R. W. (Jr.) Ritzi, P. Bobeck - Comprehensive principles of quantitative hydrogeology established by Darcy and Dupuit (796982), страница 2
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Darcy wrote using the conventional structural, petrologic, mineralogic and paleontological nomenclature ofthe time (including species names of fossils). He gave asolid scientific review of the hydrologic cycle, and of thekarst springs emanating from the Jurassic limestone nearDijon, and discussed groundwater chemistry and the role ofcarbonic acid in controlling calcite solubility. Darcy’sknowledge of the hydrogeology of the Paris basin isdiscussed further below; here we just take initial note ofthe fact that Darcy had appreciable knowledge of bothgeology and fluid mechanics to bring to bear on conceptualizing and quantifying aspects of groundwater processes.[13] Darcy and Dupuit had overlapping appointments inParis as Inspector Divisionnaire of the Corps des Ponts etChaussées, and as Chief Director of Water and Pavements.Darcy was appointed on 16 June 1848 [Lochot, 2003].Dupuit was called in May 1850 to be Darcy’s successor,presumably because of Darcy’s frail health [Caudemberg,1858; Mahyer, 1866; Hager, 2003, 2004].
By this point intheir careers, Darcy and Dupuit had both been awarded theLegion d’Honneur for their outstanding contributions tocivil engineering. The French government was choosing thebest of the Corps for its top administration. With theseappointments came a hydraulic research facility at Chaillot.From August 1849 to October 1850, Darcy conducted the2 of 14RITZI AND BOBECK: QUANTITATIVE HYDROGEOLOGYW10402experiments which revealed the important linear law between fluid motion and its driving force that exists at lowvelocities (further elucidated later by Reynolds [1883]).According to Caudemberg [1858; see Hager, 2003, p. 59]:‘‘Mr.
Dupuit, who followed Mr. Darcy in the service of thestreets and water of Paris, did not wait for the initiative ofthe administration to gracefully continue and facilitate hiscomrade’s continued important experiments.’’[14] This pipe flow work was published by Dupuit [1854]and by Darcy [1857, 1858] (publication dates are discussedfurther below). Darcy [1856, e.g., p. 93] frequently refers to thelaw (of fluid mechanics) he discovered, and writes equation (2)for pipe flow as, for water velocities above 10 cm/shL ¼Lb 2qpr5ð3ÞThe form for velocities below 10 cm/s ishL ¼Laqpr4ð4Þwhere r is radius and q is the volume discharge rate.[15] In 1855 Darcy, under failing health, was releasedfrom all appointments except research.
This freedom enabled him to conduct experiments in Dijon on flow throughporous media and through open channels. Dupuit remainedin Paris, became central to the Huassmann metamorphosisof the city, and continued to publish on political economicsand hydraulics. They were quite aware of the significance ofeach other’s work during this time, as evidenced by themutual cross citations in their publications on pipe flow,aquifer flow, and open channel flow. Indeed, as InspectorsDivisionnaires, they reviewed and reported on Corps projects throughout the country, in addition to the work of eachother [Brown, 2002].[16] We close this introduction with a final note ofimportance. The works of Darcy and Dupuit were notpresented or published in chronological order.
Darcy’srevolutionary results at Chaillot were not published until1857 (also 1858), but are extensively referred to and builtupon by Darcy [1856]. The Darcy [1856] monograph in factreviews previously unpublished, important work from thestart of Darcy’s career. It does not present knowledge in theorder in which he acquired it. The monograph was primarilyintended to be a guide to the practicing engineer developingnew municipal water supplies, and is organized as such.Thus, while Appendix D presents the equation of fluidmotion through sands, it is applied to quantitative hydrogeology much earlier in the monograph.
Dupuit’s work onquantitative hydrogeology was written at nearly the sametime as Darcy’s, and was submitted in 1857 to the Academyof Sciences as a paper for its Mémoires (more on thisbelow). Though the review was favorable, Dupuit refusedto make requested revisions, and chose instead to add it, asit was, as Dupuit [1863, chapter 8], but with an additionalsection responding to the principal issue raised by theAcademy. Our perspective is that Darcy and Dupuit’s workwas done during essentially the same period of time, and weW10402review their collective knowledge and contributions together,without further regard to chronology.[17] In the following sections, we review Darcy andDupuit’s contributions to quantitative hydrogeology in sedimentary aquifers, well hydraulics, and spring flow infractured and karst regimes. Because the English translationof Darcy [1856] by Bobeck [2004] and the work by Dupuit[1857] as reproduced by Dupuit [1863, chapter 8] are themost readily attained, we will give page numbers as in thosedocuments.2.
Quantifying Flow in Sedimentary Aquifers:Hydrogeology and Hydraulics of Artesian Wells[18] Darcy [1856, p. 122, note 56, and plate 22] reviewedthe geologic structure of the Paris Basin and depicted it incross section, as shown in Figure 1. A quintessential basincomprising Cretaceous and Jurassic sedimentary strata, it wasused as an example in the geologic textbooks of Darcy’s time[e.g., Lyell, 1834; Darcy, 1856, p. 78]. Darcy describes theconceptual hydrogeologic model, invoking the three-dimensional basin structure in relation to topography.
