Application of the Flow Net to Percolation Citation Franzini, Joseph Bernard, Jr. (1944) Application of the Flow Net to Percolation. Engineer's thesis, California Institute of Technology. doi:10.7907/2m77-cx83. https://resolver.caltech.edu/CaltechTHESIS:04152025-215456494 Abstract No abstract. Item Type: Thesis (Engineer's thesis) Subject Keywords: (Civil Engineering) Degree Grantor: California Institute of Technology Division: Engineering and Applied Science Major Option: Civil Engineering Thesis Availability: Public (worldwide access) Research Advisor(s): Unknown, Unknown Thesis Committee: Unknown, Unknown Defense Date: 1 January 1944 Record Number: CaltechTHESIS:04152025-215456494 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:04152025-215456494 DOI: 10.7907/2m77-cx83 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 17156 Collection: CaltechTHESIS Deposited By: Benjamin Perez Deposited On: 21 Apr 2025 18:52 Last Modified: 21 Apr 2025 18:53 Thesis Files PDF - Final Version See Usage Policy. 11MB Repository Staff Only: item control page

APPLICATION OF THE FLOW NET TO PERCOLATION THESIS BY: Joseph B. Franzini, Jr. Submitted as Partial Fulfillment of the Requirement,s for the Degree of Civil Engineer CALIFORNIA INSTITUTE OF TECHNOLOGY PASADENA, CALIFORNIA 1944 SYNOPSIS This paper gives a comprehensive treatment of the fundamental laws. of percolation. The problem is attacked from the analytical po.int of view and the flow-net is used to solve illustrative problems. Darcy's Equation for flow is developed, its applications and limitations are pointed out. following special eases are taken up: 'l'he aelotropic soils, variations in soil strata, and two-fluid systems. In addition, problems involving the sand filter, under-drainage , and lateral drainage are worked out. The procedures followed in solving these simpler problems may be extended to the more complicated ones of ground-water movement and the oil-production capacity of wells. It is hoped that this paper will bring to light some new ideas on the manner in which percolation problems may be analyzed and solved. ·coNTENTS Page I. Introduction II. Characteristics of Percolation A. Darcy 1 s ,Equation 3 B. The Coefficient of Permeability 7 C• Limitations 11 III. Th'? Flow Net IV. Speeial Cases , v. VI. 1 16 A. Aelotropic Soils B. Varia tior..s in Soil Strata 27 C~ Two-Fluid Systems 31 ' 25 Applications A. - Under-Drainage 34 B. Seepage through Dams 34 c. The Sand Filter 37 D. Lateral Drainage 39 Conclusions 41 Notation 42 Bibliography 44 I. Introduction Percolation is defined as the flow of liquid through a porous medium. In Civil Engineering, it refers to the movement of water through an earth mass. As percolation takes place, the water passes through the interstices between the individual soil particles. These openings, the pore spaces of the soil, may be regarded as a series of more or less tubular passages through which the water flows. Ordinarily it is assumed that the soil particles remain fixed in position. The flow may be either laminar or turbulent depending upon the conditions under, which it occurs. The resistance to flow of soil whose particles are smaller than gravel is sufficient to prevent turbulent flow, so that in the usual case of percolation, the flow is laminar. Permeability is the measure of a soil's capacity to permit percolation. Thus, a permeable soil is one through which water will pass readily, while an impermeable soil retards the flow. The rate of percolation is of importance in many engineering problems. For example, ground-water movement may be forecast if adequate data on geologic sub-strata can be secured. (1) The design of sand-filters, earth-dams, dykes, and other hydraulic structures is influenced by percolation. Permeable soils are pr·e ferred for foundations and highway sub-grades where adequate drainage is required. Impermeable soils are utilized in earth-dams and dykes where they prevent the leakage of water. (2) II. Characteristics of Percolation A. Darcy's Equation To get a picture of the flow of water through an earth mass, consider an horizontal tube filled - with an isotropic homogeneous soil, and joining two tanks of water of unequal surf"ace elevations as shown in Figurer, page 4. If the particles of soil are relatively small, and if the pressure differential is not excessive, laminar· flow will take place through the interstices. The energy equation for hydraulic flow applied between sections (1) and (2) gives: (r.1u,. + ~~ 1- t) -hf -=(PX,..