s 
 
 14. GS: 
 
 £qtol SufbOi^l 
 
 CIR ASA 
 
 1 
 
 C £ 
 
 STATE OF ILLINOIS 
 
 &^ 
 
 
 WILLIAM G. STRATTON, Governor 
 
 
 DEPARTMENT OF REGISTRATION AND EDUCATION 
 
 
 VERA M. BINKS, Director 
 
 A 
 
 
 VJ*/ 
 
 
 
 Hydraulic Fracture Theory 
 
 
 Part II. — Fracture Orientation 
 
 
 and 
 
 
 Possibility of Fracture Control 
 
 
 James M. Cleary 
 
 
 DIVISION OF THE 
 
 
 ILLINOIS STATE GEOLOGICAL SURVEY 
 
 
 JOHN C FRYE, Chief URBANA 
 
 
 CIRCULAR 252 1958 
 
 
 ILLINOIS GEOLOGICAL 
 
 
 SURVEY LIBRARY 
 
 
 JUN 10 1956 
 
HSSS ! STATE GEOLOGICAL SURVEV 
 
 3 3051 00003 8772 
 
HYDRAULIC FRACTURE THEORY 
 
 Part II. — Fracture Orientation and 
 
 Possibility of Fracture Control 
 
 James M. Geary 
 
 ABSTRACT 
 
 This study takes up the problem of hydraulic fracture mechanics, 
 orientation of fractures, and whether their control is possible. In Part 
 I, some theories on the mechanics of materials were adapted for use 
 in dealing with mechanical problems of hydraulic fracture and also to 
 help in describing conditions of stress in porous sediments. 
 
 Part II summarizes some current notions of hydraulic fracture me- 
 chanics, including my own view. It sets forth sample problems based 
 on the theory developed in Part I. 
 
 Hydraulic fracture orientation and distribution are controlled by 
 the condition of stress underground. The horizontal stress underground 
 is altered by the pore pressure. Thus the magnitude of the pore pres- 
 sure may influence the orientation and distribution of hydraulic frac- 
 tures. 
 
 INTRODUCTION 
 
 It is generally agreed that the compressive stress in the rock at the time 
 of fracture will tend to control the orientation of the fracture. Although numer- 
 ous other factors may enter into the problem the compressive stresses are prob- 
 ably a dominant influence. 
 
 Using this major premise, together with the theory stated in Part I of this 
 report, sample problems dealing with hydraulic fracture mechanics are now pre- 
 sented. A general discussion of ideas in the current literature regarding frac- 
 ture orientation also is included here. 
 
 The third part of this report, now in preparation, will deal with laboratory 
 experiments suggested by the theoretical studies presented in Parts I and II. 
 
 Definitions and Postulates 
 
 The following terms are commonly used in the discussion of hydraulic 
 fracturing. The symbols are defined in Part I. 
 
 1) The overburden pressure, cr z , is the total vertical compressive stress, 
 approximately equal to the average specific weight of overlying sediment times 
 the depth. 
 
 2) Treatment pressure, Pi, is the bottom hole pressure during the exten- 
 sion of a fracture. 
 
 3) Breakdown pressure, Pi , is the bottom hole pressure at the instant be- 
 fore fracture initiation. This may be equal to or greater than the treatment pres- 
 sure. 
 
 The following is a list of postulates to be used in the next section. Some 
 of these are given little defense and are submitted to the reader as highly reason- 
 able premises on which there is general agreement in the literature: 
 
 a) The initiation of a fracture according to (40), above, requires that 
 
 pa a t + a n 
 
 [1 ] 
 
2 ILLINOIS STATE GEOLOGICAL SURVEY 
 
 where P is the pore fluid pressure, cr t is the tensile strength, and <x n is the 
 compressive stress normal to the plane of impending fracture. If the tensile 
 strength of the rock is negligible or zero, then, according to (37) 
 
 P = cr n (42) 
 
 is a condition of impending fracture. 
 
 b) The propagation in a brittle material of a crack due to internal pressure 
 is accompanied by a high stress concentration at the edge of the crack, so that 
 only a small fraction of the tensile strength of the material, small in itself, is 
 effective in resisting the propagation of the fracture. Therefore, the treatment 
 pressure, P^, necessary to extend a fracture exceeds by a negligible amount 
 the pressure necessary to hold open the fracture, or 
 
 Pi- <r n (43) 
 
 c) When the control of joint systems and planes of weakness on the course 
 of the fracture are taken into account, the displacement of the fracture faces 
 will tend to be in the direction of the least principal compressive stress whereas 
 the fracture may follow a tortuous path whose general course would tend to be 
 normal to the least stress. 
 
 d) Local stress conditions may be created by the shape of the well bore, 
 fluid pressure gradients, or plastic deformation before the initiation of a frac- 
 ture. Such local stress fields may influence the orientation of the fracture and 
 the breakdown pressure. However, after the fracture has been initiated, the 
 presence of the well bore and the local conditions around the well bore are of 
 negligible influence and the pressure of the fluid extending the fracture will be 
 equal to the regional stress normal to the displacement of the fracture. 
 
 Evidence of Fracture Orientation from Treatment Pressures and Regional Faulting 
 
 Hubbert and Willis (1957) point out that the orientation of the least com- 
 pressive stress may be inferred from the faulting of a region. 
 
 If it appears that the observed faulting has been active up to the present, 
 it is likely that the stress condition is close to failure. The condition of failure 
 is approximated by Hubbert and Willis by means of the Mohr-Coulomb construc- 
 tion (fig. 8). 
 
 On the premise that hydraulic fractures are developed normal to the least 
 compressive stress for the region, the orientation of fractures are then known on 
 the basis of faulting. 
 
