Flow of Simple Fluids through Porous Materials GEORGEH . FANCHER AKD JAMES A. LEWIS Mineral Industries Experiment Station, The Pennsylvania State College, State College, Pa.
T
H E mechanism of t h e
A reliable method for measuring and studying
I’REYIOCS
RESEARCH
The hydraulics of fluid flow flow of fluids t h r o u g h the flow of fluids through permeable media is t h r o u g h c i r c u l a r and open p o r o u s m a t e r i a l s inpresented. Crude petroleum, water, and air are conduits have been s t u d i e d volves many problems vital to passed through seaeral arrangements of unconby many i n v e s t i g a t o r s and science and industry, particusolidated sands a n d samples f r o m m a n y sandare well understood. The work larly to the petroleum industry. stones. T h e latter are cores f r o m producing was r e v i e we d and correlated T h i s i s e s p e c i a l l y true in by W i l s o n , M c A d a m s , and Pennsylvania where secondary horizons in oil wells. The porosity and screen (14). These investigaSeltzer methods of recovery of petroanalysis of each sample are determined. The tors applied the w e l l - k n o w n leum-i. e., water flooding and data are used in the construction of a friction Fanning equation t o the data air and gas repressuring-are factor chart in terms of the properties of the fluids and plotted the friction factor, f, practiced extensively. O w i n g and the sands. The conclusions are that the flow against the Reynolds function to t h e s e m e t h o d s and to the Dus/z o n logarithmic g r a p h general recognition of the high of fluids through these porous materials closely paper and obtained the now quality of lubricants made from resembles that through pipes; that there is a familiar friction chart. P e n n s y l v a n i a - g r a d e crude condition of flow in porous systems which reChilton a n d C o l b u r n (7) oil, the petroleum industry of sembles viscous flow, another which corresponds recently made a study of the the Appalachian region today to turbulent: that the change f r o m one type to pressure drop r e s u 1t i n g when is in a flourishing condition as gas was p a s s e d through tubes far as production is concerned. the other takes place at a dejnite and repropacked with s o l i d m a t e r i a l Xatural water flooding was ducible condition for each system; and that the varying in diameter from 0.1 described by C a r l 1 (5) in 1880 permeability and conditions of flow for a n y to 1.0 inch (0.254 t o 2.54 cm.), but its importance was ignored simple fluid through consolidated and unand c o r r e l a t e d t h e i r results until r e c e n t l y 1 w h e n it h a s consolidated sands can be approximated by use with those a v a i l a b l e in the become common p r a c t i c e to literature. T h e s e data were aid nature by t h e f o r c i b l e of the relationships which are found. dotted as a f r i c t i o n factor i n j e c t i o n of w a t e r t o the igainst the modulus dup/p on producing horizon. Naturally it is essential that maximum energy be put in the oil stratum logarithmic graph paper. In this modulus u is the velocity and maximum oil be removed thereby as rapidly, as efficiently, based on gross cross-sectional area, and d the diameter of and as economically as possible. An understanding of the the packing material. A curve resembling that of the usual mechanism of flow involred obviously is of practical aid in friction factor chart for flow in pipes was so obtained. A the solution of many of the problems connected with these definite break in the curve as viscous flow changes to turbulent was indicated. The slope of the curve in the region of viscous processes. This report is concerned solely with the mechanics of simple flow was - 1 as required by theory. A reasonable hypothesis fluid flow through porous media of which oil-producing concerning the mechanism of flow through such systems was strata are examples. The experimental work has been carried advanced. on for the past three years in the petroleum research laboraA plot of deup/z against the friction factor of Fanning’s tory of the School of Mineral Industries a t this college in equation for the flow of air and natural gas through several conjunction with a study of the permeability of consolidated grades of unconsolidated sand and four sizes of lead shot was sandstone samples secured from oil well cores. The data presented by Chalmers, Taliaferro, and Rawlins (6). In so obtained have been combined with those available in this case de was defined as the mean effective pore diameter, a the literature, as well as some secured in this laboratory, hypothetical quantity equivalent t o the diameter of a capilon the flow of fluids through unconsolidated materials. lary tube which would pass the same volume of fluid in the No satisfactory data for flow through consolidated materials same time under equal drop in pressure as would a single were found in the literature. The data have been correlated series of connected pores. Using data for viscous flow only, according to the method discussed later. The correlation this term was calculated from Poiseuille’s law: is frankly empirical and probably must remain so until equations of motion, which will adequately account for each variable, are available. It would seem that equations must be hopelessly complex for such systems wherein the factors or of viscosity, density, surface forces, boundary (phase) shapes, compressibility, heterogeneity of surface, and possibly others must be included. By the use of this method a smooth continuous curve was obtained which would indicate a gradual and continuous 1 T h e laws prohibiting t h e introduction of water to an io11 eand mere transition from viscous t o turbulent flow for fluids flowing revised t o permit its use in t h e Allegheny field in New York in 1919, and in through porous media as velocities are raised. TVilde and McKean Countv, P a , in 1921 1139