+~ +f,.) * f/U)" = P~/u,. = O ; since the liquid sur:faces are in contact with the atmosphere, the gage pressures are zero. v.2- I "2, Vi_ , a l<L = .:2-a =O ; assuming the crosssectional area of the tanks is relatively large. ri'herefore: 1• -hf = }:l.. Or: hf= (}i.--r~) = Llh * The letter symbols used in this· paper are defined on page 42. • They conform essentially to American Standard Letter Symbols for Hydraulics. (3) Fi ;ru r e No. I 11 Flow of wa t e r t hr ough soil 11 (1) (Z) DATUM PLANE (Eff ective ve l ocity is defined as the r ate of diecharge q d i vide d b·r t h e cross - se ct iorn;.J. are a of t he soi l, A.) • Ener·2;y Equa t i on : ✓, O -1/ 0 -1/0 . //o • (¾+~ +t0 - ht = (¾-+ ~ +}✓ 1, - hr- =- )1-i.. hf ~ (;l' - 7u1-) hf-~ ~"'- The preceding expression indicates that the velocity of flow through the medium will be of such a magnitude that the loss of energy head will be equal to the pressure different,ial ac:.r oss· the soil. This loss of head represents loss of energy due to the resistance to flow offered by the particles of soil. This is similar to pipe friction of ordinary flow in pipe-lines. From hydraulics, it is known that the energy head loss due to friction "'' - .h...Y i~-:;- hf-f 4 m 2.~ ,.,,, in which f is the friction factor, L the length of flow, ~ ' ~ qm~the hydraulic radius, and V the average velocity of flow. In the case of laminar flow, the frie:tion factor f=~= ::.v.P where "c'' is a cons,tant depending upon the size, distribution, and compaction of the soil grains,µ is the coefficient of absolute viscosity of the percolate, and pis the density of the percolate. 11 R'' is Reynold I s_ number, the ratio of the inertia forces to the viscosity forces anting on the water. * The letter symbols used in this paper are defined on page 4 , . They conform essentially to Amerio.an Standard Letter Symbols for Hydraulics. (5) Therefore for laminar flow: L v-.. c.,u.. L _ vz. _ h.f =f~ 2t = rnVP 4-tn cµ. and 6' If 11 ki 1 7 m2.}' ~i - • L• V = A h 8q..m-z. f is defined as: Then, V= \= k f Finally; V= Je,,C I k, = ---4+,-1--c)A. where "i" is the hydraulic gradi~t, that is, the loss in head per length of flow. ']his expression is known as the Darcy Equation. It states that the effective velocity of flow 11 V11 of' a liquid through a porous medium is equal to the product of "k", the coe.f•f icient of permeability of the soil, and 11 1 11 , the hydraulic gradie.nt. The effective velocity 11 V11 in the Darcy Equation is much smaller than the actual velocity of movement Vp of the moisture through the soil pores. where "n" is the porosity of the soil. Vp = V n , Since we are interested in the quantity of liquid which passes through the soil per unit of time, we are coneerned only with 11 V11 , the effective velocity of flow. The rate of discharg e through a cross-sectional area of soil 11 A11 , is given by: • B. The Coefficient of Permeability Before g oing any further, it should be noted from the Darcy Equation, that since"i", the hydraulic gradient is an abstract number, the coefficient "k 11 has the dimensions of velocity. Therefore, the coefficient of permeabil- ity "k" of a given soil actually represents the effective velocity at which water will pass through the soil under the action of an hydraulic gradient of unity (i ; 1 ) • From the analysis in the preceding section it is seen that the coefficient of permeability e<lm.~Jl ~- cp- . This indicates that the coefficient • 1... _ of permeability varies inversely withP, the coefficient of absolute viscosity of the water, and also that it depends upon the characteristics of the .soil, such as distribution, shape, size of soil grains, and degree of compaction. Beeause of the variables involved, an analytical derivation of the functional relations.hip between 11 k 11 and the characteristics of the soil is quite impossible. The par•ticles of two distinct soils for example, may have the same "effective size" but varying (7) characteristics due to the different "uniformity coefficients 11 , or viee versa. In view of this, the coefficient of permeability of a given soil is usually foung by experimental means. Several devices have been developed to determine 11 k". Excellent descriptions of these, the constant head permeameter and the variable head permeameter, are given by Krynine 1 in his text: "Soil Mechanics 11 • In order to get good results in determiuing the coefficient of permeability 11 k 11 , great care must be used in the