R. W. (Jr.) Ritzi, P. Bobeck - Comprehensive principles of quantitative hydrogeology established by Darcy and Dupuit (796982), страница 4
Текст из файла (страница 4)
However, waterflowing outside the zone is not captured. This principle is ofcourse fundamentally important today in the design of pumpto-treat groundwater remediation systems or gradient controlwells. Focusing on the prospects of a Passy well, Dupuitstated that in knowing head at only one well at Grenelle, andthus not knowing the regional head, the potential for effects atPassy cannot be calculated.[33] During the time that Dupuit [1857] was in review bythe French Academy, the well at Passy was drilled, andcompleted with a larger well diameter.
The yield at Passywas of the order of 30 times greater than at Grenelle, andthus the Academy’s review was critical of Dupuit (andDarcy’s) contention that yield at Grenelle was essentiallycontrolled by head loss in the aquifer, not by the diameter ofthe riser pipe. Dupuit appended a new section to the original1857 monograph [Dupuit, 1863, section 152, p. 277ff.] that10 of 14W10402RITZI AND BOBECK: QUANTITATIVE HYDROGEOLOGYW10402Figure 10. (right) Vertical profile of pressure head (increasing to the right) for (middle and left) either ofthetwo containers [from Dupuit, 1857]. The leftmost container has uniform width, and intervals [l, l0,00l ] have increasingly coarser (higher k) sediment. There is pure water above00 and below these intervals.The middle container has uniform k sediment but different widths [w, w0, w ].summarized the critique and responded to it.
Figure 9 showsDupuit’s [1863] application of the principle in equations (5)and (9) to explain the differences between the well yields.Figure 9 shows that the aquifer is not ‘‘layer cake’’ uniform,and the Passy well intersects a thicker section of sandstone.Thus the plotted profiles of q versus hL for each well areboth linear, but C is smaller (more q per unit hL) for thePassy well because B is larger (and perhaps k also is larger).Dupuit pointed out that the principles of mechanics thatDarcy and he applied to permeable terrains provide a logicalexplanation for the differences. He also pointed out that theyield at Grenelle was impacted when the Passy well wasbrought into production.3.
Further Quantification of Flow ThroughSediments Including Residence Times, EffectiveConductivities, and Free-Surface Boundaries[34] Darcy [1856] began with a discussion of identifyingpublic water supplies, including a review of what he knewabout artesian wells. Dupuit [1857] was organized differently, more as a text on applying the principles of fluidmechanics to understanding flow through permeableterrains.[35] Dupuit’s [1857] introductory discussion of fluid mechanics reviewed Darcy’s linear law for low-velocity flow. Inextending the result to flow through sand, Dupuit [p.
233]reviewed Darcy’s sand column experiments and discussedthe grain size distribution of the sediment that Darcy used,noting that the porosity, 8, was 0.38. He pointed out thatqDarcy’s s was the rate over the total cross-sectional area.Dupuit pointed out that the average linear velocity of the fluidis the rate per cross-sectional area of pores and thus is givenby dividing the Darcy flow rate by 8. Dupuit [1857, p.
233]discussed the fact that there are numerous ‘‘sinuosités’’ ortortuous pathways of fluid at the pore scale, and that the realspeed is always a little more than what we project onto the(linear) axis of their general direction. Dupuit [1857, p. 249]later pointed out that the average linear velocity could be usedto determine the time for groundwater to traverse a permeableterrain over distance l:t¼8slqð11ÞDupuit [1857, p.
248] then considered the implication ofDarcy’s law for nonuniform head loss given that thepermeability or thickness of the permeable layer may vary.He considered various containers (‘‘vases’’) of sediments asanalogs. One example in Figure 10 shows his drawing oftwo containers of sediment to illustrate these effects. Theleft container has a uniform area perpendicular to flow, andlayers that systematically change to coarser-grained (higher k)sediment downward across each contact. The right containerhas homogeneous sediment, but the area perpendicular toflow systematically increases downward at each change inwidth.
He considered conservation of mass at steady state,so that in the left illustration the volume rate across theboundary of each sediment layer does not change, and onthe right, the volume rate across the transition to widerwidths does not change. The plot to the far right shows thepressure head as a function of elevation. Note that the freewater at the top and bottom of the illustrations is shown tobe at atmospheric pressure. Two important points weremade. One point was that through equation (6), we see thatincreasing k has the same effect as increasing the areaperpendicular to flow Dupuit [1857, p.