’
Wouilbine, crossed Nicola X 85
Moore (13) are in accord with t,liis view and furnisli data r~liieliindicate that the nican effective pore diameter of a given sample is independent of t,Iie fluid, as it. should be from the theory for viscoiis flow. Washhurn (12) has disciissed and iised the term. Bartell and eo-w-orkcrs (a,9) have used tho term repeatedly, and discussed it critically (4),and apparently regard it as having niore than hypotlietical significance. It is cleii brought out by tliem that in the law to ROR through porous media application of Poiiciiill it is esscntitil to corrwt, for (a) deviation of the cross seetion of the average porc f r m i eircular, (b) increased length of path in site11 a system as cuniparcd to the apparent path, (c) tlic larger pressure gradient nccresary for flow in a sinuous path, and (d) added energy consumption due to many
wliich is due-to the viscosity of the fluid under given canditioris in order to have Poiseuillc's law apply, and its value is dependent on the proper evaluation of tlie terms just discussed. Bartell and OsterhoSf (4)used a i 2 (1.57) as tlie proper magnitude of the correction for the above factors and thereby obtained a check with t.he diameter measured by the capillary rise method. Apparently it is considered that such a diameter is the actual diameter of the average pore, as it has been st,ated in the literature that results can be checked by microscopic exaniinalion and measurement. Slichter (1 1 ) has shown that in an assemblage of spliercs of equal diameters the actual path of travel is from 1.2 to 1.5 times the apparent, and that the average velocity is about 1.8 times tliat calculated as above. Chilton and Colburn (7) from tlieoretical reasoning claim tliat, in viscous flow tlirotigh porous systems, expansion and contraction account for a t least 80 per cent of the total resistance and friction due to the viscosity of the fluid for 20 per cent or less for a rat.io of the minimum to maximum crosssectional area of a pore of 0.33. Tliese figures would imply that the correction for these factors in Equation 2 should not be less t.lian 9, which would materially increase the theoretical effective pore diameter as used by various invcstigators and bring the dimensions of the t,erm to a figure vhich agrees hctter with that for pores wliich can be seen in thin sections of oil sands, and closer to the average grain size calculated from screen analyses. Figure l shows several photomicrographs of thin sections prepared f r o m t y p i c a l oil sands. Tliese have many relativcly large pores, some, in Sact, as large or larger than grain sizc. Further Fork on the microscopic study of oil sands is being pursued in this laboratory. Actually there is no such thing as the average diameter of the tortuous Row channel through a porous mediim; i t is merely a statistical plaything a t best. Equation 2 can tell only that a quantiiy, ds2,varies inversely witli A p and directly with u or QIAP. Such a conception may serve a more useful purpose, however, than merely to correlate data if, when converted to area, it can be made to approximate the actual average cross-sectional area in the average flow channel which is swept by a fluid in movement. This is obviously of frindamental importance to oil producers. Table I compares tlic effective pore diameter for a number of oil sands cnlco1atc.d from Equation 2, tlie Same term corrected for the factors listed above by use of the minimuni factor 6 6 and tlie averagc grain diameter as calculated by t,lie authors.
October, 1933
I N D U S T R I A L A N D E N G I N E E R I N G C H E 3f I S T R Y
1141
feet (12.34 meters); and Bradford crude oil through flint US~OSSOLIDATED MATERIAL.The apparatus used for the sand in 20*1 feet (6e13 meters)* In each case the material tests on unconsolidated material consisted of straight lengths was packed in l-inch (2.5-cm.) Pipe. A test of water through of a/4- and 1-inch (1.905- and 2.54-cm.) pipe varying from 1.1 20- to 30-mesh Ottawa sand in 3 feet (0.914 meter) of 0.75to 50 feet (0.335 to 15.24 meters) and containing suitable con- inch pipe was made also. nections for either spring gages or manometers a t regular interwas heid in the pipes by tightly The Bradford crude oil used had a specific gravity of 0.7687 vals. The packing fitting screens. Suitable connections depending on the fluid a t 60' F. (15.5' C.), and a viscosity of 55 seconds Saybolt used were made to these (Universal) a t 75" F. ipes. Volumes o f (2339°C.). Thewater equids were m e a s u r e d was ordinary t a p in graduated cylinders; water. v o l u m e s of g a s b y a Sargent wet test meter Great care was taken or by calibrated orifice plates. to pack the materials CONSOLIDATED SANDS. as uniformly a s The apparatus used in possible, e s p e c i a l l y t he experiments w i t h in the longer lengths consolidated s a n d s i s of p i p e . T h i s w a s shown in Figure 2. Point done by a d d i n g t h e a is a 12-inch (30.5-cm.) nipple of 6-inch (15.2material in small cm.) p i p e c a r r y i n g increments while standard cast-iron blank t