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Darcy's Law and the Field Equations of the Flow of Underground Fluids

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Darcy's Law and the Field Equations of the Flow of Underground Fluids E DARCY'S LAW and the FIELD EQUATIONS of the FLOW of UNDERGROUND FLUIDS Downloaded from http://onepetro.org/TRANS/article-pdf/207/01/222/2176441/spe-749-g.pdf by guest on 01 October 2021 M. KING HUBBERT SHELL DEVELOPMENT CO. }AEMBER A/ME HOUSTON, TEXAS T. P. 4352 ABSTRACT by direct derivation from the Navier-Stokes equation of motion of viscous fluids. We find in this manner In 1856 Henry Darcy described in an appendix to his that: book, Les Fontaines Publiques de la Ville de Dijon, q - (Nd') (P/fL)[g - (l/p) grad p] = (TE, a series of experiments on the downward flow of water through filter sands, whereby it was estahlished that is a physical expression for Darcy's law, which is valid the rate of flow is given by the equation: for liquids generally, and for gases at pressures higher than about 20 atmospheres. Here N is a shape factor q = - K (h, - h,) /1, and d a characteristic length of the pore structure of in which q is the volume of water crossing unit area in the solid, P and fL are the density and viscosity of the unit time, I is the thickness of the sand, h, and h, fluid, (T = (Nd')(P/fL) is the volume conductivity of the heights above a reference level of the water in the system, and E = [g - (1/ p) grad p] is the impell­ manometers terminated above and below the sand, re­ ing force per unit mass acting upon the fluid. It is spectively, and K a factor of proportionality. found also that Darcy's law is valid only for flow This relationship, appropriately, soon became known velocities such that the inertial forces are negligible as compared with those arising from viscosity. as Darcy's law. Subsequently many separate attempts have been made to give Darcy's empirical expression In general, three-dimensional space there exist two a more general physical formulation, with the result superposed physical fields: a field of force of character­ istic vector E, and a field of flow of vector q. The force that so many mutually inconsistent expressions of what field is more general than the flow field since it has is purported to be Darcy's law have appeared in pub­ values in all space capable of being occupied by the lished literature that sight has often been lost of fluid. Darcy's own work and of its significance. So long as the fluid density is constant or is a func­ In the present paper, therefore, it shall be our pur­ tion of the pressure only, pose to reinform ourselves upon what Darcy himself curl E = 0, E = - grad <f> did, and then to determine the meaning of his results where when expressed explicitly in terms of the pertinent physical variables involved. This will be done first by <l> = gz + r dP. J P the empirical method used by Darcy himself, and then The field of flow, independently of the force field, must satisfy the conservation of mass, leading to the Manuscript received in Petroleum Branch office on Sept. 12, 1956. Paper prepared for presentation before Darcy Centennial Hydrology equation of continuity Symposium of the International Association of Hydrology held in Dijon. France. Sept. 2'0-26. 1956. and for dual publication as part div pq = - f op/ot of the Dijon Symposium and in the Darcy Centennial Issue of Journal of Petroleum Technology. where f is the porosity, and t is the time. For steady Discu8sion of this paper is invited. Discussion in writing (3 copies) may be sent to the office of the Journal of Petroleum motion op/ot = 0, and Technology. Any discussion offered after Dec. 31. 1956, should be in the form of a new paper. div pq = 0. PInROLEl! M TRAN~ACTIO"~. AIME 222 If the fluid is also of constant density, Appareil destine a determin ... la loi div q = O. a travers Ie sable The two fields are linked together by Darcy's law, q = aE which is physically analogous to Ohm's law in elec­ tricity. Then, when E = - grad <I', q = - a grad </' • By means of the foregoing equations the flow of both homogeneous and heterogeneous fluids through porous solids becomes amenable to the same kind of analytical treatment as is already familiar in electrical and ther­ mal conduction. The relation of Darcy's work to the development of a valid theory of the flow of fluids through porous solids, is somewhat analogous to that of Faraday to the Maxwellian equations of electromagnetism. It forms a solid experimental foundation for such a field theory, Downloaded from http://onepetro.org/TRANS/article-pdf/207/01/222/2176441/spe-749-g.pdf by guest on 01 October 2021 and the errors attributed by various recent authors to Darcy appear upon closer inspection to have been those committed by the authors themselves. INTRODUCTION In Paris in the year 