Henry Darcy and His Law

Summary of Darcy's Experiments

Glenn Brown
Oklahoma State University
1/14/00
revised 4/26/00, 9/2/00

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In the following page Darcy's experiment is summarized and reviewed. Any judgmental comment is placed in italics.

Procedures

Starting late in 1855, Henry Darcy supervised a succession of experiments with the objective to determine the relation between the volumetric flow rate of water through sand beds and the hydraulic head loss. All experiments were completed in the courtyard of an unnamed hospital in Dijon. The first set was carried out with the assistance of Mr. Charles Ritter from October 29th to November 2nd. Four different packings of Saone River sand were tested, with each designated a test Series. For each Series, three to ten different experiments were performed. The inlet pressure was varied for each experiment while holding the outlet at atmospheric pressure. A second experimental set was carried out on February 17 and 18, 1856 by Ritter on a single sand packing. In this set, both inlet and outlet pressure was varied. A total of 35 experiments were reported and used in the final analysis. The Chief Engineer, Mr. Baumgarten is reported to have repeated the experiments, but the data presented does not list any duplication.

Darcy's apparatus is shown in his Plate 24, Figure 3. It was a vertical steel column with an inside diameter of 0.35 m and sealed on both ends by bolted flange plates. Its total height was reported in the text as 2.5 meters, but it is dimensioned on Figure 3 as 3.5 m. At the bottom, an outlet reservoir was created by supporting a set of screens above the bottom, which in turn supported the sand. An inlet reservoir was created by leaving a void between the between the sand and the column top. A tap at the top allowed air to be bled from the system. Water flow rate was determined by timing the effluent accumulation in a volumetric box. Supply and effluent lines were mounted on the column side, and both had valves to allow control. Water was supplied directly from the hospital house line, which induced considerable oscillations as users elsewhere turned faucets on and off.

Mercury U-tube manometers were connected to both reservoirs to provide pressure measurements. At low-flows they were read to + 1 mm, while at high flow oscillations only allowed reading to within + 5 mm. Darcy reported this represented knowing the water pressure within 26.2 mm and 1.30 m for low and high flows respectively.

Each packing used Saone River sand. Unwashed sand was used in the first two Series, the third used washed sand, and the forth used very well washed sand that was slightly larger in grain size. The sand used in the February experiments was not reported. Packing height was intentionally varied from 0.58 to 1.71 m. Sand was placed in the column by dropping it into the water filled column, which was intended to eliminate air entrapment.

Each run consisted of setting the inlet supply valve and allowing the column to reach equilibrium. Then the manometers were read and the volume flow measured over a period of 10 to 25 minutes. Most direct data measurements were not reported. Instead for the November experiments, the computed flow rate and the head loss in meters of water were reported. In the February experiments, the outlet pressure was set by an unrecorded method, and inlet and outlet heads relative to the bottom of the filter were reported.

Procedural Comments

Darcy's column was well designed even by today's standards. At 0.35 m inside diameter and up to 1.71 m packed length, it was more than adequate in size. As far as the true overall height of the column, the longest sand packing used was 1.71 m, which is most consistent with a 2.5 m total dimension. Mercury manometers are reliable, easy to use and an ideal choice for his needs.

Darcy's statement concerning pressure reading accuracy is somewhat curious. He clearly multiplied the manometer precision by twice the specific weight of mercury, (13.1). He probably was considering the maximum additive effects on the total head loss caused by taking the difference in two manometer readings. Using modern measurement theory we would predict the error as the square root of two times the specific weight of mercury, or 19 and 93 mm for the low and high flow respectively. In any case, as will be shown later, the data appears to have much better accuracy.

While Darcy used the words pressure, it is clear that he was referring to what we now term the hydraulic head at the filter base. That is the sum of gage pressure and elevations heads with the elevation datum placed at the filter base. This could not have been an accident on his part. As a graduate of L'Ecole des Ponts et Chausees he would be well versed in the Bernoulli equation. He would thus know that the difference in the Bernouli terms between inlet and outlet, (velocity, pressure and elevation) represented the energy loss though the sand. The low seepage velocity justifies the fact that he ignored the velocity term. At his highest flow, the velocity head was only 0.01 m within the inlet and outlet sections.