h e l e n g t h of p i p e flanges a t each end and containing the t i n n e d was in a n upright position and was being c o p p e r tank, d. The -zarrmi cover plate, c, and the h a m m e r e d rapidly. f, reduce possible FIGURE2. APPAR4TIJS FOR FLOWTESTSO F COVSOLIDhTED O I L S.4NDS T h e u,eight o f macontamination of liquid in d by dust and dissolved terial a d d e d w a s reair. Compressed air a t 150 pounds per square inch (10.2 atmos- corded and the resulting porosity computed from the volume pheres) could be admitted at b, and e is a vent to the atmosphere. From the liner tank, d, a brass flow line leads through the control Occupied and its density. A Satisfactory range in porosity (needle) valve, h, and thin sandstone filter, m, to the dead- was obtained by this method of Packing as Table 11indicates. weight gage, k , and the face of the sample held in the core holder, The Ottawa (Illinois) sand used was composed of welli, which, in turn, delivers to the standard 25-cc. buret. When rounded grains of quartE,in sizes and relative proportions as air is used in place of water for testing, the buret is replaced by a data the flint sand may be Sargent wet test meter. Tank g is for storage of the liquid to be given by found in the same table. The latter sand was from Len istown, used in a test. Details of brass core holder i are given in Figure 3. The sample Pa. Its component grains were quite angular as can be seen of sandstone cut to dimensions is held in a No. 7 rubber stopper in ~i~~~~ 1. The greater portion a, which in turn is wedged tightly into the core holder by means sieve and oi thumb screws attached to the brass riser connected to the Of it passed the flow line from tank d. This method of mounting and holding a was retained by the 50-mesh. dl1 sample for permeability and flow tests has been found eminently sieve tests on unconsolidated sands b satisfactory for the fluids employed and superior to the devices were made with standard 8-inch commonly in use, such as gaskets or seals of pitch, asphaltum, series sieves sealing wax, babbit, type metal, or any such material which (20*3-cm.) may contaminate the sample and seals it poorly. Upon com- shaken for 20 minutes by a Ro-Tap pletion of a test the sample can easily be removed uninjured shaker. and intact. The holder was tested for ossible leakage by submaking a series of flour tests stituting a piece of 0.75-inch (1.905-cm.~glassrod for the sandstone core and subjecting it to water under a pressure of 150 the inlet Pressure was varied for pounds per square inch. No leakage occurred overnight when each material by increments of 10 this was done. Tap c leads to the spring gage, j (Figure 2), pounds per square inch (0.68 atand the dead-weight gage, k , (Figure 4 contains details of the to maxilatter). b leads to the buret, e, t o a release valve. The rotating mosphere) from table, c, for weights is held a t the proper level for free rotation muln and back to minimum. For by adjustment of the hydraulic jack, b, which contained oil. each increment of pressure no data Compreqsed air from jet d rotates the table. The cross-sectional Tvere recorded until flow and presarea of the plunger is such as to multiply by 9 the pressure due to gradient were The neights placed on it in order to obtain the pressure per square inch curves of Figure 6 are typical of the on the face of the sample. w9-4 The spring gage is used only to establish flow at an approxi- data obtained. mate pressure, nith the valve between it and the manometer 3' DETalL OF CoNSoLID-iTED MaTER1aL* closed; afterwards this valve is opened and needle valve h adCOREHOLDER justed exactly until pressures on each side of the mercury in the and water were passed through the manometer (connections shown at a ) are equal. The dead-weight plugs or cores prepared from speclgage is calibrated nith manometers over its entire range. mens of the consolidated materials of Table I. All of these were sandstones in which crude petroleum occurs except PREPARATION .43D TESTING O F SAMPLES samples 4,6, and 17. Most of them were cores, although afew were chunks which had been bailed from various wells. The L - ~ C O N S O L I D ~ T E D . The following fluids and unconsolidated materials were used in the flow tests over a series of cores available were of t w types-(1) diamond drill cores pressure drops: Bradford crude oil through 1-mm. lead shot and (2) biscuits taken by means of the Percussion type of in 1.1 feet (0.335 meter) of pipe; air through unscreened core barrel. The first type of sample is the ideal one but Ottawa sand in 50.8 feet (15.48 meters) and also in 20.5 the second is considered by geologists to be representative feet (6.25 meters) ; water through unscreened Ottawa sand of condition s i n most cases. in 50.8 feet; Bradford crude oil through unscreened Ottawa Ceramic samples 4,6, and 17 were intended to be synthetic sand in 20.3 feet (6.19 meters); air through flint sand in sandstones to show the effect of different percentages of bond 50.7 feet (15.46 meters); water through flint sand in 40.5 or cementing material. They were made by adding 5, 10, -&PP.4R.iTUS USED
r2-.i
IKDUSTRIAL AKD ENGIKEERING CHEMISTRY
1142
and 20 per cent of clay t o portions of -200 mesh (98 per cent) potter's flint sand, forming the mixture into briquets and baking the latter a t 2370" F. (1290' C.) for 6 hours. TABLE I. SAMPLES TESTED S.4MPLE
E.LSD'
ST.LTE
grain
AV.
Ft. X 10 - 4
DIA~IETER-? Effective Corrected pore ( d e ) ( 4 7 x de! x 10-5 F t X IO-,
n.
