HONR 297Environmental ModelsChapter 2: Ground Water2.2: Physical Principles and Mathematical Models

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Water TableAs discussed in the last section, if we

dig down into the ground, we will eventually hit water.

Due to the force of gravity, water that enters the ground from surface water (such as rivers, streams, lakes, puddles, etc.) or precipitation (rain, lawn sprinklers, etc.) will move downward and fill up the open spaces within soil and rock to a certain height.

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Water TableDefinition:

◦The height to which water rises underground is called the water table.

Below the water table the ground is saturated (i.e. “full of water” in all open spaces).

We can think of the water table as the “surface” or “top” of the saturated zone.

See the next figure as well as Figure 2.1 in our textbook!

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Water Table

Courtesy USGS: http://ga.water.usgs.gov/edu/earthgwaquifer.html

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AquiferDefinition

◦An aquifer is any portion of the underground through which ground water can flow.

Note:◦There may be more than one aquifer

below the surface!◦We call the aquifer with the water

table as the upper border the water table aquifer!

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Aquifers

Courtesy USGS: http://pubs.usgs.gov/gip/gw_ruralhomeowner/gw_ruralhomeowner_new.html

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Recharge and DischargeWater can move from one aquifer to the next,

but tends to stay within a given aquifer and follows the “path of least resistance”.

Definition:◦ Areas where materials above the aquifer are

permeable enough to permit infiltration of precipitation to the aquifer are known as points of recharge (or recharge areas).

◦ Areas at which the water table intersects the land surface and water leaves an aquifer by way of springs, streams, or lakes and wetlands are known as points of discharge (or discharge areas).

◦ Source of these definitions: http://pubs.usgs.gov/gip/gw_ruralhomeowner/gw_ruralhomeowner_new.html.

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Recharge and DischargeIn Figure 2.1, sources of ground-

water aquifer recharge would be the upland pond and any precipitation from rain seeping into the ground.

A source of discharge from the ground-water aquifer would be the walls of the river.

What could be sources of recharge or discharge in the next slide?

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Recharge and Discharge

Courtesy USGS: http://ga.water.usgs.gov/edu/watercyclegwdischarge.html

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Source of Contamination?Suppose we discover

contamination in a stream or river.

The contamination may be from a factory upstream or it may be from contaminated ground water.

This in turn may be from a tank or landfill leaking into the ground water.

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Bottom LineIf there are no

identifiable sources of contamination along the shore of a river or stream if may be very difficult to identify sources of contamination!

What might be the sources of groundwater contamination in the figure on the next slide?

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Sources of Contamination

Courtesy USGS: http://md.water.usgs.gov/publications/fs-150-99/html/figure2_big.gif

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Modeling BasicsQuestion: What is a model?Answer: A model is a

representation of reality.

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Example 1What are some examples of models?

◦Maps – cities, countries, world, roads, topographies, etc.

◦Model kits – cars, planes, buildings, legos, etc.

◦Games – war (chess, Battleship, Risk, etc.), Life, Settlers of Catan, computer games (angry birds, WoW), etc.

◦Movies – Star Wars, Pirates of the Caribbean Avatar, Inception, 2012, etc.

What are the advantages, disadvantages of each?

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Mathematical ModelsDefinition

◦A mathematical model is a model that is built using the tools and substance of mathematics (including computers and computer software).

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Example 2 (A “simple” modeling project! Using the given water

bottle, estimate the number of pennies that can fit into the bottle – you have one minute!

Possible solutions …◦ How accurate are the

solutions?◦ How accurate should they

be? What if we had 15

minutes and more resources at hand to help?

What if we had as much time as we wanted?

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Mathematical ModelsExample 2 illustrates some ideas

that are common to mathematical models:◦Resource constraints – time,

equipment, money, personnel, computing power, expertise, etc. For pennies and volume, time to solve

problem.

◦Resolution – level of detail (low to high), resolution depending on given situation. For pennies and volume, how accurate is

solution?

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Mathematical ModelsJust as with all other models,

mathematical models will take what is usually a complex phenomenon and try to describe what is happening in terms of some of the factors in the situation being modeled.