field to secure soil samples which are totally undisturbed. It is of interest to note that various investigators have attempted to establish an empirical relationship between the coefficient of permeability and the soil characteristics. Charles Slichter 2 , has perhaps done more work in this field than any other man. He stat,ed permeability as a function of porosity, effective grain size, and temperature of the percolate. His expression does not give as good values for rr1-crr as- are secured by experimental means however, 2 - - Charles Slichter; 11 11/lotion of Underground w-aters n water Supply Paper, No. 67, u. S. Geolog ical Survey, U.S. Department of Interior. (8) because of the lack of uniformity present under actual conditions in the field. Some represent- ative values of the coefficient of permeability "k" are given in the following table: Table I Kind of Soil Coefficient of Permeability (Feet per day) Fine Gravel 4,000 Coarse Sand 600 Fine Sand 50 As already stated, the coefficient 11 k 11 varies inversely with JJ- , the coefficient of absolute viscosity of the liquid, which depends upon the temperature of the percolate. For example, water at 70 de grees Fahrenheit will percolate through soil fifty per cent faster than will water at 40 degrees Fahrenheit. Figure II, page 10 shows the plot of 11 k 11 vs. temperature of water. Values of µ. , the absolute viscosity coefficient of v1ater, which were used in computing "k n, were taken from 3 curves g iven in Daugherty's "Hydraulics 11 • (9) F i p:ure No• II Curv e show ing varia~ion -Of the coefficient of p8rrneab i lity with temr,e ratu re of the percolate. ,,...._ 600 T> d ~ . ~- 0... .µ Q) Q) 400 LI.. '-.-1 ~ ioo ' 40 60 'l' emperatu :re ; ~;} 80 100 De gr e es Pahrenheit Data for }J--, t he coe fficient of abs o l u te viscosity of the percolate secured from curves g iven in Dau 6 herty' s (10) 11 Hydraul i~s n. c. Limitations It should be kept in mind that in the foreg oing analysis ca pillary forces, which are of small magnitude, have been neg lected. In addition, no consideration has been g iven to the attached ground wa ter. The mo.isture which adheres to the walls of the op~nings is known as attached ground wa ter; its effect on percola tion is of minor consequence. There are definite limits to the applic:ability of Darcy's Equation. Experi- me ntal work in percolation by several investigators indica tes tha t Darcy 's Equation g ives satis ~aetory results at extremely low values of hydraulic gradient, but at high values, (i) 1), the results are inconsistent. This is accounted :for by the fact that whe n "i " - is large, the velocity may become great enough to create turbulent flow. When the flow is turbulent, the Darcy Equation is not valid, for it was derived on the bas,is of laminar flow. The criterion for the type of flow de pends on (11) the value of Reynold's number. In ordinary pipe flow, it will be reealled :from hyd.raulie:s, that the flow is laminar when R is less than 2000, otherwise the flow is turbulent. A Similar condition exists in the ease of percolation. Various men have found that the critical Reynold's number for percolation is . d,..V-p 10, where R is defined as R= µ, d being the . average grain diameter of the soil. Previously it has been stated that the loss in head due to flow through pipe-lines or L \J' percolation through soil is g iven by htf4""- 2.~• If 11 f 11 , the friction factor, as defined above, is plotted against R, Reynold 1 s number for various soils we get a series of curves similar to those of Kemler and Piggott shovm in Daugherty's 11 Hydraulics 11 • 3 Figure III, page 13 is a reproduction of "friction-factor" curves for percolation g iven by Muskat 4 which demonstrate the similarity between percolation and flow in pipe-lines. Data gathered by Tolman 5 .shows that 4 - - Muska t: "The Flow of Homogeneous Fluids 11 5 - - Tolman: 11 Ground :rater 11 (12) Fi?,ur e No. III 11 Fricti on - Factor Cur v e 0 (Pl ot of f vs . R) $"00 400 300 2.00 f 100 so 10 100 1000 ~he bre a k in the straight-line port ion 0f the curv e occurs in the vicin ity of R = 10 . Th i s ind icate s t ha t if R i s grea ter t han lQ, the flow is t u rbu lent. (1 3 ) turbulent flow occurs through soils only where the soil particles are of large diameter or where the hydraulic gradient is excessive. The following table takeri from Tolman's "Ground water" gives values of critic-al velocities in various types of soil. Table II Porosity, per cent * Grain Size