The only serious apparatus concern was the use of an unregulated inlet supply, of which Darcy was clearly aware. It would have been possible for Darcy to construct a constant head reservoir, but that would have taken more effort than the column itself. It would have also required wasting water or the use of a hand pump.

The method by which Ritter regulated the outlet reservoir pressure is suggested by Figure 3. It would be a simple matter to pressurize the outlet by restricting the outlet valve. Likewise, dropping the discharge pipe to a lower position, while positioning the valve wide open could produce a relative suction at the filter base. However, suctions larger than 3 meters were reported. Thus, the column must have been elevated greater than the 1 meter shown, or a subsurface drain must have been available.

The only point of criticism for Darcy's procedures is his placement of the sand. Dropping the sand into the water filled column probably allowed it to segregate and produce layering by size fraction with the coarsest particles on the bottom of each lift. It also probably produced a relatively low packing density that would be subject to compaction. However since he waited for equilibrium before taking any measurement and measured the filter height after the experiment, any compaction would not be obvious in the data or for that matter impact his final conclusions.

Darcy's Results

Darcy's results were reported in two tables. In the first Set, flow rate, Q varied from 2.13 to 29.4 l/min, while the head loss, H ranged from 1.11 to 13.93. Darcy computed Q/H for each experiment and noted that for a given packing, that it was near constant. He also calculated conductivity values in units of l/m2-s (10-3 m/s) for each of the first Set Series and noted that they varied due to the sand being different between each. For the second Set, Darcy again noted the near constant value of Q/H, consistent with the other experiments. Thus, he showed conclusively that the flow was a linear function of the head loss across the filter bed.

Table 1. First Set of Experiments: Outlet at atmospheric pressure

Exp #

Packed
Sand
Length, e
(m)

Test
Duration
(min)

Flow, Q
(l/min)

Inlet
Head
(m)

Outlet
Head
(m)

Head, H
(Darcy)
(m)

Head, H
(comp'td)
(m)

Q/H
(Darcy)
(l/m-min)

Q/H
(comp'td)
(l/m-min)

Darcy
Flux, q
(m/s)

H/e

Point
k
(m/s)

