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 14. GS: 
 CIR 2$ 
 
 GqtoJ -SuruJtx^ 
 
 STATE OF ILLINOIS 
 
 WILLIAM G. STRATTON, Governor 
 
 DEPARTMENT OF REGISTRATION AND EDUCATION 
 
 VERA M. BINKS, Director 
 
 Hydraulic Fracture Theory 
 
 Part III. -Elastic Properties 
 of Sandstone 
 
 James M. Cleary 
 
 DIVISION OF THE 
 
 ILLINOIS STATE GEOLOGICAL SURVEY 
 
 JOHN C.FRYE, Chief URBANA 
 
 CIRCULAR 281 
 
 ILLINOIS GEOLOGICAL 
 SURVEY LIBRARY 
 DtU 16 . 
 
 1959 
 
ILLINOIS STATE GEOU 
 
 3 3051 00003 8^ 
 
HYDRAULIC FRACTURE THEORY 
 Part III. -Elastic Properties of Sandstone 
 
 James M. Geary 
 
 ABSTRACT 
 
 This study, the last of a three-part series, was undertaken 
 primarily to evaluate certain elastic properties of the porous sand- 
 stones. Such properties may be used in calculating the changes 
 in underground stresses that take place as pore pressure changes. 
 As the underground stresses control hydraulic fracture propagation, 
 knowledge of these stresses is important in recovery of oil. 
 
 Samples of sandstone were subjected to changes in pore 
 pressure and external stresses. The resulting strains were meas- 
 ured by resistance wire gages and the elastic properties computed. 
 
 In Parts I and II of the project, the hydraulic fracturing proc- 
 ess was examined as a problem in applied mechanics, with partic- 
 ular emphasis on elasticity. It was concluded that hydraulic frac- 
 ture propagation was controlled mainly by the horizontal stress in 
 the rock and that this stress would change with the pore pressure. 
 
 The laboratory data of Part III confirm experimentally the 
 predicted horizontal stress changes. Elastic properties of the sand- 
 stone were found to vary with mean effective stress. The regular 
 variation of elastic properties with porosity shouldallow rough pre- 
 diction of elastic behavior of a given type of rock on the basis of 
 porosity. 
 
 INTRODUCTION 
 
 The induction of hydraulic fractures in buried sediments is governed in 
 large part by the stress condition of these sediments. The stress condition is in 
 turn influenced by the pore pressure. The changes in stress and strain resulting 
 from changes in pore pressure are governed by the elastic properties of the mate- 
 rial insofar as these changes are reversible. This report describes the elastic 
 behavior of sandstone, or, in other words, the reversible relationships between 
 stress, strain, and pore pressure. 
 
 Parts I and II of this study of hydraulic fracture theory were prepared as 
 part of a project set up by the Illinois State Geological Survey, in consultation 
 with the Department of Mining and Metallurgical Engineering of the University of 
 Illinois, to investigate the process and its various elements. 
 
 Part I dealt mainly with the adaptation of theories on the mechanics of mate- 
 rials to the problems of hydraulic fracture mechanics and to a description of stress 
 conditions in porous sediments. Part II was concerned with hydraulic fracture me- 
 chanics, the orientation, distribution, and possible control of fractures, and the 
 importance of pore pressure. 
 
 This final part of the study deals with laboratory experiments suggested by 
 the theoretical studies of the first two sections. Parts I and II concluded that use- 
 
 [ll 
 
2 ILLINOIS STATE GEOLOGICAL SURVEY 
 
 ful calculations bearing on the mechanical aspects of the hydraulic fracturing proc- 
 ess can be made if the elastic properties of the porous strata are known. Such cal- 
 culations would consider the influence of changes in pore pressure on the stress 
 condition of the rock around an oil well. 
 
 The elastic properties of porous granular materials are known to change as 
 the state of stress changes. The published literature gives such a diversity of 
 values for elastic properties of rocks that it did not seem practical to use published 
 values for the application considered here. For this reason an experimental inves- 
 tigation was made of the elastic properties of some sandstone samples taken from 
 drill cores. The answers to the following questions were sought in these experi- 
 ments: 
 
 1) What reasonable values of the elastic properties of sand- 
 stone, within the range of stress conditions existing in the oil res- 
 ervoir, can be established? 
 
 2) How do these properties change with stress condition? 
 
 Acknowledgments 
 
 I express appreciation to the Department of Mining and Metallurgical En- 
 gineering of the University of Illinois for financing the construction of the appara- 
 tus used in the tests; to Professor Walter D. Rose for guidance in performing these 
 experiments; to the Illinois State Geological Survey for providing samples and aid- 
 ing in the preparation of the manuscript; and to Alfred H. Bell, head of the Oil and 
 Gas Section of the Illinois State Geological Survey, Lester L. Whiting of the same 
 section, Kenneth R. Larson of the Natural Gas Storage Company of Illinois, Pro- 
 fessor H. R. Wetenkamp of the Department of Theoretical and Applied Mechanics 
 of the University of Illinois, and William J. McGuire of the Atlantic Refining Com- 
 pany, all of whom assisted in various ways. 
 
 This report is adapted from a thesis submitted in partial fulfillment of the 
 requirements for the degree of Master of Science from the University of Illinois, 
 1959. 
 
 NOTATION 
 Symbols 
 
 a - Linear coefficient of thermal expansion, l/Temp 
 
 P = Grain compressibility, in 2 /lb 
 
 e = Strain 
 
 E = Young's modulus, lb/in 
 
 e = Volume dilatation 
 
 J = Linear coefficient of pore pressure expansion, invlb 
 
 \i - Poisson's ratio 
 
 n = Fractional pore area in a plane through a porous material; the 
 porosity 
 
 a = Total stress with direction unspecified, surface pressure, or 
 mean stress, lb/in 2 . When the mean stress, <x , is used as a 
 variable, it is assumed that the stress change is hydro- 
 
ELASTIC PROPERTIES OF SANDSTONE 
 
 static. That is, the changes in each of the three prin- 
 cipal stresses are equal. 
 
 <f = Effective stress with direction unspecified, or mean effective 
 stress, lb/in^. When the mean effective stress is used as a 
 variable there is no restriction on the way in which the prin- 
 cipal effective stresses vary individually. 
 
 Ox= Total stress in the x direction, lb/in^ 
 
 a x = Solid stress in the x direction, lb/in 
 
 P = Pore pressure, lb/in^ 
 
 X = 
 
 A constant 
 
 
 z = 
 
 
 "Scr r 
 
 _8pJ 
 
 a z = constant 
 e r = constant 
 
 " TE 1 
 
 = the rate of change of the hor- 
 izontal stress with pore pres- 
 sure for the usual subsurface 
 boundary conditions. 
 
 cyclic = Two additional similar equations are obtained by cyclic inter- 
 
 change of the subscripts, implying that the given equation is 
 general for the three coordinate directions. 
 
 8 = Differential of 
 
 Subscripts: 
 
 x, y, z, r = The principal stress directions. When r is used the problem 
 is assumed to have radial symmetry with, for example, 
 
 = o\ 
 
 This is always the condition in the experi- 
 
 ments . 
 o = The initial, or base, value, 
 
 PREVIOUS WORK 
 
 Adams and Williamson (1923) measured the compressibility of rocks and 
 minerals under high confining pressure in two groups of experiments. For one set 
 of experiments they coated the rock cylinders with tin foil to prevent access of the 
 fluid to the pores. In another set of experiments they exposed the rock cylinders 
 to the pressuring fluid so that the pressure would act throughout the communicating 
 porous structure as well as on the outside surface of the specimen. 
 
 They found the compressibilities of minerals and unjacketed rock changed 
 only slightly by pressure. For example, the compressibility of quartz is reduced 
 about 1 percent by a 3, 000 psi increase in pressure. They also found that the 
 'compressibility of unjacketed rocks could be calculated with good approximation 
 by obtaining the weighted average compressibilities of the constituent mineral 
 grains. 
 
 The compressibilities of the jacketed specimens were much greater than the 
 compressibilities of the unjacketed specimens. As pressure increased, the com- 
 pressibilities of the jacketed samples decreased, rapidly at first, and approached 
 asymptotically the compressibilities of the unjacketed specimens. 
 
 Adams and Williamson reasonably explained the initially high value of the 
 compressibilities of jacketed specimens by pointing out that when the fluid is 
 
4 ILLINOIS STATE GEOLOGICAL SURVEY 
 
 excluded from the pores the pressure tended to close the pores as well as reduce 
 the grain volume. 
 
 Zisman (19 33b) studied the compressibility of rocks under moderate con- 
 fining pressures and observed in greater detail than did Adams and Williamson the 
 change in compressibility of jacketed specimens under confining pressure. 
 
 One remarkable thing about this phenomenon of changing bulk compress- 
 ibility, observed by Adams and Williamson and later by Zisman, was the fact that 
 the rocks being tested were what normally would be considered nonporous and im- 
 pervious. Most of the rocks tested were igneous. " Nonporous" limestone, sand- 
 stone, dolomite, and marble also were included. 
 
 Birch and Bancroft (19 38) shed further light on the phenomenon of the initial- 
 ly high compressibility of jacketed rocks. They pointed out that an aggregate of 
 anisotropic crystals of random orientation can exist in a hydrostatic state of stress 
 without any porosity under only one set of temperature-pressure conditions. Under 
 any other set of conditions the various crystal grains will tend to expand differently 
 in different directions, developing microscopic unequal stresses that tend to prop 
 open small fissures. 
 
 Therefore, when the dense rocks, such as granite and basalt, are re- 
 moved from the environment of temperature and pressure under which they have 
 become normalized through geologic time, their original dense structure may be 
 loosened slightly by the development of minute fissures. Birch (1943) stated that 
 a slight loosening of structure in rocks would occur with moderate heating. 
 
 The existence of these phenomena has probably best been demonstrated in 
 attempts to measure thermal expansion of rocks. Dane (1942) states, "When a 
 rock specimen is measured in the laboratory, it is found that the coefficients of 
 expansion are very different on heating and cooling, with a still different result 
 on each subsequent run. This is due to the unlike expansions of adjacent grains 
 because of differences both of composition and of orientation. As a result, when 
 a rock is heated the grains with the largest thermal dilation tend to determine the 
 apparent change in length of the whole specimen, creating internal fractures or 
 'pores'. " 
 
 If such secondary porosity can develop in compact rocks when they are re- 
 moved from their geologic environment, we must also expect that porous rocks may 
 be disrupted in a similar manner. There also may be additional alteration due to 
 treatment in the laboratory. Therefore, laboratory measurements of the elastic 
 properties of porous rocks placed under low confining pressure may be influenced 
 by secondary disruption of rock structure. However, as the contact area between 
 grains is much smaller for porous sandstones than for dense nonporous rocks, the 
 porous sandstone will have microscopic stresses at areas of grain contact that are 
 much higher than the applied external stress. Hence, moderate confining pressures 
 should be sufficient to remove slack that may have developed in the system owing 
 to removal from the natural environment. 
 
 The early workers concerned with elastic properties of rock were interested 
 principally in the interpretation of seismic wave phenomena. Zisman (19 33a) com- 
 pared measured seismic wave velocities in shallow beds with velocities (calculated 
 from statically determined Poisson's ratio, \i, and Young's modulus, E) measured 
 on rock cylinders at atmospheric pressure. He found the experimental determinations 
 of Young's modulus and Poisson's ratio too low to account for seismic wave veloc- 
 ities. Zisman also noted that E and (j. increased with the mean applied stress. 
 
 Ide (19 36) compared static and dynamic measurements of Young's modulus 
 
ELASTIC PROPERTIES OF SANDSTONE 5 
 
 on rocks at atmospheric pressure. The static measurements of E yielded the lower 
 values. 
 
 Ide found a wide discrepancy between static and dynamic measurements in 
 the same rocks and the discrepancies seemed to be greater for the more porous rocks. 
 The present investigation involves only static states of stress in porous rocks and 
 is therefore limited to static measurements. 
 
 Birch and Bancroft (19 38) measured the shear modulus of dense rocks by the 
 torsional vibration method, carrying their experiment to elevated temperatures and 
 high pressures. Their early work is not applicable to the present problem because 
 the rocks tested were almost without porosity. 
 
 Carpenter and Spencer (1941) measured the " compressibility" of high po- 
 rosity sandstones in an effort to learn whether fluid withdrawal would compress the 
 sandstone sufficiently to account for subsidence observed in certain oil fields. 
 They enclosed cylindrical samples in foil and inserted a tube to connect the porous 
 system of the specimen to the outside of the pressuring cell. As pressure was in- 
 creased on the exterior of the specimen, the volume of fluid forced from the pores 
 was measured. Carpenter and Spencer's data are converted to an appropriate form 
 and plotted with the new data reported below. 
 
 Hall (1953) and Fatt (1958a) measured what they term pore compressibility. 
 Their procedure was similar to that of Carpenter and Spencer but the volume of fluid 
 produced from the specimens resulted from a reduction in the pore pressure rather 
 than an increase in the external stress. Thus Hall and Fatt more closely duplicated 
 reservoir conditions. 
 
 In another paper Fatt (1958b) reports measurements of bulk compressibilities 
 on porous sandstones. In his experiments linear strain was measured while hydro- 
 static pressure was being applied to jacketed cylindrical specimens. He also meas- 
 ured a property he termed " pseudo bulk compressibility, " which is essentially the 
 same property measured by Carpenter and Spencer. 
 
 Fatt measured another property also obtained in the experimental work re- 
 ported here. In this report the property is expressed 
 
 Sp 
 
 e = constant or 
 
 JE 
 
 .1 - 2(i 
 
 and its physical significance is discussed. 
 
 Fatt also developed several analytical models, including an extension of 
 the sphere-pack model of Brandt (19 55), to rationalize the mechanical behavior of 
 porous rocks. 
 
 THEORY 
 
 The theory of elasticity of porous materials is here developed briefly. Ear- 
 lier works cited below cover the subject much more fully. 
 
 Biot (1941) developed a theory of elasticity of porous materials in which the 
 compressibility of the pore volume is related to the other deformation constants. 
 Later, Biot (1955) extended his theory to include anisotropy. 
 
 Gassmann (1951) also studied pore volume elasticity and the propagation of 
 elastic waves in porous materials. 
 
 Geertsma (1957) discussed elastic as well as inelastic compressibility of 
 porous materials. He gives a convenient set of connecting equations relating the 
 bulk and pore compressibilities for various boundary conditions. One of these is 
 used later in this report to convert the data of Carpenter and Spencer to the form 
 
6 ILLINOIS STATE GEOLOGICAL SURVEY 
 
 used here. 
 