CoPI-soLI D A T E D
9 10 11 12
Bradford Bradford 3rd Venango Ceramic A Robinson Ceramic B N'oodbine Wilcoxb 3rd Venango Robinson Robinson 3rd Venango
13
WlICOX
Warren 3rd Venango Robinson Ceramic B 3rd Venango Woodbine Woodbine Woodbine Woodbine Woodbine
14 15
16 17 18 19 20 21 22 23
Pennsylvania Pennsylvania Pennsylvania (20% bond) I!linois (10% bond) Texas Oklahoma Pennsylvania Illinois Illinois Pennsylvania Oklahoma Pennsylvania Pennsylvania Illinois (5% bond) Pennsylvania Texas Texas Texas Texas Texas
1.85 1.83 3.21 0.94 1.81 0.94 3.32 4.58 8.22 1.78 1.80
2.96 4.58 2.14 2.75 1.80 0.94 2.98 2.05 5.48 2.96 5.32 4.19
0.27 0.28 0.93 0.22 0.90 0.26 1.77 2.71 5.68 1.06 1.12 1.87 3.40 1.58 1.97 1.44 0.62 2.54 1.27 5.64 3.26 7.23 5.31
0.81 0.84 2.80 0.66
2 07 0.78 5.30 8.12 17.0 3.16 3.35 5.31 10.2 13 0 5.91 4.32 1.86 7.62 3.81 16.9
9.75 21.7 15 93
UNCONSOI. I D h T E D
Flint Pennsylvania 12.3 18.8 66.4 OttawaC Illinois 126 24.1 42.5 128 Ottanad Illinois 23.4 42.5 272 32.8 Lead shot ...... 90.6 a Name of geologic formation. Throughout this paper the term "sand" is used in the sense customary t o petroleum engineering and geology-i. e . , a n oil-bearing formation, usually a sandstone but frequently another rock or a n unconsolidated sand. b Sample cut perpendicular t o planes of bedding. 0 Unscreened. d Screened t o 20-30 mesh.
24 25 26 27
Vol. 23, No. 10
carefully measured. If air were to be passed thruugh a core, it was then ready for mounting in the core holder and testing. If water was to be passed through the core, it was placed in a suction flask connected t o a n aspirator. The flask wab sealed by a stopper containing a dropping funnel. After the sample ceased to evolve gas as indicated by attainment of constant pressure with the suction removed, freshly distilled water was added bj7 means of the funnel to the evacuated flask until the core was completely covered. The core was kept under xater a t atmospheric pressure f u r 4 or 5 hours, so that complete saturation was attained. The saturated sample was then mounted in the rubber stopper for test. Freshly distilled water was always used for testing a >ample. When air v a s used, it was dried. Either fluid was found to be satisfactory for all the samples tested and would yield reproducible and consistent results provided the following conditions were met: 1. The material of which the apparatus i b constructed must be insoluble, nonreactive, and inert to the fluid used, at least to the extent of adding no soluble or suspended constituent which will be sorbed by the core, or will plug the core mechanically. 2. The fluid must be pure, must be reactive physically or chemically to a minimum extent with the core, and must contain no suspended matter and a minimum amount of dissolved gas or condensable vapor. Some oil-bearing formations contain clays which can absorb water and can change in volume in so doing, in which case water would not be suitable for making a test but air, o r some liquid such as carbon tetrachloride or acetylene tetrachloride, could be used. Kone of the samples tested was of this type with the possible exception of the Bradford sand.
TABLE111. RELATION O F LEXGTH AXD .4RE.k TO FLOW The samples for testing in the apparatus were cut from RATE OF suitable specimens with either a 0.75-inch (1.905-cm.) or FLOW S.4ND LENGTH D X ~ M E T E PRR E S S U R E DROP \'OLDXE 0.375-inch (0.953-em.) special diamond-set drill mounted Cc./sec./ sq. cm. Cm. Btm. A t m . / c m . C c . / s t c . Cm. in a standard highway testing machine. The sandstone 1.83 2.72 0.792 0.428 0,155 A 3.43 samples u-ere then extracted in a Soxhlet apparatus using 0.155 2.54 1.83 2.01 0,792 0,430 carbon tetrachloride as a solvent until they were free of oil. 0.380 0.138 1.90 1.36 0.861 B 1.58 0.075 0.126 0.873 1.36 0.861 1.58 The core was cut to a length of approximately one inch (2.54 em.) by fracture, It was found possible to obtain a A similar procedure to that used with unconqolidatcd sands clean, true cross-sectional area only by fracture. Cutting of any sort plugs and glazes the wrface so that a lower was folloTved in testing the consolidated ones. A sample a-as permeability results. Brushing the surface d h a TT-ire tested over a series of pressures, data being secured at conbrush and playing a jet of compressed air on it does not stant pressure. Data for each increment of pressure were taken as the pressure was raised from minimum to maximum suffice to restore the true permeability to a cut section. All samples of sandstone except S o . 8 were cut parallel to and for the reverse. The data were not accepted unless they checked for these two methods. Some typical flon. curves the plane of bedding. The effective porosity of the samples was determined by are shown in Figure 6. Furthermore, from theory (9) the flow the method of Barnes (1) slightly modified to suit conditions, should be directly proportional to the cross-sectional area of the test piece and to the hydraulic gradient unless end or Fall and t h e data are listed in Table 11. The prepared cores were reextracted with carbon tetra- effects of appreciable magnitude inherent to the apparatus chloride and dried a t 212" F. (100" C.) in a n electric oven. or manner of test occur. The data of Table I11 for the flow After cooling in a desiccator, the diameter and length were of water are conclusively in accord with theory.