Residual
error, RE
(m/s)
1st Series

1

0.58

25

3.60

1.11

0

1.11

-

3.25

3.24

6.24E-04

1.91

3.26E-04

7.83E-05

2

0.58

20

7.65

2.36

0

2.36

-

3.24

3.24

1.33E-03

4.07

3.26E-04

1.66E-04

3

0.58

15

12.00

4.00

0

4.00

-

3.00

3.00

2.08E-03

6.90

3.01E-04

1.13E-04

4

0.58

18

14.28

4.90

0

4.90

-

2.91

2.91

2.47E-03

8.45

2.93E-04

6.63E-05

5

0.58

17

15.20

5.02

0

5.02

-

3.03

3.03

2.63E-03

8.66

3.04E-04

1.67E-04

6

0.58

17

21.80

7.63

0

7.63

-

2.86

2.86

3.78E-03

13.16

2.87E-04

2.76E-05

7

0.58

11

23.41

8.13

0

8.13

-

2.88

2.88

4.06E-03

14.02

2.89E-04

6.09E-05

8

0.58

15

24.50

8.58

0

8.58

-

2.85

2.86

4.24E-03

14.79

2.87E-04

2.86E-05

9

0.58

13

27.80

9.86

0

9.86

-

2.82

2.82

4.82E-03

17.00

2.83E-04

-2.86E-05

10

0.58

10

29.40

10.89

0

10.89

-

2.70

2.70

5.09E-03

18.78

2.71E-04

-2.58E-04

Linear k = 2.85E-04

Fit R2 = 0.993

2nd Series

1

1.14

30

2.66

2.60

0

2.60

-

1.01

1.02

4.61E-04

2.28

2.02E-04

8.22E-05

2

1.14

21

4.28

4.70

0

4.70

-

0.91

0.91

7.42E-04

4.12

1.80E-04

5.71E-05

3

1.14

26

6.26

7.71

0

7.71

-

0.81

0.81

1.08E-03

6.76

1.60E-04

-3.81E-05

4

1.14

18

8.60

10.34

0

10.34

-

0.83

0.83

1.49E-03

9.07

1.64E-04

-1.57E-05

5

1.14

10

8.90

10.75

0

10.75

-

0.83

0.83

1.54E-03

9.43

1.64E-04

-2.34E-05

6

1.14

24

10.40

12.34

0

12.34

-

0.84

0.84

1.80E-03

10.82

1.66E-04

4.92E-06

Linear k = 1.66E-04

Fit R2 =0.991

3rd Series

1

1.71

31

2.13

2.57

0

2.57

-

0.83

0.83

3.69E-04

1.50

2.46E-04

4.59E-05

2

1.71

20

3.90

5.09

0

5.09

-

0.77

0.77

6.76E-04

2.98

2.27E-04

3.57E-05

3

1.71

17

7.25

9.46

0

9.46

-

0.76

0.77

1.26E-03

5.53

2.27E-04

6.66E-05

4

1.71

20

8.55

12.35

0

12.35

-

0.69

0.69

1.48E-03

7.22

2.05E-04

-7.15E-05

Linear k = 2.15E-04

Fit R2 = 0.984

4th Series

1

1.7

20

5.25

6.98

0

6.98

-

0.75

0.75

9.10E-04

4.11

2.22E-04

2.68E-05

2

1.7

20

7.00

9.95

0

9.95

-

0.70

0.70

1.21E-03

5.85

2.07E-04

-4.56E-05

3

1.7

20

10.30

13.93

0

13.93

-

0.74

0.74

1.78E-03

8.19

2.18E-04

2.27E-05

Linear k = 2.15E-04

Fit R2 = 0.992

Table 2. Second Set of Experiments: Outlet Pressurized

Exp #

Packed
Sand
Length, e
(m)

Test
Duration
(min)

Flow, Q
(l/min)

Inlet
Head
(m)

Outlet
Head
(m)

Head, H
(Darcy)
(m)

Head, H
(comp'td)
(m)

Q/H
(Darcy)
(l/m-min)

Q/H
(comp'td)
(l/m-min)

Darcy
Flux, q
(m/s)

H/e

Point
k
(m/s)

Residual
error, RE
(m/s)

1

1.10

15

18.8

9.48

-3.6

13.08

13.08

1.44

1.44

3.26E-03

11.89

2.74E-04

-1.29E-05

2

1.10

15

18.3

12.88

0

12.88

12.88

1.42

1.42

3.17E-03

11.71

2.71E-04

-4.95E-05

3

1.10

10

18

9.8

-2.78

12.58

12.58

1.43

1.43

3.12E-03

11.44

2.73E-04

-2.65E-05

4

1.10

10

17.4

12.87

0.46

12.41

12.41

1.40

1.40

3.01E-03

11.28

2.67E-04

-8.79E-05

5

1.10

20

18.1

12.8

0.49

12.35

12.31

1.47

1.47

3.14E-03

11.23

2.79E-04

4.83E-05

6

1.10

16

14.9

8.86

-0.83

9.69

9.69

1.54

1.54

2.58E-03

8.81

2.93E-04

1.59E-04

7

1.10

15

12.1

12.84

4.4

8.44

8.44

1.43

1.43

2.10E-03

7.67

2.73E-04

-1.37E-05

8

1.10

15

9.8

6.71

0

6.71

6.71

1.46

1.46

1.70E-03

6.10

2.78E-04

2.04E-05

9

1.10

20

7.9

12.81

7.03

5.78

5.78

1.37

1.37

1.37E-03

5.25

2.60E-04

-7.63E-05

10

1.10

20

8.65

5.58

0

5.58

5.58

1.55

1.55

1.50E-03

5.07

2.95E-04

1.04E-04

11

1.10

20

4.5

2.98

0

2.98

2.98

1.51

1.51

7.80E-04

2.71

2.88E-04

3.46E-05

12

1.10

20

4.15

12.86

9.88

2.98

2.98

1.39

1.39

7.19E-04

2.71

2.65E-04

-2.60E-05

Linear k = 2.75E-04

Fit R2 = 0.994

Results Comments

Tables 1 and 2 list Darcy's results with additional calculations. In his experiments, flow and head loss range exceeded an order of magnitude, a reasonable range given the apparatus used. Two small discrepancies in his reported data can be found. Independent calculation shows that for Set 1, Series 1, Test 1 the flow per head, (Q/H) is off by 0.01 m2/min, and that for Set 2, Experiment 5 the head, (H) is off by 0.04. Those errors are highlighted in the tables, but of course are trivial. (Darcy may not have had Excel 97 to check himself.) However, in the reported conductivity values a major difference is apparent. His calculation for the forth series is clearly wrong, and should be 0.209 instead of the 0.332 l/m2-s reported.