1856 there was published by amnTUre Victor Dalmont as a part of the Libraire des Corps Imperiaux des Ponts et Chaussees et des Mines a mono­ graph by the French engineer Henry Darcy,' Inspector General of Bridges and Roads, bearing the title: "LES FONTAINES PUBLlQUES FIG. I-FACSIMILE OF DARCY'S ILLUSTRATION OF HIS DE LA VILLE DE DIJON EXPERIMENTAL ApPARATUS. (FROM Les Fontaines Puh­ Exposition et Application liques de la Ville de Dijon, A tlas, FIG. 3). DES PRINCIPES A SUIVRE ET DES FORMULES A EMPLOYER Dans les Questions 1. This consisted of a vertical iron pipe, 0.35 m in de diameter and 3.50 m in length (the figure shows 3.50 m DISTRIBUTION D'EAU but the text says 2.50), flanged at both ends. At a Ouvrage Termine was placed a horizontal screen supported by an iron Par un Appendice Relatif aux Fournitures height of 0.20 m above the base of the column, there d'Eau de Plusieurs Villes grillwork upon which rested a column, a meter or so AU FILTRAGE DES EAUX in length, of loose sand. Water could be admitted into et the system by means of a pipe, tapped into the column A la Fabrication des Tuyaux de Fonte, near its top, from the building water supply, and could de Plomb, de Tole et de Bitume." be discharged through a faucet from the open cham­ For several years previously M. Henry Darcy had ber near its bottom. The faucet discharged into a meas­ been engaged in modernizing and enlarging the pub­ uring tank 1 m square and 0.50 m deep, and the flow lic water works of the town of Dijon, and this treatise, rate could be controlled by means of adjustable valves comprising a 647-page volume of text and an accom­ in both the inlet pipe and the outlet faucet. panying Atlas of illustrations, constitutes an engineer­ For measuring the pressures mercury manometers ing report on that enterprise. were used, one tapped into each of the open chambers The item of present interest represents only a detail above and below the sand column. The unit of pres­ of the general work and appears in an appendix on sure employed was the meter of water and all man­ pages 590 to 594 under the heading "Determination of ometer readings were reported in meters of water meas­ the Law of Flow of Water Through Sand," and per­ ured above the bottom of the sand which was taken as tains to a problem encountered by Darcy in designing an elevation datum. The observations of the mercury a suitable filter for the system. Darcy needed to know manometers were accordingly eypressed directly in terms how large a filter would be required for a given quan­ of the heights of the water columns of equivalent water tity of water per day and, unable to find the desired manometers above a standard datum. information in the published literature, he proceeded to obtain it experimentally. The experiments comprised several series of obser­ A drawing of the apparatus used is given as Fig. 3 vations made between Oct. 29 and Nov. 2, 1855, and in the Atlas and is here reproduced in facsimile as Fig. some additional experiments made during Feb. 18 to 19, 1856. For each series the system was charged with lReferences given at end of paper. a different sand and completely filled with water. 223 VOL. 207. lYS6 By adjustment of the inlet and outlet valves the water k - dt, was made to flow downward through the sand at a e series of successively increasing rates. For each rate a k reading of the manometers was taken and recorded as a et I (h + e) = C - - t, pressure difference in meters of water above the bottom e of the sand. The results of two of these series, using [l is the logarithm to the base e.] different sands, are shown graphically in Fig. 2. In "Si la valeur ho correspond au temps to et h a un each instance it will be seen that the total rate of dis­ temps quelconque t, il viendra charge increases linearly with the drop in head across k I (h + e) = Z (h o + e) -- [t - to] ( 1 ) the sand of the two equivalent water manometers. e Darcy's own summary of the results of his experi­ "Si on remplace maintenant h + e et ho + e par ments is given in the following passage (p. 594): qe qoe '\ ' -k et -k-' 1 vlendra "Ainsi, en appelant e l'epaisseur de la couche s s de sable, s sa superficie, P la pression atmospher­ k ique, h la hauteur de l'eau sur ceUe couche, on lq = lqo -- (t - t,,) (2) aura P + h pour la pression a laquelle sera soumise e la base superieure; soient, de plus, P ± ho la pres- et les deux equations (1) et (2) donnent, soit la loi sion supportee par la surface inferieure, k un coef­ d'abaissement de la hauteur sur Ie filtre, so it la loi ficient dependant de la permeabilite de la couche q de variation des volumes debites a partir du temps t", Downloaded from http://onepetro.org/TRANS/article-pdf/207/01/222/2176441/spe-749-g.pdf by guest on 01 October 2021 Ie volume debite, on a q = k!...
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