TABLE 11. SCREEN (In per cent b y weight r e MESH 10 20 30 40 50 7n ..
100 140 200 270 Over 270
SIEVEOPENINQ Inch Mm. 0.0787 2.00 0.0331 0.84 0.0232 0.59 0.0165 0.42 0.0117 0.297 n.00~3 0.210 0.0059 0.149 0.0041 0.105 0.0029 0,074 0.0021 0,053 0.0021 0.053
s 4ND
2
3
.... ....
....
1
.... 3.81 57.30 17.42 12.59 8.88
__
.
.
I
.
0.15 6.57 56.04 17.99 10.13 9.12
0.24 0.62 1.39 11.74 71.54 6.90 6.95 0.62
4
....
1.14 20.32 78.54
5
6
....
.... .... .... . . .
0.59 12.84 46.67 14.61 8.77 2.94 2.60 10.98
- - - -
..
,
.... ..,.
....
1.14 20.32 78.54
....
0.29 18.43 42.09 31.93 4.79 1.31 1.16
8 .... ,.,.
....
0.17 2.44 23.23 49.28 22.45 1.77 0.44 0.22
9 3.41 17.35 24.34 30.14 14.59 4.92 2.03 1.75
0.69 0.78
.... - - - -
10
....
0.07 6.90 48.21 25.83 4.73 3.31 10.95
11
' 0 . is
9.41 43.06 30.12 3.89 2.57 10.80
-
12
. .
0.29 31.76 6,; .84 4.19 2.31 1.61
__
100.00 100.00 100 .OD 100.00 100 00 100.00 100 00 100 .oo 100.00 100.00 100.00 100 .oo Total 18.4 22.3 19.5 15.9 11.94 37.8 19.7 20.3 37.0" 12.3 16.9 12.5 Porosity, yo Av. grain diam inch 0.00222 0.00220 0.00358 0.00113 0.00217 0.00113 0.00398 0.00550 0.00986 0.00214 0.00216 0.00356 0.00549 0.00902 0.00544 0.0140 0.0250 0.00287 0.0101 0.00551 0,00287 0.00569 0.00910 0.00564 Av. grain diam:: cm. Average. ~
a
7 .... .... ....
~
INDUSTRIAL AND E N G I S E E R I N G
October, 1933
CETERLIIXATION OF
GRAIS SIZE
CHEMISTRY
1143
The average grain diameter was calculated from the screen analysis by the formula:
The portion of each sample immediately surrounding the 0.75-inch plugs was reduced as nearly as possible to (3) grain size, and a tcreen analysis was made of it. The procedure for this is inportant as pointed out by Nutting (IO) in order to obtain coherent results. The sample of sandstone The values of n and d for the various screens may be found in was broken into pieces the size of a bean as gently as possible Table IV. The number of grains, n,for any average diameter by means of a vice. These fragments were carefully ground of opening, d,, was calculated by assuming that the grains were spherical and of an average density of 2.65. The latter figure was obtained experimentally. That this assumption is not greatly in error has been proved for the larger screen sizes by actual count under the microscope. Several counts indicated that the actual number of grains was 10 per cent less than the calculated. This is due to divergence of shape from spherical, to the large assortment of sizes obtained on a sieve even with careful screening, to faulty separation, and t o the fact that the microscopical count is only a statistical average in which it becomes increasingly difficult to secure an average count-i. e., to count enough grains-as the absolute size decreases. Further data concerning the smaller sizes and for more samples are being secured in this laboratory. The term d, in Equation 3 is the arithmetic mean of the size of openings in any two consecutive screens.
r
i
NUMBEROF GRAINSFOR ANY SCREEN SIZE
TABLE IV.
FIGURE4.
(Screen size,
DEAD-WEIGHT PRESSURE GAGE
DETAIL O F
JfESH
in a mortar until they passed a 30-mesh screen. Approximately 225 grams were sieved on the Ro-Tap shaker for 20 to 55 minutes, depending on the average size of the material, the longer time being necessary for the finer materials. The portion retained on each sieve was then rubbed between the fingers to separate aggregate grains and the several portions Tvere carefully examined under a binocular microscope to make certain that such separation was as complete as possible. Frequently further treatment was found to be needed. The screen analyses of the samples are listed in Table 11. The screen analysis of an oil sand is an important test, and, if care and proper precautions are taken, reliable and reproducible results can be achieved. The Bradford sand was so tightly and strongly cemented that it was exceedingly difficult t o separate its constituent grains. Soaking in water and mild chemical solutions did not help. Undoubtedly a complete separation was not realized, which tended to increase the average grain diameter. Some individual grains inevitably were fractured by the attempt3 to separate them which would decrease the average grain diameter. It is believed that the effect of the latter was not great enough for compensation and that the average grain diameter for the Bradford sand as given in Table I 1s too large.