 Lubinski (1954) discussed a method for the solution of problems in elas- 
 ticity that takes into account the effect of pore pressure and pore pressure gradi- 
 ents. 
 
 Cleary (1958b) considered the relation of earth stresses to hydraulic frac- 
 turing, using a method somewhat modified from Lubinski's. The theory of elastic- 
 ity used is somewhat limited as it assumes linearity of the stress-strain relation. 
 The fact that the stress-strain relation for porous materials is nonlinear is well 
 demonstrated in the experiments reported here. 
 
 Compressibility 
 
 The unit change in volume per unit change in the pressure applied to the 
 boundary is the compressibility. For porous materials, however, several bound- 
 aries may be defined- the external boundary, the pore boundaries, or the external 
 and pore boundaries taken together. Several volumes also may be defined - the 
 gross volume, the volume of the grain material, or the pore volume. Several types 
 of compressibility may be defined for porous materials, depending on what volume 
 the pressure is changing. We will distinguish between the pressure acting on the 
 gross external boundary, <x , and the pore pressure, P. 
 
 Grain Compressibility 
 
 If an isotropic porous material is immersed in a fluid under pressure, the 
 pressure acts on the outer boundary and throughout the connected pore boundaries, 
 so An = AP. 
 
 A change in the pressure results in a change in the gross volume of the mate- 
 rial. The grain compressibility, (3, is the slope of the volume dilatation versus 
 pressure and is defined by the derivative 
 
 .8<x. 
 
 (cr- P) = constant 
 
 where the volume dilatation, e, is brought about by equal changes in the external 
 stress, a , and the pore pressure, P. 
 
 Linear strain, e, is equal to one-third the dilatation, e. Thus, what will 
 be called here the linear compressibility is equal to one-third the volume com- 
 pressibility. Therefore 
 
 (a - P) = constant 3 ^' 
 
 Sa 
 
 Experimental data will be interpreted in terms of the linear compressibility because 
 strains rather than volume changes are measured directly by the strain gages. 
 
 The linear grain compressibility defined by equation (1) is independent of 
 the geometry of the porous structure. It depends directly on the material making 
 up the porous structure. 
 
 Bulk Compressibility 
 
 If in another experiment the specimen to be tested is given a thin, imper- 
 meable coating and the fluid pressure applied, again a uniform strain is produced. 
 This time the pressure, a , is applied to the external surface and excluded from 
 
ELASTIC PROPERTIES OF SANDSTONE 
 
 the internal pore surfaces. The slope of the strain-versus-pressure curve in this 
 case is the bulk compressibility of the material. The bulk compressibility 
 
 Se 
 
 Sa 
 
 constant 
 
 can be expressed in terms of the other bulk elastic constants. 
 
 Se 
 
 8<r 
 
 - P = constant 
 
 1-2|JL 
 
 L E J 
 
 according to the connecting equation given by Timoshenko and Goodier (1951, p. 10). 
 E is Young's modulus for the bulk material, and \± is Poisson's ratio for the bulk 
 material. Linear strain is equal to one-third the volume strain; therefore the lin- 
 ear bulk compressibility is 
 
 "S el 1-2 
 
 (2) 
 
 P = constant 
 
 Hooke's Law 
 
 Porous materials such as rock and ceramics have often been treated like the 
 more conventional nonporous elastic materials. It is customary to ignore the fact 
 that the normal stresses are forces per unit area that act discontinuously over a 
 plane passed through the material. A fractional area, n, of a plane passed through 
 the material is occupied by voids; hence, the stresses are average forces per unit 
 gross area. However, even for nonporous materials we must deal with average 
 stresses because there are always microscopic stress fluctuations in real elastic 
 materials. 
 
 The conventional form of Hooke's law is 
 
 e v = 
 
 ox 
 E 
 
 t ( ^y + 
 
 <t z ) (cyclic) 
 
 (3) 
 
 The stress, a 
 
 <r 
 
 is the force per unit gross area acting on the solid portions 
 of a plane passed through the material and will 
 be called the solid stress. The stress, a , is 
 the total force per unit gross area acting on a 
 plane passed through the material and will be 
 called the total stress. The relation between 
 these two kinds of stress is illustrated in fig- 
 ure 1 and given by 
 
 nP 
 
 cr x = cr x 
 
 + nP (cyclic) 
 
 (4) 
 
 Equation (3) is a special case of Hooke's Law 
 for porous materials when the pore pressure is 
 n p g.1 zero. However, the presence of fluid pressure 
 
 in the pores introduces a strain in addition to 
 Fig. 1. - Forces acting in two princi- the strain that is due to the solid stresses, 
 
 pal directions on an element of which may be visualized in the following way. 
 porous material. The pore pressure, P, while acting on the total 
 
8 
 
 ILLINOIS STATE GEOLOGICAL SURVEY 
 
 internal boundary of our unit cube in figure 1, is excluded from the solid portions 
 of the external boundary. The effect of such a condition is the same as having 
 the unit cube entirely immersed in a fluid at a pressure, P, and then having a 
 superposed tension normal to the solid portions of the external surfaces of magni- 
 tude -[(l-n)P]. This equivalence is indicated in figure 2. 
 
 -(l-n)P 
 
 1 1 1 1 
 
 f 1 1 
 
 nP — 
 
 IllUllilll 
 
 - — — 
 
 .(l-n)P*- — 
 
 ^nP 
 
 ;i-n)P 
 
 f t t t t t t t t t i 
 P 
 
 -(l-n)P 
 Fig. 2. - Diagram showing equivalence of pore pressure distribution 
 on an element of porous material to the complete immersion of 
 the element, together with a superposed tension on the external 
 solid boundary of intensity: -(1-n) P. 
 
 The strain due to complete immersion in a fluid under pressure is the same 
 in all directions and according to (1) is the linear grain compressibility times the 
 change in pressure 
 
 (|,p 
 
 The strain due to a tension of -(l-n)P on the solid portions of the external 
 boundaries of the unit cube is the linear bulk compressibility, defined by equation 
 (2), times the normal tension at the surface 
 
 -(1-n) ( ^N P 
 
 The strain due to the presence of fluid pressure is therefore 
 
 | -d-n) 
 
 (l-2u| 
 
 (5) 
 
 The expression for the total strain is then obtained by adding (5) to the right hand 
 term in (3) 
 
 - _ °x n / ' ' \ 
 
 e x _ "£— ~ "£ vcr y + a z ) - 
 
 (1-n) 
 
 (1-2(jl) (3 
 
 P (cyclic) (6) 
 
 This is a form of Hooke's law for porous materials taking into account the strain 
 due to pore pressure. It was obtained by Lubinski (1954) and is in terms of the 
 solid stress. 
 
 A form of Hooke's law for porous materials in terms of the total stress will 
 be obtained from (6) by substituting for the solid stresses, a^, cry, and a' z , 
 
ELASTIC PROPERTIES OF SANDSTONE 9 
 
 their expressions in terms of the total stresses and pore pressure (4). It then becomes 
 
 e v = 
 
 o~x 
 E E 
 
 - ■£- (cr v + a- z ) - 
 
 l-2n P 
 
 "T — "3" 
 
 P (cyclic) (7) 
 
 Cleary (1958a) noted several advantages in using this form of Hooke's law 
 for porous material. The porosity term does not appear in the expression. The 
 stress is continuous across an impermeable boundary to the porous system. No 
 body force due to fluid motion through the porous material exists in terms of the 
 total stress. Consequently, the analogy suggested by Lubinski (1954) between 
 thermal stresses and stresses resulting from pore pressure distribution is more 
 conveniently applied. 
 
 The analogy between thermal stresses and stresses due to pore pressure 
 becomes evident by noting that the strain due to pore pressure is uniform in all 
 directions, as is the strain due to temperature changes. The coefficient of P in 
 (7) has a physical significance analogous to the coefficient of thermal expansion. 
 Therefore, it is natural to define the coefficient of P as a single property of the 
 porous material. Thus 
 
 J 
 
 l-2>i P 
 
 (8) 
 
 where J is the coefficient of pore pressure expansion. If we apply this definition, 
 (7) becomes 
 
 e x =^-£ (<^y+^z) -JP (cyclic) (9) 
 
 For zero porosity, J equals zero and the pore pressure term in (9) vanishes. The 
 total stress becomes identical to the solid stress and (9) becomes Hooke's law 
 for isotropic nonporous materials. 
 
 Effective Stress 
 
 The effective stress is a component of the total stress that tends to compact 
 a porous structure, change its resistance to shear, and change its elastic prop- 
 erties by altering its microscopic pore structure. The effective stress is exten- 
 sively used in the field of soil mechanics (Terzaghi and Peck, 1948, p. 52). It 
 is stated in terms of the total stress and the pore pressure 
 
 a x = cr x - P (cyclic) (10) 
 
 It is useful to separate the effective stress from the hydrostatic component of the 
 total stress because changes in elastic properties as stress condition changes can 
 be attributed to changes in the effective component. The hydrostatic component, 
 numerically equal to the pore pressure, has negligible effect on the pore geometry 
 and elastic properties. 
 
 EXPERIMENTAL PROCEDURE 
 
 Apparatus 
 
 The pressure cell used in the experiments consisted of a thick, steel cyl- 
 inder with end closures held in the cylinder by the ram or a large hydraulic press. 
 
10 
 
 ILLINOIS STATE GEOLOGICAL SURVEY 
 
 The core was cemented to the 
 lower closure of the cylinder and a float- 
 ing piston was mounted on the upper end 
 of the core (fig. 3)„ Figure 4, an ex- 
 ploded view of the apparatus, shows the 
 steel cushion plate, 1, that distributes 
 the load from the ram of a hydraulic 
 press to 2, a large brass plunger. A 
 short piston, 3, separates the axial and 
 radial fluid pressures. The test cylin- 
 der, 4, shows connections, 5 and 6, 
 through which the axial and radial fluid 
 pressures were controlled and measured. 
 The core, 7, is mounted on the bottom 
 closure, 8, of the cylinder together with 
 the tubing connected to the closure and 
 communicating with the pores of the 
 specimen. 
 
 Figure 5 shows the apparatus as- 
 sembled in the hydraulic press. Tubing 
 connections from top to bottom control 
 the axial stress, the radial stress, and 
 the pore pressure. The wires connecting 
 to the strain-indicating instrument are 
 also visible. 
 
 r 1 1 
 
 Fig. 3. - Core mounted on the lower 
 closure of the test cylinder, with 
 floating piston mounted on upper 
 end . 
 
 ill 
 
 Fig. 4. - Exploded view of apparatus, 
 
 Strain Measurement 
 
 Strains in the rock samples were 
 measured by means of SR-4, A-12-2, 
 bonded wire resistance strain gages. A 
 Baldwin type L portable strain indicator 
 was used to read strain in the gage wire. 
 
 SR-4 gages consist of a thin 
 strand of fine cupro-nickel gage wire 
 soldered to larger leads. The gage wire 
 and leads are cemented with nitrocellu- 
 lose cement to a paper backing. The 
 gage wire normally has a protective 
 coating of red felt lightly cemented over 
 the top, but this felt covering was strip- 
 ped from the gages for the experiments 
 so that the gage could be smoothly paint- 
 ed over with plastic after it was mounted. 
 
 Most SR-4 gages contain a grid 
 of gage wire. At the bends a small frac- 
 tion of the wire is perpendicular to the 
 gage axis. This introduces an error 
 
ELASTIC PROPERTIES OF SANDSTONE 
 
 11 
 
 because the wire senses some of the strain perpendicular to the gage direction. 
 Ordinarily, this error is unimportant and is ignored. For the present experiments, 
 however, this error would be significant and difficult to correct, so the single 
 strand A- 12-2 gage was adopted. 
 
 Another advantage in using the A-12-2 gage is its relatively long gage length 
 (1 5/8 inches), which permits the measuring of strain over an appreciable portion 
 of the specimen and thus averages out microscopic strain fluctuations. 
 
 To determine the effect of pressure on the readings, gages were mounted 
 on high-purity aluminum and copper rods and subjected to hydrostatic pressure up 
 to 8,000 pounds per square inch. Strain was recorded as a function of the pressure. 
 
 Had the gages functioned ideally and given the actual strains in the alu- 
 minum and copper as a function of pressure, the linear compressibilities of the two 
 metals would have been measured. For copper this is 
 
 86 
 
 and for the aluminum 
 
 8<r 
 
 Se 
 
 = 1.68 x 10 / pound per square inch 
 
 jjTj: — = 3. 14 x 10 / pound per square inch 
 
 In actual fact the slope of the strain pressure curve indicated by the gage 
 was for the copper 
 
 8* 
 
 ST 
 
 = .72 x 10 / pound per square inch 
 
 and for the aluminum 
 8e 
 
 8^ 
 
 = 2. 23 x 10 / pound per 
 
 square inch 
 
 These results were given the fol- 
 lowing interpretation. 
 
 Assume that if the gage were 
 mounted on a completely incompressible 
 sample the gage would indicate a strain, 
 even though no change in length oc- 
 curred. This indicated strain would be 
 due solely to the action of the fluid 
 pressure on the gage. If, on the other 
 hand, the gage is mounted on a com- 
 pressible specimen it will indicate the 
 actual strain in the specimen due to the 
 action of the fluid pressure on the spec- 
 imen plus a superposed apparent strain 
 due to the action of the pressure on the 
 gage. This fictitious strain divided by 
 the pressure it took to produce it we will 
 call the gage pressure coefficient. 
 
 Subtracting the measured from 
 the actual compressibility of copper we 
 obtain as the gage coefficient .9 6 x 10"^ 
 (indicated tensile strain) inches squared 
 
 Fig. 5. - Assembled apparatus in hy- 
 draulic press . 
 
12 ILLINOIS STATE GEOLOGICAL SURVEY 
 
 —8 
 per pound. For aluminum the experimental value of the coefficient is .91 x 10 
 
 inches squared per pound. The significance of this correction is discussed later 
 in this report. 
 
 The pressure coefficient determined here for the A- 12-2 gage is not the same 
 as those reported in the literature for other gages. Apparently the pressure coef- 
 ficient varies considerably and must be obtained by experiment for the particular 
 gage used. 
 
 Accuracy 
 
 Relative pressure measurements were made to an accuracy of about ±10 
 pounds per square inch. The error in the resulting stresses due to error in pressure 
 measurement is about the same. Error in pressure measurement generally amounted 
 to about ±2 percent of the stress and pressure increments measured in the experi- 
 ments. 
 
 The error due to packing friction is about 1 percent of the pressure difference 
 across the packing of the floating piston. The axial stress always was achieved by 
 increasing the pressure from some lower value before taking a reading. As a result, 
 the packing friction would always act in the same direction. 
 
