G.O. Brown - Henry Darcy and the making of a law (796978), страница 5
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Several topics are discussed including geology,water quality, well development, sand production, andrecharge from precipitation. Building on his 1834 report,he predicted well drawdown as a function of well-pipefriction and aquifer filtration losses, and showed how itwould vary depending on which process dominated. Thistime, however, his theoretical hypotheses were compared todetail field data. Ten examples of artesian well productionrates are presented, where the flow was measured as afunction of the discharge elevation.
Figure 4 shows hisschematic of the measurement and one of his data sets takenin 1847. Flow was measured volumetrically as the dischargepoint was raised above the ground surface. In the chart, flowin liters per minute are plotted vertically against the elevation plotted horizontally (with increasing elevation drawnto the left.) Two different measurement dates are shown inthis example. A clear linear trend in flow is evident in thisand the other data sets.
In this case, the linear trend isreasonable as Darcy noted the pipe losses were in the region11 - 8BROWN: HENRY DARCY AND THE MAKING OF A LAWFigure 4. Spring flow measurements. Detail from Darcy [1856, Plate 22].of Equation 9, or in modern terms, the pipe flow waslaminar. Similarly, since each data set was collected relatively rapidly, the aquifer cone of depression would nothave changed, thus the net hydraulic gradient would remainproportional to the discharge elevation. These data, hisprevious pipe research, and the sand column experimentsto follow led Darcy to conclude that the wells were suppliedby conduits that were either filled with sand or very small insize.[38] By only measuring well drawdown, Darcy was notable to observe the cone of depression that would beexpected even in a fractured aquifer.
Thus he continued tothink of linear conduits and not radial flow. This limitationis understandable since the difficulty of drilling would havemade observation wells out of the question. However,Darcy by then clearly understood that most groundwateroccurred within a porous matrix. He wrote, ‘‘We understandthat drilled wells more often bring to the surface water thathas infiltrated through layers of sand than water circulatingin natural cavities. In fact, the vertical section of the latter isnecessarily limited, whereas water-bearing sands may havean almost infinite surface area.’’[39] Appendix, Note D contains Darcy’s review andanalysis of water treatment filters.
Only a small portionhas been republished by Hubbert [1969] or translated by R.Freeze [Freeze and Back, 1983]. The first part reviewsexisting natural and artificial filters in England and France,as he had done previously [Darcy, 1834]. This time, however, the analysis includes flow per unit area of the bed,indicating Darcy was now applying continuum concepts tofilters. He next demonstrated the application of his law, theproof of which was to follow, to artificial filters with verticalflow.
With its first introduction, he again noted the similarity of the law to flow in small pipes with low velocities.[40] The best was held for the end of Note D; the columnexperiments. Starting late in 1855, Darcy, with the assistance of Charles Ritter, completed the two sets of tests. Set 1consisted of four series, with each series having a differentsand packing. For each series, three to ten different experimental trials were performed at increasing flow rates.
Theinlet pressure was varied for each experiment while holdingthe outlet at atmospheric pressure. Set 2 was carried out inearly 1856 by Ritter on a single sand packing and both inletand outlet pressures were varied. In total, 35 trials werereported.[41] Darcy’s apparatus is shown in Figure 5. It was avertical steel pipe with an inside diameter of 0.35 m andsealed on both ends by bolted flange plates. A total heightof 2.5 m was reported in the text, but it is dimensioned onthe illustration as 3.5 m. At the bottom, an outlet reservoirwas created by supporting a set of screens 0.2 m above thebottom. An inlet reservoir was created by leaving a voidbetween the sand and the column top, and a tap at the topallowed air to be bled from the system. Water was supplieddirectly from the hospital house line on the left, whichinduced considerable oscillations as users elsewhere turnedfaucets on and off.
Flow was from the top down, and thedischarge was determined by timing the effluent accumulation in a volumetric box one-meter square at the bottom.Mercury U-tube manometers mounted on the right allowedthe calculation of the hydraulic head-on either side of thesand. Both supply and effluent lines had valves to allowcontrol. In the set 2 experiments, the outlet pressure was setat both positive and negative gage pressures by unrecordedmethods, the details of which may only be conjectured. Itwould be a simple matter to pressurize the outlet byrestricting the outlet valve, while dropping the dischargepipe to a lower position with the valve wide open couldproduce a relative suction at the filter base.
However,suctions up to 3.6 m of water were reported. Thus thecolumn must have been elevated greater than the one-metershown, or a vacuum pump used.[42] Each test used Saône River sand with variousdegrees of washing. Darcy described the packing process:‘‘The sand was placed and packed in the column, whichbeforehand had been filled with water, so that the sand filtervoids contained no air, and the height of sand was measuredat the end of each series of experiments, after the passage ofwater had suitably packed it.’’ Packing height was variedfrom 0.58 to 1.71 m.