 The most common type of faulting in sedimentary basins is normal faulting, 
 in which case the least normal stress is horizontal and the greatest stress is 
 vertical. McGuire et al. (1954) pointed to normal faulting on the Gulf Coast as 
 evidence that the least compressive stress was horizontal and because treatment 
 pressures were less than would support the overburden they concluded that frac- 
 tures were vertical. 
 
 If a strong parallelism of normal faulting exists, then, according to Hubbert 
 and Willis, hydraulic fractures should be vertical and parallel also. If this is 
 the case it should be possible to establish line drive waterfloods in some areas 
 by connecting input wells with vertical fractures, an interesting possibility. 
 
 Hubbert and Willis' conclusions were based heavily on the assumption 
 that horizontal fractures require treatment pressures equaling the total over- 
 burden load. This assumption was also made by McGuire et al. (1954) and 
 
HYDRAULIC FRACTURE THEORY 3 
 
 Walker (1949). The following argument is based on this assumption: Treatment 
 pressures are consistently less than overburden pressures. The treatment pres- 
 sures must be greater than overburden pressure in order to induce a horizontal 
 fracture. Therefore, fractures are consistently vertical. 
 
 Those who take the contrary view will argue that horizontal fractures can 
 be induced with pressures less than the average overburden pressure, because 
 the overburden pressure is not evenly distributed and, therefore, horizontal 
 fractures might be induced along paths of least overburden load. This mechanism 
 was suggested by Yuster and Calhoun (1945) to explain the low parting pressures 
 obtained in waterflooding. The fractures at that time were generally assumed to 
 be horizontal. 
 
 Certainly we must admit that a certain unevenness in the vertical stress 
 distribution exists, as few things are uniform in nature; but it is difficult to be- 
 lieve that the strength of sedimentary rocks is sufficient to sustain such large 
 scale unevenness in overburden pressure distribution through geologic time at 
 depths of several thousand feet, as would be required to explain the deficit be- 
 tween the usual treatment pressures and overburden pressures. One may also 
 question why such distribution should exist if it could exist. This view that the 
 rocks are not strong enough was taken by McGuire et al. (1954), and the Russian 
 workers Zeltov and Kristionovich (1955). 
 
 Another objection arising when uneven overburden pressure is used to ex- 
 plain low treatment pressures is the likelihood that a well will be drilled in an 
 area of higher than average overburden pressure as often as in areas of low over- 
 burden pressure. Unequal overburden distribution might perhaps be important 
 for shallow depths, which were the circumstances for which the idea was first 
 suggested by Yuster and Calhoun (19 45). 
 
 Proponents of the view that horizontal fractures are in fact produced with 
 lower than overburden treatment pressures will point to gamma ray logs that give 
 a high reading at a single spot in the hole after the injection of radioactive sand, 
 thus indicating a horizontal fracture even though the treatment pressure was less 
 than the average overburden pressure. 
 
 There are two (and perhaps more) ways that such evidence could be mislead- 
 ing. We may have a vertical fracture in which the radioactive sand has been over- 
 flushed away from the hole at all but a single point, or we may have a vertical 
 fracture inclined to the hole at a small angle and intersecting the hole within a 
 distance of a few feet. It is assumed that when we use the term vertical we are 
 using it loosely and include fractures inclined some 30° to the well bore. 
 
 One evidence for the production of horizontal fractures at less than over- 
 burden pressure is the reported propagation of fractures from an input well to a 
 producing well during waterflooding such as described by Yuster and Calhoun. 
 This feat is easy to visualize with horizontal fractures, but difficult if vertical 
 fractures are assumed. I am not certain of the ease with which this experiment 
 can be duplicated, so for me the matter is unsettled. 
 
 Control of Fractures 
 
 In spite of the evidence that regional stresses control the orientation of 
 hydraulic fractures, those who maintain that fractures may be controlled by well 
 bore and fracture treatment conditions also have a case. The ability of the fluid 
 to penetrate the rock, and boundary conditions such as the shape of the hole and 
 the thickness of the bed, will influence the local stress pattern that develops 
 
ILLINOIS STATE GEOLOGICAL SURVEY 
 
 1 
 
 \ 
 
 7777^ 
 
 before breakdown occurs, and may con- 
 trol the path of initial breakdown. 
 
 A great deal of significance has 
 been attached to the degree of fluid 
 penetration and its effect on orienta- 
 tion of hydraulic fractures. 
 
 An experimental study by Scott, 
 Bearden, and Howard (19 53) demon- 
 strated rather graphically what the 
 effect of fluid penetration may be. 
 They fractured drill cores with both 
 penetrating fluids and an oil-base 
 mud whose effective filtration rate 
 was assumed to be zero. These tests 
 seemed to indicate that a penetrating 
 fluid would rupture the core along the 
 bedding planes and that a nonpenetrat- 
 ing fluid broke the rock along the axis 
 of the bore hole. Largely on the basis 
 of this evidence, Clark and Reynolds 
 (1954) proposed a process, "Vertifrac", 
 which consisted essentially of fracture 
 treatment with a nonpenetrating fluid. 
 
 When a nonpenetrating fluid is 
 used the shape of the hole clearly de- 
 fines the shape of the pressurized 
 region that delivers the parting force to 
 the rock until breakdown occurs. 
 Figure 13 illustrates a well bore to be 
 fractured with a nonpenetrating fluid. 
 The pressure of the injected fluid is 
 
 confined to section A, by an expandable packer at the top and at the bottom by 
 a gravel pack. The fluid in the hole cannot penetrate A significantly because 
 of mud cake built on the rock walls. When the fluid pressure is applied, it is 
 confined to a cylindrical region defined by these boundaries. 
 