 The error in strain measurement varied widely. The strain-indicating in- 
 strument could be read to an accuracy of ±1 micro-inch per inch. The gage accu- 
 racy as stated by the manufacturer is ±1 percent of the strain measured. This last 
 error is systematic and would not contribute to the scatter of the data points. 
 
 The values of the gage pressure coefficients (see above) probably are sig- 
 nificant only in the first decimal place, and for convenience the mean experimental 
 
 value of 
 
 _p 
 .9 35 x 10 inches squared per pound 
 
 is rounded to 
 
 1 x 10 inches squared per pound 
 
 This correction for changes in pressure on the gage generally is small com- 
 pared to the strain measured in the experiment. 
 
 The most serious errors originating in the strain gages result from a shift 
 in the gage zero. Zero shift can result from any of the following causes. 
 
 Temperature changes may seriously affect the gage zero but were not im- 
 portant in these experiments. The room temperature was fairly constant and the 
 apparatus too massive to allow important temperature changes to take place over 
 the time interval of a single experiment. 
 
 Another important source of difficulty is electrical leakage around the gage 
 due to moisture in the system. Fifty million ohms resistance should be maintained 
 between the gage circuit and ground. 
 
 Zero shift of the gage also may be caused by creep of the gage wire relative 
 to its bonding material when the wire is under strain. 
 
 A shift in the gage zero, probably from this last cause, at times prevented 
 correlation of strains except within a limited range of stress in which the gage was 
 
ELASTIC PROPERTIES OF SANDSTONE 13 
 
 allowed to become " normalized. " An example of this is found in the strain data 
 from the z gage sample 5, table 2. In going from one set of data to the next, the 
 strains did not change systematically with the increased confining stress. After 
 the gage was allowed to relax within a given range of stress, however, the gage 
 would behave reversibly within a limited range of this stress. The trouble could 
 have been avoided entirely by more thorough curing of the plastic cement. 
 
 Perry and Lissner (1955) provide a useful source of information on the tech- 
 nique and limitations of resistance wire strain gages. 
 
 Samples 
 
 Sample 1 was Spar Mountain (" Rosiclare" ) Sandstone, heavily cemented 
 with calcite and having a porosity of 1 percent. 
 
 Sample 2 was Spar Mountain ("Rosiclare") Sandstone, calcite cemented, 
 with 4 percent porosity. 
 
 Sample 3 was Spar Mountain ("Rosiclare") Sandstone, calcite cemented, 
 of 7 percent porosity. 
 
 The Spar Mountain ("Rosiclare") is Mississippian in age and a part of the 
 Valmeyer Series in Illinois. 
 
 Sample 4 was a conglomerate of unknown age consisting of quartz grains 
 1 to 2 mm in size. The quartz grains have secondary quartz growth such that all 
 the crystal faces are well developed and visible. This rock is extremely competent, 
 almost pure quartz, and had a porosity of 9 percent. 
 
 Sample 5 was St. Peter Sandstone, medium-grained, very friable when wet, 
 almost pure quartz, Ordovician in age, and had a porosity of 16 percent. 
 
 Sample 6 was Galesville Sandstone, coarse-grained, nearly pure silica, 
 soft enough to crush in the hand when wet, Cambrian in age, and had a porosity 
 of 22 percent. 
 
 Sample Preparation 
 
 Sandstone specimens with a fair degree of homogeneity were selected from 
 drill cores, and finished to right cylinders 3.45 inches in diameter and 5 inches 
 long. 
 
 Two coats of thermosetting plastic were painted on the cores. The first 
 coat was applied as lightly and quickly as possible and then cured. This pro- 
 cedure kept the imbibition of the plastic into the specimen to a minimum. The 
 coatings were from .01 to .02 inch thick. 
 
 After the second coat of plastic had been applied and cured, a very light 
 lathe cut was made with a carbide tool to obtain a smooth surface for gage appli- 
 cation. 
 
 Plastic also was used in applying the gages. One gage was mounted par- 
 allel to the axis of the specimen, and another perpendicular to the axis. A one- 
 half inch entry hole at the bottom of the sample was packed with copper wire to 
 prevent possible collapse or spalling of the sandstone. 
 
 The mounted cores were subjected to a vacuum and then kerosene was al- 
 lowed to enter the core under atmospheric pressure. With this fairly incompress- 
 ible fluid filling the pores, a uniform pore pressure distribution could be attained 
 in a short length of time when pressure was applied to the core. 
 
 Figure 3 shows a mounted specimen cemented to the bottom closure of the 
 
14 
 
 ILLINOIS STATE GEOLOGICAL SURVEY 
 
 
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ELASTIC PROPERTIES OF SANDSTONE 15 
 
 test cylinder. A short brass piston is cemented to the top of the core. By control- 
 ling the fluid pressure above and below the short piston at the top of the core, the 
 axial and radial stresses in the specimen were controlled. 
 
 The strain-measuring gages were cemented to the core. The soldered wire 
 leads were carried out through the base. 
 
 EXPERIMENTAL RESULTS 
 
 Application of Hooke's Law 
 
 It must be emphasized that equation (9), Hooke's law for porous materials 
 derived earlier, is dependent on linearity of the relation between stress and strain. 
 
 In the succeeding discussion, equation (9) is used to relate one elastic 
 property to another and to relate elastic properties to experimental conditions and 
 measurements. 
 
 Because the experiments show the elastic properties are themselves func- 
 tions of the condition of stress, the stress-strain relation is clearly not linear. 
 Therefore, some justification of the use of equation (9) is required. 
 
 When sufficiently small changes in stress and strain are observed in a 
 porous material the changes will appear to be linear and follow Hooke's law. 
 Therefore, it is legitimate to apply (9) within a limited range of stress condition. 
 We will therefore assume a set of elastic properties applies to each limited range 
 of stress condition. 
 
 To describe the stress condition completely it is necessary to specify the 
 three principal stresses ( <x x , cry, and <x z ) and the pore pressure (P). But, as was 
 pointed out on page 9, it is the effective components of these stresses defined 
 by equation (10) that can influence the elastic properties of the porous material. 
 
 It would be extremely complicated to express the elastic properties in 
 terms of the effective stresses ( a Xl c?y, and cr z ) taken in detail. Therefore, it 
 is convenient to assume that the elastic properties can be specified in terms of the 
 mean effective stress given by 
 
 + o- v + cr 7 
 
 (11) 
 
 Applying equation (10), together with the experimental condition that 
 u x = cry = a r , equation (11) becomes 
 
 + 2 cr r 
 
 - P (12) 
 
 The experimentally determined elastic properties are subsequently meas- 
 ured and plotted as a function of the mean effective stress, <x , defined by equation 
 (12). To assume that the influences of the three principal effective stresses on the 
 elastic properties can be combined in this manner is a convenient simplification 
 not theoretically justified. The main justification of this assumption is the regu- 
 larity with which the experimental values of the elastic properties can be plotted 
 against the function, a . 
 
 Table 1 summarizes the experiments performed and the increment ratios 
 defining the properties measured. 
 
16 ILLINOIS STATE GEOLOGICAL SURVEY 
 
 Young's Modulus 
 
 Table 2 contains the data for the experimental determination of Poisson's 
 ratio, (jl, and Young's modulus, E. 
 
 In this experiment the pressure of the fluid in the annulus of the apparatus 
 was held constant while the pressure above the floating piston at the top of the 
 specimen was varied. The horizontal stress, cr r , in the specimen was equal to the 
 annular pressure. The axial stress was obtained from the relation 
 
 = PlM - P2< a l- a 2> 
 Z a2 
 
 where a z is the axial stress, p^ is the pressure of the fluid above the floating 
 piston, P2 is the pressure in the annulus, a^ is the cross section of the cylinder 
 and a2 is the cross section of the specimen. 
 
 The values of cr r and & z were set at initial magnitudes and strain readings 
 taken. <r z was then changed to a new value holding cr r constant and new strain 
 readings taken. o~ z was then returned to its former value and strain again meas- 
 ured. If everything worked perfectly and the specimen behaved elastically the 
 final strains measured would be equal to the initial strain readings. 
 
 The data in table 2 reveal that the strains do not in general return to their 
 initial values when the specimen is returned to the initial stress condition. These 
 deviations in the strain are of the same order as the expected errors in the experi- 
 ment so they could not be examined quantitatively, but they reveal that a small 
 amount of inelastic set appears to take place with elastic deformation. Inelastic 
 deformation occurs to a greater extent in the direction of the axial stress. 
 
 In several cases in which the inelastic strain was excessive the stress 
 cycle was repeated. Repeating the stress cycle generally resulted in more purely 
 elastic strains. 
 
 An example of partly inelastic behavior is shown in the data for sample 5, 
 trial 3, data set 1, in table 2. In trials 1 and 2 the core was loaded so that the 
 axial stress was greater than the radial stress. In trial 3 the axial stress was less 
 than the radial stress. 
 
 Under conditions of trials 1 and 2 the core tended to shorten, and the data 
 indicate that the core underwent a small amount of inelastic shortening in adjust- 
 ing to the stress conditions. 
 
 Under the conditions of trial 3 the core tended to lengthen and a noticeable 
 inelastic deformation occurred as the core adjusted itself to the new set of deform- 
 ing stresses. 
 
 Because of this large inelastic strain the stresses were repeated to get a 
 better record of the elastic strain. The core had become somewhat adjusted upon 
 the first application of stress so that the strain resulting from the second appli- 
 cation was more elastic. 
 
 Trial 3 was performed on the fifth sample, using an entirely different stress 
 combination, to obtain some verification of the hypothesis that mean effective 
 stress alone may be sufficient for describing changes taking place in the elastic 
 properties as the condition of stress changes. 
 
 In figure 6 the experimental values of the modulus of elasticity are plotted 
 against mean effective stress for all the samples on which this determination was 
 made. The points plotted for sample 5 are distinguishable according to whether 
 the condition <x z > <x r or a z < cr r existed during the course of the experiment. 
 
ELASTIC PROPERTIES OF SANDSTONES 
 
 Table 2. - Determination of Young's Modulus, E, and Poisson's Ratio, /a 
 
 17 
 
 Data 
 set 
 
 ^r 
 lbs 
 sq 
 
 per 
 inch 
 
 e r 
 
 *z 
 
 e r € z 
 
 a 
 
 E 
 
 micro 
 per 
 
 -inch 
 inch 
 
 micro-inch 
 per inch 
 
 lbs 
 sq 
 
 per 
 inch 
 
 
 
 
 
 SAMPLE 
 
 1 - porosity 
 
 1 % 
 
 
 1 
 
 500 
 500 
 500 
 
 1460 
 2680 
 1460 
 
 6175 
 6200 
 6175 
 
 5981 
 5764 
 5980 
 
 
 1020 
 
 5.6 x 10* 
 
 2 
 
 1200 
 1200 
 1200 
 
 2000 
 3520 
 2000 
 
 6072 
 6113 
 
 6070 
 
 5899 
 5670 
 5887 
 
 
 1710 
 
 6.7 x 10 ( 
 
 3 
 
 1000 
 1000 
 1000 
 
 2930 
 5970 
 2930 
 
 6122 
 6195 
 6122 
 
 5732 
 5327 
 5698 
 
 
 2150 
 
 7.8 x 10 f 
 
 4 
 
 2800 
 2800 
 2800 
 
 3040 
 5770 
 3040 
 
 5890 
 5958 
 5882 
 
 5746 
 5432 
 5742 
 
 
 3340 
 
 8.7 x 10* 
 
 5 
 
 3000 
 3000 
 3000 
 
 4850 
 8180 
 4850 
 
 5908 
 5989 
 5908 
 
 5541 
 
 5188 
 5528 
 
 
 4170 
 
 9.6 x 10 ( 
 
 6 
 
 3800 
 3800 
 3800 
 
 5970 
 8710 
 5970 
 
 5849 
 5913 
 5852 
 
 5422 
 
 5155 
 5428 
 
 
 4820 
 
 10.2 x 10* 
 
 7 
 
 5400 
 5400 
 5400 
 
 5490 
 8540 
 5490 
 
 5670 
 5742 
 5670 
 
 5548 
 5225 
 5539 
 
 
 5940 
 
 9.6 x 10* 
 
 8 
 
 4800 
 4800 
 4800 
 
 6770 
 
 9820 
 6770 
 
 5757 
 5812 
 5754 
 
 5377 
 5091 
 5373 
 
 SAMPLE 
 
 3 - porosity 
 
 5540 
 7 % 
 
 10.7 x 10* 
 
 1 
 
 500 
 500 
 500 
 
 1465 
 2690 
 1465 
 
 5785 
 5793 
 5785 
 
 4784 
 4522 
 4784 
 
 
 1020 
 
 4.68 x 10* 
 
 2 
 
 1200 
 1200 
 1200 
 
 2000 
 3520 
 2000 
 
 5589 
 5620 
 5589 
 
 4735 
 4447 
 4715 
 
 
 1720 
 
 5.48 x 10* 
 
 3 
 
 1000 
 1000 
 1000 
 
 2940 
 6000 
 2940 
 
 5650 
 5705 
 5650 
 
 4453 
 3970 
 4465 
 
 
 2180 
 
 6.27 x 10* 
 
 4 
 
 2800 
 2800 
 2800 
 
 3030 
 5780 
 3030 
 
 5286 
 5358 
 5280 
 
 4645 
 4192 
 4668 
 
 
 3330 
 
 5.93 x 10* 
 
 5 
 
 3000 
 3000 
 3000 
 
 4850 
 8200 
 4850 
 
 5300 
 5382 
 5301 
 
 4373 
 3869 
 4373 
 
 
 4170 
 
 6.65 x 10 f 
 
 ,116 
 
 ,184 
 
 .231 
 
 ,234 
 
 ,232 
 
 .225 
 
 ,199 
 
 ,031 
 
 ,111 
 
 ,113 
 
 ,162 
 
 ,162 
 
ILLINOIS STATE GEOLOGICAL SURVEY 
 Table 2. - Continued 
 
 Data 
 set 
 
 lbs 
 
 per 
 
 e r 
 
 € Z € T € Z 
 
 cr 
 
 E 
 
 H- 
 
 micro 
 
 -inch micro-inch 
 
 lbs 
 
 per 
 
 
 
 sq 
 
 inch 
 
 per 
 
 inch per inch 
 
 sq 
 
 inch 
 
 
 6 
 
 3800 
 
 5960 
 
 5205 
 
 4270 
 
 
 7.26 x 10 6 
 
 
 
