R. W. (Jr.) Ritzi, P. Bobeck - Comprehensive principles of quantitative hydrogeology established by Darcy and Dupuit (796982), страница 3
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Thus it isbetter to conceptualize a circular (in areal view) flowdomain of radius r and thickness B, and radial flow inwardtoward the well so that equation (6) is written with s as acylindrical cross-sectional area perpendicular to flow:q ¼ k ð2prBÞdhðrÞdrð7ÞHe integrated this expression over an arbitrary distancebetween r1 and r2 which gave the following expression[Dupuit, 1857, p.
261]:hL ¼ hðr2 Þ # hðr1 Þ ¼Figure 2. ‘‘Artesian Well at Grenelle’’ wood engravingpublished in an 1879 encyclopedia (uncopyrighted andfreely distributed by http://www.antiqueprints.com). Thiswell, with upgrades made by Dupuit in the early 1850s, wasone of the major municipal supplies of water for the SWside of Paris, along with the Perrier pumping station atChaillot, which supplied water from the Seine. The externalstructure as shown here, designed by the architect Delaportein 1841, was demolished in 1903.(note: we have replaced Darcy’s sum of symbols for thepressure and elevation components of head loss with thesingle, combined, hL).
Here s is the total cross-sectional areaperpendicular to flow, and k is a coefficient unnamed byDarcy, today called the hydraulic conductivity, which hesaid depends on the permeability of the sand. As elucidatedlater, the coefficient also depends on the density andviscosity of the fluid. This is correctly reflected by Slichter’s[1899] writing and is evident in connecting equation (6)with Poiseuille’s [1841] equation. Darcy [1857] wrote thathe and Poiseulle arrived at the same expression underdifferent circumstances, at which point Darcy may have hada fuller understanding of his unnamed coefficient, whichDupuit [1857] referred to as Darcy’s ‘‘coefficient of thenatural layer.’’ Hubbert [1940] would later rigorously definepermeability itself, and by also defining the hydraulicpotential, would raise the understanding of each term in (6)to a higher level of sophistication.! "qr2ln2pkBr1ð8Þ(note we have replaced Dupuit’s symbols with thoseconsistent with Darcy’s and the equations above; forexample, Dupuit uses m0, which is equal to [kB]#1).Figure 6 shows Dupuit’s plot of this spatial function (in hisplot, H # y gives hL, today commonly called thedrawdown), from the well at radius r1 = rw out to theradius of influence at steady state, r2 = L.
Importantly, hereDupuit first showed that the head loss in the aquifer awayfrom the well forms a cone of depression. Many U.S.textbooks incorrectly attribute equation (8) to Thiem [1906].[28] With this development, Dupuit [1857] determinedthe constant C in Darcy’s equation (5) when there is no wellloss. If h0 equals the background head at r = L, and h1equals the head in the well at r = rw, then q0 = 0 and atsteady state, with conservation of mass across the wellscreen, q1 equals the q in equation (8), and (h0 # h1) isthe drawdown, hL, in equation (8). This gives the constant Cin equation (5) asC¼'r (1wln2pkBLð9ÞAlmost a century later, Theis [1935] would show that undertransient flow:C ðt Þ ¼1W ðuÞ4pkBð10Þwhere W(u) is a dimensionless function of time and thediffusivity of the aquifer.
Thus ln(rw/L) in Dupuit’s equationis the large time limit of W(u)/2 when at apparent steadystate or when fixed-head boundaries are reached. Thus, theframework for well hydraulics was established by Darcy[1856] and Dupuit [1857], penecontemporaneously with thepublication of equations (5) and (8). Equation (5) is still inconventional use today to determine well losses from step-5 of 14W10402RITZI AND BOBECK: QUANTITATIVE HYDROGEOLOGYW10402Figure 3. Plots of data collected from systematically changing the height of discharge for artesian wells[from Darcy, 1856] (see Figure 4a). Each plot represents a different artesian well.
The abscissa iselevation of the discharge orifice, decreasing to the right, and the ordinate is the discharge rate (increasingupward; see Figure 4b). Each vertical line represents a datum from an experiment.drawdown pumping tests [cf. Driscoll, 1986, p. 556,equation (16.8)].[29] Dupuit [1857, p.
261] noted that equation (8) revealsthat the shape of spatial head loss between rw and L isconical independent of the k or B of the layer, but themagnitude of q per unit head loss out in the aquifer islinearly proportional to k and/or B. Furthermore, Dupuit[1857, p. 267] pointed out that while cones of depressionmust exist around the wells at Grenelle and Tours whichDarcy [1856] had discussed, the kB is larger at Tours than atGrenelle, and if Grenelle had the same kB then discharge perhead loss would increase by fifteenfold.6 of 14W10402RITZI AND BOBECK: QUANTITATIVE HYDROGEOLOGYW10402Figure 4. (a) Enlargement of the left well in Figure 1 but with the symbols of Darcy’s [1856] equation(our equation (5)) added.