in mm. 0.2 1.0 10.0 25 65.,400 13.,100 1.,300 30 50.,800 10,200 1.,000 35 40.,500 8,100 800 40 32.,700 6,500 700 * Velocities are in feet per day If the velocity of flow through soil is less than that shown in the above table, the flow will be laminar and Darcy's Equation will apply. Our discussion up to this point has been concerned only with homogeneous isotropic soils. As long as the soil particles are of regular shape and uniformily distributed., the permeabil- (14) ity factor 11 k 11 will be independent of direction. Appreciable stratification however, or the presenc·e of flat grains, deposited in a parallel fashion, will result in a lower resistance to ' flow in one direction than in another. This gives rise to speeial eases which will be eonsidered later. In nature, we also run across instance.---s where there are layers of different soils of various permeabilities. This problem will also be taken up in a later section of this paper. (15) TII. The Flow Net The- hydraulic flo w-net is a graphical representation of the direction of flow and of the potential head available at any point within a flow system. It consists of flow lines and contours of equal potential superimposed on the cross-section through which the flow is taking place. In the case of steady two-dimensional fl.ow, two families of curves may be derived mathematically 6 by means of the continuity equation written as a differential equation. Mathematical solution indicates that. these families of curves intersect each other at right angles. There are actually an infinite number of flow lines and equipotential lines in a flow system. It is convenient, however, to choose only a limited number of these lines such that the rate of flow dq between each pair of flow lines is equal, and such that the potential drop dh between successive potential lines is the same. The interval between the flow lines is made equal to the interval between _the potential lines thus forming a series of 6 - - Hunter Rouse: "Fluid Mechanics" (16) squares. In the vicinity of' a bend, the squares will be distorted but they will become more nearly perfect as the number of lines is increased. Mathematically speaking, they l;:l.pproach true squares as they become infinitesimal in size. The flow-net is usually constructed by free-hand sketching, with gradual adjustment and correction until the flow lines and potential lines "intersect at right angles and form curvilinear squares. From the principle of continuity as applied between eaeh pair of flow lines, the velocity must .vary inversely with the spacing. Where the flow is in a curved path the position of the flow lines may be adjtjsted and checked by the relationship that ( ~ =~) 6 in which IIY is the velocity at any point and ris the radius of curvature of the flow line at that. point. A graphical method of checking the spacing of the flow-lines is described in a paper by H. Alden Foster 7 in the May 1944 issue of "Proceedings of the American Society of Civil Engineers". 6 - - Hunter Rouse: 11 Fluid Mechanics 11 (17) It will now be sho v,rn how the r a te of percolation can be computed from the flo w-net . Fie;ure IV,page 19 represents a portion of a correctly constructed flo w-ne t for the c-.ase of flow through soil. If the distance between the lines of equal potential is dL, the hydraulic g radient is ,_ =-£~, where dh. is the head loss betwBen succesaive potential lines. For a unit width, the area through which the f lovv takes place would be l il!ldD where dD is the distance between flow-lines . The rate of flo w between any t wo flow- lines as g iven by the Darcy Equation is dt=- A~L=:cl.Dit~~ where is the coefficient of permeabi lity. 11 k 11 Since the head loss from one potentia l line to another is equal, the drop dh from one to another is g iven by dh=- ~ where h is the total potential drop p through the medium under consi deration and hp is the number of increments into which the total potentia l dr op is divided . Subst i tuting this value for dh in the previous e qua tion, the h. follo wing expression results: d\-== d.Dic. k nr d"'vihere d q is the r a te of flo 1.,v between any t wo successive flo w lines. (18) By construction, dD Fi gure rv (Flow Net and Dis charge) ~A.. "" di.. clA I h °'d..J..."' d.h.. ,1.,=- - - d.L • ' ~D: d.L J BY CONSTRUCTION has been made equal to dL, therefore~~~h._ p The total percolation through the medium under consideration would then be i=~sdiwhere n. 5 is the number of' spaces between the flow lines. • J.S The rate of total percolation therefore g 1.:.