U.S. Standard)
.kV. DIAMETER (X
10-20 20-30 30-40 40-50 50-70 70-100 100- 140 140-200 200-270 Over 270
GRUNSPER
Cm. 142.0 71.5 50.4 35.8 25.4 18.0 12.7 8.89 6.35 2.67
Inches 55.90 28.15 19.85 14.10 10.00 7.10 5.00 3.50 2.50 1.05
2.52 1.97 5.62 1.57 4.40 1.23 3.52 I . 03 2.81 3.80
ISTERPRETATIOS O F
x X x
X
X
X
X
x
GR4M
102 10s 103 10' 104 10' 10s
IO'
X 108 X 10:
DATA
The data secured from 800 tests have been used in the construction of Figure 7 and are too voluminous for tabulation here. The data were inserted in Fanning's equation.
and the value off calculated. The term L is the length of the sample core used, d the average diameter computed by Equation 3, and U the apparent velocity obtained from the equation,
u = -Q
(5)
A
It has been found that the velocity calculated by this equation results in as good correlation as the more complicated
ANALYSESOF SANDS tamed on the reepective sieve, 13
14
15
16
17
18
19
20
21
22
23
. . .
....
0.17 2.44 23.23 49.28 22.45 1.77 0.44 0.22
...
....
0.04 7.33 58.26 20.05 9.26 5.06
....
0.05 0.92 22.70 64.70 6.44 3.10 2.09
. . .
....
0.07 5.86 49.28 18.61 9.00 3.68 2.46 11.04
- - - - - _ _ -
....
....
....
....
1.14 20.32 78.54
0.09 1.68 24.03 60.83 9.61 2.27 1.49
....
0.14 2.11 10.01 20.64 42.17 13.28 6.74 4.91
....
2.25 30.78 54.15 12.15 0.50 0.11 0.06
....
....
0.98 14.90 68.99 10.74 3.00 1.39
1.77 36.59 53.92 5.43 1.25 0.39 0.35 0.30
- - - _ _ -
..
0.21 8.25 58.23 27.80 4.17 1.09 0.25
100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 16.3 19.2 21.4 20.6 33.2 21.9 23.8 26.9 27.7 22.1 28.8 0.00550 0.00256 0.0033 0.00216 0.00113 0.00357 0.00241 0.00658 0.00355 0.00638 0.00504 0.0140 0 00650 0.00838 0 00549 0.00287 0.00907 0.00612 0.0167 0.00902 0.00162 0.00128
24
....
0.01 0.77 11.93 58.94 24.47 3.16 0.63 0.06
0.02 0.01 100.00 38.5 0.0148 0 0376
"j
....
0.41 96.66 2.72 0.07 0.09 0.02 0.03
27
26
.... ....
100.00
.... ....
.... ~
100.00 30.9 0.0290 0 0737
100.00 34.5 0.0282 0 0717
_
....
34.5 0.0394 0,100
_
INDUSTRIAL AND ENGINEERING CHEMISTRY
1144
term corrected for porosity. Since either is scarcely an approximation to the actual mean velocity through a pore, it is simpler to use the former. The friction factor has been
+,ATMOSPHERES
PER CM
CURVES OF UNCONSOLIDATED FIGURE5 . FLOW SANDS SANPLE
T E M P .OF T E S T (' C.) 68 (20) 68 (20) 48 (8.9) 54 (12.2) 68 (20)
O F .
6C
D E
VISCOSITY, E Centipoises 1.01 6.70 1.36 1.24 6.70
plotted against the modulus, dup/p, on logarithmic graph paper using data for flow tests for all the materials listed in Table I. Corresponding work of Chalmers, Taliaferro, and Rawlins (6) was kindly made available and was added to the data for unconsolidated materials together with that of Cloud (8). Certain assumptions as to the conditions of test for the latter were made. It is apparent that an excellent correlation and alignment of the data for unconsolidated sands were secured by this method. While a line could be drawn for each sample, the average line as drawn is sufficiently accurate notwithstanding the differences in porosity, relative sizes, and angularity of the grains composing the samples. The line for lead shot is displaced considerably from that for sands chiefly by virtue of the difference in average diameter and relative smoothness of the shot. These results are remarkable when it is realized from the definition of the terms d, L, and U, the strictly empirical nature of the assumptions which were made. Clearly, the evidence is that the flow of fluids through sands and other unconsolidated solids resembles the flow of fluids in pipes, and that the same laws of hydrodynamics may be applied as far as our knowledge of the effect of the various related factors which control permits. The relative displacement of this line from that which Chilton and Colburn (7) obtained and the relative displacement of the data for the various materials from each other is due to two factorsnamely, increased roughness and angularity, and differences in the distribution of sizes of solids. There seems to be a definite zone in which the type of flow may change with some abruptness from viscous to turbulent. This change was observed qualitatively in the laboratory by filling a glass tube 3 feet (0.914 meter) in length with shot, placing it in a vertical position with a two-hole stopper in the bottom, and flowing water through the tube and injecting a stream of fluorescein solution into the main current. The course taken by the dye was observed a t different rates of flow. At low rates the streamers of dye in passing around a shot, followed approximately a hexagonal path with little diffusion of color. At a certain rate the type of flow changed
Vol. 25, No. 10
