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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
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 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!.... [h + e += ho ] qui se "Si k et e etaient inconnus, on voit qu'il faudrait e deux experiences preliminaries pour faire disparaitre ' k " reduit a q = k !....(h + e) quand ho = 0, ou lorsque de I a secon d e Ie rapport lllconnu -. e e la pression sous Ie filtre est egale a la pression at Translating Darcy's statements into the notation mospherique. which will subsequently be used in the present paper, "II est facile de determiner la loi de decroissance what Darcy found and stated was that, when water de la hauteur d'eau h sur Ie filtre; en effet, soit flows vertically downward through a sand, the volume dh la quantite dont cette hauteur s'abaisse pen of water Q passing through the system in unit time is dant un temps dt, sa vitesse d'abaissement sera given by h, - h, h, - h, dh . I" ' 'd d = KA ---- orby - KA--' (1) - --;]j; malS equatIon CI essus onne encore pour Q I ' Z ' and the volume crossing unit area in unit time by ceUe vitesse l'expression h, - h, h, - h, q k Q/ A = q = K --Z- , or by - K . --[- - = v = - (h + e) . s e (2) dh k h ) d" dh "On aura d onc -- = - ( + e' ou -:-c---c- where K is a factor of proportionality, A the area of dt e ' (h + e) cross section and I the thickness of the sand, and h, and h, the heights above a standard reference elevation of water in equivalent water manometers terminated above 30r------~.------~ and below the sand, respectively. Writing Eq. 2 in differential form gives FIRST SERIES OCTOBER 29-NOVEMBER 2,1855 q = - K (dh/dl) . (3) Soon after the publication of Darcy's account of these experiments, the relationship expressed by Eqs. 1 to 3 became known, appropriately, as Darcy's law. It has subsequently come to be universally acknowl edged that Darcy's law plays the same role in the theory of the conduction of fluids through porous solids as Ohm's law in the conduction of electricity, or of Four ier's law in the conduction of heat. On the other hand, Darcy's own statement of the law was in an empirical form which conveys no insight into the physics of the phenomenon. Consequently, during the succeeding cen tury many separate attempts were made to give the statement of the law a more general and physically sat isfactory form, with the result that there appeared in the technical literature a great variety of expressions, many mutually contradictory, but all credited directly or indirectly to Henry Darcy. DROP IN HEAD ACROSS SAND (METERS) It has accordingly become recently the fashion, when FIG. 2-GRAPHS COMPILED FROM DARCY's TABULAR some of these expressions have been found to be DATA ON HIS EXPERIMENTS OF OCT. 29 TO Nov. 2, physically untenable, to attribute the error to Darcy 1855, AND OF FEB. 17-18, 1856, SHOWING LINEAR RE himself. In fact one recent author, in disclliising a sup LATION BETWEEN FLOW RATE AND DIFFERENCES IN posed statement of Darcy's law which is valid for hori HEIGHTS OF EQUIVALENT WATER MANOMETERS. zontal flow only, has gone so far as to explain that PETROLEl'M TR.\:\SAl:'fIO:\S, AIME 224 ometers, at an axial distance , apart, are connected by flexible rubber tubing, we can determine the validity of Darcy's law with respect to the direction of flow. With the apparatus vertical and the flow downward at a total rate Q, the manometer difference h, - h1 will Q- have some fixed value f':,h. Now, keeping Q constant and inverting the column so that the flow will be ver tically upward, it will be found that f':,h also remains constant. Next, setting the column horizontal, f':,h still h Z remains constant. In this manner we easily establish that Darcy's law is invariant with respect to the direc tion of the flow in the earth's gravity field, and that for a given f':,h the flow rate Q remains constant whether the flow be in the direction of gravity or opposed to it, or in any other direction in three-dimensional space. This leads immediately to a generalization for flow REFERENCE ELEVATION in three-dimensional space. At each point in such space FIG. 3--ApPARATUS FOR VERIFYING DARCY'S LAW FOR there must exist a particular value of a scalar quantity FLOW IN VARIOUS DIRECTIONS. h, defined as the height above a standard elevation datum of the water column in a manometer terminated Downloaded from http://onepetro.org/TRANS/article-pdf/207/01/222/2176441/spe-749-g.pdf by guest on 01 October 2021 at the given point. The ensemble of such values then Darcy was led to the commission of this error by re gives rise to a scalar field in the quantity h with its at stricting his experiments to flow in a horizontal direc tendant family of surfaces, h = constant. In such a tion. scalar field water wiIl flow in the direction perpen On this centennial occasion of Darcy's original pub dicular to thc surfaces, h = constant, and at a rate lication, it would appear to be fitting, therefore, in given by stead of merely paying our respects to Darcy in the q = -- K grad h . (4) form of an empty homage, that we first establish un equivocally what Darcy himself did and said with respect Continuing our empirical experimentation, we find to the relationship which bears his name; second, try that when we change either of the fluid properties, den to ascertain the generality and physical content of the sity or viscosity, or the geometrical properties of the relationship and to give it a proper physical expression; sand, Eq. 4 still remains valid but the value of K and third, attempt to see how this fits into a general changes. In particular, by varying one factor at a time, field theory of the flow of fluids through porous solids we find in three-dimensional space. Kexp ,f (5) The first of these objectives has already been ac K ex tilL, ( . complished; the second and third will now be given our attention. where p is the density and fL is the viscosity of the fluid. Likewise, if we use a number of geometrically similar sands which differ only in grain size, we find that THE PHYSICAL CONTENT OF DARCY'S LAW Kexd', (6) As we have seen heretofore, what Darcy determined where d is a length such as the mean grain diameter, was that, when water flows vertically downward through which characterizes the size scale of the pore structure a sand, the relation of the volume of water crossing unit of the sand. area normal to the flow direction in unit time, to the thickness of the sand, and to the difference in heights Introducing the results of Eqs. 5 and 6 into Eq. 4 of equivalent water manometers terminated above and then gives below the sand, is given by the following equation: q = K'd' (plfL)( - grad h), . (7) h t - hJ hJ - h t in which K' is a new factor of proportionality contain '1 = K -,- , or, q = - K -1- (2) ing all other variables not hitherto explicitly evaluated. This remains, however, an empirical equation devoid of where K is "a coefficient depending upon the per dynamical significance since there is no obvious reason meability of the sand." why the flow of a viscous fluid through a porous solid Questions immediately arise regarding the generality should be proportional to a dimensionless quantity, of this result. Would it still be true if the water flowed - grad h. would be effected in the relationship if some different liquid characterized by a different density and viscosity This deficiency can be eliminated when w,~ introduce upward through the sand? or horizontally? What changes the equation relating the manometer height h to the were used? In what manner does K depend upon the dynamical quantities, gravity and pressure. At any point permeability of the sand, or upon its measurable sta P within the flow system, characterized by elevation z tistical parameters such as coarseness and shape? And and manometer