 3800 
 
 8710 
 
 5262 
 
 3882 
 
 4970 
 
 .152 
 
 
 3800 
 
 5960 
 
 5202 
 
 4265 
 
 
 
 
 7 
 
 5400 
 
 5470 
 
 4980 
 
 4434 
 
 
 8.18 x 10 6 
 
 
 
 5400 
 
 8500 
 
 5033 
 
 4051 
 
 5920 
 
 .144 
 
 
 5400 
 
 5470 
 
 4980 
 
 4409 
 
 
 
 
 8 
 
 4800 
 
 6760 
 
 5075 
 
 4248 
 
 
 7.81 x 10 6 
 
 
 
 4800 
 
 9820 
 
 5135 
 
 3839 
 
 5960 
 
 .137 
 
 
 4800 
 
 6760 
 
 5087 
 
 4215 
 
 SAMPLE 4 - porosity 
 
 9 % - 
 
 Trial 1 
 
 
 1 
 
 500 
 
 500 
 
 1332 
 
 1259 
 
 
 2.43 x 10 6 
 
 
 
 500 
 
 1070 
 
 1340 
 
 1022 
 
 690 
 
 .025 
 
 
 500 
 
 500 
 
 1336 
 
 1254 
 
 
 
 
 2 
 
 1000 
 
 1000 
 
 1203 
 
 1070 
 
 
 3.70 x 10 6 
 
 
 
 1000 
 
 1570 
 
 1209 
 
 911 
 
 1190 
 
 .049 
 
 
 1000 
 
 1000 
 
 1200 
 
 1060 
 
 
 
 
 3 
 
 1500 
 
 1500 
 
 1118 
 
 947 
 
 
 4.10 x 10 6 
 
 
 
 1500 
 
 1070 
 
 1122 
 
 803 
 
 1566 
 
 .043 
 
 
 1500 
 
 1500 
 
 1114 
 
 938 
 
 
 
 
 4 
 
 2000 
 
 2000 
 
 937 
 
 932 
 
 
 5.05 x 10 6 
 
 
 
 2000 
 
 2570 
 
 945 
 
 808 
 
 2190 
 
 .08 
 
 
 2000 
 
 2000 
 
 935 
 
 910 
 
 
 
 
 5 
 
 2500 
 
 2500 
 
 878 
 
 847 
 
 
 5.38 x 10 6 
 
 
 
 2500 
 
 3070 
 
 889 
 
 730 
 
 2690 
 
 .094 
 
 
 2500 
 
 2500 
 
 880 
 
 826 
 
 
 
 
 6 
 
 3000 
 
 3000 
 
 830 
 
 768 
 
 
 5.76 x 10 6 
 
 
 
 3000 
 
 3570 
 
 848 
 
 660 
 
 3190 
 
 .076 
 
 
 3000 
 
 3000 
 
 840 
 
 750 
 
 
 
 
 7 
 
 3500 
 
 3500 
 
 818 
 
 690 
 
 
 6.56 x 10 6 
 
 
 
 3500 
 
 4070 
 
 840 
 
 598 
 
 3690 
 
 .126 
 
 
 3500 
 
 3500 
 
 829 
 
 680 
 
 Trial 2 
 
 
 
 
 1 
 
 500 
 
 500 
 
 857 
 
 1800 
 
 
 2.83 x 10 6 
 
 
 
 500 
 
 1070 
 
 860 
 
 1594 
 
 690 
 
 .022 
 
 
 500 
 
 500 
 
 854 
 
 1790 
 
 
 
 
 2 
 
 1000 
 
 1000 
 
 740 
 
 1638 
 
 
 3.52 x 10 6 
 
 
 
 1000 
 
 1570 
 
 749 
 
 1468 
 
 1190 
 
 .062 
 
 
 1000 
 
 1000 
 
 738 
 
 1622 
 
 
 
 
 3 
 
 2000 
 
 2000 
 
 588 
 
 1415 
 
 
 4.79 x 10 6 
 
 
 
 2000 
 
 2570 
 
 596 
 
 1292 
 
 2190 
 
 .063 
 
 
 2000 
 
 2000 
 
 589 
 
 1407 
 
 
 
 
ELASTIC PROPERTIES OF SANDSTONE 
 
 19 
 
 Table 2. - Continued 
 
 Data 
 set 
 
 lbs 
 
 per 
 
 f r 
 
 micro 
 
 e z 
 
 -inch 
 
 e r € z 
 
 micro-inch 
 
 
 sq 
 
 inch 
 
 per 
 
 inch 
 
 per inch 
 
 4 
 
 3000 
 
 3000 
 
 470 
 
 1269 
 
 
 
 3000 
 
 3570 
 
 482 
 
 1172 
 
 
 
 3000 
 
 3000 
 
 472 
 
 1263 
 
 
 5 
 
 3500 
 
 3500 
 
 420 
 
 1209 
 
 
 
 3500 
 
 4070 
 
 432 
 
 1113 
 
 
 
 3500 
 
 3500 
 
 422 
 
 1206 
 
 
 6 
 
 4000 
 
 4000 
 
 370 
 
 1152 
 
 
 
 4000 
 
 4570 
 
 380 
 
 1076 
 
 
 
 4000 
 
 4000 
 
 372 
 
 1153 
 
 
 7 
 
 4500 
 
 4500 
 
 328 
 
 1109 
 
 
 
 4500 
 
 5070 
 
 337 
 
 1028 
 
 
 
 4500 
 
 4500 
 
 328 
 
 1110 
 
 
 8 
 
 6000 
 
 6000 
 
 179 
 
 962 
 
 
 
 6000 
 
 6570 
 
 189 
 
 890 
 
 
 
 6000 
 
 6000 
 
 180 
 
 963 
 
 
 lbs per 
 sq inch 
 
 H- 
 
 3190 6.07 x 10 .117 
 
 3690 6.03 x 10 .116 
 
 4190 7.45 x 10 .144 
 
 4690 7.00 x 10 .110 
 
 6190 7.86 x 10 .131 
 
 SAMPLE 5 - porosity 16 % 
 Trials 1 and 2, cr > cr 
 
 Trial 1 
 
 Trial 2 
 
 1500 
 
 1500 
 
 1353 
 
 1434 
 
 1360 
 
 935 
 
 
 
 
 1500 
 
 2050 
 
 1375 
 
 1270 
 
 1375 
 
 782 
 
 1591 
 
 3.80 
 
 X 
 
 1500 
 
 1500 
 
 1355 
 
 1410 
 
 1358 
 
 915 
 
 
 
 
 2000 
 
 2000 
 
 1240 
 
 1362 
 
 1245 
 
 868 
 
 
 
 
 2000 
 
 2550 
 
 1262 
 
 1228 
 
 1265 
 
 738 
 
 2091 
 
 4.30 
 
 x 
 
 2000 
 
 2000 
 
 1241 
 
 1355 
 
 1242 
 
 860 
 
 
 
 
 2500 
 
 2500 
 
 1135 
 
 1325 
 
 1131 
 
 821 
 
 
 
 
 2500 
 
 3050 
 
 1154 
 
 1200 
 
 1150 
 
 715 
 
 2591 
 
 4.84 
 
 x 
 
 2500 
 
 2500 
 
 1135 
 
 1322 
 
 1130 
 
 820 
 
 
 
 
 3000 
 
 3000 
 
 1003 
 
 1415 
 
 1028 
 
 785 
 
 
 
 
 3000 
 
 3550 
 
 1023 
 
 1300 
 
 1040 
 
 680 
 
 3091 
 
 5.00 
 
 X 
 
 3000 
 
 3000 
 
 1005 
 
 1420 
 
 1028 
 
 782 
 
 
 
 
 3500 
 
 3500 
 
 930 
 
 1475 
 
 922 
 
 770 
 
 
 
 
 3500 
 
 4050 
 
 945 
 
 1382 
 
 941 
 
 660 
 
 3591 
 
 5.30 
 
 X 
 
 3500 
 
 3500 
 
 928 
 
 1490 
 
 927 
 
 765 
 
 
 
 
 4000 
 
 4000 
 
 831 
 
 1470 
 
 833 
 
 715 
 
 
 
 
 4000 
 
 4550 
 
 844 
 
 1375 
 
 850 
 
 620 
 
 4091 
 
 5.80 
 
 X 
 
 4000 
 
 4000 
 
 830 
 
 1470 
 
 833 
 
 716 
 
 
 
 
 4500 
 
 4500 
 
 
 
 747 
 
 684 
 
 
 
 
 4500 
 
 5050 
 
 
 
 765 
 
 580 
 
 4591 
 
 5.35 
 
 X 
 
 4500 
 
 4500 
 
 
 
 746 
 
 683 
 
 
 
 
 5000 
 
 5000 
 
 
 
 652 
 
 640 
 
 
 
 
 5000 
 
 5550 
 
 
 
 664 
 
 544 
 
 5091 
 
 5.74 
 
 X 
 
 5000 
 
 5000 
 
 
 
 651 
 
 640 
 
 
 
 
 10" 
 
 i cr- 
 
 ier 
 
 10" 
 
 10" 
 
 10 
 
 10" 
 
 10 
 
 131 
 
 .172 
 
 ,167 
 
 .141 
 
 .150 
 
 .160 
 
 .11 
 
 .13 
 
20 ILLINOIS STATE GEOLOGICAL SURVEY 
 
 
 
 
 
 Tabic 
 
 ! 2. - 
 
 Continued 
 
 
 
 Data 
 
 r 
 lbs 
 
 a 
 
 z 
 
 per 
 
 € 
 
 r 
 
 z 
 
 r 
 
 Z 
 
 a 
 
 E 
 
 H- 
 
 set 
 
 micro 
 
 -inch 
 
 micrc 
 
 i-inch 
 
 lbs 
 
 per 
 
 
 
 sq 
 
 inch 
 
 per 
 
 inch 
 
 per 
 
 inch 
 
 sq 
 
 inch 
 
 
 9 
 
 5500 
 
 5500 
 
 
 
 566 
 
 597 
 
 
 6 
 
 
 
 5500 
 
 6050 
 
 
 
 585 
 
 500 
 
 5591 
 
 5.56 x 10 
 
 .197 
 
 
 5500 
 
 5500 
 
 
 
 565 
 
 601 
 
 
 
 
 10 
 
 500 
 
 500 
 
 1640 
 
 1440 
 
 
 
 
 A 
 
 
 
 500 
 
 1050 
 
 1700 
 
 1135 
 
 
 
 591 
 
 1.85 x 10 
 
 .17 
 
 
 500 
 
 500 
 
 1660 
 
 1415 
 
 
 
 
 
 
 11 
 
 1000 
 
 1000 
 
 1495 
 
 1278 
 
 
 
 
 6 
 
 
 
 1000 
 
 1550 
 
 1520 
 
 1060 
 
 
 
 1091 
 
 2.82 x 10° 
 
 .12 
 
 
 1000 
 
 1000 
 
 1498 
 e r 
 
 1230 
 e r 
 
 Trial 
 e z 
 
 3, tr x > 
 € z 
 
 <J 
 
 Z 
 
 
 
 1 
 
 1000 
 
 1000 
 
 1496 
 
 1478 
 
 1230 
 
 1315 
 
 
 2.18 x 10 6 
 
 
 
 1000 
 
 450 
 
 1440 
 
 1432 
 
 1570 
 
 1572 
 
 900 
 
 .165 
 
 
 1000 
 
 1000 
 
 1472 
 
 1479 
 
 1318 
 
 1321 
 
 
 
 
 2 
 
 1500 
 
 1500 
 
 1349 
 
 
 1208 
 
 
 
 6 
 
 
 
 1500 
 
 950 
 
 1312 
 
 
 1379 
 
 
 1419 
 
 3.22 x 10 
 
 .200 
 
 
 1500 
 
 1500 
 
 1343 
 
 
 1209 
 
 
 
 
 
 3 
 
 2000 
 
 2000 
 
 1218 
 
 
 1116 
 
 
 
 A 
 
 
 
 2000 
 
 1450 
 
 1190 
 
 
 1260 
 
 
 1919 
 
 3.82 x 10° 
 
 .097 
 
 
 2000 
 
 2000 
 
 1211 
 
 
 1128 
 
 
 
 
 
 4 
 
 2500 
 
 2500 
 
 1108 
 
 
 1055 
 
 
 
 A 
 
 
 
 2500 
 
 1950 
 
 1082 
 
 
 1180 
 
 
 2419 
 
 4.36 x 10° 
 
 .203 
 
 
 2500 
 
 2500 
 
 1109 
 
 
 1063 
 
 
 
 
 
 5 
 
 3000 
 
 3000 
 
 1006 
 
 
 997 
 
 
 
 6 
 
 
 
 3000 
 
 2450 
 
 988 
 
 
 1105 
 
 
 2919 
 
 5.19 x 10 
 
 .151 
 
 
 3000 
 
 3000 
 
 1002 
 
 
 1000 
 
 
 
 
 
 6 
 
 3500 
 
 3500 
 
 908 
 
 
 948 
 
 
 
 6 
 
 
 
 3500 
 
 2950 
 
 890 
 
 
 1050 
 
 
 3419 
 
 5.45 x 10 
 
 .158 
 
 
 3500 
 
 3500 
 
 904 
 
 
 950 
 
 
 
 
 
 7 
 
 4000 
 
 4000 
 
 810 
 
 
 885 
 
 
 
 6 
 
 
 
 4000 
 
 3450 
 
 790 
 
 
 989 
 
 
 3919 
 
 5.45 x 10 
 
 .188 
 
 
 4000 
 
 4000 
 
 808 
 
 
 890 
 
 
 
 
 
 
 
 
 
 SAMPLE 
 
 ; 6 - porosity 
 
 22 % 
 
 
 
 
 
 
 e r 
 
 € z 
 
 € T 
 
 € z 
 
 
 
 
 1 
 
 500 
 
 550 
 
 1764 
 
 1770 
 
 
 
 
 A 
 
 
 
 500 
 
 1470 
 
 1807 
 
 297 
 
 
 
 670 
 
 1.87 x 10 
 
 .102 
 
 
 500 
 
 550 
 
 1749 
 
 1812 
 
 
 
 
 
 
 2 
 
 1000 
 
 1410 
 
 1580 
 
 1355 
 
 
 
 
 6 
 
 
 
 1000 
 
 2950 
 
 1648 
 
 886 
 
 
 
 1390 
 
 3.16 x 10° 
 
 .146 
 
 
 1000 
 
 1410 
 
 1575 
 
 1390 
 
 
 
 
 
 
 3 
 
 2000 
 
 2810 
 
 1295 
 
 1050 
 
 
 
 
 6 
 
 
 