-wen b y: a.= ns ..![ d.0, If = le.rt l'ts ne • This equation may be used whenever sufficient information is available for the construction of a flow net. The accuracy of the results increases with the number of divisions in the flow net. As long as the flow remains laminar, the same flow-net applies equally will for any rate of discharge, because the pattern of the flow lines depends not upon the velocities, but upon the geometry of the flow boundaries. It has long be en known that percolation through soil is analagous to the propagation of electricity. By virtue of this, percolation problems can be studied by electrical means. In studying percolation by the electric analogy, a model is constructedo The apparatus usually consists of a shallow tray with plate-glass bottom and sides, filled to a depth of one inch (20) or less with an electrolyte. Copper plates under different potentials are used to represent the "soureerr and i'sink" of the f'low system. When the current passes from one terminal to the other, the trajectories described are exactly the same as streamlines in the case of percolation. By use of proper electrical equipment, the equipotential lines may be determined and the resulting flow net may then be constructed. 'I'he applicability of the electric analogy may be verified by comparing Ohm's law with the Darcy Equation. Ohm's law expresses the fundamental relation for the flow of electric current while Darcy's equation gives the fundamental relation f0r percolation. Ohm's Law expressed by an equation isl"'~ in which., I I =- current in amperes., a.measure of quantity of f'low / E = pressure in volts., and R = resistance in ohms. This equation is in terms of resistance. The Darcy Equation on the other hand., is in terms of conductance. Since electrical conductance, K', (21) is the reciprocal of resistance, Ohm's law may also be written I= EK'. The conductance K 1 varie.s directly with specific. conductivity k 1 , with the area A, and inversely with the l I /\ length L, then K =Ii 1: • 1-' f\ f:: Therefore 1 = N- \... ~ • If I is the quantity of flow, k' the coeffieient of flow, Ethe potential, L the length of flow, and A the area, this equation is ananagous to the Darcy Equation. That is, <1, = P\ ~;., (Darcy Equation) I= A~lt) (Ohm's Law) In addition to the electric analogy technique, models are used to study percolation problems. By this method a glass-walled soil model is constructed of the system to be studied, and the flow-lines are determined by introducing dye at various points in the system. The soil used in the model is usually many times more permeable than the soil of the prototype since in the prototype the velocities are usually extremely low. All discussion up to this point has dealt with two dimensional flow only, but all statements made so far apply (22) equally well to three dimensional flow. Figure V, page 24 shows the plot o:f a flow-net for percolation under a dam. Beneath the diagram are shown computations for determining the ra t ,e. of percolation under the dam by use of the Darcy Equation. The soil sub-layer has been assumed. to be homogeneous and isotropic:, meaning that the water will flow equally well in all directions. (23) Figure No. V Flow Net for Percolation Under a Dam wit h Computat i ons for Rate of Percolatione I I I I' ,, , I ,j > I I I >f j ~ I MPl:RVIOVS I I > >t ' ;t > I > I ; '---+ r . \\ H I ,,, ,, l I I I \ I 1 I ~ I } I I I I ;;;; > ~ > ST/(/.Arl/ s_J Yl5_= ( 11 p =IS Su?.POS£ : k = 200 h = 4-0 ~ G G7 FT. .I>AY FT Cl/15/C ,PE~ ( !?.4 ) PER FEET P~ll. .DIIY, FOOT OF IIWDTH TV. Special Cases A. Aelotropic Soils Aelotropic soils are those which have a lower resistance to flow in one direction than in another. Such a condition is found in nature where considerable stratification has occured or where the soil consists of · flat grains deposited in a ,parallel fashion. The permeability of a. soil in the horizontal direction kx is oftentimes five or ten times greater than that in the vertical direction kr• Conditions of this sort require a modification of the flow net, merely a linear distortion. Samsioe, a Swedish scientist, proposed that the horizontal dimensions be reduced by the factoriE The procedure for constructing such a flow-net is to first plot the flow-net- for the reduced structure and then to plot the vertical ordinates of this flow-net under the corresponding points of the natural-shaped, nonreduced struc:ture. On the bas is of such a flow-net, the rate of discharge is computed by means of the following expression: . \' = It ~r fk.J,; Figure VI, page 26, shows the procedure for such a problem. (25) F~gure No. VI Given: Treatment for Aelotr op ic Soils le):, = 4 Je.1' {It%"' = 1/2. Reduced Scale Natu ra l Scale i .