height h, the pressure is given by the finally, what physical expression can be found which hydrostatic equation properly embodies all of these variables? p = pg (h - z), The answer to most of these questions can be de from which termined empirically by an extension of Darcy's original experiment. Jf, for example, we construct an apparatus h = (plpg) -t z . (8) such as that shown in Fig. 3, consisting of a movable and cylinder with a rigid sand pack into which two man- --- grad h = - (l/pg) grad p- grad z (9) VOL. 207,19.36 Multiplying both sides of Eg. 9 by g then gives is a measure of the energy per unit mass, or the poten - g grad h = - (1 I p) grad p - g grad z tial, of the water in the system at the point at which (10) the manometer is terminated. A manometer is thus seen With the z-axis vertical and positive upward, then to be a fluid potentiometer, the potential at every point grad z is a unit vector directed upward, so that - g grad being linearly related to the manometer height h by z is a vector of magnitude g directed downward. Des Eq. 14. Then, if we let E be the force per unit mass, ignating this by g, Eg. 10 becomes or the intensity of the force field acting upon the fluid, we have - g grad h = g - (lIp) grad p, . (11) E = - grad (I> = - g grad h = g - (lIp) grad p, (l5) in which each of the terms to the right represents, both and, by substitution into Eq. 12, Darcy's law may be in direction and magnitude, the force exerted upon expressed in any of the following equivalent forms unit mass of the fluid by gravity and by the gradient of (Fig. 4): the fluid pressure respectively; the fluid flows in the q = (Nd') (pilL) grad (P = aE, direction of, and at a rate proportional to, their re sultant, - g grad h. Hence the dynamical factor g has q = (Nd')(pIIL) g grad h = a [g - (lIp) grad pJ , evidently been concealed in the original factor K and q = (Nd') (pilL) g grad h = (alp) rpg - grad pl, must still be present in the residual factor K'. Intro (16) ducing this explicitly, we may now write where q = (Nd')(pIIL) ( - g grad h), (12) a = [(Nd') (pilL)] Downloaded from http://onepetro.org/TRANS/article-pdf/207/01/222/2176441/spe-749-g.pdf by guest on 01 October 2021 in which N is a final factor of proportionality. Dimen is the volume conductivity of the system. In the last sional inspection shows that N is dimensionless, and a of Egs. 16, the hracketed term physical review indicates that no dynamical variables have been omitted, so that N must he related to the Ipg - grad p 1 = pE = H . (17) only remaining variahle, namely the shape of the represents the force H per unit volume. passages through which the flow occurs. Since shape When we compare Darcy's law in the form: is expressed by angular measurement and angles are dimensionless, rLIL], then N must be a dimensionless q = aE = -- a grad (l' , shape factor whose value is constant for systems which with Ohm's law: are either identically, or statistically, similar geo i = a e E" = - a e grad V, metrically. By identical similarity is meant similarity in the strict Euclidean sense: all corresponding angles where i is the current density, a" the electrical con equal, and all corresponding lengths proportional. By ductivity, E" the electrical force-field intensity, and V statistical similarity is meant that two complex geo the electrical potential, the physical as well as the metrical systems which may not be identically simi mathematical analogy between Darcy's law and Ohm's lar as a microscopic scale are still indistinguishable as law becomes immediately apparent. to shape on a macroscopic scale~for example, two sets of randomly packed uniform spheres. THE PROBLEM OF PERMEABILITY Now that we have achieved a complete physical state The term (-- g grad h) can also he written in the form ment of Darcy's law, it remains for us to define what - g grad h =- grad (gh), . ( 13) shall be meant by the permeability of the system. It will be recalled that Darcy stated that the factor where gh represents the amount of work required to K is "a coefficient depending upon the permeability lift a unit mass of water from the standard datum of of the sand." Following this it has often been the cus elevation outside the system to the height h of the water tom, especially among ground-water hydrologists, to in the manometer. Then, since the additional work re define the permeability of a system to be synonymous quired to transport the water down the water-filled with K. But, as we have seen, the factor K is the tube of the manometer to its terminus is zero, it fol lumped parameter, lows that K = (Nd') (pilL) g, gh = (T> (14) comprising the geometrical properties of the sand, the dynamical properties of the fluid, and even the accelera tion of gravity. Consequently, if K is taken as a meas ure of the permeability, it will be seen that the same sand will have different permeabilities to different fluids. During recent years there has been a convergence of opinion toward the conclusion that permeability / / should be a constant of the solid independently of the / / fluids involved. If we accept this view, then it is seen 9 I I that the only property of the solid affecting the rate of / I I flow is the geometrical factor. I I k=Nd', ~IG. 4~RELATION OF FORCE PER UNIT MASS. E, TO which we may accordingly define to be its permeability. THE PRIMARY FORCES g AND - (lIp) GRAD p; AND OF Then since N is dimensionless and d is a length, it fol THE FLOW VECTOR q TO E. lows that the dimensions of permeability are [V]; and in any consistent system of units, the unit of permea T bility is the square of the unit of length. In practice the magnitude of this quantity, for a given porous solid, is determined hydrodynamically by flowing a liquid through the solid, and measuring all variables except k, and then solving Darcy's law for k: k = Nd' = qjJ,- pg - grad p , which, when the vector quantities are resolved into their f (P) components in the flow direction s, becomes k = ___ ~L_. (18 ) AV --- pg, - ap/as FIG. 5-METHOD OF DEFINING POINT VALUES OF MACRO It is found in this manner that for randomly packed, SCOPIC QUANTITIES ILLUSTRATED WITH THE POROSITY f. uniform spheres of diameter d, the value of the shape factor N is approximately 6 X 10'. Then, for a pack of uniform spheres of any size, the permeability will Few permeability measurements are accurate to with be approximately in 1.3 per cent and, when several specimens from the same formation are measured, the scatter is much Downloaded from http://onepetro.org/TRANS/article-pdf/207/01/222/2176441/spe-749-g.pdf by guest on 01 October 2021 k = (6 X 10-') d'. greater than this. Consequently, for all ordinary com If d, for different packs, is allowed to vary from about putations, the approximate conversions: 10-' to 10-' cm, corresponding to the approximate range 1 darcy""" 10- 8 cm', of grain sizes from fine silts to coarse sands, the per 6 1 md""" 10-" cm', meability will vary from about 10-" to 10- cm', which is also approximately the range of the permeabilities of are more accurate than the permeability data available; the corresponding clastic sediments. though if the data warrant it, the more precise conver In view of the fact that magnitudes of permeabilities sion can of course be used. of rocks are remote from that of the square of any unit of length in common use, there is some advantage DERIVATION OF DARCY'S LAW FROM in having a practical unit such that most measured NAVIER-STOKES EQUATION values fall within the range 1 - 10,000 practical units. If such a practical unit is to fit into a consistent sys Having thus achieved the desired generalization and tem of measurement without awkward conversion fac a proper physical statement of Darcy's law by an ex tors, then it must also be a submultiple of the funda tension of the empirical method which Darcy himself mental unit