 What part of the axial thrust is delivered to the surrounding rock is uncer- 
 tain. The upward thrust is transmitted diffusely to the rock walls by the ce- 
 mented casing. The downward thrust is transmitted by shear forces distributed 
 over the surface of contact between the wall of the hole and the gravel, and also 
 against the bottom of the hole by the gravel. Such a diffuse distribution of the 
 axial thrust leads one to suspect that the change in the axial stress within the 
 section being considered is small compared to the change in the tangential 
 stress due to the fluid pressure applied directly to the walls of the hole. In 
 fact, it seems fairly certain that the initial breakdown must occur along the axis 
 of the well bore. The example cited above roughly describes the "Vertifrac" 
 process, which will be further dealt with in a sample problem. 
 
 A tool for the control of fracture orientation was reported recently by Gilbert 
 et al. (1957). It consists of an assembly of shaped charges that in effect pro- 
 duce an oriented notch in the wall rock. When the fracturing pressure is applied, 
 the fracture is alleged to propagate from the notch in a controlled orientation be- 
 cause of the oriented stress pattern that develops before breakdown. 
 
 Fig. 13. - Fracturing with a "non- 
 penetrating fluid." The pressure 
 is confined to the well bore by 
 mud filter cake built on the walls. 
 
HYDRAULIC FRACTURE THEORY 5 
 
 If an oriented crack is produced by this tool, it is easy to believe that the 
 fracturing fluid might enter and open a fracture in this plane in preference to a 
 plane normal to the least stress along which no pre-existing fracture exists. 
 
 As it may be possible to control the fracture as it is first propagated from 
 the well bore by a locally altered stress field, and as we recognize the tendency 
 of the fracture to lie normal to the least principal stress, we are faced with the 
 question of whether the fracture will continue in its "controlled" path or will 
 align itself to the regional stress field as it is extended. 
 
 In my opinion, a fracture induced in a non-preferred orientation but follow- 
 ing a tortuous path due to variously oriented planes of weakness will have a 
 slight selectivity in choice of path that would gradually orient the general path 
 taken by the fracture into the preferred plane. On the other hand, if the frac- 
 ture is initiated along a single dominant plane of weakness, such as a bedding 
 plane, it is reasonable that the fracture may continue in this plane in spite of 
 a regional stress field that would favor vertical orientation. 
 
 In any case, fractures can be extended only by pressures greater than the 
 compressive stress in the direction of fracture displacement. Therefore "con- 
 trolled fractures" should be verified by the treatment pressures it took to produce 
 them. 
 
 SAMPLE PROBLEMS 
 Fracturing with a non-penetrating fluid 
 
 The well bore subjected to internal fluid pressure, where the fluid is pre- 
 vented from filtering out into the surrounding rock by a mud filter cake, has 
 often been treated as a cylinder of infinite wall thickness subjected to internal 
 and external pressure. 
 
 The Lame formulae and special cases of the Lame formulae follow; 
 
 o- re r e 2-P i r i 2-lii^( P ._cr re ) 
 
 r, 
 
 T 
 
 ^? 
 
 cr r 
 r e e 
 
 9 o r i 2r e 2 , x H4) 
 
 ri 2 
 
 <x„ = cr„ = constant 
 
 z ^o 
 
 Letting r e -*oo we obtain the special case of (44) for a cylinder of infinite 
 wall thickness 
 
 ^6 = ^r e "-fr(Pi - <?r e ) d 
 
 -. 2 
 
 °r = °r e + ~pr ( Pi " °"r e ) e 
 
 a z - a z Q = constant 
 where rj is the bore radius, Pi is the fluid pressure in the well bore, and a Te is 
 the regional horizontal stress. For simplicity we will assume that the two prin- 
 cipal horizontal stresses are equal to a r . We will further assume that the 
 
6 ILLINOIS STATE GEOLOGICAL SURVEY 
 
 tensile strength of the rock is negligible, so the condition of impending tensile 
 failure will be P = <r n , where P is the pore pressure and a n is the total stress 
 normal to the plane of failure. 
 
 The problem will be to determine the bottom hole pressure, Pi, necessary 
 for breakdown and the pressure necessary to extend the fracture in the vertical 
 plane. 
 
 For breakdown to occur along the axis of the well bore, a necessary condi- 
 tion is P = (Tq. Substituting P for <Xq in (44d) and letting r equal rj gives us 
 
 Pi = 2 a TQ - P 
 
 as the well bore breakdown pressure. But according to (41) 
 
 're^ro+h-^P 
 
 So 
 
 Pi= 2cr ro + 2[^]P-P (45) 
 
 Let -^ — 1 equal .7, the value used previously for a porous sand. Then 
 
 Pi = 2 cr ro + . 4 P 
 
 where 0"r o is the regional horizontal stress at zero pore pressure. 
 
 Equation (45) indicates that for a porous sand the breakdown pressure will 
 increase with increasing pore fluid pressure. For a hard sand JE 
 and the breakdown pressure is 1 - h 
 
 Pi=2<r ro -P 
 
 After breakdown the system changes. The fluid pressure then acts within 
 the fracture and according to postulate (d) the fluid pressure falls to the value of 
 the regional compressive stress normal to the fracture, or simply Pi = o" r . 
 Letting JE _ and using equation (41) this becomes 
 
 Pi = a r Q + . 7 P 
 
 and for the hard sand, letting JE _ 
 
 1 - y. 
 
 Pi = ov = constant 
 
 These results are plotted in figure 14 for arbitrary values of P and Pi, denoting 
 either the treatment pressure or breakdown pressure, P, the pore fluid pressure 
 and 0"rn' t ' ie regional horizontal stress, when P equals zero. 
 
 The theory presented here governing breakdown pressure is not likely to 
 have much practical application because actual breakdown pressures must be 
 governed by many capricious unknowns such as the effective tensile strength of 
 the rock imperfections in the hole, and possibly plastic yield of the rock around 
 the hole . 
 
