 1-D domain into equal cells

of heterogeneous K

… …h1 h2 h3 hi-1 hn

x x x x x x x

head specified: and Constant

22

1no

1ii1/2i

1-ii1/2-i

hhx

KKKKKK

…hi hi+1

x x

Solve for the head at each node gives n equations and n unknownsThe head at each node is an average of the head at adjacent cells weighted by the Ks

ho hn+1

Spe

cifie

dH

ead

Spe

cifie

dH

ead


2-D, Steady State, Uniform Grid Spacing, Finite Difference Scheme Divide the 2-D domain into equally

spaced rows and columns of heterogeneous K

ha ho hb

hd

hc

x

x

x

2

222

dood

cooc

boob

aooa

KKKKKKKKKKKK

Ka

Kc

Kb

Kd

Kd

x x x

Solve for ho

ha ho hb

hcKa

Kc

KbKd


Incorporate Transmissivity: Confined Aquifers multiply by b (aquifer thickness)

22

222222

ddoodood

ccoocooc

bbooboob

aaooaooa

bKbKTTTbKbKTTTbKbKTTTbKbKTTT

x x x x x

Ko KbKa

ba bo bb

Solve for ho

ha ho hb

hcKa

Kc

KbKd

x x x x x

Ko KbKa

ha ho hb


Incorporate Transmissivity: Unconfined Aquifers b depends on saturated thickness which is head

measured relative to the aquifer bottom 2

222

dood

cooc

boob

aooa

hhhhhhhhhhhh

Solve for ho


2-D, Steady State, Isotropic, HomogeneousFinite Difference Scheme

ha ho hb

hd

hc

x

x

x

x x x

Solve for ho

Basic Finite Difference Design

Discretization and Boundary Conditions

Grids should be oriented and spaced to maximize the efficiency of the model

Boundary conditions should represent reality as closely as possible

Basic Finite Difference Design (cont.)

Discretization: Grid orientation Grid rows and columns should line up with as many rivers,

shorelines, valley walls and other major boundaries as much as possible


Discretization: Variable Grid Spacing Rules of Thumb

Refine grid around areas of interest

Adjacent rows or columnsshould be no more than twice (or less than half) as wide as each other

Expand spacing smoothly Many implementations of

Numerical models allowOnscreen manipulation of Grids relative to an imported Base map


Boundary Conditions Any numerical model must be bounded on

all sides of the domain (including bottom and top)

The types of boundaries and mathematical representation depends on your conceptual model

Types of Boundary Conditions Specified Head Boundaries Specified Flux Boundaries Head Dependant Flux Boundaries

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