 2000 
 
 4360 
 
 1344 
 
 611 
 
 
 
 2530 
 
 3.62 x 10 
 
 .119 
 
 
 2000 
 
 2810 
 
 1292 
 
 1025 
 
 
 
 
 
 
ELASTIC PROPERTIES OF SANDSTONE 
 
 21 
 
 Table 2. - Continued 
 
 Data 
 set 
 
 a r 
 
 °z 
 
 € t 
 
 € z 
 
 micro 
 
 *z 
 
 -inch 
 
 a 
 
 E 
 
 
 
 H- 
 
 lbs 
 
 per 
 
 micrc 
 
 -inch 
 
 lbs 
 
 per 
 
 
 
 
 
 sq 
 
 inch 
 
 per 
 
 inch 
 
 per 
 
 inch 
 
 sq 
 
 inch 
 
 
 
 
 4 
 
 3900 
 
 4100 
 
 760 
 
 900 
 
 
 
 
 
 
 10 6 
 
 
 
 3900 
 
 5640 
 
 799 
 
 517 
 
 
 
 4220 
 
 4.18 
 
 X 
 
 .106 
 
 
 3900 
 
 4100 
 
 760 
 
 871 
 
 
 
 
 
 
 
 
 5 
 
 4500 
 
 5560 
 
 640 
 
 580 
 
 
 
 
 
 
 10 6 
 
 
 
 4500 
 
 7100 
 
 688 
 
 171 
 
 
 
 5110 
 
 3.93 
 
 X 
 
 .133 
 
 
 4500 
 
 5560 
 
 632 
 
 545 
 
 
 
 
 
 
 
 
 6 
 
 5000 
 
 5490 
 
 510 
 
 600 
 
 
 
 
 
 
 10 6 
 
 
 
 5000 
 
 7030 
 
 545 
 
 238 
 
 
 
 5420 
 
 4.28 
 
 X 
 
 .107 
 
 
 5000 
 
 5490 
 
 503 
 
 595 
 
 
 
 
 
 
 
 
 7 
 
 5500 
 
 5740 
 
 390 
 
 577 
 
 
 
 
 
 
 10 6 
 
 
 
 5500 
 
 6970 
 
 412 
 
 289 
 
 
 
 5760 
 
 4.41 
 
 X 
 
 .093 
 
 
 5500 
 
 5740 
 
 382 
 
 560 
 
 
 
 
 
 
 
 
 Somple 6 ••• 
 
 Sample 5 0" z >0> + + + 
 CT, < 0". XXX 
 
 Sample 4 trial I ooo 
 trial 2 OOO 
 
 Sample 3 AAA 
 
 Sample I odd 
 
 From Zisman (1933) zzz 
 
 I 
 
 3 4 5 6 7 
 
 Modulus of elasticity, E — 10 lb /in 
 
 Fig. 6. - Modulus of elasticity as a function of the mean ef- 
 fective stress. 
 
22 
 
 ILLINOIS STATE GEOLOGICAL SURVEY 
 
 If there is any difference in the traces made by the two sets of points, the difference 
 lies within the scatter of the data. In this case at least, the hypothesis has been 
 verified. Within experimental error it is sufficient to consider changes in elastic 
 properties as a function of the mean effective stress. 
 
 The values of Young's modulus plotted in figure 6 correlate fairly well with 
 porosity in spite of the fact that the samples vary widely in age and description. 
 Two of the points are from the work of Zisman (19 33a). Zisman's data were obtained 
 by testing at atmospheric conditions a dense quartzitic sandstone. 
 
 Poisson's Ratio 
 Poisson's ratio is the ratio of the radial strain over the axial strain 
 
 Ae, 
 
 * Ae : 
 
 resulting from a change in the axial stress, Acr z . 
 
 In figure 7 the experimental determinations of Poisson's ratio are plotted. 
 
 Sample 6 Sample 5 Sample 4 Sample 3 Sample I 
 
 Trial I & 2 ••• 
 Trial 3 xxx 
 
 Trial I ••• 
 
 Trial 2 xxx 
 
 
 6 
 
 
 • 
 • 
 
 
 5 
 
 
 • 
 
 
 
 
 
 -O 
 
 
 
 • 
 
 o 
 o 
 
 g 
 
 4 
 
 
 
 1 
 
 ib 
 
 
 
 
 CD 
 
 3 
 
 
 
 a> 
 
 > 
 
 2 
 
 
 • 
 
 <D 
 
 c 
 o 
 
 QJ 
 
 5 
 
 1 
 
 
 • 
 
 • 
 
 i i 
 
 Zisman 
 1933 
 
 
 
 
 
 J 
 
 .1 
 
 
 
 .1 
 
 .2 .1 .2 .1 .2 
 Poisson's ratio — dimensionless scale 
 Fig. 7. - Poisson's ratio plotted as a function of mean effective stress. 
 
 Data points are widely scattered because the increments of measured horizontal 
 strain are not large compared with the deviations in radial strain caused by inade- 
 quate control of the radial stress. An error of 10 pounds per square inch would in- 
 troduce a radial strain error of about 3 micro-inches per inch, and the strain in- 
 dicator could be read only within ±1 micro-inch per inch at best. Therefore, the 
 measured horizontal strain increment of about 10 micro-inches per inch was not 
 large compared to the probable errors. The value of Poisson's ratio does not appear 
 
ELASTIC PROPERTIES OF SANDSTONE 
 
 23 
 
 Table 3. - Experimental Values of Poisson's Ratio, fi 
 
 Sample 
 
 Mean value 
 
 for 
 
 <x<2000 lb/in 
 
 Mean value 
 
 for 
 
 o 1 >2000 lb/in 
 
 Mean of all 
 experimental values 
 
 1 
 
 .15 
 
 .218 
 
 .185 
 
 3 
 
 .071 
 
 .145 
 
 .108 
 
 4 
 
 .045 
 
 .109 
 
 .077 
 
 5 
 
 .147 
 
 .168 
 
 .157 
 
 6 
 
 .124 
 
 .112 
 
 .118 
 
 to change a great deal from one specimen to the next nor does it vary a great deal 
 with the condition of stress in a single specimen. 
 
 There is evidently a relation between Poisson's ratio and the mean effective 
 stress. Table 3 shows the mean values obtained when the mean effective stress 
 was less than two thousand pounds per square inch and the mean values under 
 stress conditions greater than two thousand pounds per square inch. With one ex- 
 ception the mean values are greater for conditions of greater mean effective stress. 
 
 Obvious cor- 
 relation between the 
 value of Poisson's 
 ratio and the porosity 
 does not appear. 
 
 Linear Bulk 
 Compressibility 
 
 Strains due to 
 changes in the exter- 
 nal stress, cr , meas- 
 ured while the pore 
 pressure, P, is held 
 equal to zero, are re- 
 corded in table 4. 
 
 The strain gag- 
 es are exposed to a 
 change in fluid pres- 
 sure from one reading 
 to the next, making 
 it necessary to apply 
 the gage pressure co- 
 efficient. As explained 
 earlier in this report, 
 the influence of fluid 
 pressure on the gage 
 produced an indicated 
 strain in addition to 
 the actual strain in the 
 specimen. 
 
 £ 1000 
 
 c 800 
 o 
 
 600 
 
 500 
 
 400 
 
 
 • 
 
 
 /• 
 
 Sample 1 
 
 / 
 
 
 ,/ 
 
 
 /" 
 
 
 / 
 
 
 i" 
 
 
 y 
 
 Trial 1 ••• 
 
 / 
 
 
 7 
 
 Trial 2 *** 
 
 
 
 *• 
 
 
 / 
 
 
 / 
 
 
 / 
 
 
 / 
 
 / 
 
 / 
 
 / 
 
 
 •x 
 
 / 
 
 / 
 
 
 / 
 
 / 
 
 2 4 6 
 
 Compressive strain, e - 
 
 10 
 in /in 
 
 2 4 6 8 10 
 
 Compressive strain, e — 10 in /in 
 
 Fig. 8. - Compressive strain is plotted against the mean 
 effective stress in figures 8 to 12. The data in table 
 3 show tensile strain as positive. The plotted strain 
 readings have been converted so that compressive 
 strain is positive and stress-versus- strain plots as a 
 straight line on logarithmic paper. This was done by 
 subtracting all the strain readings of a single trial 
 from a constant greater than the largest reading. The 
 constant was obtained by trial and error. In figure 8 
 strain versus mean effective stress is measured by 
 the r and z gages on sample 1. 
 
24 
 
 ILLINOIS STATE GEOLOGICAL SURVEY 
 
 Table 
 
 4. - Determination of 
 
 Bulk 
 
 Linear Compressibility 
 
 E 
 
 
 
 SAMPLE 1 - 
 
 porosity 1% 
 
 
 
 
 Trial 1 
 
 
 
 
 Trial 2 
 
 
 a 
 
 e r 
 
 € z 
 
 a 
 
 € T 
 
 € z 
 
 lbs per 
 
 micro-inch 
 
 micro-incl 
 
 n 
 
 lbs per 
 
 micro-inch 
 
 micro-inch 
 
 sq inch 
 
 per inch 
 
 per inch 
 
 
 sq inch 
 
 per inch 
 
 per inch 
 
 1000 
 
 6012 
 
 6073 
 
 500 
 
 6090 
 
 6101 
 
 2000 
 
 5891 
 
 5900 
 
 
 2000 
 
 5879 
 
 5899 
 
 4000 
 
 5674 
 
 5698 
 
 
 4000 
 
 5692 
 
 5693 
 
 6000 
 
 5527 
 
 5519 
 
 
 6000 
 
 5520 
 
 5528 
 
 8000 
 
 5368 
 
 5356 
 
 
 8000 
 
 5361 
 
 5385 
 
 10000 
 
 5191 
 
 5240 
 
 
 7000 
 
 5440 
 
 5455 
 
 9000 
 
 5260 
 
 5302 
 
 
 5000 
 
 5593 
 
 5605 
 
 7000 
 
 5410 
 
 5448 
 
 
 3000 
 
 5760 
 
 5758 
 
 5000 
 
 5570 
 
 5591 
 
 
 1000 
 
 5975 
 
 6040 
 
 3000 
 
 5744 
 
 5751 
 
 
 
 
 
 1000 
 
 5980 
 
 6028 
 
 SAMPLE 2 - porosity 4% 
 
 SAMPLE 4 - porosity 
 
 <T 
 
 e r 
 
 e z 
 
 a 
 
 e r 
 
 e z 
 
 lbs per 
 
 micro-inch 
 
 micro-inch 
 
 lbs per 
 
 micro-inch 
 
 micro-inch 
 
 sq inch 
 
 per inch 
 
 per inch 
 
 sq inch 
 
 per inch 
 
 per inch 
 
 400 
 
 2775 
 
 10127 
 
 500 
 
 850 
 
 1790 
 
 1000 
 
 2507 
 
 9899 
 
 1000 
 
 729 
 
 1620 
 
 2000 
 
 2286 
 
 9819 
 
 2000 
 
 568 
 
 1391 
 
 3000 
 
 2158 
 
 9654 
 
 3000 
 
 441 
 
 1236 
 
 4000 
 
 2040 
 
 9475 
 
 3500 
 
 386 
 
 1172 
 
 5000 
 
 1958 
 
 9367 
 
 4000 
 
 331 
 
 1112 
 
 6000 
 
 1840 
 
 9201 
 
 4500 
 
 283 
 
 1064 
 
 7000 
 
 1740 
 
 9067 
 
 6000 
 
 119 
 
 902 
 
 8000 
 
 1630 
 
 8966 
 
 
 
 
 9000 
 
 1522 
 
 8814 
 
 
 
 
 8000 
 
 1650 
 
 8949 
 
 
 
 
 7000 
 
 1850 
 
 9078 
 
 
 
 
 6000 
 
 1825 
 
 9219 
 
 
 
 
 5000 
 
 1959 
 
 9333 
 
 
 
 
 4000 
 
 2081 
 
 9489 
 
 
 
 
 2000 
 
 2343 
 
 9800 
 
 
 
 
 400 
 
 2606 
 
 10385 
 
 
 
 
ELASTIC PROPERTIES OF SANDSTONE 
 
 25 
 
 Table 4. - Continued 
 
 SAMPLE 5 - porosity 1( 
 
 Trial 1, 
 
 a = a 
 r z 
 
 Trial 2, a 
 
 = <r 
 
 r z 
 
 a 
 
 € T 
 
 a 
 
 e r 
 
 lbs per 
 
 micro-inch 
 
 lbs per 
 
 micro-inch 
 
 sq inch 
 
 per inch 
 
 sq inch 
 
 per inch 
 
 500 
 
 1640 
 
 500 
 
 1657 
 
 1000 
 
 1472 
 
 1000 
 
 1482 
 
 1500 
 
 1339 
 
 1500 
 
 1344 
 
 2000 
 
 1220 
 
 2000 
 
 1223 
 
 2500 
 
 1110 
 
 2500 
 
 1105 
 
 3000 
 
 974 
 
 3000 
 
 998 
 
 3500 
 
 894 
 
 3500 
 
 889 
 
 4000 
 
 790 
 
 4000 
 
 793 
 
 
 
 4500 
 
 701 
 
 
 
 5000 
 
 601 
 
 
 
 5500 
 
 510 
 
 Trial 3, a 
 
 < a 
 
 
 
 Tr 
 
 ial 4, a < 
 
 a 
 
 
 r 
 
 z 
 
 
 
 
 r 
 
 z 
 
 a 
 
 
 e r 
 
 
 
 <x 
 
 
 € r 
 
 lbs per 
 
 
 micro-inch 
 
 i 
 
 
 lbs per 
 
 
 micro-inch 
 
 sq inch 
 
 
 per inch 
 
 1684 
 
 
 
 sq inch 
 
 
 per inch 
 
 693 
 
 
 693 
 
 
 1701 
 
 1193 
 
 
 1497 
 
 
 
 1193 
 
 
 1508 
 
 1693 
 
 
 1358 
 
 
 
 1693 
 
 
 1358 
 
 2193 
 
 
 1240 
 
 
 
 2193 
 
 
 1243 
 
 2693 
 
 
 1127 
 
 
 
 2693 
 
 
 1123 
 
 3193 
 
 
 991 
 
 
 
 3193 
 
 
 1008 
 
 3693 
 
 
 908 
 
 
 
 3693 
 
 
 904 
 
 4193 
 
 
 802 
 
 
 
 4193 
 4693 
 5193 
 5693 
 
 
 808 
 718 
 612 
 528 
 
 
 
 SAMPLE 
 
 6 
 
 - porosity 22% 
 
 
 
 
 a 
 
 
 
 € x 
 
 
 e z 
 
 
 
 lbs per 
 
 micro-inch 
 
 i 
 
 nicro-inch 
 
 
 
 sq inch 
 
 
 P' 
 
 bt inch 
 
 
 per inch 
 
 
 
 500 
 
 
 
 1685 
 
 
 1794 
 
 
 
 1000 
 
 
 
 1502 
 
 
 1589 
 
 
 
 2000 
 
 
 
 1205 
 
 
 1312 
 
 
 
 3000 
 
 
 
 938 
 
 
 1098 
 
 
 
 4000 
 
 
 
 676 
 
 
 895 
 
 
 
 5000 
 
 
 
 433 
 
 
 695 
 
 
 
 6000 
 
 
 
 189 
 
 
 500 
 
 
 
 7000 
 
 
 
 -23 
 
 
 310 
 
 
26 
 
 ILLINOIS STATE GEOLOGICAL SURVEY 
 
 For example, if the specimen is 
 subjected to a thousand pounds per 
 square inch increase in external pres- 
 sure, the indicated compressive strain 
 on the instrument is less than the ac- 
 tual value by one thousand times the 
 gage pressure coefficient or ten micro- 
 inches per inch. 
 