 The treatment pressure, on the other hand, has little to do with the hole 
 condition (we are not taking fluid-flow resistance into account), and it is not 
 dependent on the tensile strength of the rock, according to postulate b. It is 
 therefore much more predictable. 
 
 Note in figure 14 that the treatment pressure for the soft sand rises steadily 
 with .7 P . The treatment pressure for the hard sand is constant for P < a TQ and 
 increases with P for the interval P > ov , according to (37). 
 
HYDRAULIC FRACTURE THEORY 
 
 14 
 
 
 
 
 13 
 
 
 
 ^..-"" 
 
 12 
 
 
 
 ^. .'-•"' ..••■' 
 
 I 1 
 
 ~ 
 
 
 .-'' 
 
 
 ,» 
 
 
 1 
 
 
 
 
 .• 
 
 10 
 
 \ 
 
 
 1 .- ' 
 (45) .-' / 
 
 
 \ 
 
 
 \ 
 
 , 9 
 
 " \ 
 
 
 \ 
 
 1 
 
 
 \ 
 
 / 
 
 ; 3 
 
 
 
 \ 
 
 f 7 
 
 > 
 
 - 
 
 
 ■ ■■ \ / 137) 
 
 6 
 
 
 
 (41) •- / 
 \ / 
 
 \ / 
 
 ■ 5 
 
 
 
 \/ 
 
 ^r 
 
 
 t 
 
 
 
 
 
 (41) 
 
 4 
 
 
 
 Porous sandstone 
 breakdown pressure 
 
 3 
 
 
 
 
 2 
 
 
 
 Hard sandstone 
 breokdown pressure 
 
 1 
 
 1 
 
 
 treatment pressure 
 
 _i i i i i 1 1 1 1 
 
 I 23456789 10 
 
 Pore pressure 
 
 Fig. 14. - Treatment and breakdown pres- 
 sures as a function of the pressure of the 
 pore fluid pressure for the "non-penetra- 
 ting fluid" case for a porous and a non- 
 porous bed. 
 
 The value of cr r used in figure 14 is the same for each sand. Its value in 
 the general case would be different so the relative positions of the curves for 
 the two sands have no significance. 
 
 "Thin bed" radial flow problem 
 
 Two cases of fracturing with a penetrating fluid will be considered. The 
 first of these involves sustained injection of fluid through casing perforations 
 into a thin bed of sand bounded by soft shales. Assume that in the shales 
 
 e 
 
 r r = a z and in the sand cr TQ = <tq q < a z 
 
 When the fluid has been injected for an extended period of time, a pressure 
 gradient will have developed that can be described approximately by the steady 
 state radial flow equation 
 
 Pi+ ^f) in(rAi) 
 
 (46) 
 
 This formula applies specifically to two dimensional radial flow from a 
 cylindrical source but it also will give the pressure distribution around a cased 
 and perforated well, except when r/ri is small if the value of q used is the 
 radius of an open hole well of equivalent conductivity. 
 
8 ILLINOIS STATE GEOLOGICAL SURVEY 
 
 The condition "thin bed" used to describe the problem means that the dimen- 
 
 sionless group , _ 
 
 d P 
 
 «] 
 
 dP 
 is small in magnitude. T is the bed thickness and J — is the gradient of the 
 
 radial strain, assuming free expansion of the elements of the system. The bed 
 
 in question is constrained between two semi-infinite regions. The restraining 
 
 shear stresses , r r , delivered to the bed by these regions, are proportional to 
 
 the bed thickness, T , and strain gradient, r d_P ] , which would result in the 
 
 b £] ■ 
 
 absence of restraint. Therefore, if T or |d_P| became small, the bed may be 
 
 I dr I 
 considered entirely controlled by the regions above and below, and horizontal 
 displacements would be prevented. 
 
 In the present problem, the shear stresses at the bed contact necessary to 
 prevent horizontal movement would be of the order 
 
 A e r„_ dp- 
 
 b' £] 
 
 2 1 - fl 
 
 Letting n = ^ ft. , r p = 100 ft. , r^- = .7, (Pj - PJ = 1,000 psi, and T = 2 ft. 
 x i. e i — ix a 
 
 Then, for r = 1 ft. , x « 133 psi, and for r = 2 ft. , r « 66 psi. So, except for 
 
 perhaps very near the well bore, r r is small and horizontal restraint must be 
 
 nearly complete. 
 
 A layer of paint on a block of steel subjected to thermal stresses would be 
 a good analogy to the condition assumed to exist in this problem. It is safe to 
 say that in such a case the areal distribution of the paint is controlled by the 
 steel. 
 
 Horizontal restraint by adjacent rock layers establishes the condition that 
 the components of horizontal strain due to the pore-pressure gradient, €q and 
 € r *, equal zero. It also is known that o" z * = 0. These are the same conditions 
 that gave us (41). Substituting (41) in (4 6) gives 
 
 ILf n .J!b 
 
 0" r = CTq = O" 
 
 e = 'ro + 7^ Mi^T^ <■*!>] <«> 
 
 This equation gives the horizontal stress as a function of the radius, except 
 perhaps within the region r < 3 q where the pressure gradient is uncertain and 
 the assumption of zero horizontal strain questionable. 
 
 If Pi is increased very gradually, the local pore pressure is increased and 
 along with it the horizontal stress, according to (47). As a result, the "treat- 
 ment" pressure for vertical fractures is increased. 
 
 Eventually the condition that P = ctq is reached locally. Thenceforth further 
 increases in P must be accompanied by an equal increase in cTq. But if P = ctq ( 
 then Pi >Cq, the condition of breakdown (42). If a vertical fracture does form it 
 will terminate in the shales because in the shales <Tq = <r z >P^. If the sand is 
 thin enough, openings of vertical fractures may be prevented by the horizontal 
 restraint of the shale boundaries. Whether or not vertical fractures occur, it 
 will still be possible to continue injection into the bed until P> <x z , at which 
 time a horizontal fracture is induced. 
 