of the form: employed, let us now see if the same result can be de rived directly from the fundamental equation of Navier 1 practical unit = 10" fundamental units. and Stokes for the motion of a viscous fluid. In the cgs system with the fundamental unit the (centimeter)" the optimum value of the exponent n MACROSCOPIC AND MICROSCOPIC SCALES would be about 12, or In order to do this we must first distinguish between 1 practical unit = 10" cm'. the two size scales, the macroscopic and the micro Regrettably the unit of permeability used almost scopic, on which the phenomena considered are to be universally in the petroleum industry, for which the viewed. name "darcy" has been pre-empted, was defined origin The macroscopic scale, which is the one we have ally in terms both of an incomplete statement of Darcy's been using thus far, is a scale that is large as compared law:'" with the grain or pore size of the porous solid. On this scale the flow of a fluid through a porous solid is seen k=-~. (19) as a continuous phenomenon in space. However, when ap/as we are dealing with macroscopic quantities which have and an inconsistent system of measurement. The per particular values at each point in space, but which meability k is defined to be 1 darcy when q = 1 (cm"/ may vary with position, it is necessary for us to define cm') /sec, jJ- = 1 cp, and ap/as = 1 atmosphere/cm. more clearly what is meant by the value of a macro In the complete Darcy's law of Eq. 18, the factor scopic quantity at a given point. pg" except when the motion is horizontal, is of com This can be illustrated with the concept of porosity. parable magnitude to ap/os, and so cannot be ignored. Suppose that we are interested in the porosity at a Consequently, since it is not practical to measure pg, particular point. About this point we take a finite in atmospheres/cm, it follows that permeabilities, ex volume element !c,. V, which is large as compared with pressed in darcys, cannot be used in a proper statement the grain or pore size of the rock. Within this volume of Darcy's law without the insertion of a numerical fac element the average porosity is defined to be tor to convert pg, from cgs units into atmospheres/cm. - !c,.V, The only alternative is to convert permeabilities ex f = !c,.V ' (20) pressed in darcys into the cgs unit, the cm'. For this where !c,.V, is the pore volume within !c,.V. We then conversion allow !c,. V to contract about the point P and note the 1 darcy = 0.987 X 10' cm', value of 7 as !c,. V diminishes. If we plot 7 as a function which is within 1.3 per cent of the submultiple, 10' of !c,. V (Fig. 5), it will approach smoothly a limiting cm'. value as b. V diminishes until !c,. V approaches the grain 227 YO!.. 207,1956 or pore size of the solid. At this stage f will begin to Next, consider the microscopic flow through a macro vary erratically and will ultimately attain the value of scopic volume element /I, V of sides /I, x, /l,y, and /l,z. either 1 or 0, depending upon whether P falls within The void space in such an element will be seen to be an the void or the solid space. intricately branching, three-dimensional network of flow channels, each of continuously varying cross section. A However, if we extrapolate the smooth part of the fluid particle passing through such a system will follow curve of 7 vs /I, V to its limit as /I, V tends to zero, we a continuously curving tortuous path. Moreover, the shall obtain an unambiguous value of f at the point speed of the particle will alternately increase and de P. We thus define the value of the porosity f at the crease as the cross section of the channel through which point P to be it flows becomes larger or smaller. Such a particle will accordingly be seen to be in a continuous state of accel f P) = extrap lim /l,V f (21) ( /l,V~O /l,V' eration with the acceleration vector free to assume any possible direction in space. where extrap lim signifies the extrapolated limit as ob tained in the manner just described. Consider now the forces which act upon a small vol ume element dV of this fluid. By Newton's second law By an analogous operation the point value of any of motion other macroscopic quantity may be obtained, so that hereafter, when such quantities are being considered, dm a = 2;dF, . (27) their values wiII be understood to be defined in the where dm is the mass of the fluid, a the acceleration, and foregoing manner, and we may state more simply: 2;dF is the sum of all the forces acting upon the fluid Downloaded from http://onepetro.org/TRANS/article-pdf/207/01/222/2176441/spe-749-g.pdf by guest on 01 October 2021 lim - contained within dV. There are many ways in which these Q (P) = /I, V ~ 0 Q (/I, V) , forces may be resolved, but for present purposes it wiII or be convenient to resolve them into a driving or impelling force dF and a resistive force arising from the viscous Jim - u Q (P) /l,S ~ 0 Q (/I,S) , (22) resistance of the fluid element to deformation, dFv • Eq. 27 then becomes or dm a = dFd + dF, . (28) Q (P) = -~ Q (/I,s), J /l,s ~ 0 By the principle of D'Alembert we may also introduce where the quantity of interest is a function of a volume, a force dF" = - dm a, which is the inertial reaction of an area, or a length, respectively. the mass dm to the acceleration a, and with this substitu tion Eq. 28 becomes The microscopic scale, on the contrary, is a scale commensurate with the grain or pore size of the solid, dFu + dF" + dF, = O. . (29) bTlt still large as compared with molecular dimensions Of these forces, dFd , which is imposed from without or of Hie motional irregularities due to Brownian or and does not depend primarily upon the motion of the molecular movements. fluid, may be regarded as the independent variable. The forces dF" and dFv both owe their existences to the fluid MICROSCOPIC EQUATIONS OF MOTION motion, and their effect is to impede that motion. Let us next consider the steady, macroscopically recti The relation of the separate terms of Eq. 29 to the linear flow of an incompressible fluid through a porous externally applied forces and the fluid motion are given solid which is macroscopically homogeneous and iso by the equation of Navier and Stokes, which in vector tropic with respect to porosity and permeability. We form may be written as follows: shall then have the fluid flowing with a constant macro scopic flow rate q under a constant impelling force per p [g - (1/ p) grad p] dV = p (Dv /Dt) dV unit of mass E, and in virtue of the isotropy of the - f.L [v' v + (1/3) vV'v] dV (30) system, we shall have in which v is the microscopic velocity, and p = - (1/3) (~x ~y ~z) is the microscopic pressure at a point. The q = ~E (23) + + ~'s are normal components of microscopic stress. where ~ is an unknown scalar whose value we shall seek The expression Dv/ Dt is the total derivative with re to determine. spect to time of the velocity v, and is equal to the accel Then, choosing X-, y-, and z-axes, eration a. This can be expanded into q = i qx + j qy + k q, , (24) (lv E = i Ex + j Ey + k E, , }. Dv/Dt = iit+ v'vv, and qx = ~Ex, ~ qy :: ~Ey, . (25) q, - ~E,. where i, j, and k are unit vectors parallel to the X-, y-, and z-axes, respectively, and the subscripts signify the corresponding scalar components of the vectors. Further, there will be no loss of generality, and our analysis will be somewhat simplified, if we choose the x-axis in the macroscopic direction of flow. Then q = i qx ; E = i Ex , { and (26) FIG. 6-MICROSCOPIC VIEWS OF Two DYNAMICALLY qy = q, = 0 ; Ey = E, = o. \. SIMILAR FLOW SYSTEMS. PETROLEUM TRANSACTIONS, AIME 22H in which the term ov/ot signifying the rate of change of for a dimensionless factor of proportionality, provided the velocity at a particular point is zero for steady motion. the flow field is kinematically similar for different rates The expression V·v, in the