abruptly and the streamers of dye seemed ;O bounce from one shot to another in a chaotic path with c3nsequent complete diffusion of color. This type persisted for all rates of flow in excess of the critical. The behavior of fluids in consolidated 3ands is even more interesting and remarkable. Consolidatetl sands differ from unconsolidated chiefly by the amount and type of cementing material which they contain. It is possible to have a small or large quantity of bonding material evenly distributed over each sand grain, in which case it serves mainly to increase the diameter of the sand grain. It is also possible to have the same amounts unevenly distributed, thereby greatly decreasing the number of pores and the diameter of some flow channels, and hardly affecting others. The Wilcox sand, samples 8 and 13, is an example of a sand rather lightly bonded and evenly distributed. The Bradford sand, samples 1 and 2 (Table I), is an example of a more strongly, but evenly cemented sand. On the other hand, the Third Venango sand, samples 3 and 9, is exceedingly variable in distribution and amount of bond. The determination of the amount of bond seems to be impossible, but qualitatively it can be estimated in many cases from the screen analysis, the porosity, and an examination of the sample under the miscroscope. The data are for both air and water, and in most cases the data for each fluid in a particular sand overlap and define the same line. Naturally, turbulent flow was not attained in the less permeable sands even with air but was for the more permeable ones. Furthermore, it was attained a t a definite range of values for the modulus, d U p / p , unique for each sand. In contradistinction to the unconsolidated sands, a definite line is obtained for each consolidated sand, which is displaced to the right and above the former by an amount proportional to the smallness of the average value of d, the amount and arrangement of cementing material, and the degree of angu-
+,ATMOSPHERES
PER CM.
FIGURE6. FLOWCURVESOF CONSOLIDATED OIL S.4NDS SAND
1 Bradford 2 Bradford 13 Wilcox 18 3rd Venango 20 Woodbine 22 Woodbine 23 Woodbine
T E M P .OF TEST F. (a C.) 80 (26.7) 71 (21.7) (25.6) is (24.4) 76 (24.4) 76 (24.4) 76 (23.3) 74
VISCOSITY,E Centipoise 0.859 0.972 0.884 0,907
0.907 0.907
0.931
larity of the sand grains. Of these, the amount and arrangement of cementing material is by far the more significant. This is shown by the data for unconsolidated material in which only the effect of the other two factors may be noted. As has been stated, there is some displacement for these materials in accordance with these factors but not enough to prevent the definition of an average line for all of the materials. The effect of the amount of bond is convincingly demonstrated by the relative positions of the three synthetic sandstones (ceramic samples 4, 6, and 17, Table I) on Figure 7 .
October, 1933
I N D U S T R I A L A N D E N G I N E E R I S G C H E hl I S T R T NOMENCLATURE ~
FRICTION FACTOR, N O UNITS 3 2 2 FEET PER SEC PER SEC DIAMETER OF AVERAGE GRAIN FEET PRESSURE DROP LBS FER S Q F T LENGTH OF CORE F E E T F L U I D DENSITY LBS P E R CUBIC F T APPARENT VELOCITY,
td&Zo,','?L
8
FEET PER SEC
1145
~-
~ S A M P L ENO SAND I ZONSOLIDATEC I I iBRADFORD BRADFORD 3 R D VENANGO CERAMIC A
%
12.5 123 I69 37 0 20 3 37 8 19 7 I59 119
p -ABSOLUTE VISCOSITY 0000872 2 . LBS PER SEC PER FT
195 +
1
W T O m
GAS U S BUREAU
1 1 I
332
I
221
1
219 238 269 277 288
h
1
18.4 22.3 16.3 19.2 214 206
385 309
~
1
0
FIGURE 7. FRICTION FACTORS FOR ALL SIMPLE FLUIDSTmomn POROUS MATERIALS
These samples mere made of -200 mesh flint sand with 5, 10, and 20 per cent of clay bond, respectively. Each curve is the result of check tests on triplicate samples. Neglecting the slight differences caused by the relative concentration of the silica and clay phases during the firing process on the
final physical properties of the sample, the displacement from the line for unconsolidated materials and from each other is proportional to the amount of cementing material. It likewise is evident, other factors allowed for, that sample 23 of the Woodbine sand is only slightly bonded, that the
1146
INDUSTRIAL AND ENGINEERING CHEMISTRY
other samples of Woodbine, with the exception of 7 , are but little more bonded, that a great number of sandstones are bonded somewhat similarly, and that the Bradford sand is strongly bonded. The relative positions of the curves for samples 8 and 13 of the Wilcox sand illustrate nicely the effect of arrangement of bonding material (Figure 7). In this case the amount of bond is small and the grains of sand are rather uniform and fairly smooth; however, sample 8 was cut perpendicular to the planes of bedding, and sample 13 parallel to them. The characteristics of the grains are shown in Figure 1. The effect of heterogeneity, both of size of components and of arrangement of bonding material, is best shown by the relative positions of samples 3 and 9 of the Third Venango sand. These samples are taken not only from the same geological formation but from the same well a t close intervals, yet they are widely displaced. There is some difference in grain diameter, sample 9 being coarsely conglomeratic; but, from the evidence presented by the porosities of the two samples, their appearance under the microscope, and the distribution of sizes of grains, the chief factor causing displacement is the arrangement of the bond. Sometimes other factors may affect flow. One of these may be mineral composition or the effect of some particular constituent ordinarily not present in the average sample