 In summary, there seem to be three possible sequences resulting from sus- 
 tained injection into a "thin bed". 
 
HYDRAULIC FRACTURE THEORY 9 
 
 1) when 0" ro < <r z and cr T increases according to (41) until cr r > <x z and a hor- 
 izontal fracture is formed; 
 
 2) when cr r < cr z andog increases according to (41) until P = cr r . Then cr r 
 increases with P until a horizontal fracture is formed after a T > cr z ; or 
 
 3) when cr r increases according to (41) until P = cr r . A vertical fracture is 
 induced but cr r continues to increase as P is increased. Eventually cr r > <x z and 
 a horizontal fracture can be formed. 
 
 "Thick bed" radial flow problem 
 
 In contrast to the previous problem, imagine that fluid has been injected 
 into a thick bed of sandstone for a short period of time over a long open-hole 
 section. A fluid pressure gradient has been established in a relatively small 
 cylindrical region around the hole. 
 
 The first part of the problem is the determination of the stress components 
 o"q* / o"r*' an< ^ <r z* resu lting from the pressure inside the well, Pi, a uniform 
 pore pressure, P e , and the regional stresses cr z and a r . Two more sets of 
 stress components will then be obtained, due to the pore pressure gradient be- 
 tween r^ and r e which goes from (P^ - P e ) at r. to zero at r e . 
 
 The solution to the first set of components is the same as that obtained in 
 the first sample problem. Applying (44 c, d, and e) 
 
 r-2 
 
 <r r *-cr + -V-( p i-<%) («) 
 
 From (41) 
 
 l e r 
 
 a = <T 
 Z Z 
 
 *V. = °V. + T-. 7 p e 
 
 Similarly J E 
 
 ov. = cr_ + ■; P 
 
 1 -fi 
 JE 
 
 e e " r o ' i -/i ±e 
 
 Substituting in (48 
 
 2 
 
 °0* - % + I^Pe + ^T ( ^o + T^ P e - Pi) 
 
 r .2 
 
 a- * = cr + T 1J -Pe " ~V( a r + T 1 "^ P e " p i> (49) 
 
 r r Q 1 -fi e r Z x o l -fi e i 
 
 and * 
 
 a z " ^Zo 
 
 neglecting the uncertainties of the axial thrust of the fluid column in the well 
 bore. 
 
 The second part of the solution is obtained by applying the analogy between 
 the stresses due to pore pressure gradients and those due to unequal heating. 
 
 When a long circular cylinder is subjected to radial forces that are inde- 
 pendent of z, the problem is one of plane strain in the region sufficiently removed 
 from the ends of the cylinder. 
 
10 ILLINOIS STATE GEOLOGICAL SURVEY 
 
 Timoshenko (1934, p. 372) gives the solution to the thermal stresses in a 
 long, thick-walled cylinder that are due to steady state radial heat flow. In 
 applying the analogy we substitute P, the pore pressure, for T, the temperature, 
 and J, the coefficient of pore pressure expansion for, a, the coefficient of thermal 
 expansion. Lubinski (1954) applies this formula to the problem of radial flow in 
 a thick porous cylinder. Once we have made these substitutions, Timoshenko "s 
 solution applied to our problem becomes 
 
 'r~- H^][fefe]{*. H^,* ^- 7-1} 
 
 Letting r = r e , the stress components at the outer surface of the cylinder are 
 
 a = 
 
 l e 
 
 e 'i 
 ** ** 
 
 z e 9 e 
 
 Timoshenko 's equations (50) are applicable when the long, thick cylinder is 
 free to expand and lengthen under the influence of thermal, or, in our case, pore- 
 pressure gradients. 
 
 In the problem under consideration, lengthening of the cylinder is prevented 
 in order to maintain the plane strain condition, at the same time maintaining 
 continuity with the material outside r . In addition, lateral expansion is partly 
 restrained. 
 
 The axial strain of the unrestrained cylinder, e z , is determined by the 
 
 stresses at the outer surface, <j z **, erg **, and oy **, by the application of 
 
 Hooke's Law (26) and as cr_ ** - e a a ** and oy ** = 
 
 ^•0 oe *e 
 
 «^"-«8."" ( 4* 1 *^" (52 » 
 
 It is now necessary to find the set of stress components cr z ***, °q # an d 
 
 /•ir -i- -ir Jf 
 
 r e that will satisfy the following conditions when imposed on the surface of 
 the freely expanded cylinder: 
 
 1) an axial strain, - € 7 ** , will result, restoring the cylinder to its original 
 length; 
 
 2) a tangential strain, *g e ***, will result such that [e e ** + e Qe ***] is the 
 
HYDRAULIC FRACTURE THEORY 11 
 
 tangential strain at the bore of a thick cylinder representing the region outside 
 r subjected to an internal pressure cr r **, and 
 
 3) f0 e *** is the tangential strain on the surface r = r e produced by the ex- 
 ternal pressure a r *** and axial stress cr z **. 
 
 The first condition, -e z ** = e z ***, in terms of the stress components is 
 
 The second condition is satisfied by 
 
 te ** , ***, -(1 +11 ) *** , _ . 
 
 Uq ** +€p. ) = N - r ' cr r (54 
 
 °e We E r e 
 
 ** 
 
 Where (cq +e fi ) * s tne tangential strain at r = r e in the thick cylinder 
 
 r e - r - 00 due to the internal pressure cr r ***, 
 
 The final condition is merely a statement of Hooke's Law 
 
 *** 
 
 *** Qe U , *** ***> /rr\ 
 
 € 6 e = ~T~ - E ( % + ^ e ) (55) 
 
 for the component of tangential strain, €q ' , resulting from stress components, 
 o"q ' , o" r ' , °~ z , acting on the surface of the cylinder q < r < r e . 
 