last term of Eq. 30, is the 0/ flow. divergence of the velocity; and for an incompressible fluid this also is zero. CRITERIA OF SIMILARITY Since the flow being considered is the steady motion of Consider two flow systems consisting of two geometri an incompressible fluid, Eq. 30 simplifies to cally similar porous solids through which two different p [g - (l/p) gradp] dV = p (v'Vv) dV fluids are flowing. The criterion of geometrical similarity - p, (V'v) dV . (31) is that if 1, and I, are any corresponding lengths of the two systems, then for every pair of such lengths TRANSFORMATION FROM MICROSCOPIC TO 1,11, = I,. = const . (38) MACROSCOPIC EQUATIONS OF MOTION The criterion of kinematic similarity is that if V, and V, Eq. 31 expresses the relation of the fluid velocity and are the velocities at corresponding points in the two sys its derivatives in a small microscopic volume element to tems, the two velocities must have the same direction and the applied force dFu acting upon that element. If we their magnitudes the ratio could integrate the three terms of this equation with re spect to the volume, over the macroscopic volume 6 V, v,/v, = v" = const . and then convert the results into equivalent macroscopic Then, since the forces dF, and dFv acting upon a fluid variables, our problem would be solved. In fact the inte element are each determined by the velocities and the Downloaded from http://onepetro.org/TRANS/article-pdf/207/01/222/2176441/spe-749-g.pdf by guest on 01 October 2021 gration of the first two terms presents no difficulty. The fluid density, or viscosity, and dF" is determined by dF, total driving force on the fluid content of the volume 6 V, and dF" if the fluid motions of the two systems are kine as obtained by integrating the microscopic forces dFd , is: matically similar, all corresponding forces will have the same directions and their magnitudes the same ratio Fd = J dFd = J (pg - grad p) dV = (Fig. 6). /6V Thus (pg - grad p) /6 V . (32) (dF.). _ (dF,), _ (dF,,), where grad p is the volumetric average of the microscopic (39) (dFu), - (dF,), - (dF ), grad p over the fluid volume /6 V. d Since only two of the three forces are independent,. we From the macroscopic equations, the driving force F d is given by: need to consider the ratios of only the first two, and, by reciprocation, Fd = p/6VE = p[g - (l/p) gradp]/6V. (33) Then by combining Eqs. 32 and 33, dFa) (dF.) (40) ( dFv ,= dF,. I' (pg - grad p) dV = (pg - grad p) 6 V J f indicating that for each system the ratio of the inertial to /6V the viscous force must be the same. = p/6V [g - (l/p) gradp], (34) From Eq. 31 from which it is seen that the macroscopic grad p is equal (dFa), p, (v'Vv), dV, . to the volumetric average, grad p, of the microscopic (41 ) (dF,,), ;. (v'Vv), , grad p. dV ' Integrating the inertial term: (dFvL p" (V'v), dV, (42) (dF ), F, = J dF, = p J V'Vv dV v /6V In Eq. 41 V'Vv expands into the sum of a series of = i p J (u GU/ox + vou/oy + terms, each of the form u au/ox, which is a velocity /6 V w ou/oz) dx dy dz squared divided by a length. In Eq. 42 V'v expands into a (35) + j p J (u ov lox + v ilv joy + series of terms, each of the form o'u/ox', which is a veloc /6V wov/oz) dxdydz ity divided by the square of a length. Then, since the + k p J (u ow/ox + v ow/oy + ratios of all corresponding velocities and of all corre /6V wow/oz)dxdydz. sponding lengths of the two systems are constant, we may Since there is no net gain in velocity with macroscopic choose any suitable velocity and any convenient length. distance, each separate integral of the expanded form of We accordingly choose for the velocity the micro Eq. 35 is equal to zero, and we obtain for the volume ele scopic flow rate q whose dimensions are [L' L-' r'l, or ment 6V, [Lr']. For the characteristic length we choose d, which must be some convenient statistical length parameter of Fa = J dF = 0 (36) a the microscopic geometry of the system. With these sub Then, in virtue of Eq. 36, stitutions the first two terms of Eq. 39 become J dF,. = - p, J V'v dV = - F" . (37) p, q,' d,' p" q, d, (43) /6V /.1., q, d, ' Ordinarily the evaluation of and, by reciprocation, Eq. 40 becomes J .v'v dV p, q,' d,' P1q,'d,' /6V would require a detailed consideration of the geometry or of the void space through which the flow occurs and of q,d, the flow field within that space. This difficulty can be (44) circumvented, however, and the integral evaluated except p,,/p, VOL. 207,1956 The dimensionless quantity (qd) I (fLl p) is the Reyn pIg - (lIp) grad p] jf':,V = - r dF, = ~~2 jf':,V, olds number R of the system, which, as seen fr;Jm its deri vation, is a measure of the ratio of the inertial to the vis or cous forces of the system. Our criterion for kinematic q = Nd' (plfL) [g - (lIp) gradpl, (51) similarity between the two systems thus reduces to the which is the derived Darcy's law in the same form as requirement that Eq. 16 deduced earlier from empirical data. R,=R, . (45) DISCUSSION OF DARCY'S LAW Now let us specialize the two systems by making The direct derivation of Darcy's law from fundamen d, = d" p, = P" fL' = fL' , tal mechanics affords a further insight into the physics of which is equivalent to requiring the same fluid to flow the phenomena involved over what was obtainable from through the same porous solid at velocities, q, and q" the earlier method of empirical experimentation. It has However, when these values are substituted into Eq. 44, long been known empirically, for example, that Darcy's we obtain law fails at sufficiently high rates of flow, or at a Reyn olds number, based on the mean grain diameter as the q, = q" characteristic length, of the order of R = I.' indicating that, in general, when the same fluid flows At the same time one of the most common statements through a given porous solid at two different rates, the made about Darcy's law has been that it is a special case resulting flow fields cannot be kinematically similar. of Poiseuille's law; and most efforts at its derivation Downloaded from http://onepetro.org/TRANS/article-pdf/207/01/222/2176441/spe-749-g.pdf by guest on 01 October 2021 However, since dFa is proportional to q' and dFv to q, have been based upon various models of capillary tubes then as q is decreased dFa diminishes much more rapidly or of pipes. It also has been known since the classical than dFv • Consequently there must be some limiting value studies of Osborne Reynolds' in 1883 that Poiseuille's of q = q*, or of R = R *, at and below which the inertial law fails when the flow makes the transition from laminar force dFa is so much less than the viscous force dFv that to turbulent motion, so the conclusion most often reached the effect of the former is negligible as compared with the as to the cause of the failure of Darcy's law has been that latter. For flow in this domain we may then write the motion has become turbulent. From what we have seen, this represents a serious mis dF dF ; v = - d interpretation and lack of understanding of Darcy's law. and the force ratios become In the Darcy flow each particle moves along a continu ously curvilinear path at a continuously varying speed, (dFv ), _ (dFd ), (46) and hence with a continuously varying acceleration; in (dFv), - (dF ), , d the Poiseuille flow each particle moves along a rectilinear whereby kinematical similarity is maintained for all rates path at constant velocity and zero acceleration. There of flow q < q*. fore, instead of Darcy's law being a special case of Poi seuille's law, the converse is true; Poiseuille's law is in INTEGRATION OF VISCOUS FORCES fact a very special case of Darcy's law. Another special With this result established let us now return to the case of Darcy's law is the rectilinear flow between paral lel plates. integration of dFv over the volume element!f':, V. Consequently deductions