(In Figure 5, however, the sandpacking is shown as only 0.25 m.) A test run consisted ofsetting the inlet valve, allowing the column to reachequilibrium, reading the manometers and measuring theflow over 10 to 25 minutes. The only serious apparatusBROWN: HENRY DARCY AND THE MAKING OF A LAW11 - 9Figure 5. Column apparatus [Darcy, 1856, Figure 3, Plate 24].concern was the use of an unregulated supply, for whichDarcy had no alternative.
Dropping the sand into the waterfilled column probably allowed it to segregate by sizefraction with the coarsest particles on the bottom of eachlift. It would also produce a low packing density subject tocompaction. However, since he waited for equilibriumbefore taking any measurement and measured the filterheight after the experiment, any compaction would nothave impacted his conclusions.[43] In set 1, the flow rate varied from 2.13 to 29.4 L/min,while the head loss ranged from 1.11 to 13.93 m. The resultslead Darcy to observe, ‘‘It thus appears that for sand ofcomparable nature, one can conclude that flow volume isproportional to the head loss and inversely related to thethickness of the layer traversed.’’ He then proposed equation (1), calculated conductivity values in units of L/m2 s(103 m/s) and noted that they varied due to the differencein the sand used.
For set 2, Darcy again noted the near11 - 10BROWN: HENRY DARCY AND THE MAKING OF A LAWFigure 6. Darcy’s sand column experimental results. Lines show best linear fit.constant value of Q/hL, consistent with the other experiments. Thus he showed conclusively that the flow was alinear function of the head loss across the filter bed and notthe actual water pressure.[44] Figure 6 plots Q versus the gradient, hL/L and a leastsquares linear fit for each set. As can be seen, the data arequite good and the linear regression, R2 is greater than 0.98for each.
Comparing the measured conductivity with thetype of sand does not provide a clear trend. This wouldsupport the hypothesis that there was grain segregationduring filling. Independent checks of his reported dataconfirm his calculations with one exception. As noted byFreeze [1994], in set 1, series 4 the conductivity should be0.209 instead of the 0.332 L/m2 s reported. There is anotherpossibility for the apparent error. The filter height may havebeen 2.70 m instead of the recorded 1.70 m. A 2.70 m bedheight is consistent with 0.322 L/m2 s and seems a morereasonable experimental variation from the earlier series,but it would require a total column height of 3.5 m as shownon the plate.[45] Even with careful translation, Darcy’s writings canbe easily misinterpreted for a variety of reasons.
Like otherauthors of this period, he used the words pressure, load andcharge rather imprecisely and the reader must infer fromthe equations if he meant pressure head, hydraulic head orhead loss. The text and figures also contain a number ofeditorial errors, which may be due to Darcy’s poor healthor the distance between Dijon and the Paris publisher.However, his equations and analysis were completelycorrect and he introduced many of the concepts we usetoday including, the conductivity, ‘‘a coefficient dependenton the permeability of the layer,’’ and the Darcy flux, v,defined as ‘‘Q/A = v’’.[46] Darcy ended the appendix with a solution for theunsteady flow through a filter with a declining depth ofwater on the upper surface and a constant atmosphericpressure on the bottom.
He would use the solution in thefollowing section, Note E, which attempted to explain thebehavior of artesian wells. His solution is now known asthe falling head problem, and consisted of combining (1)with continuity, expressing head and time as differentialsto allow for the temporal reduction in flow, separatingvariables and integration.
The result rearranges to themodern convention islnyþLK¼ ðt to Þ;yo þ LLð11Þwhere y is the depth of water above the filter, t is the timeand the subscript o indicates the initial variable values.Darcy’s solution anticipates the solution for a falling headpermeameter and other problems of this type. Moreimportantly, he was the first to apply equation (1) tounsteady flow.[47] Unexpectedly, on 3 January 1858, Darcy died ofpneumonia while on a trip to Paris.
He had just been electedto the French Academy of Sciences and his last work, whichpresented the first modern design for the Pitot tube, waspublished posthumously [Darcy, 1858; Brown, 2001].While Darcy’s death was tragic, his research was continuedand built on by others in the Corps. His protégé, Bazinexpanded and completed the open channel investigations[Darcy and Bazin, 1865]. In 1861, Dupuit an associate andsuccessor as Chief Director for Water and Pavements forParis, submitted a groundbreaking report [Dupuit, 1863].BROWN: HENRY DARCY AND THE MAKING OF A LAW11 - 11Figure 7.
Dupuit’s solution for radial flow [Dupuit, 1863, Figure 69] (History of Hydraulics Collection,Iowa Institute of Hydraulic Research).This work expanded on an earlier edition of the same title.Principle among its new findings was the solution for steadyradial flow to a well, using Darcy’s law. One of Dupuit’sillustrations of aquifer drawdown is presented in Figure 7.Dupuit had thus overcome the overly simplified assumptionof discrete groundwater conduits, and had introduced modern aquifer analysis.
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