 From Lame's thick-cylinder formulae (44), the relation between the tangential 
 stress, o-Q ***, at the surface, r = r e , due to an external pressure, cr r *** , is 
 
 ^•-■vlfrri] (56) 
 
 r e ~ r i 
 From (52), (55), and (54) 
 
 0"„ ** 
 
 -OX *** 0" r *** UCT- *** 
 
 z„ - + -; — — - (57) 
 
 J e 1 - [i 1 - p. l _ /i 
 
 From (5 6) and (57) 
 
 r.J- + ri 2 
 
 1+ ~2 2 
 
 re - ri 
 
 
 o" ** = - + a- *** ( 58 ) 
 
 z e (1 - fi) 1 -/z z e ^ aj 
 
 From (53) and (56) 
 
 *** 9 2 
 
 0" 
 
 ** = 
 
 r^[ 1+ rrrfy-l «-r*** (59 
 
 L i 
 From (58) and (59) 
 
 2 2 
 
 r r e £ + ri z I 
 
 tr 2 *** = 1 + — 5 VI a r *** ( 60 ) 
 
 ^•e L r ' - r.2 J e 
 
 e i 
 
12 ILLINOIS STATE GEOLOGICAL SURVEY 
 
 and from (60) and (56) 
 
 2 2 
 
 %*** = [ ^2 " ^2 +1 Ke*** (61 
 
 re + 1 
 
 and substituting (60) and (61) in (57) 
 
 o- ** = _ a *** ( 62 
 
 z e z e 
 
 The components cr z *** , a x *** , <Tq*** due to the surface pressures, cr z *** 
 and c *** 7 are obtained as follows. To maintain the plane strain condition, 
 
 e 
 
 <t *** is applied uniformly across the top of the cylinder, so for q < r < r, 
 z e 
 
 a z *** = & *** ( 63) 
 
 z e 
 
 and for r e < r < co 
 
 cr z *** = 0. (64) 
 
 The radial stress, <x r ***, for the region r e < r < co, due to the internal pressure 
 <t, ***, applying the Lame formula (44) is 
 
 r 2 
 o- *** = _£_ *** ( 65 ) 
 
 1 r 2 x e 
 
 Similarly, the tangential stress for r e < r < oo is 
 
 re 2 
 a ■ *** = - S— a- *** (66) 
 
 W j-Z J e 
 
 For the region rj < r < r e , again applying the Lame formula 
 
 *«■*** «v" -ir-d 1 -^] (67) 
 
 r e ~ r i r 
 
 and 
 
 r 2 r 2 
 
 -0-* = %*** r 2 G r 2 [ 1+ 7r] (68) 
 
 r e ~ r i 
 
 Summarizing these results, we have for the region r^ < r < co 
 
 2 2 
 
 -r* = [i-7rH% + (i 1 ^K] + ^ p i (49) 
 
HYDRAULIC FRACTURE THEORY 13 
 
 For the region rj < r < r e 
 
 2 
 
 For the region r e < r < CO 
 
 cr r ** = 0; erg** = 0; a z ** = 
 
 By substituting for cr z *** and cr T *** their equivalents in terms of <x z ** using 
 (61) and (62), then substituting for cr z ** using (51), we obtain for the region 
 r e < r < oo 
 
 , *** = _ i r JLE_i r p i- p e ir 2r i 2 in Ee. _ i r e 2 - ^ 2 
 
 a r 4 Ll -/i JUn(r e /r i )jL I - e 2 - r .2 ^ n r 4 N r 2 
 
 n 2 (69) 
 
 *** _ I r TE 1 r Pi - p e 1 r 2r i . £e_ .1 r e 2 - r j 2 
 
 a ' 4 Ll -/J L-6n(r e / ri ) J L 2 - 2 *" Fj X J r 2 
 
 and for the region q < r < r 
 
 t— -ib^][S^r][^i<-»]['-S] 
 
 2 2 
 
 i[i^][fc(r^][^r^-2 fa ^-- 1 ][ 1 + ^r] 
 
 2 
 t- 1 I - T E ] r P i - p e 1 [ 2r i ^e_ ] 
 
 " " 2 Ll -/J L -en(r e / ri ) J L r 2 - r .2 *" q M 
 
 •kifk =r 
 
 By combining the components <r_ n ** and cr ,, *** for the region 
 
 r, y , z r, y , z 
 
 r i < r < r e , the stresses due to the fluid pressure gradient are 
 
 a **** = u ** + a *** = — I— Li_ 1 1 — * §_| J p„ _§_ 
 
 [l^][fo(r7r^r n 7 
 
 (-j- **** — 0- ** -i- rr *** 
 Z Z Z 
 
 equations (71) 
 
14 ILLINOIS STATE GEOLOGICAL SURVEY 
 
 In the region r > r_ the components a ** are zero, so for this region 
 
 the stresses <x r Q **** equal the components a ***. The total stress 
 
 ''"'■, i" , 9 , z 
 
 components are given by 
 
 cr r = cr r * + a **** 
 
 £To= cr fi * + o- **** (72) 
 
 and for region r < r < go 
 
 a 2. = a Z 
 
 a r = cr T * + a ** 
 
 o- e = o- e * + o-q** (73) 
 
 cr = cr 
 
 z z 
 
 Figure 15 shows a plot of the stresses around a well bore due to the steady- 
 state flow of fluid from the well into the surrounding formation. The pressure in 
 the porous stratum grades from the well-bore pressure Pi to the undisturbed for- 
 mation pressure P e at a distance r e from the bore axis. The bore axis coincides 
 with the vertical cr axis in figure 15. 
 