concerning the Darcy-type J dFv = - fL [i J \J'u dV + j J \J'v dV flow made from the simpler Poiseuille flow are likely to !f':,V !f':,V !f':,V 2 be seriously misleading. The deduction that the Reynolds + k J \J w dV] . (47) number at which Darcy's law fails is also the one at which !f':,V turbulence begins is a case in point. We have seen that the In virtue of the fact that, by our choice of axes, qy and cause of the failure of Darcy's law is the distortion that qz are both zero and there is no net flow in the y- or z results in the flowlines when the velocity is great enough direction, the last two integrals to the right are both zero, that the inertial force becomes significant. This occurs at and Eq. 47 simplifies to a very slow creeping rate of flow which, for water, has the approximate value of J dFv = - i fL J \J'u dV ( 48) !f':,V !f':,V q* r-..' :d r-..' (l X 10-' cm' sec-') (lId) , From our earlier discussion, so long as the flow remains kinematically similar for different rates, the quantity \J'u, corresponding to R" = 1, when d is the mean grain which is a velocity divided by the square of a length, is diameter. related to the macroscopic parameters by Thus when d = 10-' cm Darcy's law fails at a flow rate q r-..' 1 cm/see. \J'u = a (qld') , (49) Since this represents the threshold at which the effects where a is a dimensionless constant of proportionality for of inertial forces first become perceptible, and since tur the element dV but has a different value for each different bulence is the result of inertial forces becoming predom element of volume. inant with respect to resistive forces, it would be inferred Substituting Eq. 49 into Eq. 48 then gives that the incidence of turbulence in the Darcy flow would occur at very much higher velocities or at very much - i fL (qld') J a dV = - ~~2 !f':,V, higher Reynolds numbers than those for which linearity !f':,V between the flow rate and the driving force ceases. That (50) this is in fact the case has been verified by visual obser vations of the flow of water containing a dilute suspension where 1IN is the average value of a over!f':, V. of colloidal bentonite through a transparent cell contain Substituting this result into Eq. 37, we obtain ing cylindrical obstacles. This system, when observed in PETROLEUM TRANSACTIONS, AI ME 230 polarized light, exhibits flow birefringence which is sta isotropic system, would also be the average value for a tionary for steady laminar flow but highly oscillatory volume. when the motion is turbulent. Observations of only mod To attempt to do this in detail would be a statistical erate precision indicate that the incidence of turbulence undertaking beyond the scope of the present paper. As a occurs at a Reynolds number of the order of 600 to 700, first approximation, however, we may simplify the prob or at a flow velocity of the order of several hundred times lem by assuming: that at which Darcy's law fails. 1. That all the gaps are equal and of width 2A, EXAMINATION OF THE SHAPE-FACTOR N where I is the average half-width of the actual In both procedures used thus far, the factor N has gaps. emerged simply as a dimensionless factor of proportional 2. That through each gap the velocity profile sat ity whose magnitude is a function of the statistical geo isfies the differential equation metrical shape of the void space through which the flow occurs. For systems which are either identically or sta 3'u/oy' = o'u/oy' = - C . (55) tistically similar geometrically, N has the same value. 3. That the total discharge through the averaged Beyond this we have little idea of the manner in which gaps is the same as that through the actual N is related to the shape or of what its numerical magni gaps. tude should be, except as may be determined by experi ment. Let us now see if the value of N, at least to within The velocity profile for the averaged gaps can then be obtained by integrating Eq. 55 with respect to y. Taking an order of magnitude, can be determined theoretically. Downloaded from http://onepetro.org/TRANS/article-pdf/207/01/222/2176441/spe-749-g.pdf by guest on 01 October 2021 a local origin of coordinates at the middle of the gap, Since we have already seen in Eq. 50 that l/N = -;, and integrating Eq. 55 twice with respect to y, we obtain where a is the factor of proportionality between the Cy'/2 Ay B, . (56) macroscopic quantity q/tf and the microscopic quantity u = - + + V'u, it follows that in order to determine the magnitude in which A and B are constants of integration. Then, of N we must first determine that of the average value of supplying the boundary conditions, u = 0 when y = V'u. For this purpose, with the fluid incompressible, the ± A, gives macroscopic flow parallel to the x-axis, and the inertial forces negligible, only the x-component of the Navier o = - c?: 2/2 ± AI + B Stokes Eq. 31, from which o'u o'u o'u) A 0 ; B '/2 . pgx - op/ox = - jJ- ( ox' + oy' + oz" (52) = = CA Substituting these into Eq. 56, we obtain might be considered. When this is integrated with respect to the volume over the fluid space If'::, V, it becomes u = (C/2) (A' _ y') (57) o· as the equation of the parabolic profile of the aver (pgx - op/ox) If'::,V = - jJ- [Ilv a:' dV + aged velocity across the averaged gap. The mean value, ;;, of u across this gap is given by J o'u J o'u d 1 (53) If'::,V oy' dV + If'::,V oz' V ,\ I Of the three integrals to the right, the first, which rep u = (l/A) rudy = :"f(A' - y') dy = cA'/3. J 2A. resents the expansion of the fluid in the x-direction, is o 0 zero; and from symmetry, the last two, both being the (58) integrals of derivatives with respect to axes at right angles to the flow, are equal to each other. In consequence, Eq. Then, replacing C by (l/2jJ-) (pgx - op/ox) from Eqs. 55 and 54, we obtain 53 is reduced to the simpler form: - I' p ;---..,...... ~~~.,. (pg, - op/ox) If'::,V = - 2jJ-J (o'u/oy') dV u = 6-;;' [gx - (l/p) op/ox] (59) = - 2jJ- (02U/oy') If'::, V . This can be converted into terms of the macroscopic in which o'u/oy' is the average value ofo'u/3y' throughout velocity, q., by noting that for a macroscopic length the macroscopic volume element. Solving this for o'u/oy' of line I normal to the flow direction then gives (o'u/oy') = - (l/2jJ-) (pg, - op/3x). (54) or Our problem now reduces to one of attempting to qx = 2;; (J..,\jl) = ;; I , (60) determine the average value of 0'U/oy2 for the system in where I is the porosity. terms of the kinematics of the flow itself. If we extend any With this substitution Eq. 59 becomes line through the system parallel to the y-axis, this line will pass alternately through solid and fluid spaces. At each _ /A' p _ qx - T . -; . [gx (l/p) op/ox] (61) point on the line in the fluid space, there will be a particu lar value of the x-component u of the velocity, which will or, more generally, be zero at each fluid-solid contact, but elsewhere will have finite, and usually positive values, giving some kind of a fA' p [g - (l/p) grad p] (62) velocity profile across each fluid gap. If this profile for q=T·-;· each gap could be determined, then we could also com which is a statement of Darcy's law that IS valid to pute o'u/oy' at each point on the line and thereby deter the extent of the validity of the averaging approxima mine o'u/3y' for the line, which, in a homogeneous and tion used in its derivation. 231 VOL. 207, 1956 In this, it will be noted, that the geometrical factor a18' = ;;18' = area of hemisphere = 27rr' = 2 (f16) A' represents the permeability, so that area of diametral plane TrY'!. • (69) k~ (f16) A' = N,A' or Introducing this result into Eq. 67 then gives N;:~ /16 (63) I = 2/113 . (70) where N:;: is the shape factor corresponding to I as the A method for measuring 13 has been described by characteristic length of the system. Brooks and Purcell,' but for present purposes the data on randomly packed uniform spheres, for which 13 can DETERMINATION OF A be computed, will suffice. For such a system, with n The mean half gap-width A along a linear traverse spheres per unit volume, can be determined in either of two ways. The most ob area of spheres 17 • 47rr' 3 (1 - f) vious way is by direct observation by means of micro 13 = total volume Il 47rr' r meter measurements along rectilinear traverses across a plane section of the porous solid. 