~h ~ iit.~ = ~ ~J-,;:~ s B. Variations in Soil Strata There are instances in which a soil mass is composed of successive layers of material which have different coefficients of permeability. Let us consider first the case in which the flow is parallel to the bedding planes. Figure VIIa, page 29 shows a diagram for such a condition. Let k,, k 1 , -----lc"'- be the coefficients of permeability of the individual layers, and let, B, , B 1 , -----B"' be their respective thicknesses. Also let k 1 represent the average coefficient of permeability parallel to the bedding planes. For such a case, the hydraulic gradient i must be the same everywhere since the same potential drop occurs over identical lengths in each layer. ·The discharge per unit width through any layer is given by: q = A (V) = B (ki) where B is the thickness of the layer, kits coefficient of permeability, and i the hydraulic gradient common to all layers defined as ( i = t ). The total rate of dis charge through the mass would be: ~ T = JJ, k, i, -t B2,le2, [,,~ t- - - - - B"" k h . i.'1. And the average discharge velocity, is .B,k,i., -1- 152 .k~ i2- -t ---- Bl-tie,,,_ in , - - -B,-+B2. - -+ = k ~ -----.8"'Con~uently, the average coefficient of permeability parallel to the bedding planes k 1 is given by: k I == ~le,+ Bz.le2...,.. -- - ~ .811,kn_ ..B, t- B 2. ..,. - - - .l3;z. (27) A second case to be considered is one in which the flow is at right angles to the bedding planes. Figure VIIb, page 29 shows a diagram for such a condition. Let k" represent the average coefficient of permeability at right ang les to the bedding planes. Let i, , i,, -----in represent the hydrauliC' gradients in the different layers and let h,, hz, -----h~ represent the potential head losses through the various layers. For this case, the discharge velocity v must be everywhere the same since the flow is continuous. The discharge velocity in any layer· is g iven by: v = ki. But as stated above, -----k 11.i k • Now v And = le, i., B, +5'2..+---- B"' h = i.,B, +L,.B'l.-+ - - - - L.., B"" Therefore; /1.l'(EB) 2 h. = Je.;_ (B,"t8 1 +-·-B1i) = l, B, + L,.l\_ -t --- {,,., 8 11 .B,/. /k -,. • 1 e t us cons 1.· der the Asano th er i• 11us t ra t ion, ' 13:,./ 7k -t --,_ ease of flow through the earth-mass shown in Figure VIII, page 30. In this case, the flow takes place in a direction that is essentially parallel to the bedding plane between the t -v:10 layers considered as such. One of these layers is homogeneous, while the other, is composed of two different mat.erials. The construction of the flow-net for this case indicates that. if the flow were unr·e stricted , there would be a slight horizont.al drift to the flow. (28) ~low Parallel to Bedding Planes Figure No • VI Ia. r i i i ~ / L V / / / / ' t • ® k, / / / kz ,/ I ; I • - •, . .. ' ... . . . . ' / / _' / .I • / _,,,,--- \ \ '. / / I I ,,,,✓-: / (!) / / / .. .. . . . . .-. ' . . ' .. ..- =-· . .<. .. .' . ... .... @"3' . . . . . ... .. . - . .. ... -. . ...... ·k :' . . " ,J .' . . ~ :' . . ... . . ~'; / / i l _;, : . --.. ' .., • J,, ~- . . .. - / / .•,... ..,.,.,, I / B.3 ---1---1/ k -t- ----.B,..,le n. .13,k, + l32.Z. 8 .,. 8 A ...,.. - - -.B 1-1 I F i hd,7ure No • VI Ib Flow at Right An6 les to Bedding Planes ! ! ! t ..B, (Z) •,• .,_ • • • ':' © • • ~:l, , • t , I ' •' • • ~ • . • _'. ·/·:•_·': << -~ ·, \ .: .. / •..---~, ·:\.:. . ··.-.'_<:r·.· :,~:: ~ ( 2 9) Anaiysis of Complex System Figure No. VIII The su perimposed flow-net demonstraies the horizo n tal drift i m1)osed on the flow. ! l l l ~t----t---11 ~I = 1-e'.' = ~. L.., t-lz.. L/1 -t- l1/i It, ~= kL B,~1 t-B~k . .B, -r B2.. I 1=A1ti. -== (B, + l:>?.