 The assumed set of conditions are shown in the upper right hand corner of 
 the figure. 
 
 The components, a* „ , given by equations (49) are due to the well-bore 
 pressure. The components, ov.**q* z , given by equations (71) are due to the 
 pore pressure gradients. The resultant total stress components, <j t p. z , are 
 shown as solid lines. 
 
 DISCUSSION AND CONCLUSIONS 
 
 In the example given as a review of Part I, the relation for shear strength as 
 a function of pore fluid pressure (36) was plotted on the diagram, figure 11. 
 The shear criterion was not taken into consideration in later problems, because 
 I considered that enough had been said in light of present knowledge. Future 
 workers may want to investigate further and somehow obtain usable values for 
 <}> the angle of internal friction, appropriate to the application used here. 
 
 Another type of behavior that has been ignored here is the inelastic consoli- 
 dation that may take place in a porous material upon reduction in pore pressure. 
 This certainly is important in dealing with silts and clays, but the inelastic com- 
 pression of consolidated sands is probably not too important within the range of 
 pressures encountered in the oil fields. For discussions of this type of behavior 
 the reader is referred to Terzaghi and Peck (19 48) and Geertsma (19 5 6). 
 
 I should like to put some emphasis on the phenomenon described in the last 
 part of the review problem in Part I, the opening of vertical joints due to an in- 
 crease in the value of the pore-fluid pressure after it has become equal to the 
 least horizontal stress. It can be seen from figures 11 and 12 that the condition 
 P = cr x is more likely to occur in a bed of low porosity. 
 
 The basic assumption that large masses of rock will not sustain an "effective" 
 tensile stress is highly reasonable and the conditions necessary for the occurrence 
 
HYDRAULIC FRACTURE THEORY 
 
 15 
 
 a (lbs./in. ) 
 7000 
 
 6000 
 
 5000 
 
 4000 
 
 3000 
 
 2000 
 
 1000 
 
 \ 
 
 \ 
 
 \ 
 
 \ P 
 
 \ 
 \ \ 
 
 \ v 
 \ 
 
 \ 
 
 / N 
 
 la**** 
 
 > 
 
 2 3 
 
 r (feet) 
 
 -4— - 
 
 Fig. 15. - Stresses due to the steady -state radial 
 flow of fluid from a well into a thick porous 
 bed. 
 
16 ILLINOIS STATE GEOLOGICAL SURVEY 
 
 of the process, i.e. , that one of the horizontal stresses be considerably less 
 than the vertical stress, must be fairly widespread. It thus seems likely that 
 the existence of this phenomenon may be fairly common. 
 
 A distinction must be made between a hydraulic fracture that is opened be- 
 tween two blocks of material actively forced apart by fluid pressure and a distri- 
 bution of vertical fractures developed when the requisite pore pressure acts 
 within a region. The latter fractures are analogous to shrinkage cracks that might 
 develop in a plate of brittle material clamped between semi-infinite regions and 
 then cooled. 
 
 The conditions for the production of these "shrinkage fractures " are about 
 the same as those required for the induction of hydraulic fractures, i.e. , con- 
 dition (42). 
 
 Therefore, the production of these fractures generally should be accompanied 
 or preceded by a hydraulic fracture extended from the injection well. However, 
 in the situation described in the thin bed problem of Part II where a vertical 
 fracture is confined within horizontal shale boundaries and the shales prevent 
 horizontal displacements of any appreciable magnitude, it seems possible that 
 an open vertical joint system might develop in the vicinity of the injection well 
 unaccompanied by a major hydraulic fracture. 
 
 The main point is that these "shrinkage fractures" should exist, as the neces- 
 sary condition for their existence is not unusual in the vicinity of an injection 
 well. 
 
 The application of fracture theory in the field will be greatly facilitated by 
 the technique of direct bottom-hole pressure measurement described in a paper 
 by Godbey and Hodges at the October 19 57 meeting of the AIME in which they 
 give the results of bottom-hole pressure measurements during fracture treatment. 
 Their investigation showed that the bottom-hole pressure during the extension of 
 the fracture was fairly steady during treatment, irrespective of fluid properties 
 and pumping rate, while the surface pressures varied widely. 
 
 The investigators also point to the great difficulty in obtaining the bottom- 
 hole pressure by correcting the surface pressure for hydrostatic head and friction 
 losses in the tubing because of the non-Newtonian flow characteristics and the 
 changing density of the fluids. 
 
 The bottom-hole pressure during treatment is the key variable and its direct 
 measurement should greatly facilitate the application of hydraulic fracture theory 
 in the field. 
 
 Waterflood injection wells should be another important source of data for the 
 study of pressure-parting phenomena. Good records of injection rate vs. input 
 pressure may be kept, and the critical injection pressure resulting in pressure 
 parting may be measured fairly accurately. Dickey and Andresen (1946) discuss 
 in detail the technique for detecting pressure parting with input data. 
 
 These authors state, "Apparently the critical pressure is lower in the early 
 life of a well, than after a considerable volume of water has been injected into 
 the surrounding sand. " This is consistent with the predicted increase in vertical 
 parting pressure with formation fluid pressure given by equations (41) and (37). 
 
 I believe that horizontal fractures require pressures approximately equal to 
 the overburden load. The treatment pressures required for the production of verti- 
 cal fractures should vary with the pressure of the fluid in the formation in ac- 
 cordance with the theory presented here. Therefore, success in the control of 
 
HYDRAULIC FRACTURE THEORY 17 
 
 fracture orientation by some special procedure should be substantiated or refuted 
 by study of the bottom-hole treatment pressures required for fracture extension. 
 
 The problems given in Part II are intended to demonstrate the sorts of calcu- 
 lations that may be made with the theory presented, and specific conclusions 
 should not be drawn from them. 
 