1-1'-3- 6(1 - f) Of greater theoretical interest, however, is an indirect . (71) method due to Corrsin." Instead of a line, let a rec d tangular prism of cross-sectional area 8', where 8 can where d is the sphere diameter. be made arbitrarily small, be passed through a porous Downloaded from http://onepetro.org/TRANS/article-pdf/207/01/222/2176441/spe-749-g.pdf by guest on 01 October 2021 Then, introducing this result into Eq. 70, we obtain solid which is macroscopically homogeneous and iso tropic. This prism will pass alternately through solid for packs of uniform spheres segments and void segments. Let n be the number of - f each which is traversed per unit length. Then the num A = 3(l _ f) . d (72) ber of intersections with the solid surface per unit and length will be 2n, and if 7; is the average area of the solid surface cut out by the prism at each intersection, (73) the total area per unit length, df3, will be df3 = 2n 7;. (64) COMPARISON WITH EXPERIMENTAL DATA If a parallel family of such prisms is made to fill all In order to compare the value of Nr, correspond space, the number per unit area perpendicular to the ing to the characteristic length A, with No, correspond axis of the prisms will be 1/8', and the solid surface in ing to the sphere diameter d, we must first establish tersected per unit volume will be the relation between N, and N". This can be done by 13 = 2n (7;18') . (65) noting that, by definition, In addition, the total length of the void spaces per NiX: = Nd a' = k, unit length of line will be where k is the permeability of the system. Consequently 2n I = /, (74) or (66) Then, introducing the vahle of a'IA' from Eq. 73 into Eq. 74, gives for a pack of uniform spheres Substituting the value of n from Eq. 66 into 65 and solving for r then gives N, = 9(1; /)' . No . (75) - f a A= 13 . 8' (67) 5.00 ~--'----'----'---r--""-----'---r--' Since / and 13 can be measured, the value of I could be determined if ;;18' were known. The latter can be determined in the following man 4.50 I---+--+--;r---+--+/~---t---+--j ner: A homogeneous and isotropic distribution of the internal surface S, inside a macroscopic space, implies ! 4.00 1----+---+/ that if all equal elements dS of the surface were placed without rotation at the center of a reference sphere, J their normals would intersect the sphere with a uni ~ 3.50 form surface density. Then, with the normals fixed "~ • HYDROGEN in direction, if the surface elements were all moved • NITROGEN equal radial distances outward, at some fixed radios ~=---+--+---I----+-l "' gt~x~~~ they would coalesce to form the surface of a sphere. If 2.50 L--L.-o.L.2-L-...J .-'--L-oL.6---1~0J..... s:--L-..LI. o,,--.l....-..J, '-=. 2--'----"J...... :--' the prisms of cross-sectional area 8', parallel to a given O line, were then passed through this sphere, the aver RECIPROCAL MEAN PRESSURE (ATMJ-I age value of a18' would be FIG. 7-VARIATION AS A FUNCTION OF lip OF THE ApPAR 7 ~a ENT PERMEABILITY OF A GIVEN SOLID AS DETERMINED al - = ~8' ; . (68) BY THE DIFFERENT GASES. THE VALUE OF 2.75 MD, AS (lIp) TENDS TO 0, DIFFERS BUT SLIGHTLY FROM THAT :!nd, when the summation includes one whole hem OF 2.55 MD OBTAINED USING A LIQUID (AFTER L. J. isphere, KLINKENBERG, API Drill. and Prod. Prac., 1941). PETROLEI'M TRANSACTIO:\S, AI ME 232 Th&vame- M N" for well-rounded quartz sands, be noted that each is of the form: velocity/(lengthr. screened to nearly uniform sizes, has been determined We have already seen that by the author's former research 'assistant, Jerry Con ner. Using packs of different uniform sands with mean o'u/oy' "" - 3 ;i/,\,' "" - q/ A' , grain diameters ranging from 1.37 X 10-' to 7.15 X indicating that the magnitude of this term is determined 10-' cm, Conner made seven independent determina by the fact that large variations of u in the y-direction tions of N". The average value obtained was take place within the width of a single pore. Compar N,I = 6.0 X 10' , . (76) able variations of u in the x-direction, however, due to the expansion of the fluid, occur only in fairly large with individual values falling within the range between macroscopic distances. Consequently, we may write 5.3 X 10-' and 6.7 X 10-'. o'u/ox' "" 0' qlox' "" q/l' Conner did not determine the porosity, but the aver age value of the porosity of randomly packed, uniform where I is a macroscopic distance. The ratio of the two spherical glass beads found by Brooks and Purcell was terms is accordingly 0.37. Inserting this value of I, and Conner's value of ~ A' N , into Eq. 75, then gives for the equivalent experi (81) d o'u/oy' ;::::: 7 mental value of Nr: N;:: = 26.2 Nd = 1.57 X 10.2 (77) Then, since l' > > :\', it follows that the additional frictional drag caused by the divergence term is negli Comparing this experimental result with the approxi gible, and this term may be deleted from the equation. Downloaded from http://onepetro.org/TRANS/article-pdf/207/01/222/2176441/spe-749-g.pdf by guest on 01 October 2021 mate theoretical result of Eq. 63, it will be seen that We conclude, therefore, from this approximate analy N. (theoretical) 116 sis, that Darcy's law in its differential form is the N;:: (observed) 1.57 X 10-' = 4 . (78) same for a gas as for a liquid, provided that the flow Since our object at the outset was merely to gain behavior 01 a gas in small pore spaces, other than ex some insight into the nature of the shape-factor N pansion, is similar to that 01 a liquid. occurring in Darcy's law, no particular COncern is to It has been conclusively shown, however, by L. J. be felt over the discrepancy in Eq. 78 between the ob Klinkenberg' that the two flows are not similar, and served and the theoretical values. All that this really that, in general, kg, the permeability to gas based on indicates is that the system of averaging required should the assumed validity of Darcy's law for gases, is not be better than the oversimplified one actually used. For equal to k" the permeability to liquids; and, in fact, a more complete analysis, account needs to be taken of is not even a constant. the frequency distribution of the half gap-width '\', and In the case of the flow of a liquid through small also of the functional relation between -;; and ,\,. The pores, the microscopic velocity v becomes zero at the fact that our approximate analysis yields a result in er fluid-solid boundary; for gas flow, on the contrary, ror by only a factor of 4 makes it appear promising there exists along the boundary a zone of slippage that if account is taken of the variability of ,\, and of of thickness 8, which is proportional to the length of -;; as a function of '\', much better approximations may the mean-free path of the molecules. Consequently the be obtainable. gas velocity does not become zero at the boundaries, and the frictional resistance to the flow of gas is less DARCY'S LAW FOR COMPRESSIBLE FLUIDS than that for a liquid of the same viscosity and macro scopic velocity. Our analysis thus far has been restricted to the flow of incompressible fluids for which the divergence term, Since 8 is proportional to the mean-free path, it is - (/3) p. \l\l . v, could be eliminated from the Navier also approximately proportional to lip. Consequently Stokes equation. For the flow of a compressible fluid, when the gas permeability, kg, of a given porous solid this term must be retained, and with the flow