249]. The other pointis that the effective k for flow across layers, !k, is theharmonic mean. Dupuit [1857, p. 248] derived for the leftillustration in Figure 10:000ðl þ l þ l Þ!k¼l l0 l00þ þk k 0 k 00ð12Þwhere l, l0, l00 are the layer thicknesses, as noted. Thepressure head profile and a careful reading of Darcy’sAppendix D, show that both Dupuit and Darcy understood11 of 14RITZI AND BOBECK: QUANTITATIVE HYDROGEOLOGYW10402that the law of linear head loss applied over negative orpositive gauge pressures.
If we are to critique Figure 10, it isclear from equation (6) that the gradient should be greatestacross l and least across l00, opposite from what is shown. Itseems that Dupuit would have been clear on this, and othershave pointed out numerous drafting errors in the plates ofDarcy [1856], so perhaps this was also a drafting error.[36] For 1-D flow or such ‘‘container’’ flow, Darcy[1856, Appendix D, p. 459, equation (1)] also appliedconservation of mass to consider transient changes in waterlevel, a distance x above to top of a uniform sand column ofheight l, so thatqðtÞðx þ l Þ #dx¼k¼sldtð13Þwhich, after separation of variables, and integration from hoto h over interval to to t gives:klnðhðt Þ þ l Þ ¼ lnðh0 þ lÞ # ðt # t0 Þlð14ÞToday we call this the falling-head permeameter equation.Darcy had Ritter perform the falling-head test with their (nowwell known) permeameter apparatus containing sand, andwith only water.
The linear law was verified for the sand filledtests, and the equivalent transient equation based on theparabolic law, which Darcy also gave, was also verified withtests using only water.[37] In discussing quantification of hydrogeologic attributes, Darcy [1856, p. 121, notes 44 and 45] shows greatfamiliarity with the wide range of flow conditions oneencounters in nature, including (1) groundwater filtrationthrough clastic sediments and sedimentary rocks, (2) filtration through fractures or solution networks, that behave inaggregate as do sedimentary formations, and (3) highvelocity, stream-like flows in large karst conduits.
Accordingly, he discussed using the linear and parabolic forms ofequation (14) for transient changes, in this context, toevaluate whether springs are supplied by sand-like porousflow versus pipe-like cavity flow, while invoking questionsabout the famous spring at Nimes. Thus, Darcy extendedquantitative hydrogeology beyond steady state considerations. Darcy was aware of phenomena that were difficult toexplain including intermittent and cyclical spring discharge,largely out of phase with rainfall events.
He discussed theapplication of the basic principles of groundwater mechanics to explain these phenomena.[38] All of Dupuit’s ideas are applied to both water table(‘‘surface libre’’) and confined (‘‘surface forcée’’) aquifers.Sadly, Dupuit is perhaps known to many practicing U.S.hydrogeologists only for how he treated the existence of afree-surface upper boundary (the water table) in unconfinedgroundwater flow. He considered horizontal 1-D or 2-Dradial flow and let s = hw where w is width perpendicular toflow.
For 1-D flow, substituting into (4) and integrating for ahead drop between ho and h, over a distance from 0 to L,Dupuit [1857, p. 236] wroteq¼#$kw h20 # h2L2ð15ÞW10402Dupuit [1857, p. 255] presented the corresponding form ofequation (8) from the same approach. The approach isconventional today when the vertical components ofvelocity can be ignored. The slight of only recognizing thisone contribution in U.S. textbooks is made worse by the factthat the idea is referred to as the Dupuit-Forchiemerapproach, though Forchiemer’s work did not appear until30 years later.4.
Discussion[39] Darcy [1856] is an amalgamation of sections ofprose, each with distinctly different character. Some sections are very much like a handbook on methods of watersupply. Some sections present Darcy’s untested ideas (a newriver filter) or novel approaches that recently came intoDarcy’s awareness (wind-driven water pumps). Some sections review Darcy’s knowledge of the philosophy ofnatural history as it relates to springs and the hydrologiccycle, including quotes from Greek scholars.
Some sectionsare developed like a research paper, presenting derivationsof equations and discussing new findings. Some sections arelike essays, such as in describing water quality, publichealth, and the need for free water distribution. Darcy’svernacular style is consistent with someone who, withfailing health, is trying to get all of his important thoughtsrecorded to paper.[40] In contrast, Dupuit [1857] is organized more like ascientific text. The organization is clearly described withinthe introduction: First the relevant principles of fluid mechanics are covered and, as per Darcy, applied to porousmedia.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