 In a general way it has been shown that treatment pressures for the production 
 of vertical fractures may change with formation fluid pressure. Consequently, 
 the tendency to fracture a given zone may change with formation fluid pressure. 
 The fluid pressure distribution and bed thickness are important considerations. 
 In general one can conclude that sustained injection before breakdown in a thin 
 zone would be most conducive to horizontal fracture formation whereas a short 
 period of injection in a thick bed, or zero fluid loss before breakdown, should 
 lead to vertical fractures. It may be, however, that the orientation of fractures 
 is dominated by the regional streses. If control of fractures is possible, the 
 bottom-hole treatment pressure obtained should reflect the orientation of the 
 fracture. If horizontal fractures are obtained, the bottom-hole treatment pressure 
 should be about equal to the calculated overburden pressure. If the fracture is 
 vertical the treatment pressure should be about equal to the least horizontal 
 stress, which is normally less than the overburden pressure. 
 
18 ILLINOIS STATE GEOLOGICAL SURVEY 
 
 REFERENCES 
 
 Biot, M. A., 1941, General theory of three dimension consolidation: Jour. 
 Applied Physics, v. 12, p. 155-164. 
 
 Bridgman, P. W. , 1947, Effect of hydrostatic pressure on the fracture of brittle 
 substances: Jour. Applied Physics, v. 18, p. 246-258. 
 
 Clark, J. B. , 1949, Hydrafac process: Oil and Gas Jour., v. 13, no. 5, p. 31. 
 
 Clark, R. C. , Jr. , and Reynolds, J. J. , 1954, Vertical hydraulic fracturing: Oil 
 and Gas Jour. , v. 53, no. 14, p. 104. 
 
 Cleary, J. M. , 1958, Elastic properties of sandstone: Univ. 111. Master's 
 Thesis. 
 
 Dickey, P. A. , and Andresen, K. H. , 1946, The behavior of water input wells: 
 Drilling and Production Practice, 1945, p. 34-58, API, New York. 
 
 Gassmann, F. , 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. 
 
 Gilbert, Bruce, Neill, George, and Clark, Roscoe, 1957, Fracture initiation can 
 be controlled: Oil and Gas Jour., v. 55, no. 31, p. 64-67. 
 
 Godbey, J. K. , and Hodges, H. D. , 1957, Pressure measurements during formation 
 fracturing operations: Paper presented at Annual Fall Meeting AIME, Dallas, 
 Texas, October 6-9. 
 
 Grebe, J. J., 1943, Tools and aims of research: Chem. Eng. News, v. 21, 
 p. 2004. 
 
 Grebe, J. J., and Stoesser, S. M. , 1935, Increasing crude production 20,000,000 
 barrels from established fields: World Petroleum, August, p. 473-482. 
 
 Griggs, D. T., 1938, Deformation of single calcite cyrstals under high confining 
 pressures: Am. Mineralogist, v. 23, no. 1, p. 28-33. 
 
 Handbook of Physical Constants, 1942, Geol. Soc. America Spec. Papers 36. 
 
 Handin, John, and Hager, R. V., 1957, Experimental deformation of sedimentary 
 rocks under confining pressure: Geol. Soc. America Bull., v. 41, no. 1, 
 p. 1-50. 
 
 Hubbert, M. K. , 1951, Mechanical basis for certain familiar geologic structures: 
 Geol. Soc. America Bull., v. 62, part 2, p. 355. 
 
 Hubbert, M. K. , and Willis, D. G. , 1957, Mechanics of hydraulic fracturing: 
 Jour. Petroleum Tech. , v. IX, no. 6, p. 153-166. 
 
 Lubinski, Arthur, 1954, Theory of elasticity for porous bodies displaying a 
 
 strong pore structure: Proc. Second U. S. Natl. Cong. Appl. Mech. , p. 247. 
 
 McGuire, W. J., Jr., Harrison, Eugene, and Kieschnick, W. F. , Jr., 1954, 
 
 The mechanics of formation fracture induction and extension: Jour. Petroleum 
 Tech. , v. 6, no. 10, p. 45. 
 
HYDRAULIC FRACTURE THEORY 19 
 
 Poollen, van, H. K. , 1957, Theories of hydraulic fracturing: Colorado School of 
 Mines Quart., v. 52, no. 3. 
 
 Scott, P. P., Bearden, W. G. , and Howard, G. C. , 1953, Rock rupture as 
 affected by fluid properties: Jour. Petroleum Tech. , v. V, no. 4, p. 111. 
 
 Terzaghi, Karl, and Peck, R. B. , 1948, Soil mechanics in engineering practice: 
 J. Wiley and Sons, New York. 
 
 Timoshenko, S. , 1934, Theory of elasticity: McGraw-Hill Book Co., New York. 
 
 Walker, A. W. , 1949, Squeeze cementing: World Oil, v. 129, no. 6, p. 87. 
 
 Yuster, S. T. , and Calhoun, J. C. , 1945, Pressure parting of formations in 
 
 water flood operations, Parts 1 and 2: Oil Weekly, v. 117, March 12, p. 34, 
 March 19, p. 34. 
 
 Zheltov, Y. P. , and Kristianovich, S. A. , 1955, Hydraulic fracture of an oil 
 
 bearing stratum: Izvest. Akad. Nauk S.S.S. R. , Otdel, Tekh. Nauk, no. 3, 
 p. 3-41. 
 
 Zisman, W. A. , 19 33, Compressibility and anisotropy of rocks at or near the 
 earth's surface: Nat. Acad. Sci. Proc. , v. 19, no. 7, p. 680-86. 
 
 Illinois State Geological Survey Circular 252 
 19 p., 3 figs., 1958 
 

 CIRCULAR 252 
 
 ILLINOIS STATE GEOLOGICAL SURVEY 
 
 URBAN A ^ffs*' lu