parallel is determined with the same gas at a number of differ to the x-axis, ent mean pressures, the resulting values of kg, when _ _ r[4 o'u (j'u o'u 1( o'v o'w)] F" - p. J 3 ox' + (Jy' + oz' + 3 oxoy + oxow dV , (79) of whidh the last two terms are equal to zero, leaving . f( 4 0'£1 o'u o'u) f" = - p. 3 ox' + oY' + oz' dV, or F,I = - 20 [(2/3) (o'ulox') + (o'uloy')] I!':,.V (80) Here, o'u/ox' represents the gradient of the divergence of the velocity in the x-direction, or the rate of the fluid expansion. Should this term be of the same order of magnitude as o'u/oy', and if the flow of a gas through the porous system is otherwise similar to that of a liquid, then the viscous resistance to a gas should be greater than that for a liquid of the same viscosity. FIG. 8-THE POTENTIAL AS A LINE INTEGRAL OF THE To compare the two terms o'u/ox' and o'u/oy', it will FIELD OF FORCE E. 233 VOL. 207, 1956 plotted as a function of l/p, give a curve which is ap range of temperatures and pressures normally encoun proximately linear with l/p. Moreover, different gases, tered in the earth to drillable depths, having different mean-free paths, give curves of differ p = const. (83) ent slopes. The limiting value of kg as l/p ~ 0, or as For gases, on the other hand, we shall have an equa p becomes very large, is also equal to k" the permea bility obtained by means of a liquid (Fig. 7). tion of state In view of this fact it is clear that, in general, the p = f(p, T) . (84) flow of gases through porous solids is not in accord where T is the absolute temperature. ance with Darcy's law. However, from Klinkenberg's The vector E has a particular value at each point in data, at pressures greater than about 20 atmospheres space and the ensemble of such values comprises its (2 X 10' dynes/cm', or 300 psi), the value of kg differs vector field. The criterion of whether this field has a from k 1 by less than I per cent. Therefore, since most potential, that is to say, of whether oil and gas reservoir pressures are much higher than this, it can be assumed that gases do obey Darcy's law E = - grad The establishment of Darcy's law provides a basis = V X g - V X [(l/p) Vp]. Downloaded from http://onepetro.org/TRANS/article-pdf/207/01/222/2176441/spe-749-g.pdf by guest on 01 October 2021 upon which we may now consider the field equations As is well known, the gravity field is irrotational that must be satisfied by the flow of fluids through even without the assumption that g = const, so that porous solids in general, three-dimensional space. This problem is complicated, however, by the fact that the V X g = O. Also - V X [(l/p) Vp] = - V(1/p) X \lp fluids considered may be either of constant or of var iable density; that one or several different fluids, either - (l/p) V X Vp = - v(l/p) X Vp. intermixed or segregated into separate macroscopic spaces, may be present simultaneously; and that all de grees of saturation of the space considered are possible. grad p~ The problem of dealing with such cases becomes Ps I tractable when we recognize that the behavior of each I different fluid can be treated separately. Thus, for a I specified fluid, there will exist at each point in space I capable of being occupied by that fluid a macroscopic I force intensity vector E, defined as the force per unit I mass that would act upon a macroscopic element of I the fluid if placed at that point. In addition, if the I fluid does occupy the space, its macroscopic flow rate I at the given point will be indicated by the velocity vec tor q, the volume of the fluid crossing unit area nor mal to the flow direction in unit time. We shall thus have for any fluid two superposed Es fields, a field of force and a field of flow, each inde pendently determinable. The equations describing the grad p properties of each of these fields, and their mutual in PI terrelations, comprise the field equations of the system; these in turn, in conjunction with the boundary condi tions, determine the nature of the flow. THE FIELD OF FORCE We have seen already that the force per unit mass is given by E = g - (l/p) grad p , (l5) and the force per unit volume by H = pE = pg - grad p . (l7) Either of these force vectors could be used, and either / / is determinable from the other, but before choosing g / one in preference to the other, let us first consider the / properties of their respective fields, of which the most / important for present purposes is whether Or not the / field has a potential. To simplify our analysis we will / make the approximations that / g const , (82) = FIG. 9-FoRCE VECTORS E AT THE SAME POINT CORRE and for chemically homogeneous liquids under the SPONDING TO FLUIDS OF DIFFERENT DENSITIES. PETROLEUM TRANSACTIONS, AIME 2S4 Consequently where the integral from p .. to P is taken along any path E = - 'V(1lp) X 'Vp = 'Vp X 'V(llp) , curl s. Then by setting PETROLEIIM TRAIXSACTIO:>iS, AIME 236 lines of force will coincide; if the solid is anisotropic, a The migration of petroleum and natural gas, through will be a tensor and E and q will then differ somewhat an otherwise water-saturated underground environment, in direction except when parallel to the principal axes from an initial state of high dispersion to final positions of the tensor. of concentration and entrapment, constitutes an exam Limiting our discussion to isotropic systems, by tak ple of the latter kind. ing the curl of Eq. 105, we obtain PROPERTIES OF THE COMBINED FIELD OF FLOW curl q = - V X aV = - Va X V - aV X V . Although the present paper does not permit of their Then, since the last term to the right is zero, this be elaboration, practically everything that is now known comes concerning the flow of fluids in a porous-solid, three curl q = - Va X V , (106) dimensional space is deducible from the foregoing field which is zero only when a is constant throughout the equations. The equations for the flow of an incom field of flow. pressible fluid are of the same form as those for the Therefore, in general, steady conduction of electricity. Consequently the well known solutions to the electrical equations can be ap curl q =1= 0 , (107) plied directly to analogous situations in fluid flow. The and this circumstance precludes the derivation of the unsteady flow of a compressible fluid, when the force flow field from an assumed velocity potential, for, with field is irrotational, is closely analogous, although some Downloaded from http://onepetro.org/TRANS/article-pdf/207/01/222/2176441/spe-749-g.pdf by guest on 01 October 2021 the exception of the flow of a fluid of constant density what more complex, to the unsteady conduction of heat and viscosity in a space of constant permeability, no and the solutions of the heat equations can be adapted such function exists. with some modification to the analogous fluid-flow The flow of a given fluid through a porous solid problems. incompletely saturated with that fluid is equivalent to The fluid phenomena which are unique and have no flow through a solid of reduced permeability, because counterpart in other more familiar field theory are the space available to the flow diminishes as the sat those involving multiple fluids. If we consider the flow uration decreases. For saturations greater than some of two immiscible fluids of unequal densities, such as critical minimum value, the flow obeys Darcy's law water and oil, interspersed in the same macroscopic subject to the permeability having this reduced, or so space, we have seen from Eq. 97 that, in general, the called, "relative-permeability" value. Thus, with two force fields E, and E, of the two fluids will lie in the interspersed but immiscible fluids in the same macro same vertical plane, but will have divergent directions, scopic space, that for the less dense fluid being upward with respect q, :: a" E, :: a,., [g - (1/ p,) grad p] ,} .. (108) to