 The strain values in table 4 
 have been corrected for the changes in 
 pressure on the gages. In figures 8 
 through 12, the stress-versus-strain 
 is plotted on logarithmic paper from 
 the data in table 4. The data have 
 been adjusted to fit an approximately 
 straight line. The empirical equation 
 of each curve was then obtained by as- 
 suming that the relation between stress 
 and strain takes the form 
 
 X 
 
 
 
 
 // 
 
 8000 
 
 
 
 ,",* 
 
 6000 
 
 Sample 
 
 2 
 
 — IK 
 
 /V 
 
 "= 4000 
 
 
 
 M m 
 
 \ 
 
 
 
 // 
 
 1 
 
 
 
 /* 
 
 lb 
 
 
 
 / 
 
 5 2000 
 
 
 
 
 | 1000 
 
 
 
 
 0) 
 
 
 
 
 c 800 
 
 
 
 
 o 
 
 
 
 
 
 
 
 
 2 
 
 
 
 
 600 
 
 
 
 
 400 
 
 ■ mJ L_ 
 
 
 1 1 I 
 
 2 4 
 
 Compressive strain, 
 
 6 
 
 ■ 10"' 
 
 8 10 
 in /in 
 
 Fig 
 
 e = K 
 
 1 v 
 
 + K, 
 
 (13) 
 
 9. - Strains measured by z and r gages 
 are plotted versus the mean effective 
 stress for sample 2. 
 
 The constant K2 is a quantity 
 added to each measured value of e to adjust the particular curve to make it plot as 
 a straight line. The application of the first few hundred pounds per square inch 
 of pressure resulted in rather large and unreproducible strains, so strains near zero 
 pressure were not recorded. 
 
 The first derivative of the empirical equation (13) gives the bulk linear com- 
 pressibility as a function of the mean effective stress. Thus 
 
 S°~ 
 
 P = constant 
 
 - K^*" 1 * 
 
 (14) 
 
 6000 
 
 / 
 
 Sample 4 / 
 '/ 
 
 
 / 
 
 4000 
 
 i 
 I 
 
 
 / 
 
 2000 
 
 / 
 
 / 
 
 / 
 
 
 '• 
 
 1000 
 800 
 
 / 
 
 7 
 
 
 600 
 
 400 
 
 - / 
 
 / 
 
 
 4 6 
 
 Compressive strain, 
 
 in /in 
 
 4 6 8 10 
 
 Compressive strain, e — 10 in /in 
 
 Fig. 10. - Strain measured by z and r gages are plotted 
 versus the mean effective stress for sample 4. 
 
 By equation (11) off equals 
 Scr if P is a constant. Scr 
 has been substituted for 
 Sa on the left side of 
 (14). 
 
 The constants K, 
 and X for (14) are given 
 in table 5. The empiri- 
 cal equation (14) is plot- 
 ted in figure 13. 
 
 The data of Car- 
 penter and Spencer (1941) 
 also are plotted in figure 
 13. Carpenter and Spen- 
 cer measured, while keep- 
 ing the pore pressure con- 
 stant, the change in pore 
 volume as a function of the 
 external stress. It was 
 
ELASTIC PROPERTIES OF SANDSTONE 
 
 27 
 
 8000 
 6000 
 
 4000 
 
 Sample 5 
 
 / 
 
 / 
 
 o z -o, -- 550 lb /in 2 ?" 
 
 / 
 
 / 
 
 4 
 
 6 
 
 8 
 
 10 
 
 2 
 
 4 
 
 6 
 
 8 
 
 10 
 
 15 
 
 Compressive 
 
 strain, 
 
 t — 
 
 10 in /in 
 
 
 Compressive 
 
 strain, 
 
 e — 
 
 io" 
 
 in / in 
 
 Fig 
 
 11 . - Strains measured by the r gage are plotted versus 
 the mean effective stress for sample 5. For the right- 
 hand curve the condition a z - a r = 550 lb/in^ is main- 
 tained. For the left-hand curve the condition a z = a T is 
 maintained. This last condition also obtains for the 
 curves in figures 7 to 10. 
 
 Sample: 
 
 2 4 6 8 10 2 4 6 8 10 20 
 
 Compressive strain, e — 10 in /in Compressive strain, e — 10 in /in 
 
 Fig. 12. - Strains measured by z and r gages are plotted 
 versus the mean effective stress for sample 6. 
 
 Table 5. - Empirical Constants for Equation (14) 
 12 4 5 
 
 Gage: 
 
 r 
 
 z 
 
 r 
 
 z 
 
 r 
 
 z 
 
 r 
 
 z 
 
 r z 
 
 K x x 10 6 
 X 
 
 4.86 
 
 ■ .5-85 
 
 2.22 
 .664 
 
 1.11 
 .763 
 
 .596 
 
 .852 
 
 7.70 
 .558 
 
 8.56 
 .555 
 
 2.50 
 .740 
 
 
 1.42 1.065 
 .814 .826 
 
28 
 
 ILLINOIS STATE GEOLOGICAL SURVEY 
 
 i 
 
 O 
 </) 
 0) 
 
 I 
 
 3 
 
 u 
 
 en 0) 
 
 <U 3 
 
 t- — < 
 
 *■* O 
 
 <" c 
 
 <u - 
 
 > <u 
 
 o 
 
 en 
 
 a; 
 u 
 
 c 
 
 0) 
 
 a 
 
 CO 
 
 X> 
 
 c 
 
 (0 
 
 10 
 
 f0 
 
 ■e"8 
 
 a) 
 
 <o 
 
 CD 
 
 >* c 
 
 w n 
 
 Q) O 
 u 
 Q. 10 
 
 E 
 o 
 u 
 
 10 
 X> 
 
 0) 
 
 s - 
 
 S S 
 ■3 o 
 
 3 X! 
 _q CD 
 
 x: 2 
 
 i 
 
 
 u !/ PIOOOI — ^ 'SS3J4S illp3||9 UD3^ 
 
ELASTIC PROPERTIES OF SANDSTONE 
 
 29 
 
 therefore necessary to convert their data to the form giving linear bulk compress- 
 ibility rather than what they called "pore compressibility" as a function of the mean 
 effective stress. 
 
 Their data were plotted on logarithmic paper and an empirical equation was 
 obtained in a manner similar to that described above. This empirical equation took 
 the form of (13) and is stated 
 
 n = K^a + K2 
 
 where n is the cumulative change in porosity. 
 
 The first derivative gives the rate of change of porosity with mean effective stress 
 
 Sn 
 
 = KnXa^" l) (15) 
 
 -P = constant 
 
 On the left sideSa has been replaced by 8<x. Geertsma (1957) gives the follow- 
 ing identity connecting the change in porosity with external stress and the bulk 
 volume dilatation with external stress 
 
 S e T S n 
 
 L8°-J 
 
 P = constant 
 
 So" 
 
 (16) 
 
 constant 
 
 In isotropic materials the linear strains, e, due to a hydrostatic change in stress, 
 are equal to one-third the volume dilatation, e. By dividing the right side of (16) 
 by three and replacing e with e on the left, the relation between the bulk linear 
 compressibility and the " pore compressibility" measured by Carpenter and Spencer 
 is obtained as 
 
 ' Se" 
 
 P = 
 
 : constant 
 
 1 
 = 3 n 
 
 r Snl 
 
 .So-. 
 
 Lso-J 
 
 P = constant ° 
 
 (17) 
 
 From (15) and (17) the empirical formulae of the bulk linear compressibilities 
 were obtained from Carpenter and Spencer's data. These formulae are plotted in 
 figure 13. 
 
 Two correlations are evident in figure 13. The compressibility decreases 
 with decreasing porosity and decreases with mean effective stress. 
 
 Carpenter and Spencer's data show this with greater regularity and indicate 
 a higher compressibility for a given porosity than the data presented herein. 
 
 Zisman's data for compressibility of a quartzitic sandstone also are plotted 
 in figure 13. 
 
 Note that the low porosity specimens seem to approach asymptotically the 
 theoretical compressibility of quartz, which is about 6.2 x 10"° inches squared 
 per pound. 
 
 Another effort was made to confirm the hypothesis that only the mean effec- 
 tive stress need be considered in describing the changes in elastic properties as 
 the condition of stress changes. This time different stress combinations were used. 
 Four trials are given for sample 5 in table 4, in two of which a r was held equal to 
 cr z and the strain was measured as a function of c, In the other two trials cr r < a z 
 as the mean stress was varied and the strain measured. Examination of figure 11 
 will show that the curves for mean effective strain versus stress are about the same 
 for the four trials. 
 
30 ILLINOIS STATE GEOLOGICAL SURVEY 
 
 Linear Grain Compressibility 
 
 A single quartz crystal, being anisotropic, exhibits two linear compressibil- 
 ities. The one parallel to the major axis is 4.95 x 10~8/ pounds per square inch 
 and that perpendicular to the axis is 6.85 x 10" / pounds per square inch. 
 
 A nonporous aggregate of randomly oriented quartz grains behaving as a con- 
 tinuous framework (no relative movements at the grain contacts) would exhibit a 
 compressibility equal to the mean compressibility of a single crystal. For quartz 
 this is 6.2 x 15""/ pounds per square inch. 
 
 Table 6 gives the data obtained for determination of the linear grain compress- 
 ibilities of two samples. The stress-strain relation is linear. Table 7 summarizes 
 the result. 
 
 The fluid pressure to which the gages were exposed changed during the course 
 of this experiment so that it was necessary to apply the gage pressure coefficient 
 correction. The data given in table 6 have been corrected for the pressure on the 
 gage. 
 
 The mean values of the compressibilities measured axially and horizontally 
 in the two specimens were a little greater than the theoretical mean value for quartz. 
 The values measured along the axis differed markedly from the values measured with 
 the horizontal gages indicating a nonrandom grain orientation. This is not surprising 
 but it does suggest the use of compressibility measurements as a tool in the deter- 
 mination of the distribution of grain orientation in sandstone. 
 
 In trial 1 on sample 4 of the external pressure, cr was held at a value greater 
 than the pore pressure, P. This was also true of the procedure used with sample 6. 
 This procedure should insure a closer fulfillment of the requirement that the porous 
 material behave as a unit without relative movements at grain contacts. 
 
 Some difference was noted between the value of the compressibility obtained 
 in trial 1 of sample 4, in which the total stress was greater than the pore pressure, 
 and the value obtained in trial 2, in which the external stress was kept equal to 
 the pore pressure. 
 
 Coefficient of Pore Pressure Expansion 
 
 The coefficient of the pore pressure term in (9) is defined in terms of the 
 other elastic properties by (8) 
 
 It follows from (9) that 
 
 
 r = 
 
 [ 1 - 2u 3 1 
 E 3 
 
 Up. 
 
 a= 
 
 = "J 
 
 constant 
 
 (18) 
 
 The product of J and the change in the pore pressure gives the strain due to 
 changes in the pore pressure, just as the product of the coefficient of thermal ex- 
 pansion and a temperature change gives the strain due to the termperature change. 
 For this reason J was labeled the coefficient of linear pore pressure expansion. 
 
 The measurement of J was accomplished by holding the externally applied 
 stresses, <x z and cr Tt constant and measuring the strains versus pore pressure. 
 
 The data for the determination of J are given in table 8. The mean effective 
 stress is calculated from the stress, a , and the pore pressure, P, for each stress 
 condition. 
 
ELASTIC PROPERTIES OF SANDSTONE 
 
 31 
 
 Table 6. - Determination of Linear Grain Compressibility, £> 
 
 3 
 
 
 
 SAMPLE 
 
 4 - porosity 9% 
 
 
 
 
 Trial 1, o~ 
 
 > P 
 
 Trial 
 
 2, a = P 
 
 Trial 
 
 3, cr = P 
 
 cr 
 
 P 
 
 € z 
 
 o~=P 
 
 e z 
 
 cr=P 
 
 e r 
 
 lbs per 
 
 lbs per 
 
 micro-inch 
 
 lbs per 
 
 micro-inch 
 
 lbs per 
 
 micro-inch 
 
 sq inch 
 
 sq inch 
 
 per inch 
 
 sq inch 
 500 
 
 per inch 
 1642 
 
 sq inch 
 
 per inch 
 
 1000 
 
 500 
 
 1272 
 
 6000 
 
 1349 
 
 2000 
 
 1500 
 
 1193 
 
 1000 
 
 1615 
 
 5000 
 
 1423 
 
 3000 
 
 2500 
 
 1125 
 
 2000 
 
 1562 
 
 4000 
 
 1493 
 
 4000 
 
 3500 
 
 1071 
 
 3000 
 
 1510 
 
 3000 
 
 1564 
 
 5000 
 
 4500 
 
 1041 
 
 4000 
 
 1452 
 
 2000 
 
 1633 
 
 6000 
 
 5500 
 
 990 
 
 5000 
 
 1397 
 
 1000 
 
 1703 
 
 5000 
 
 4500 
 
 1040 
 
 6000 
 
 1343 
 
 1000 
 
 1701 
 
 4000 
 
 3500 
 
 1080 
 
 
 
 2000 
 
 1628 
 
 3000 
 
 2500 
 
 1142 
 
 
 
 3000 
 
 1558 
 
 2000 
 
 1500 
 
 1182 
 
 
 
 4000 
 
 1489 
 
 1000 
 
 500 
 
 1237 
 
 
 
 5000 
 6000 
 
 1419 
 1350 
 
 SAMPLE 6 - porosity 22% 
 
 cr 
 
 P 
 
 € r 
 
 e z 
 
 lbs per 
 
 lbs per 
 
 micro-inch 
 
 micro-inch 
 
 sq inch 
 
 sq inch 
 
 per inch 
 
 per inch 
 
 500 
 
 
 
 1650 
 
 1769 
 
 1000 
 
 500 
 
 1603 
 
 1739 
 
 2000 
 
 1500 
 
 1540 
 
 1697 
 
 3000 
 
 2500 
 
 1466 
 
 1635 
 
 4000 
 
 3500 
 
 1381 
 
 1581 
 
 5000 
 
 4500 
 
 1305 
 
 1512 
 
 6000 
 
 5500 
 
 1230 
 
 1425 
 
 7000 
 
 6500 
 
 1163 
 
 1366 
 
 Sample 4 
 Sample 6 
 
 Table 7. - Experimental Values of the Linear 
 Grain compressibility, |J, lb/in 
 
 3 
 
 z gage 
 cr>P 
 
 5.3 x 10" 
 
 z gage 
 Q-= p 
 
 5.6 x 10" 
 6.1 x 10" 
 
 r 
 
 gage 
 
 
 a - 
 
 : P 
 
 
 7.2 
 
 x 10" 
 
 ■8 
 
 7.4 
 
 x 10' 
 
 ■8 
 
32 ILLINOIS STATE GEOLOGICAL SURVEY 
 
 Table 8. - Determination of the Coefficient of Pore Pressure Expansion, J. 
 
