Predicting Hydraulic Conductivity and Flow Rate of a Saturated Porous Media: A Modeling Perspective

By: Veatasha H. Dorsey

ENSC 4400 Environmental Modeling and Systems Dynamics

Instructor: Garrett Love, PhD

May 5, 2011

Introduction

The rate at which water flows through a porous media, or hydraulic conductivity, provides many useful

applications to our understanding of the hydrologic cycle. It is difficult to infer results from hydraulic

conductivity from measurements made at the sample level in application to a more realistic scenario.

Furthermore, incidences of error and disturbances are high when collecting hydraulic conductivity data. For

those reasons it is challenging to characterize the hydraulic conductivity of an aquifer, or even a small cross-

sectional area through direct laboratory experimentation. Modeling through STELLA offers the ability to

manipulate different parameters measured at the sample level (soil column) in order to simulate a real-world

dynamic system such as a groundwater aquifer. Policy can also be inferred from the model in the areas of rate

of recharge, groundwater use, depletion and transport of contaminants. In the area of groundwater

contamination hydraulic conductivity is an important soil property because the size of pore spaces largely

determine the transport and potency of the contaminant.

Materials and Methodology

Empirical Method

In this experiment, the saturated hydraulic conductivity or coefficient of permeability, K, was measured using the constant head method with a formula

K = VL/Ath

wherein V is the collected volume of water (cm3), L is the length of the soil column (cm), A is the cross-

sectional area of the column, t is the total time required to get volume (s) and h is the hydraulic head.

Essentially, the volume, V, of water that flows through the column during time, t, determines the saturated

hydraulic conductivity.

Limitations to this experiment primarily lie within the difficulty of determining the hydraulic conductivity of a

saturated media. The movement within media, inflow of water, and air pockets present in the medium

collectively impede on accuracy.

Systems Dynamics Modeling Method

The essence of systems dynamics lies in its focus on feedback mechanisms and rates of change over time. With

the empirical data, or saturated hydraulic conductivity of coarse-grained soil, collected utilizing the constant

head permeability, this portion of the study attempts to model the dynamic system by altering certain

parameters such as the change in head (∆h), cross-sectional area and length. Changing the aforementioned parameters could essentially allow for the simulation of falling head permeability used for fine grained soils. The formula for the falling head permeability is

K = aL/At * ln(h0-h1)

wherein a is the area of the burette, h0 is the initial height of water, h1 is the final height of water and t is the

time required to get a head drop of ∆h. The falling-head test essentially consists of measuring change in head

and quantity of flow over time.

The goal of this model is to attempt to simulate for conditions with falling head permeability (fine-grain soils)

by using the saturated hydraulic conductivity estimate found using constant head permeability (coarse-grain

soils). The ability to manipulate the flow (Q) by changing specified parameters: hydraulic conductivity (K),

cross-sectional area (A), head loss (∆h) and length (L) and verify the results mathematically using relevant

formulas will demonstrate a successful model. The flow equation used for the STELLA model is

Q = KA ∆h/L

STELLA modeling is a technology that can be used to better understand the relationships between components

of any system. It will also enable us to project changes into the future, subject to changes today. The stocks, or

units which will change in time, is the exited water (volume) out of the tube in the permeameter and the

hydraulic head. The flow into the “exited volume” stock (Figure.1) indicates a rate of change which is

coincidentally termed the “flow of water into the graduated cylinder (stock)”. The changeable parameters

feeding into the flow are the variables in the equations already listed.

Fig.1 Stella© Model for Constant and Falling Head Permeability

Stella Equation and “Switches/Dials”

Needless to say, one unit that is not included as a parameter in the STELLA model is time. Flow rates and

hydraulic conductivities are in terms of volume and length per unit time respectively. Consequently, the

functions used to compute flow rates will be different for different time units. Calculating flow per day (time

step of one day) will differ from flow per minute (time step of one minute). A time step that progressively

moves toward the termination at designated “length”, or DT, should have the same dimension as time. If DT is

too large the model becomes numerically unstable. Likewise, if DT is too small it will take longer to run the

model. In this study I used the common modeling DT values ≈ 0.5, 0.25, .0125.

The instrumentation used in the experimental portion of this study was a constant/falling head permeameter and two graduated cylinders.

Results

Constant Head Method Experimental DATA

Volume (v) Time (t) Flow Q (V/t) hydraulic conductivity K

50 9.7 5.154639175 0.097081698

100 17.3 5.780346821 0.108866181

150 26.9 5.576208178 0.105021465

200 33.6 5.952380952 0.112106246

250 42 5.952380952 0.112106246

Slope = 5.8529 0.110232637

50 7.5 6.666666667 0.071150097

100 14.75 6.779661017 0.072356031

150 21.75 6.896551724 0.073603549

200 29.25 6.837606838 0.072974459

250 36.5 6.849315068 0.073099415

Slope = 6.8449 0.073052295

50 3.7 13.51351351 0.086533902

100 7.25 13.79310345 0.088324259

150 11 13.63636364 0.087320574

200 14.6 13.69863014 0.087719298

250 18.2 13.73626374 0.087960285

Slope = 13.709 0.087785702

50 2.1 23.80952381 0.095290309

100 4.2 23.80952381 0.095290309

150 6.3 23.80952381 0.095290309

200 8.3 24.09638554 0.096438385

250 10.4 24.03846154 0.096206562

Slope = 23.996 0.096036623

cross-sectional area (A) head loss (∆h) length (L)

45.6 17 14.6

30

50

80

Constant Head System (Comparative Graph)

Number K Q

1 0.11023 5.8529

2 0.07305 6.8449

3 0.08779 13.709

4 0.09603 23.996

Falling Head Permeability (Comparative Graph)

Number K Q

1 0.11023 5.8529

2 0.07305 6.8449

3 0.08779 13.709

4 0.09603 23.996

Falling Head Permeabilities at Various Head Lengths (standard h0=0.05cm)

Head cm Experimental Stella Relative

TIME (s) TIME (s) Error

0 0 0 -----------

17 13 13.24 1.81%

30 24 23.81 0.80%

50 41 39.24 4.49%

80 64 62.8 1.91%

Head length vs. Time

90

Vo

lum

e 80

70

60

50

He

ad

/Ex

ite

d

40

30 Head/Exited Volume vs. Time

20

10

0

0 10 20 30 40 50 60 70

Time (s)

Conclusion

Based on the indicated results and parameters used in the model, this study concludes that the model is indeed

efficacious in predicting hydraulic conductivities at various head lengths. The volume vs. time graphs as well as

the comparative graphs (constant head permeability) displays a linear relationship found under a constant head

system, wherein the exited volume is the changing variable and the other parameters remain constant. Under

falling head conditions, the theory stands in the representation of the comparative graphs shown above. The rate

of flow, Q, under falling head conditions, displays decay, which is the inverse of the natural log function present

in the falling head calculation.

k

Soil Type cm/sec

Clean Gravel 1.0 to 100

Coarse Sand 0.01 to 1.0

Fine Sand 0.001 to 0.01

Silty Clay 0.00001 to 0.001

Clay Less Than 0.000001 Hydraulic Conductivities for various soil/clay types. Average K for experiment using coarse grained aquarium gravel (0.091776814)

Some potential sources of error in conducting both the constant and falling head experiments are air trapped in

sample or sample not 100% saturated; soil was washed from the sample; some of the head loss occurred in the

apparatus rather than in the sample; not starting and stopping stop watch at correct point; sample settling during

test; sample disturbed by flowing water at inlet and significant figures.