Groundwater Systems D Nagesh Kumar, IISc Water Resources Planning and Management: M8L3 Water Resources System Modeling

Groundwater Systems

D Nagesh Kumar, IIScWater Resources Planning and Management: M8L3

Water Resources System Modeling

Objectives


To briefly learn about groundwater hydrology

To discuss about the governing equations of groundwater

systems

To discuss simulation and optimization of groundwater

systems

Introduction


Groundwater management: Deals with planning, implementation, development

and operation of water resources containing groundwater

Numerical-simulation models have been used extensively for understanding the flow

characteristics of aquifers and evaluate the groundwater resource

Simulation models attempt various scenarios to find the best objective

Optimization models directly consider the management objective taking care of all

the constraints

Groundwater Hydrology


Subsurface water is stored underneath in subsurface formations called aquifers Unconfined aquifer: Upper surface is water table itself Confined aquifer is confined under pressure greater than the atmospheric Confined aquifer may be confined between two impermeable layers An aquifer serves two functions:

Storage and Transmission.

Storage function: Exhibited through porosity Φ, specific yield Sy and storage coefficient S.

Transmission function: Exhibited through the permeability property (coefficient of permeability K).

Groundwater Hydrology…


Porosity: Measure of the amount of water an aquifer can hold

Specific yield: Water drained from a saturated sample of unit volume

Specific retention: Water retained in the unit volume

Porosity is the sum of specific yield and specific retention

Storage coefficient: Volume of water an aquifer releases or stores per unit surface

area per unit decline of head

Permeability: Measure of the ease of movement of water through aquifers

Coefficient of permeability or hydraulic conductivity: Rate of flow of water

through a unit cross-sectional area under a unit hydraulic gradient

Groundwater Hydrology…


Transmissivity T: Rate of flow of water through a vertical strip of unit width extending the saturated thickness of the aquifer under a unit hydraulic gradient

T = K b for a confined aquifer where b is the saturated thickness of aquifer T = K h for a confined aquifer where h is the head (saturated thickness)

Darcy’s law Flow thorough an aquifer is expressed by Darcy’s law

Can be expressed as (1)

where v is the velocity or specific discharge, l is the length of flow along the average direction and is the rate of headloss per unit length. Then, the total discharge, q is (2)

lhKv

lh

lhKAAvq

Darcy’s law states that flow rate through a porous media is proportional to the head loss and inversely proportional to the length of flow path

Simulation of Groundwater Systems


Governing equations: Darcy’s law in terms of transmissivity is

for confined aquifers (3)

for unconfined aquifers (4) Considering a two-dimensional horizontal flow as shown by a rectangular control

volume element.

lh

bTv

lh

hTv

q2

q4

q1

q3

Δy

Δx

i i+1i - 1

j - 1

j

j +1

Simulation of Groundwater Systems…


Governing equations:

General equations of flow can be expressed as: The flow discharge q = Av for four sides

(5)

(6)

(7)

(8)

where A = Δx. h for unconfined case or A = Δx. b for confined case and assuming

constant transmissivities along the x and y directions. are the hydraulic gradients at

sides 1,2 ,… respectively.

q2

q4

q1

q3

Δy

Δx

i i+1i - 1

j - 1

j

j +1

11

xhyTq x

22

xhyTq x

33

xhxTq y

44

xhxTq y



Governing equations: The rate at which water is stored or released in the element is

(9)

where S is the storage coefficient of the element. The flow rate for constant net withdrawal or recharge for time Δt

(10) Applying continuity law,

(11) Substituting eqns. 5 – 10 in above eqn, and dividing by Δx Δy and simplifying, the

final form of eqn. 11 will be

(12)

where W = q / Δx Δy.

45

thyxSq

tqq 6

654321 qqqqqq

WthS

yhT

xhT yx

2

2

2

2



Governing equations: The above equations can be written in finite difference form and solved for each

rectangular element. Partial derivatives in eqns. 5-9 can be expressed in finite difference form as,

(13)

These can be substituted in eqn. 12 and solved using finite element methods.

thh

th

yhh

yh

yhh

yh

xhh

xh

xhh

xh

tjitji

j

tjitji

j

tjitji

i

tjitji

i

tjitji

1,,,,

,1,,,

4

,,,,

3

,,1,,

2

,,,,1

1

Optimization Model


Optimization models for hydraulic management for groundwater have been

developed based on three approaches: Embedding approach: Equations from a simulation model are incorporated into an

optimization model directly Optimal control approach: Simulation model solves the governing equations, for

each iteration of the optimization. It works on the concept of optimal control theory Unit response matrix approach: A unit response matrix is generated by running the

simulation model several times with unit pumpage at a single node. Total

drawdowns are then determined by superpositions. In this lecture, we will discuss only about the embedding approach for steady-state

one-dimensional confined and unconfined aquifers.

Embedding approach: Steady state one-dimensional confined aquifer


Consider a confined aquifer with penetrating wells and flow in one-dimension

From eqn. 12, the governing equation is

where (14)

q1 q2 q3

▼

▼

Δx

Δx

Δx

Δx

h0

h4

xTW

xh

2

2

0

th

Embedding approach: Steady state one-dimensional confined aquifer…


Eqn. (14) can be expressed in finite difference form as (using central difference

scheme)

(15) Optimization problem can be stated as

Maximize (16)

where i is the number of wells. Subject to

(17)

where Wmin is the minimum production rate for each well.

The pumpage can be finally determined from the relation qi = W.Δ xi2.

x

iiii

TW

xhhh

2

11 2

i

ihZ

00

min

i

i

ii

Wh

WW

Example: Steady state one-dimensional confined aquifer


Formulate an LP model for the above confined aquifer for maximum heads. The

boundaries have a constant head h0 and h5. The distance between the wells is Δ x.

Solution:Objective function: Maximize Subject to: Acc. to eqn. 15

And acc. to eqns. 17

The unknowns are h1, h2, h3, h4 and W1, W2, W3, W4.

4321 hhhhZ

54

2

43

3

2

432

2

2

321

01

2

21

2

02

02

2

hWTxhh

WTxhhh

WTxhhh

hWTxhh

4,...,104,...,10

min4321

iWih

WWWWW

i

i

Embedding approach: Steady state one-dimensional unconfined aquifer


Governing equation can be written as

(18)

where T = Kh.

Substituting w=h2 and assuming K is constant, the finite difference form can be

written as

(19)

KW

xh

WxhT

x x

22

22

KW

xwww

xw iiii 22

211

2

2

Embedding approach: Steady state one-dimensional unconfined aquifer…


Optimization problem is to

Maximize (20)

where i is the number of wells.

Subject to

(21)

After solving the above problem, the heads

i

iwZ

00

min

i

i

ii

Ww

WW

ii wh

Example: Steady state one-dimensional unconfined aquifer


Formulate a LP model to determine the steady state pumpages of an unconfined

aquifer shown below

Solution

Optimization problem is

Objective function: Maximize

q1 q2 q3

▼

▼h0 h4

Δx Δx Δx Δx

4321 wwwwZ

Example: Steady state one-dimensional unconfined aquifer…


Subject to:

Acc. to eqn. 19

And acc. to eqns. 21

The unknowns are w1, w2, w3, w4 and W1, W2, W3, W4.

54

2

43

3

2

432

2

2

321

01

2

21

22

022

022

22

wWKxww

WKxwww

WKxwww

wWKxww

4,...,104,...,10

min4321

iWiw

WWWWW

i

i


Thank You

Documents

Groundwater Systems D Nagesh Kumar, IISc Water Resources Planning and Management: M8L3 Water Resources System Modeling