-) ~ ~,,_ (30) ~ L. -t-L,. c. Two Fluid Systems The occurence of ground waters in the vicinity of oil-bearing sands gives rise to a problem which involves two liquids of different viscosities. This problem is of great concern in the field o.f oil-production technology. Sufficient data and proper analysis will enable one to predict the effect of the movements of ground-waters on the oilproduction capacity of nearby wells. a very simple case of 11 In this section water-eneroachment 11 shall be considered merely to show the general technique for analyzing such a problem. Let us consider a portion of a homogeneous medium through which two liquids of different viscosities are flowing. Figure YX, page 33. See To simplify this illustration let us assume that the potential drop across the soil remains constant as the interface between the two liquids passes through it. The time required for the interface to pass through the sand may be found as follows: Let kw represent the coefficient of permeability of the sand in the regions where water is present. Let lc 0 represent the coefficient of permeability of sand in the regions where oil is present. (31) From previous discussion in this paper, (See page 7), it will be recalled that the coefficient of permeability k is inversely proportional to the coefficient of absolute viscosity of the percolateµ; that is, L_ 8inl"J> _ _c Kc,µ. - µ. where C is a constant depending on Hence lrtw =~wand the properties of the soil. ~o-=- ~ 0 where JJ-w and }lo are the coefficients of absolute viscosity of the sand with water and with oil respectively. It will be : recognized that this condition, ,. the flow of two fluids through andhomogeneous soil is analagous to the flow of one fluid through layers of different soils at right ang.les to the bedding planes. (Discussed on page 28). Consequently the effective coefficient of permeability k 111 for this two fluid . . b s ys t em is given y: : -X/it. L+ (L-Y-L -- --...--L---r--~---t.,j.1, -t (L-~) t':2. I ~~ C C Refer to Figure IX, page 33. k"' =- __c_•L _______ Now -y., J-A- I + l- }t 2.. - 1l M:t. But V = k'." iv = ~' h./L klll J-t_ = 1\)- ott a... t; = ( cL-t- = 1l,M, +Lti:z.. - Y-f-J.,. J J ,o- Finally o ~ ( C,..h.. ' - t == ~ch. ~.+ f<..1-1 - ~ L ( ,M, + M2.) Z( ( hfL) This gives the time required; for the interfae:e to pass through the soil mass. Note that the time given by the above expression for an homogeneous soil, (JJ.-, = J-1.l , 1s : (Y;, t = CL Fio-ur e No . IX g 1\~o Fluid System for homogene ous soil and constant gradient. l l ---l ----------------- ------- 0 __,,,... ~ ----- ---- ~ ------- --------- ------- --------- ------k=£ -,,,U, ------------ ----- ----- ---- - -----REGION I ---: ----- /_./ I ----- __,,,... --- ----- INTERF,4CE ---- - --------- ------ _:.--- ---h., L z A!= _f_ REG/ON 2 ~2. - ,-m l l l The time :required for the inte r face to pass throug h the soil mas s is g iven by: " ~ v. Applications A. Under-Drainage 'rhe flow-net may be applied to various types, of percolation problems. In this section, it shall be applied to the underdrain shown ln Figure x, page 35. The flow-net for this ease is super-imposed on the section and computations for the rate of drainage are shown · below the diagram. The shape of the flow-net for this case depends on the relationship between a, the width of the ba~in1.and b, the distance the drain is located beneath the soil surface. Figure XI, page 36 shows the effect of varying the ratio of a to b. Where bis small, more drainage tends to take plaee from the C'entral portions of the bas in, while a large value of' b tends to induce practically uniform drainage over the basin area. • B. Seepage through Dams The solution of the very important percolation problem of seepage through earth dams is greatly facilitated by construction of the flow-net. The general procedure is quite similar to that shown in Figure V, page 24 for seepage beneath a dam. When the section of the dam is composed of materials of different permeabilities the flow-net must be adjusted by methods already discussed. (34) FifTure Ho. X Flow-Net for t he Under-D:raln T i, 1 1e. = 100 FEET PER DAY Computations for Rate of Drainage : 240 cubic feet per daj per foot. Fi gryure No. XI . Changes i n the Flow-Net induced by varj_ations of U 1.e ratio b/a. i - - - - - - - - - - - - ~ a... H,-.0 .Shallow Drain . - . - - - - - - - - - - - - - - a. - - - - - - --- - - - - T l b Deep Drain C. The Sand Filter As another interesting illustration of percolation supposing we wish to find the rate of discharge through a sand filter whose dimensions are twenty feet by twenty feet. Let it be composed of three layers of different material as shown in Figure XII, page 38 . . The thickness B, of the first layer is two feet, its coefficient of permeability k, is fifty feet per da·y . The second layer is six feet. thick and has a coefficient of permeability k~ of two hundred feet per day. Let the bottom layer be two feet t h ick composed of fine gravel with a coefficient of permeability of one .tfi6t:i§l:trui feet per day. According to the principles given in section TV B, page 28, the e.