There is another possibility for the apparent error in Series 4 conductivity. It may be that the reported filter height, e for this experiment of 1.70 m is wrong, and that it was really 2.70 meters. A 2.70 meter bed height is consistent with the reported 0.322 l/m2-s, and seems a more reasonable experimental variation from the earlier series, but would require a total column height of 3.5 m as shown in his plate.

It is possible that two columns were actually used. A 2.5 meter column for the initial set of experiments, and a 3.5 meter column for the second set. The fabrication of a second column would also explain some of the delay between the first and second experiment sets.

Besides Darcy's original data, Tables 1 and 2 provide the Darcy flux, q, the hydraulic gradient, H/e and a single experiment calculation of the conductivity, k. At the bottom of each Series, a least squares fit for the conductivity using all the data and the corresponding R2 value is listed. Figure 1 plots q versus H/e and the linear fits for each series. As can be seen the data is quite good, and all R2 are greater than 0.984. Figure 2 presents a plot of the relative residual error in q for the linear fits shown. It was computed using,

RE = (qm - qp)/qm

Where RE is the relative error, qm is the measured Darcy flux and qp is the predicted flux based on the fitted k and measured H/e. Errors are small, most are less than 5% and the largest errors are associated with the low flows as could be expected. Only one data point, Series 1, Experiment 10 had an error consistent with Darcy's error estimate. It appears the actual error in the manometer readings were smaller. Overall, there is no clear trend in the error, indicative of an unbiased fit. However, Series 1 shows a trend from positive to negative as H/e increases. That trend may have resulted from bed compaction during the test.

 

Figure 1. Darcy's data plotted as q versus H/e.

 

Figure 2. Relative residual error in q.

The fitted conductivity values are similar to the values for Series 1, 2 and 3. Series 4 can of course be interpreted more than one way. Comparing the computed conductivity's with the type of sand does not provide a clear trend in k. Series 1 and 2, which had the same unwashed sand, had the highest, (2.85 x 10-4 m/s) and lowest, (1.66 x 10-4 m/s) k values, respectively. The washed sand on Series 3 and 4 had the same intermediate value, (2.15 x 10-4 m/s). The second Set had a similar conductivity of 2.75 x 10-4 m/s. The lack of a clear trend in k would indicate there was segregation of sand grains that produced layered packing with variable conductivity.

Darcy's Solution to the Falling Head Problem

Darcy ended the appendix with a solution for the flow through though a filter with a falling depth of water on the upper surface. He would use the solution in the next Appendix section, Note E, which attempted to explain the behavior of artesian wells. His solution is now known as the falling head problem. Slightly rearrange it is,

where h is the height of water above the filter, t is the time and the subscript o indicates the initial variable values.

Comment on the Falling Head Solution

This solution does clearly demonstrates a good mathematical ability. A point of confusion may arise with the hydraulic potential across the bed. Darcy is correct in defining it as the sum of h and e. That is, h is not the head, it is only the pressure component, (pressure divided by the specific weight) at the top filter surface, while e in the log term results from the elevation component.

Darcy's solution anticipates the traditional solution for a falling head permeameter,

with A being the reservoir area and a the column area. Thus, Appendix D not only defines the Law, but also provides the two standard methods, (constant head and falling head) used to measure conductivity.

 Concluding Comments

Henry Darcy's testing and analysis was simple, but clearly well thought out, carefully performed and theoretically complete. While he himself may not have known of its eventual full impact, it is still appropriate to give him full attribution for discovering the Law. Not only was his analysis substantially correct, but with the falling head problem he provided the first analytical solution to a complex saturated flow. Thus, he not only discovered the Law, but also showed how to use it.

Finally, Appendix D, demonstrates that Darcy just didn't stumble onto the Law. He obviously knew what he was doing. In the Appendix he demonstrates,

It would be refreshing if everyone in the ground water field today had those basics down.

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