(b) The relation between the plot of Darcy [1856] in Figure 3 and the plot ofDupuit [1857] in Figure 5.[30] Both Darcy [1856, p. 88] and Dupuit [1857, p. 262]showed that because the second part of the LHS ofequation (5) is negligible among the artesian wells considered, increasing the radius of the well will have anegligible effect on flow rates.
In support of this, Darcycomputed the large-radius limit for the Grenelle well showing it would give negligible additional flow compared to thecurrent diameters of the riser pipe sections. Dupuit followedFigure 5. Difference between linear and parabolic behavior in the artesian well experiments [fromDupuit, 1857]. The relationship to the plots in Figure 3 is given in Figure 4b. Dupuit’s symbols are notconsistent with Darcy’s. If head loss occurred primarily because of high-velocity flow up the riser pipe,the second term on the left-hand side (LHS) of equation (5) would dominate, giving the parabolic curve.If head loss occurred primarily because of low-velocity seepage through the aquifer sand, the first term onthe LHS of equation (5) would dominate, giving the straight line behavior and maximum-limit discharge(‘‘limite des débits’’).
The data in Figure 3 all show the straight line behavior.7 of 14W10402RITZI AND BOBECK: QUANTITATIVE HYDROGEOLOGYW10402Figure 6. Plot from Dupuit [1857] showing that the radial seepage through sands, toward the well,causes head loss with a cone of depression in the piezometric level (‘‘niveau piézométrique’’).by discussing how the area of the intake was reduced whilerepairing an accident, with no affect on the flow rate.
Dupuit[1857, p. 261] stated:The proportionality of the volume rate of flow to head difference (asper the constant C) has been demonstrated by numerous experimentsdone on various artesian wells, including the Grenelle well, amongothers. This is discussed in Mr. Darcy’s work, which we have cited.Thus, there is a rather remarkable confirmation of the formulas wehave derived.[31] After presenting how to quantify aquifer head lossaround an artesian well, Dupuit [1857, p.
265] stated that anew well was to be drilled at Passy, about 3 km from theGrenelle well, in the same formation. He stated that questions on the diameter of the well were not important, in lightof the previous development. He stated that if it were drilledinto the same formation, with same thickness, permeabilityand initial head, it should provide essentially the same yieldregardless of having a similar diameter versus a much largerdiameter. He argued that the more important question waswhether or not a flowing well at Passy will lower the yieldat Grenelle. This led to an important presentation of ideasabout well interference and capture zone geometries. Here,Dupuit was following up on Darcy’s [1856, p. 105] discussion of the ‘‘reciprocating’’ effect of multiple artesianwells at Tours (Figure 3).
If one well were shut off, yields atFigure 7. An aquifer stratum that is recharged on the right, at the edge of the Paris basin, and which isexposed downgradient in the wall of one of the major river valleys (e.g., the Seine), creating springdischarge [from Dupuit, 1857]. This scenario is discussed by Darcy [1856, note 56]. Note that theregional piezometric gradient steepens over the interval where the stratum is thinner, illustrating steadystate relationships Dupuit explored using Darcy’s law. The figure is used as the starting point for Dupuit’sanalysis of reciprocal effects (superposed drawdown) among proximal wells tapping the same stratum.8 of 14W10402RITZI AND BOBECK: QUANTITATIVE HYDROGEOLOGYW10402Figure 8. Capture zone geometry for a well discharging from an aquifer with a regional gradient [fromDupuit, 1857].nearby wells increased. Dupuit noted that the wells at Tourswere closer than the distance between Passy and Grenelle,but that interference was a possibility.[32] Dupuit [1857, p.
267] then gave an interesting analysis of the superposition of the drawdowns created bymultiple, flowing wells. He began the analysis by consideringmultiple, fully penetrating wells in an aquifer cross sectionwith no width into the plane of the section as in Figure 7, (orequivalently, as he stated, with the case of wells that are likedrains extending infinitely into the plane).
Here the consideration of superposition is fairly trivial, because the head ineach upgradient well is a boundary condition for flow to thedowngradient well. Importantly, Dupuit [1857] (p. 270,section titled ‘‘Reciprocating influence of artesian wells inthe same layer’’) stated that the natural system is not like that,but rather one must consider a multitude of sections extending radially from each well.
As shown in Figure 8, Dupuit[1857, Figure 75] made the important logical step of superposing the cone of depression created by any one well ontothe regional flow field, showing that the zone of influence(conventionally called a capture zone geometry today) willbe asymmetric around a well. Here Dupuit states that inside9 of 14W10402RITZI AND BOBECK: QUANTITATIVE HYDROGEOLOGYW10402Figure 9. Heterogeneity in the sand formations tapped by the artesian wells at Grenelle and Passy [fromDupuit, 1857]. Note that the thicker formation tapped by the Passy well gives a smaller slope (less headloss per unit discharge) in the orifice elevation versus discharge plot.of the zone of influence, head is affected.
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