---i'fee:tive permeability k" for flow at right angles to the planes is: k"= and 10 .07 '1.. - FEET rER 'l>AY 138 v = k .. i V Therefore = k" i - 1'38 ( ~ ) = SS: 2... q = A(v) _ 20(20)(55.2) _ 22,080 cubic feet per day. In instances of this sort where the geometry is extremely simple and symmetrical, the flow-net does not facilitate the computations. 'I'hG Sand Fil tcr Ffr:ure No. 12 Refer to Page 37 !-- - = ---- - __-_-_ __-_ ___ ---==~--====-'= --= -ih.= 4' Depth of Hater· , h _, assumed to be constant. ,, k_ = 13.-8 FT/DAY X = 4/io (38) D. Lateral Drainage As a final illustration of the application of the flow-net to percolation let us conside:,r briefly the case of lateral drainage pictured in Figure XIII, page 40. Z2. Supposing k = 200 feet per day, z,-= 40 feet, = 10 feet, L = 60 feet, w = 100 feet,. The rate at which drainage originally occurs may be readily computed from the flow-net as follows: o_ = ,kz. .!'!L = 200 (40} : ::::. -§..3.33 CUBIC FEET l'E~ .DAY {T np I To determine the time required for the watersurface in the reservoir to drop from elevation 40 feet to 10 feet, a simple integration is required as follows: In any element of time dt, let the level of the reservoir drop the distance dz. The hydraulic gradient i = z/L. The quantity of water entering the face of the soil bank in any interval of time dt is given by: dQ:::: /}ki dr = Z k 1 t:/L But in the time dt the volume of water lost from the reservoir is: dQ = -ttr d~ 1 Therefore; -ur de = From which t ~ - w-Ljl, d~ h '2z. ;e Evaluating t == urL (Z, -22 ) k.. .a2. g 2, 22. And finally, t= (39) /OO(<.o} Jo 2 00 ,40 (/oj = · .;e..~ -,,_I ;DAYS Fi 9n·ur e No~ XII I Lat er a l Dra ina ge L ~ -====-·~-====--=--,--____ Z, rt __ R_ ESEllVOIR. Z: IMP£RVIOVS VI. Conclusions The preceding discussion demonstra~es that the flow-net when properly applied, may greatly facilitate the solving of the more complicated problems of pereolation. Several special cases and particular applieations have been considered to point out the procedures that must be resorted to in order to solve the various complex problems that are confronted in the field. These may be extended to the more complicated problems of ground-water movement, influent seepage, and the oil-production capacity of wells. It is hoped that this paper clarifies some of the principles of percolation and that it also presents some new ideas in the methods by which percolation problems may be solved. (41) NOTATION The following letter symbols, used in this paper, conf'orm essentially to American Standard Letter Symbols for Hydraulics. A = cross-sec·tional area of sec·tion through which flow takes plac"e. B = thic·kness of layer through which flow occurs. = a constant. f = hydraulic friction-factor. c g -= h = potential head. hf= i acceleration of gravity. head loss due to frictional res·istance. = hydraulic gradient, loss in head per unit of length. = coefficient of permeability. L = length of flow path. k m = hydraulic radius, ratio of cross-sectional area to wetted perimeter. n = porosity of a soil. n p = number of spaces between equipotential lines in a given flow net. n s = number of spaces between flow lines in a given flow net. p .. hJdrostatie: pressure. q = rate of dis charge . (42) R Reynold's Number, the ratio of inertia forces to viscosity forces acting on a moving liquid . .I' = density of a liquid. __µ. = coeffieient of absolute viscosity of a liquid. = average effective velocity of flow. vP = actual veloe·ity of liquid through the soil v pores. w = weight per uni~ volume of a liquie.. z = elevation head. (43) BIBLIOGRAPHY 11 1 - D.P.Krynine: Soil Mechanics", McGraw-Hill Book Company, Inc., New York, 1941 2 - Charles Slichter: "Motion of Underground waters 11 , Water Supply Paper, No. 67, U .S .Geological Survey, U.S.Department of Interior "Hydraulics 11 , 3 - R. L.Daugherty: McGraw-Hill Book Company, Inc. 11 New York 11 1937 4 - M. Muskat: "The Flow of Homogeneous Fluids through Porous Media", McGraw-Hill Book Company, Inc. 11 New Yol!'k, 1937 5 - C .F. Tolman: 11 Ground water", Mc:Graw-Hill Book Company, Inc., New York, 1937 6 - Hunter Reuse: "Fluid Mechanics for Hydraulic Engineers 11 , McGraw-Hill Book Company, Inc., New York, 1938 7 - H. Alden Foster: "Construction of the Flow Net for Hydraulic Design", Proceedings of the American Society of Civil Engineers·, May 1944 (44)