that for the more dense. As a consequence the two q2 - a,., E, - a" [g - (1/P2) grad p] , fluids will drift in these respective directions, and, in response to suitable impermeable barriers, will tend to where a" and a,., are the relative conductivities of the two fluids. become completely segregated. It will be noted that, except for vertical motion, Once this segregation is achieved, for any further E, and E, are not parallel. Consequently the two force steady motion of either of the fluids, the interface fields and flow fields will be, in general, transverse to will appear as an impermeable barrier. Across this in one another. terface, neglecting minor pressure differences due to capillarity, the pressure in the two fluids must be the Pioneer work on relative permeability as a function same. Then, in case neither of the fluids is in motion, of saturation was done on the single fluid, water, by the interface will have to be horizontal, with the less L. A. Richards' in 1931. Subsequently studies of the dense fluid uppermost, since this is the only inclination simultaneous flow of two or more fluids were initiated of the surface for which the pressures on opposite sides by Wyckoff and Botset,1O and by Hassler, Rice, and can be the same. Leeman" in 1936. Since that time many other such stud When either or both of the fluids is in motion, how ies for the systems water-ail-gas have been published. ever, in a nonvertical direction, the vectors (1/ p,) grad One flaw which has been common to most of these p and (1/ p,) grad p will be inclined from the vertical, multifluid experiments has been that the experimental and the corresponding equipressure surfaces will be in arrangements and their interpretation were usually based clined from the horizontal by the same amounts. At the upon the premise that the ,flowlines of the various interface every equipressure surface in one system must match that of the same value in the other. components are all parallel and in the direction - grad p. Since this is far from true, there is some ques At the. same time, the flowlines in each system must tion of the degree of reliability of the results of such be parallel to the interface so that the (E" E 2 )-plane experiments, Even so, the existing evidence indicates must be tangent to the interface. Consequently the in that for saturations greater than some critical minimum terfacial surface and the flow patterns in the two sys the flow obeys Darcy's law in the form given in Eqs. tems must mutually adjust themselves until these condi 108. For saturations less than this limit, there should tions are simultaneously satisfied before a steady state still be a general drift of discontinuous fluid elements in of flow becomes possible. the direction of the field E, but with, as yet, no well Of particular interest is the special case of this gen defined relationship between q and E. eral situation wherein one of the two fluids is in motion 237 VOL. 207, 1956 and the other is completely static. In this case the centennial occasion, we stated at the outset that it equipressure surfaces in the static fluid are horizontal should be our endeavor: and surfaces equal difference of pressure are equally 1. To show unequivocally what Darcy himself did spaced; whereas, in the moving fluid the equipressure and stated with respect to the relationship which now surfaces are inclined downward in the direction of the bears his name, and to give his results a more general, horizontal component of the flow. Their lines of inter but still equivalent, physical formulation. section at the interface must accordingly be horizontal. Also, since the vector, grad p, lies in the same vertical 2. To derive Darcy's law directly from the funda plane as the vector E of the flowing fluid, it follows mental Navier-Stokes equation of motion of viscous that the horizontal component of the flow direction fluids. must be parallel to the direction of steepest slope of the 3. To develop, in at least their primitive forms, the interface. principal field equations of the flow of fluids through If we consider a vertical plane parallel to the flow porous solids. direction and perpendicular to the interface, then in this These objectives have now been accomplished, and plane the surfaces of constant pressure will appear as the result is that, despite a number of troublesome lines of constant pressure, inclined in the flowing fluid complexities such as those arising from thermal con and refracting into the horizontal across the interface in vection and from waters of variable salinity, the field Downloaded from http://onepetro.org/TRANS/article-pdf/207/01/222/2176441/spe-749-g.pdf by guest on 01 October 2021 the static fluid. Let the flowing fluid be denoted by the theory of the flow of underground fluids is capable of I subscript 1 and the static fluid by the subscript 2. Then being brought into the same kind of a comprehensive along the interface in the direction of the flow unification as that already achieved for the more fa (op/os), = cap/os), . (l09) miliar phenomena of electrical and thermal conduction. In the static fluid In closing it is pertinent to reiterate that Darcy's (op/os), = - p,g sin e , (110) empirical formulation: and consequently depends only upon the density p, q = K (hI - h,)/l, and the angle of slope e, where e is positive when the which, as we have seen is valid for flow of a homogen slope is upward in the flow direction. eous liquid in any direction, is physically equivalent to In the flowing fluid, since the expressions: E = g, - (l/p,) (oP/os)" q = - (Nd') (p/p..) g grad h, } then q = + (Nd') (p/p..)[g - (l/p) grad p] . (op/os), = p,gs - p,E } From these it follows that fluids can and do flow in any (lll) = - p,g sin e + p, Igrad PETROLEUM TRANSACTIO~S, AIME REFERENCES 6. Corrsin, Stanley: "A Measure of the Area of a Homogeneous Random Surface in Space," Quart. 1. Darcy, Henry: Les Fontaines Publiques de la Ville Appl. Math. (1954-1955),12,404. de Dijon, Victor Dalmont, Paris (1856). 7. Brooks, C. S., and Purcell, W. R.: "Surface Area 2. Wyckoff, R. D., Botset, H. G., Muskat, M., and Measurements on Sedimentary Rocks," Trans. Reed, D. W.: "The Measurement of the Permeabil AIME (1952), 195,289. ity of Porous Media for HomogeneoJ.:.3 Fluids," 8. Klinkenberg, L. J.: 'The Permeability of Porous Rev. Sci. Instruments (1933),4,394. Media to Liquids and Gases," API Drill. and Prod. Prac. 1941 (1942),200. 3. A.P.I. Code No. 27, "Standard Procedure for De termining Permeability of Porous Media" (Tenta 9. Richards, L. A.: "Capillary Conduction ot Liquids tive), 1st Edition (Oct., 1935), American Petro Through Porous Mediums," Physics (1931), 1, leum Institute. 318. 4. Fancher, G. H., Lewis, J. A., and Barnes, K. B.: 10. Wyckoff, R. D., and Botset, H. G.: "The Flow of "Some Physical Characteristics of Oil Sands," Gas-Liquid Mixtures Through Unconsolidated Sands," Physics (1936), 7, 325. Pennsylvania State College Mineral Industries Ex periment Station Bulletin 12 (1933), 65. 11. Hassler, Gerald L., Rice, Raymond R., and Lee Downloaded from http://onepetro.org/TRANS/article-pdf/207/01/222/2176441/spe-749-g.pdf by guest on 01 October 2021 man, Erwin H.: "Investigations on the Recovery of 5. Reynolds, Osborne: "An Experimental Investiga Oil from Sandstones by Gas Drive," Trans. AIME tion of the Circumstances Which Determine (1936), 118, 116. Whether the Motion of Water Shall Be Direct or Sinuous and of the Law of Resistance in Parallel 12. Hubbert, M. King: "The Theory of Ground-Water Channels," Phi/os. Trans. Royal Soc. London Motion," Jour. Geol. (1940),48,785. (1883), 174, 935; or Papers on Mechanical and 13. Hubbert, M. King: "Entrapment of Petroleum Physical Subjects, University Press, Cambridge Under Hydrodynamic Conditions," Bull. AAPG (1901), Vol. II, 51. (1953),37, 1954-2026. *** 239 VOL. 207, 1956