 SAMPLE 5 - porosity lb% 
 
 0" 
 
 P 
 
 € r 
 
 € z 
 
 a 
 
 lbs per 
 
 lbs per 
 
 micro-inch 
 
 micro-inch 
 
 lbs per 
 
 sq inch 
 
 sq inch 
 
 per inch 
 
 per inch 
 
 sq inch 
 
 7000 
 
 100 
 
 315 
 
 488 
 
 6900 
 
 7000 
 
 400 
 
 348 
 
 528 
 
 6600 
 
 7000 
 
 1750 
 
 483 
 
 632 
 
 5250 
 
 7000 
 
 2950 
 
 625 
 
 740 
 
 4050 
 
 7000 
 
 3880 
 
 740 
 
 830 
 
 3120 
 
 7000 
 
 4780 
 
 875 
 
 946 
 
 2220 
 
 7000 
 
 5500 
 
 980 
 
 1032 
 
 1500 
 
 7000 
 
 5750 
 
 1020 
 
 1060 
 
 1250 
 
 7000 
 
 
 
 310 
 
 520 
 
 7000 
 
 
 SAMPLE 6 - porosi 
 
 ty 22% 
 
 
 
 
 Trial 1 
 
 
 
 (J 
 
 P 
 
 € T 
 
 € 2 
 
 a 
 
 lbs per 
 
 lbs per 
 
 micro-inch 
 
 micro-inch 
 
 lbs per 
 
 sq inch 
 
 sq inch 
 
 per inch 
 
 per inch 
 
 sq inch 
 
 6000 
 
 100 
 
 270 
 
 443 
 
 5900 
 
 6000 
 
 1000 
 
 380 
 
 531 
 
 5000 
 
 6000 
 
 2000 
 
 554 
 
 675 
 
 4000 
 
 6000 
 
 3000 
 
 720 
 
 805 
 
 3000 
 
 6000 
 
 4000 
 
 ' 904 
 
 955 
 
 2000 
 
 6000 
 
 5000 
 
 1102 
 
 1149 
 
 1000 
 
 6000 
 
 5500 
 
 1223 
 Trial 2 
 
 1295 
 
 500 
 
 a 
 
 P 
 
 € t 
 
 e z 
 
 a 
 
 lbs per 
 
 lbs per 
 
 micro-inch 
 
 micro-inch 
 
 lbs per 
 
 sq inch 
 
 sq inch 
 
 per inch 
 
 per inch 
 
 sq inch 
 
 7000 
 
 
 
 106 
 
 410 
 
 7000 
 
 7000 
 
 1200 
 
 189 
 
 483 
 
 5800 
 
 7000 
 
 2100 
 
 325 
 
 590 
 
 4900 
 
 7000 
 
 3000 
 
 473 
 
 708 
 
 4000 
 
 7000 
 
 4000 
 
 675 
 
 833 
 
 3000 
 
 7000 
 
 5000 
 
 822 
 
 985 
 
 2000 
 
 7000 
 
 6000 
 
 1043 
 
 1195 
 
 1000 
 
 7000 
 
 6600 
 
 1428 
 
 1230 
 
 400 
 
ELASTIC PROPERTIES OF SANDSTONE 
 
 33 
 
 \ 
 
 a e 
 
 ? 4 
 
 Sampled Sample 6 Sample 5 Sample 6 
 
 r r z 
 
 I 
 
 10 in /lb 
 
 Fig. 14. - The coefficient of pore pressure expansion, J, and the bulk 
 
 l-2i 
 
 are plotted against the mean ef- 
 
 linear compressibilities, 
 
 fective stress. Equation (8) predicts that corresponding curves 
 should differ by a constant equal to the grain linear compressibil- 
 ity. (3/3. The figure shows that corresponding curves (indicated 
 by arrows) are spaced horizontally at nearly constant intervals. 
 The intervals are a little greater than the mean linear compressi- 
 bility of quartz, 6.2 x 10 8 in 2 /lb. 
 
 The strains versus the mean effective stress were plotted on logarithmic 
 paper. The strains given in table 8 were subtracted from a larger number selected 
 so that compressive strain would read positive and the points would fall most nearly 
 on a straight line. An empirical equation having the same form as (13) was then ob- 
 tained from the plotted points. The first derivative of equation (13) is 
 
 s 
 
 8<? 
 
 J cr= constant 
 
 = K!X<f (X 
 
 1) 
 
 (19) 
 
 This is about the same method used in handling the data for bulk linear compressi- 
 bility. 
 
 It follows from (11) that if cr is a constant then 
 
 SP = -So ; 
 
 So, by substituting -§P for 8<x in the left-hand member of (19), we see that the 
 right-hand member of this equation represents values of J in terms of the mean 
 effective stress. 
 
34 
 
 ILLINOIS STATE GEOLOGICAL SURVEY 
 
 Table 9. - Determination of f JE 
 
 Ll-2/J 
 
 SAMPLE 4 - porosity 9% 
 
 lbs per 
 sq inch 
 
 lbs per 
 
 sq inch 
 
 lbs per 
 sq inch 
 
 JE 
 l-2/i 
 
 3000 
 3700 
 4310 
 4970 
 5860 
 6700 
 
 
 1550 
 2690 
 3760 
 4920 
 6000 
 
 Trial 1 
 
 i! 
 
 2570 
 
 .45 
 
 1880 
 
 .54 
 
 1410 
 
 .62 
 
 1070 
 
 .77 
 
 820 
 
 .78 
 
 Trial 2 
 
 4000 
 4430 
 4810 
 5210 
 5940 
 6900 
 
 
 1200 
 2200 
 3210 
 4420 
 5900 
 
 } 
 
 3610 
 2920 
 2300 
 1760 
 1260 
 
 .36 
 .38 
 .40 
 .52 
 .65 
 
 Trial 3 
 
 1800 
 2570 
 3260 
 4360 
 5670 
 
 
 1380 
 2390 
 3775 
 5340 
 
 }- 
 
 1490 
 
 1030 
 
 730 
 
 460 
 
 .56 
 
 .78 
 .79 
 .84 
 
 Si 
 SP 
 
 J cr - constant 
 
 -iqxff 
 
 (x- 1) 
 
 The empirical equation (19) giving J as a function of the mean effective stress is 
 plotted for each sample in figure 14. 
 
 According to (8) the values of J differ from the bulk linear compressibility 
 by a constant, (3/3. To compare theory with experiment, the corresponding ex- 
 perimental curve of the linear bulk compressibility is plotted with J. Note that the 
 differences between the curves are not far from the mean value of the linear grain 
 compressibility of quartz, 6.2 x 10" . 
 
 Surface Pressure Preventing Expansion Due to Pore Pressure 
 
 When an elastic porous material is contained within immovable boundaries, 
 a change in the pore pressure is accompanied by a change in the boundary stress 
 necessary to prevent expansion. By imposing the condition that the three principal 
 
ELASTIC PROPERTIES OF SANDSTONE 
 Table 9. - Continued 
 SAMPLE 5 - porosity \b% 
 
 35 
 
 lbs per 
 sq inch 
 
 lbs per 
 sq inch 
 
 a 
 
 lbs per 
 sq inch 
 
 JE 
 1-2/z 
 
 2000 
 3110 
 3950 
 5000 
 5530 
 6000 
 6470 
 
 175 
 1670 
 2875 
 4190 
 4860 
 5410 
 6000 
 
 Trial 1 
 
 1632 
 
 .74 
 
 1250 
 
 .70 
 
 940 
 
 .80 
 
 740 
 
 .79 
 
 630 
 
 .85 
 
 530 
 
 .80 
 
 Trial 2 
 
 4000 
 4750 
 5350 
 5750 
 6200 
 6600 
 
 140 
 1285 
 2270 
 2840 
 3610 
 4140 
 
 3660 
 
 .52 
 
 3270 
 
 .61 
 
 3000 
 
 .70 
 
 2750 
 
 .59 
 
 2520 
 
 .76 
 
 Trial 3 
 
 5000 
 
 125 
 
 5730 
 
 1275 
 
 6370 
 
 2380 
 
 6720 
 
 2910 
 
 4660 
 
 .64 
 
 4220 
 
 .63 
 
 3900 
 
 .66 
 
 strains are constant, 
 
 it follows from (9) that 
 
 "So-" 
 
 Sp 
 
 JE. 
 
 e = constant 
 IE 
 
 l-2|x 
 
 is seen by recalling the analogous 
 
 The physical significance of y_2 u 
 thermal stress condition. A change in the temperature of an element confined 
 within immovable boundaries is accompanied by a change in the pressure pre- 
 venting the expansion of the element. The ratio between the change in the pres- 
 sure and the change in the temperature is the function a E i where a is the 
 coefficient of thermal expansion. 
 
 In order to measure the property 
 
 JE. 
 
 1-2(JL 
 
 l-2|x 
 
 the pore pressure, P, 
 
 and external 
 
 stress, a , were varied in such a way that the strain in one gage remained con- 
 stant. To the extent that the material is anisotropic, a small amount of strain 
 would have occurred in the other gage direction. The theoretical condition of zero 
 strain in all directions was therefore only approximately maintained. 
 
36 
 
 ILLINOIS STATE GEOLOGICAL SURVEY 
 
 Table 9. - Continued 
 SAMPLE 6 - porosity 22% 
 
 lbs per 
 sq inch 
 
 lbs per 
 sq inch 
 
 a 
 
 lbs per 
 
 sq inch 
 
 JE 
 1-2 fj. 
 
 2000 
 3110 
 3950 
 5000 
 5530 
 6000 
 6470 
 
 175 
 1670 
 2875 
 4190 
 4860 
 5410 
 6000 
 
 Trial 1 
 
 } 
 
 } 
 
 Si 
 
 1630 
 
 .74 
 
 1250 
 
 .70 
 
 940 
 
 .80 
 
 740 
 
 .79 
 
 630 
 
 .85 
 
 530 
 
 .80 
 
 Trial 2 
 
 4000 
 4750 
 5350 
 5750 
 6200 
 6600 
 
 140 
 1285 
 2270 
 2840 
 3610 
 4140 
 
 3660 
 
 .52 
 
 3270 
 
 .61 
 
 3000 
 
 .70 
 
 2750 
 
 .59 
 
 2520 
 
 .75 
 
 5000 
 5730 
 6370 
 6720 
 
 125 
 1275 
 2380 
 2910 
 
 Trial 3 
 
 4660 
 
 .64 
 
 4220 
 
 .63 
 
 3900 
 
 .66 
 
 The experimental values of the function 
 
 JE. 
 
 are given in table 9 . Ex- 
 
 perimental error resulted in a large scatter in these values. 
 
 Fatt (1958b) also evaluated this property indirectly from his compressibility 
 
 data. His values of &£ are consistent with those given here. How- 
 
 L8*J e = constant 
 ever, his measurements are carried all the way to zero stress where the value of 
 
 M 
 
 s^ 
 
 e = constant 
 
 approaches one. The measurements in this report were not 
 
 carried to effective stress much below five hundred psi. 
 
 Rate of Change of Horizontal Stress With Pore 
 Pressure in a Porous Sedimentary Bed 
 
 If a porous material is restrained from horizontal movement and a constant 
 vertical stress is maintained, the relationship between the pore pressure and the 
 horizontal stress is obtained by imposing these stated conditions on (9). The 
 following relation can then be obtained . 
 
ELASTIC PROPERTIES OF SANDSTONE 
 Table 10.- Determination of Z 
 
 37 
 
 lbs per 
 sq inch 
 
 lbs per 
 sq inch 
 
 lbs per 
 sq inch 
 
 lbs per 
 sq inch 
 
 4700 
 4700 
 4700 
 4700 
 4700 
 4700 
 
 SAMPLE 4 - porosity 
 
 3000 \ 
 
 3500 1300 J 1 
 
 4000 2520 X J 
 
 4500 3420 Jl . 
 
 5000 4130 1 J 
 
 5250 4580 J 
 
 3080 
 
 .39 
 
 2150 
 
 .41 
 
 1430 
 
 .56 
 
 960 
 
 .71 
 
 630 
 
 .56 
 
 SAMPLE 6 - porosity 22^ 
 Trial 1 
 
 5700 
 
 3500 
 
 
 
 5700 
 
 3670 
 
 500 
 
 5700 
 
 4030 
 
 1000 
 
 5700 
 
 4750 
 
 2000 
 
 5700 
 
 5320 
 
 3000 
 
 }- 
 
 4040 
 
 .34 
 
 3720 
 
 .72 
 
 3340 
 
 .72 
 
 2760 
 
 .57 
 
 
 
 Trial 2 
 
 6200 
 
 3700 
 
 100 \ 
 1100 J 
 
 6200 
 
 4220 
 
 6200 
 
 4900 
 
 2100 \ 
 
 6200 
 
 5400 
 
 2800 J 
 
 h- 
 
 4100 
 
 .52 
 
 3500 
 
 .68 
 
 3050 
 
 .71 
 
 Scr r 
 
 L Sp 
 
 The function 
 
 8 o- r 
 
 8P 
 
 = constant 
 <j z ~ constant 
 
 = constant 
 
 = constant 
 
 JL. 
 