Rechargeoccurs where formations are exposed at higher elevationsaround the basin rim (Figure 1). Where the entrenched valleysof the major rivers expose these formations at lower elevations, springs occur (see also Figure 7).[19] The well on the left in Figure 1 represented thefamous Grenelle artesian well in Paris, at the basin center(Figure 2). The Grenelle well drew from a Cretaceoussandstone formation that is confined by much thicker,low-permeability strata (as reflected in Darcy’s cross section). The well at the right represented other artesian wellslocated out toward the basin margin, such as at Tours.[20] Brown [2002] discussed that in Darcy and Dupuit’stime, because of the expense and difficulty in drilling wells,monitoring wells did not exist.
Thus, piezometric differences (head loss) in the aquifer away from an artesian wellcould not be measured, and hydrogeologic processes associated with flow to a well had to be deduced from singlewell experiments.[21] Figure 3 has 10 plots that Darcy constructed fromdata collected in single-well experiments in which theelevation of the discharge orifice (the abscissa in the plots)of an artesian well was changed, and the resulting change inthe volume rate of flow from the well (the ordinate in theplots) was quantified (Figure 4a).
Darcy [1856, p. 93] askedwhy these plots are linear in each case and answered thequestion by applying his law of linear versus parabolic headloss, to differentiate between the head loss occurring because of groundwater flowing in the aquifers versus flow upthe riser pipes. Darcy pointed out that groundwater generally circulates at velocities less than 10 cm/s and so headloss in sands probably follows the linear law, whereas flowin the riser pipe occurs at velocities above 10 cm/s and thehead loss is probably parabolic.[22] So that these respective head losses could be quantified, Darcy [1856, p. 94] superposed them in deriving thefollowing formula (using his symbols):3 of 14ðh1 # h0 Þ þ%&$b1 #22¼ C ðq0 # q1 ÞHq#Hq1 10 0p2 r 5ð5ÞW10402RITZI AND BOBECK: QUANTITATIVE HYDROGEOLOGYW10402Figure 1. Stratigraphy and structural geology of the Paris basin [from Darcy, 1856].
The well on the left(representing the famous artesian well at Grenelle (Figure 2)) taps a Cretaceous sandstone (the‘‘Greensand’’ in Darcy’s time) that is interstratified with thicker chalk strata. The well on the right(representing artesian wells at Tours, 150 km SW of Paris) taps a deeper Jurassic sandstone interstratifiedwith vuggy limestone (‘‘cornbrash’’ in Darcy’s time).where the ‘‘0’’ designates a higher discharge orifice elevationand ‘‘1’’ designates a lower elevation (see Figure 4a), h is theaquifer pieziometric height measured from ground surface[Darcy, 1856, p. 90, and p.
122, note 53], H is the length of theriser pipe measured from the base of the well within theaquifer [Darcy, 1856, p. 91], r is the riser pipe radius (Darcy[1856, p. 82 and p. 84]). As per Dupuit [1854], Darcy uses theharmonic mean for an effective radius when, as at Grenelle,the riser pipe varies in radius, and C is a constant [Darcy,1856, p. 94], considered further below.[23] The formula is written consistent with the axes of theplots in Figure 3, and the first part of the left-hand side(LHS) is the difference in aquifer head loss at the wellcreated by changing the elevation of the discharge orifice,and the second part is the difference in the well loss fromflow up riser pipes of corresponding, different lengths.[24] In presenting this equation, Darcy discussed the factthat if head loss was entirely due to flow up the riser pipe(as if the aquifer were infinite and capable of supplying flowq to the well with no measurable aquifer head loss relativeto prewell conditions) then the first term on the LHSdisappears, and the plots in Figure 3 would be parabolicinstead of linear.
Dupuit [1857, Figure 71] plotted a representation of linear versus parabolic behavior as shown inFigure 5 (also see Figure 4b; the orientation and symbols arechanged from Darcy’s). In considering the lack of theparabolic behavior in the data plotted in Figure 3, Darcy[1856, p. 87] observed:But for the most part, it is not like that [i.e., well loss in the riser pipedoes not dominate, so parabolic behavior is not observed], and I willnow consider the opposite limiting case, that is, where the artesian wellencounters the impermeable ceiling of a [lower] sandy layer in whichthe [groundwater] has conditions analogous to that of water passingthrough a [sand] filter.[25] Note that in the last phrase, Darcy makes a clearstatement of the fact that flow through sand filters (aspresented by Darcy [1856, Appendix D]) is analogous toflow through sedimentary aquifers. His focus at this point inthe monograph is that seepage in aquifer sands occurs atlow velocities, and in this case the second term on the LHSof equation (5) should disappear, giving the linear relationship indeed seen in the plots of Figure 3.[26] Thus, Darcy has reciprocating references betweenthis section of the monograph and the better known Appendix D, where he showed that head loss in low-velocityflow through sands indeed followed his linear law.
Theequation he wrote for fluid motion through sands, with thehead loss between two points separated by a distance l, is[Darcy, 1856, p. 458]:4 of 14q ¼ kshLlð6ÞW10402RITZI AND BOBECK: QUANTITATIVE HYDROGEOLOGYW10402[27] Dupuit [1857] made a very important step in understanding what happens in the aquifer distal to the well, byusing equation (6) to represent flow in the aquifer towardthe well, and linking it through conservation of mass withequation (5) to represent head loss at the well. Doing soallowed the determination of Darcy’s constant C and gavenew insight into aquifer processes. Prior to the derivation,Dupuit discussed that cross sections as in Figure 1 lead oneto incorrect thoughts of unidirectional flow, whereas flowwill actually radially converge on an artesian well.
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