of sand. Thus the position of sample 5 , a specimen of the Robinson sand from Illinois, is plainly anomalous, judging from its porosity, screen analysis, and other characteristics. It is abnormally low in permeability. Examination under the microscope revealed the presence of laminations of mica which projected into open pores and probably served as baffle plates which greatly interfered with or prevented flow through them. Notwithstanding these anomalies, sands from regions geographically as far removed as Pennsylvania, Illinois, Oklahoma, and East Texas, differing widely in mineralogical composition, geologic history, and other important characteristics, are arranged on the basis of flow tests, screen analyses, and porosity in consistent positions on Figure 7. I n fact, with many consolidated sands it is now possible from a simple screen analysis to predict with reasonable accuracy the position of the line representing viscous flow through that sample. This may be done with more precision if other qualitative information which frequently may be available is considered-namely, the porosity, distribution of sizes of grains, appearance of the sand under the microscope, and its geologic history and relationships. The region of viscous flow is of greatest interest to the petroleum engineer and producer because fluid velocities are low in the underground strata, except possibly in the immediate vicinity of an oil well in flush production. The average rate of travel of water in the Bradford field under usual conditions of operation has been estimated to be in the neighborhood of 3 inches (7.62 cm.) per day. In any event, one experimental point in the viscous region plus a screen analysis defines the line because the slope for each is -1. I n the case of unconsolidated sands, since a single line on the chart adequately represents all the sands, a screen analysis is sufficient information to enable all necessary calculations concerning rate of flow and pressure drop to be made. The equations representing the line in the viscous flow region for any sample shown on the chart may be calculated from the equation
The value of the constant for each sample is listed in Table V.
Vol. 2 5 . x o . 10
TABLE V. VALUEOF CONSTAST C FOR EACHSAMPLE c SAMPLE r: C SAMPLE
ShhlPLE
615,000 558,000 103,000 80,200 50,000
10 11
47,100
15 16
23,500 22,500 18,100
12 13
17,800
14
28,600 28,200 “8,100
15,400 14,800 12,100 11,200 9,240
17
18
19 20 21 22 23 All unconsolidated sands Lead shnt
(
8480 5700 4780 3890 3510 1700
585
Permeability is a property of sandstones xhich is of considerable value and interest. Its value may be calculated for any sandstone or loose sand from the data presented by Figure 7. The property usually is measured by the evaluation of the coefficient in the equation for Darcy’s law (9) namely:
or
From Equation 4 (Fanning’s): _L= _ _gd
(97
ap 2 p p Substituting Equations 9 and 5 in 8,
Substituting Equation 6,
,.
For viscous flow,
Permeability (consistentmetric units)
=
1.9028 X IO3 X k (13)
Consequently, from the screen analysis of a sand and the properties of the fluid a t the temperature of flow, it is possible to compute the permeability of a sand to that fluid by assuming desired dimensions and a rate of flow. This is, of course, the permeability of the stone to a fluid in terms of that fluid. If it is desirable to choose a standard fluid and to have for purposes of comparison the relative permeability for any fluid in terms of the standard, it is necessary t o multiply this permeability coefficientby the ratios of the viscosities of the fluids concerned. Water is frequently used as a standard fluid for this purpose. Likewise, it is apparent that by assuming suitable dimensions and a rate of flow, the pressure necessary to pass a fluid through a sand is readily evaluated by use of data from Figure 7 and the physical properties of the sand and fluid, provided the flow is rectilinear. Equations for radial flow which involve the permeability coefficient are in common use by petroleum engineers; consequently, calculation of the coefficient by Equation 12 with proper correction factors for differences in units used enables flow for this condition to be evaluated. SUMMARY
1. An accurate method of studying the flow of fluids through permeable media has been described. 2. The flow of fluids through unconsolidated and consolidated sands resembles that through pipe. There is a definite change from viscous to turbulent flow as velocities increase. 3. A chart (Figure 7) representing the change in friction factor with the modulus, d p u / p , has been plotted and we11 represents the data for the flow of fluids through porous materials.
.,
(:3)
~.
.
Bsrtali. 1'. E., and Miller.
k. I,.,
Ixu. Exn. Cam& 20, 7 3 8 4 2
( I Wh) .
(.I) Ilartcll, 12. IS., arid OaterlmR. I7. J . , J . Pliys. Cheni., 32, 1 5 s - 7 1 (10%).
E."Goology oi the Oil Itwiions of Warren. Yenawo, Clarion, and Jiutler Count,ior," 2nd Gcol. Survey 01 Pa.. Vol.
( 5 ) C:vll. 3.
I1 = inside dirrm. of pipe, in = friction factor, dimerisi~,nlcss o = zccrleratian of EravitY. 32.2 ft./ser
I, =
,n = = P = .%p = ,t
brtueen faces of core,