 (20) 
 
 that was expressed in terms of the other 
 
 elastic properties, 
 
 JL 
 
 , in Cleary (1958a, b) will here be given the designation, 
 
 Z, except when necessary to express it in the derivative form. The conditions of 
 
 horizontal restraint and constant vertical stress are those generally assumed for the 
 
 petroleum reservoir. Integration of (20) gives the expression for the horizontal 
 
 stress underground in terms of the pore pressure, P, and the horizontal stress when 
 
 the pore pressure is zero, c r . 
 
 r o 
 
 o- r = a r + ZP (21) 
 
38 
 
 ILLINOIS STATE GEOLOGICAL SURVEY 
 
 m - u 
 
 dimensionless 
 
 <r z - const. 
 (, = const. 
 
 da v 
 
 Fig. 15. - The empirical equations for— -, where a = constant, 
 
 dP z 
 
 e r = constant, as a function of the mean effective stress are 
 plotted for samples 1, 4, 5, and 6. 
 
 The property Z was measured by maintaining constant horizontal strain and axial 
 stress and measuring the horizontal stress as a function of the pore pressure. The 
 data and calculated values of Z are given in table 10. 
 
 Unfortunately the errors inherent in the apparatus introduced large errors in 
 this particular experiment. These errors resulted from difficulties in controlling ac- 
 curately three pressure readings and the reading on the strain indicator at the same 
 time. 
 
 One of the principal reasons for performing these experiments was to obtain 
 values of Z. Therefore, empirical equations of this function have been obtained 
 from some of the other experimental data. 
 
ELASTIC PROPERTIES OF SANDSTONE 
 
 39 
 
 Z is expressed in terms of other properties by 
 
 Z = 
 
 l-2u 
 
 lTiJ 
 
 1 - 
 
 1ZL 
 
 The bulk linear compressibility, 
 
 "]p = 
 
 S^ 
 
 (22) 
 
 P = constant_ 
 
 , has been expressed in terms of 
 
 8<x J P = constant 
 
 the mean effective stress by (14). The empirical equation (14) together with the 
 mean experimental values of Poisson's ratio and the mean compressibility of quartz 
 can be substituted in (22) to obtain empirical formulae of Z as a function of the mean 
 effective stress for samples 1, 4, 5, and 6. These curves are plotted in figure 15. 
 The empirical equations obtained for Z take the form 
 
 Z = 
 
 8o- r 
 
 SP 
 
 J e r = constant 
 o~z = constant 
 
 1 
 
 + C 
 
 = X 
 
 (23) 
 
 In figure 15 it is evident that Z decreases with increasing mean effective stress and 
 that its value decreases with the porosity of the material. The porosities of sam- 
 ples 1, 4, 5, and 6 are respectively 1, 9, 16, and 2 2 percent. 
 
 FRACTURING 
 
 The elastic behavior of sedimentary rock is important to the understanding 
 of hydraulic fracture mechanics because the deformation involved in separating the 
 fracture faces during fracture induction is very probably an elastic deformation. In 
 addition, the stresses resisting the propagation of the fracture through the rock are 
 in part the result of elastic behavior. It is this second relationship between the 
 elastic behavior and earth stresses that we concern ourselves with here. 
 
 We can with reasonable assurance make the following three statements 
 concerning the extension of hydraulic fractures. 
 
 1) The pressure necessary to open and extend a hydraulic fracture approx- 
 imates the stress in the rock normal to the fracture. 
 
 2) Fractures will tend to form in planes normal to the least compressive 
 stress in the rock. 
 
 3) The horizontal stress in porous rocks increases with the magnitude of 
 the pore pressure. 
 
 Under certain conditions the increase in the horizontal stress as the pore 
 pressure increases is governed by the elastic behavior of the material. The con- 
 ditions necessary for elastic behavior to obtain are discussed by Cleary (1958a). 
 If the stress condition of the rock does lie within the elastic range, then the change 
 in the horizontal stress with pore pressure is governed by equation (21) 
 
 a r = a r + ZP 
 1 x o 
 
 This is equation (41) in Cleary (1958a). 
 
 It is necessary to assume that Z is a constant in order to apply equation (21), 
 The variation in the value of Z has now been measured as the value of the mean 
 effective stress changes, therefore we will now obtain an equation to replace (21) 
 and take into account the changing value of Z. 
 
 Equation (23) is the empirical equation expressing Z as a function of the 
 mean effective stress. We could in principal start with this equation and, by im- 
 posing the proper boundary conditions and integrating, obtain the horizontal stress 
 as a function of the pore pressure. The irrational exponent X would make this dif- 
 
40 
 
 ILLINOIS STATE GEOLOGICAL SURVEY 
 
 ficult in practice, however, so to simplify matters let X equal one and assume a lin- 
 ear relation between Z and the mean effective stress. Such a straight line relation 
 is indicated by the dashed line approximating the experimental values of Z for sam- 
 ple 5 in figure 15. This line will be referred to in a numerical example below. 
 
 The derivative form of Z expressed as a linear function of the mean effective 
 stress is 
 
 Scr r 
 
 SP 
 
 cr z = constant 
 e r = constant 
 
 = C i + C 2 a 
 
 (24) 
 
 where C^ and C2 are experimentally determined constants „ 
 
 Substituting the right-hand member of equation (12) we obtain 
 
 Sov 
 
 LSP J 
 
 cr z = constant 
 e r = constant 
 
 + 2oy 
 
 (25) 
 
 Integration of this equation yields 
 
 P - 
 
 C x + 3/2 
 C 2 
 
 <j z 
 3 
 
 C 3 e 
 
 2/3 C 2 P 
 
 (26) 
 
 This gives us the change in the subsurface horizontal stress as pore pressure changes 
 under the subsurface boundary conditions of constant vertical stress and horizontal 
 strain. 
 
 For a specific example assume that a buried porous strata has the elastic 
 properties of sample 5. In figure 15 approximate values of Z for sample 5 are in- 
 dicated by a dashed line, which is a plot of equation (24). The values of C^ and 
 C2 for this particular sample are, respectively, 0.66 and 1.75 x 10"^. 
 
 For example, let <r z , a constant for all values of P, be equal to 5,000 
 pounds per square inch. The vertical stress, a z , can be computed from the depth 
 of burial of the stratum and the mean density of overlying sediments. 
 
 Let the horizontal stress equal 3,000 pounds per square inch when the pore 
 pressure is 2,000 pounds per square inch. It is possible to obtain such information 
 by measuring, respectively, the bottom hole pressure necessary to open a fracture 
 and the bottom hole shut-in pressure. 
 
 To obtain the horizontal stress from the fracturing pressure, as just sug- 
 gested, one assumes that the fracture is vertical. If the fracturing pressure is less 
 than the overburden pressure, as it is in our example, then it may be supposed that 
 the fracture is vertical. 
 
 The assumed set of values given in the previous paragraphs have been sub- 
 stituted in (26) to compute the curve (26) plotted in figure 16. 
 
 Restrictions on the underground stress conditions due to the negligible ten- 
 sile strength of large masses of sedimentary rock also are shown in figure 16„ They 
 are 45-degree and vertical lines indicating that the horizontal stress, <x r , can not 
 be less than the pore pressure, P, whereas P can not be greater than the vertical 
 stress, cr z . 
 
 The curvature of line (26) in figure 16 is not great and could have been well 
 approximated by linear relation (21), assuming an average value of Z was selected. 
 
ELASTIC PROPERTIES OF SANDSTONE 
 
 41 
 
 Pore pressure, P — 1000 lb /in 
 
 Fig. 16. - The change in horizontal stress with 
 change in pore pressure, calculated for a 
 hypothetical buried porous stratum that has 
 the properties of sample 5. 
 
 DISCUSSION AND CONCLUSIONS 
 
 The hypothesis which holds that it is sufficient in measuring elastic proper- 
 ties of sandstone to consider the mean effective stress has been tested for validity 
 by plotting the experimentally determined properties against mean effective stress 
 and relating one property to another for a given value of the mean effective stress. 
 Within the limited accuracy of these experiments, it appears that the elastic prop- 
 erties may be treated as functions of the mean effective stress. 
 
 As mean effective stress increased, the following regular variation in the 
 elastic properties was observed: 
 
 1) The modulus of elasticity increased. 
 
 2) The bulk linear compressibility decreased. 
 
 3) Pois son's ratio increased, but only slightly. 
 
 The rate of change of the horizontal stress in a buried porous stratum as the 
 pore pressure changes has been called Z. Under subsurface conditions, as the 
 pore pressure increases the value of Z increases. Z must always be less than one. 
 
 The elastic properties varied from one sample to the next as a moderately 
 regular function of porosity. In general as the porosity decreased 
 
 1) the modulus of elasticity increased, 
 
 2) the bulk linear compressibility decreased, 
 
 3) Poisson's ratio showed no regular variation, and 
 
 4) Z approached zero. 
 
 If the specimens had been more closely related geologically, a more regular 
 variation in the elastic properties with changes in porosity would have been observed. 
 
42 ILLINOIS STATE GEOLOGICAL SURVEY 
 
 Fatt (19 58b) has stated that intergranular material such as clay may contrib- 
 ute little or nothing to the elastic structure of the material and yet serve to reduce 
 the porosity. Therefore, the presence of soft intergranular material will fog some- 
 what the correlation between the elastic properties and porosity. 
 
 These experiments have served to show that sandstones have fairly predict- 
 able elastic behavior. Further experimental work may make it possible to predict 
 the elastic behavior of buried porous sediments on the basis of lithology, porosity, 
 geologic history, and stress condition. 
 
 The increase in the horizontal stress in porous strata as pore pressure in- 
 creases, predicted in Parts I and II, has been confirmed experimentally. 
 
 Hydraulic fracturing pressures generally are less than the computed over- 
 burden pressure. It may therefore be supposed that the fractures are vertical. If 
 this is true the fractures are then controlled by the horizontal stress. If the pore 
 pressure is changed within a given zone, the horizontal stress changes and along 
 with it the zone's tendency to fracture. Because the change in the stress may dif- 
 fer from one zone to the next, the vertical extent of the fracture or fractures may be 
 influenced by the pore pressure. 
 
 These considerations may be of importance in secondary recovery operations 
 as well as in the stimulation of primary oil production. 
 
ELASTIC PROPERTIES OF SANDSTONE 43 
 
 REFERENCES 
 
 Adams, S. H., and Williamson, E. D. , 1923, The compressibility of minerals and 
 rocks at high pressure: Franklin Inst. Jour., v. 195, p. 475-529. 
 
 Biot, M. A., 1941, General theory of three dimensional consolidation; Jour. Appl. 
 Physics, v. 12, p. 155-164. 
 
 Biot, M. A., 1955, Theory of elasticity and consolidation for a porous anisotropic 
 solid: Jour. Appl. Physics, v. 26, p. 182-185. 
 
 Birch, F. , 1943, Elasticity of igneous rocks at high temperature and pressure: 
 Geol. Soc. America Bull., v. 54, p. 263-285. 
 
 Birch, F., and Bancroft, D. , 1938, Effect of pressure on the rigidity of rocks, Parts 
 I, II; Jour. Geology, v. 46, no. 1, p. 59-87, and no. 2, p. 113-141. 
 
 Brandt, H., 1955, Study of the speed of sound in porous granular media: Trans. 
 Amer. Soc. Mech. Engrs., v. 22, p. 470-486. 
 
 Carpenter, C. B., and Spencer, G. B. , 1941, Measurement of the compressibility 
 of consolidated oil bearing sandstones: U. S. Bur. Mines Rept. Inv. 3540. 
 
 Cleary, J. M., 1958a, Hydraulic fracture theory, Part I. - Mechanics of materials: 
 Illinois Geol. Survey Circ. 251. 
 
 Cleary, J. M., 1958b, Hydraulic fracture theory, Part II. - Fracture orientation and 
 possibility of fracture control: Illinois Geol. Survey Circ. 252. 
 
 Dane, E. B. , Jr., 1942, Handbook of physical constants: Geol. Soc. America 
 Spec. Paper 36, p. 28. 
 
 Fatt, I., 1958a, Pore volume compressibilities of sandstone reservoir rocks: Jour. 
 Petroleum Technology Tech. Note 2004, v. X, no. 3. 
 
 Fatt, I., 1958b, Compressibility of sandstones: Am. Assoc. Petroleum Geologists 
 Bull., v. 42, no. 8, p. 1924. 
 
 Gassmann, Fritz, 1951, Uber die elastizitat poroser Medien; Naturforschenden 
 Gesellshaft Vierteljahrsschrift, Zurich, v. 96, no. 1, p. 1. 
 
 Geertsma, J., 1957, Effect of fluid pressure decline on volume changes in porous 
 rocks: Jour. Petroleum Technology, v. IX, no. 12, p. 331. 
 
 Hall, H. N. , 1953, Compressibility of reservoir rocks: Jour. Petroleum Technology 
 Tech. Note 149, p. 17. 
 
 Hubbert, M. K. , and Willis, D. G., 1957, Mechanics of hydraulic fracturing: 
 Jour. Petroleum Technology, v. LX, no. 6, p. 15 3-166. 
 
 Ide, J. M., 1936, Elastic properties of rocks, a correlation of theory and experi- 
 ment: Natl. Acad. Sci.Proc, v. 22, no. 8, p. 482-496. 
 
 Lubinski, Arthur, 1954, Theory of elasticity for porous bodies displaying a strong 
 pore structure: Proc. Second U. S. Natl. Cong. Appl. Mech., p. 247. 
 
 Perry, C. C, and Lissner, H. R. , 1955, The strain gage primer: McGraw-Hill 
 Book Co., New York. 
 
 Terzaghi, Karl, and Peck, R. B., 1948, Soil mechanics in engineering practice: 
 Wiley & Sons, New York. 
 
44 ILLINOIS STATE GEOLOGICAL SURVEY 
 
 Timoshenko, S., and Goodier, J. M., 1951, Theory of elasticity: McGraw-Hill 
 Book Co., New York. 
 
 Zisman, W. A., 19 33a, Young's modulus and Poisson's ratio with reference to geo- 
 physical applications: Natl. Acad. Sci. Proc, v. 19, no. 7, p. 653-665. 
 
 Zisman, W. A., 19 33b, Compressibility and anisotropy of rocks at or near the earth's 
 surface: Natl. Acad. Sci. Proc, v. 19, no. 7, p. 680-686. 
 
 Illinois State Geological Survey Circular 28J) 
 44 p., 16 figs., 10 tables, 1959 
 

 CIRCULAR 281 
 
 ILLINOIS STATE GEOLOGICAL SURVEY 
 
 URBAN A ■*H* 1 "