Most of the time, not all factors can be used in a model, as it would become far too complicated.

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Diagram of the Modeling Process

Real-world Phenomen

on

Mathematical Model

Real-world Conclusion

s

Mathematical

Conclusions

Abstraction(simplify)

Solve

Solution

Equation, etc.

Interpretation (test or compare)

Modeler or Experimenter -->Two ways to reach conclusions about what happens in the real world!

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Model vs. ExperimentConclusions about the real world

can be made by experimenting or modeling.

Each method has advantages and disadvantages.

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Experiment Real-world trials lead to

precise results. Not all real-world

phenomena can be studied with experiments.

For example,◦ Effects of a nuclear power

plant failure.◦ How much exposure to a toxic

substance is fatal to humans.◦ Conditions at the center of the

sun.◦ Exact extent of groundwater

contamination. Cost prohibitive, not moral,

impossible to perform.

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ModelA way to predict

outcomes of real-world phenomena in a cheaper and safer fashion.

The problem is to find a model that actually describes the phenomena accurately and in a useful way!

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How modeling usually worksDesign ModelSolve ModelDraw ConclusionsReflect Reality?

◦Yes Done, model is useful!◦No Refine model design.

Note:◦A big assumption here is that the

model can actually be solved!

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A Simple Groundwater Model DiagramWhen trying to

construct a groundwater model, a good diagram to keep in mind as a starting point is a diagram like that in Figure 2.2 in our text or the figure on this slide!

What might be a problem with this approach?

Courtesy USGS: http://pubs.usgs.gov/gip/gw_ruralhomeowner/gw_ruralhomeowner_new.html

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A More Complex Groundwater Model DiagramModels like those on

the last slide or Figure 2.2 in our text may be too simple for an initial model

The underground composition may be much more complicated, such as in the diagram on this slide or Figure 2.3 in our text!

Courtesy USGS: http://md.water.usgs.gov/publications/fs-150-99/html/figure5_big.gif

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“Real-life” Complexities - SoilLooking at Figure 2.2, we’d expect

that as we dig down from the surface, we’d first encounter a layer of unconsolidated (i.e. loose) materials known as soils, followed by a layer of rock, known as bedrock.

Types of soil include gravel, sand, silt, clay, and organic materials.

Soils vary in particle size, chemical composition, presence of organic matter, etc.

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“Real-life” Complexities - BedrockJust like soil, bedrock can be made up of

different types of rock.Examples of sedimentary rock, which are

made up of compacted sediments laid down by ancient bodies of water, include shale, sandstone, siltstone, and limestone.

Igneous rock is formed by melting and heating deep below the earth’s surface – examples include granite and obsidian.

See Figure 2.3 in our text!Bedrock Geology of Indiana

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“Real-life” Complexities – AquifersGround water is able to flow

through both soil and rock – the “ability” to flow is dependent on the makeup of the individual components of each.

An aquifer can be made up of soil, rock, or a combination of both.

Typical aquifer materials include sand or gravel (soils) and sandstone, limestone, or fractured granite (rocks).

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“Real-life” Complexities – Aquicludes and AquitardsDefinitionAn aquiclude blocks the flow of water.

An aquitard resists the flow of water.Examples of materials that block or

resist water flow include clay or fine silt (soils) and salt, shale, or non-fractured granite (rocks).

Note that even within a specific type of material such as limestone, sandstone, sand, or gravel, water flow rate or ability to flow may vary.

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References USGS

◦ http://ga.water.usgs.gov/edu/earthgwaquifer.html◦ http://pubs.usgs.gov/gip/gw_ruralhomeowner/gw_

ruralhomeowner_new.html◦ http://ga.water.usgs.gov/edu/watercyclegwdischar

ge.html◦ http://md.water.usgs.gov/publications/fs-150-99/h

tml/figure2_big.gif◦ http://md.water.usgs.gov/publications/fs-150-99/h

tml/figure5_big.gif IGS

◦ http://igs.indiana.edu/Bedrock/ Michael Olinick, An Introduction to

Mathematical Models in the Social and Life Sciences

Charles Hadlock, Mathematical Modeling in the Environment, Chapter 2, Section 2.2