A Review of Hydraulic Fracture Modeling—Part I General Concepts, 2D Models, Motivation for 3D...

D. A. Mendelsohn Department of Engineering Mechanics,

Ohio State University, Columbus, Ohio 43210

A Review of Hydraulic Fracture ModelingPart I: General Concepts, 2D Models, Motivation for 3D Modeling This is Part I of a two-part paper which reviews in depth the literature on the modeling of the propagation of large hydraulic fractures in underground rock formations. Part I presents a general formulation of the problem, its geometry and only the most fundamental and unrestrictive physical assumptions. The two basic two-dimensional models of constant height (rectangular) fractures which formed the core of modeling efforts from 1960 through the late 1970's are discussed in detail followed by a brief review of the effects of fluid diffusion in the fractured medium on crack propagation. The recent field and laboratory observations which have shown that nonrectangular fractures can occur quite often are discussed, as are their impact on the modeling effort, specifically the need for three-dimensional models of crack propagation in layered environments. Part II of the paper deals with the three-dimensional modeling efforts and some fundamental crack and fracture mechanics problems related to the vertical growth of a hydraulic fracture.

1 Introduction Over the past thirty years the process of hydraulic frac-

turing of underground formations has found widespread geotechnical applications. One of the major applications is the creation of very large fractures to improve the flow rate of oil and/or gas from low permeability rock. Due to the high cost of fracturing fluid and proppant, it is crucial that as much of the fracture as possible remain in the hydrocarbon layer, almost routinely requiring an a priori estimate of the fracture area in contact with the hydrocarbon layer [1]. While stimulation of water wells [2] does not require as large fractures, there has been an increased interest in controlling or predicting the growth of the fracture. Hydraulic fractures are also being used to create geothermal energy reservoirs in hot dry rock masses [3] which consist of large fractures in the hot rock with inlet and outlet wells for water to be heated and circulated back to the surface. Here, too, the prediction of growth behavior is essential to calculating the performance and efficiency of the reservoir. Hydraulic fracturing is also used in liquid waste disposal [4], and if the waste is toxic, it is essential to know the shape of the fracture to avoid contact with nearby permeable formations. Hydraulic fracturing is also a useful tool in stress-state determination [4, 5] and may become routine prior to creating large fractures [1], or designing mines or large underground structures. Undesirable instances of hydraulic fractures occur in solution mining,

Contributed by the Rock Mechanics Committee of the Petroleum Division for publication in the JOURNAL OF ENERGY RESOURCES TECHNOLOGY. Manuscript received by the Petroleum Division, September 27, 1983; revised manuscript received February 28, 1984.

enhanced oil and gas recovery, or around dams and other structures surrounded by saturated rock or soil. The engineering analysis of the propagation of a hydraulic fracture in underground rock formations is therefore of extreme interest in a variety of applications.

Since in most of the applications mentioned the least principal compressive tectonic stress is horizontal, it is assumed that the hydraulic fractures are planar and are oriented vertically, Fig. 1. Section 2 begins with the description of the transient three-dimensional problem of determining at any time the shape of the planar fracture and the fluid pressure profile in the fracture. This is a complex problem requiring the coupling in time of solutions to problems in fluid mechanics, solid fracture mechanics, and heat transfer. In the following, a "hydraulic fracture simulator" or "simulator" is taken to mean any com-putational scheme designed to trace the growth or evolution of a fluid-filled crack in a rock-type medium. While the main objective of research in this area is the creation of more ac-curate simulators, much attention is also paid in this review to more fundamental investigations into the nature of fracture growth in the hydraulic fracture environment.

Most of the currently used simulators for the design of large industrial hydraulic fractures are based on two types of two-dimensional models, both assuming a vertical fracture of constant height, i.e., the crack contour in Fig. 1 forms a rectangle. A discussion of these basic models in Section 3 is followed by a brief discussion of poro-elastic effects involved in the initiation and extension of a fluid-driven crack in a porous medium, Section 4. While this general review of

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Table 1 Constituent problems in the analysis of hydraulic fracturing

Problem Quantities related on solution (and parameters) Fluid flow in fracture Fluid rheology Heat transfer Flow into formation Crack stress analysis Fracture mechanics Sand transport

w,q(p0 or q0 ,n or n' and K') q,yp,T(ixox n' arx&K') q,T(z,T0,TR,KR,KF) p,qy (,k,\iox n' andA") p,w,

w(x , t )

Fig. 2 Geometry of the CGDD rectangular fracture of constant height confined between bounding layers

difference between the areal extent and the narrow width of the crack, the fluid and sand transport problems are taken to be independent of y, reducing these to two-dimensional problems in the area of the fracture. However, the stress problems for the crack are in general three dimensional. Due to the difference in time scales between the fluid and solid problems, the solid problem is usually taken to be quasi-static. See the papers by Veatch, et al. [1] for general historical and technical overviews of hydraulic fracture modeling and design.

3 Two-Dimensional Rectangular Models Most current industrial fractures are designed by simulators

based on one of two "first generation" models with the common assumption of a constant height fracture which extends in the x direction only (z (x, f) forms a rectangle of area HL(t), Figs. 2, 3). Each of these models assumes no fluid flow in the vertical direction, however, they differ significantly in their mechanical assumptions.

Christianovic and Zheltov [6, 7] developed a model of an infinite homogeneous and isotropic elastic solid subjected to plane strain loading in horizontal planes (xy), and containing a crack in the region -L(t) < x < L(t), -H/2 < z < H/2 and - w(x, t)/2 < y < w(x, t)/2. One half length of such a crack is shown in Fig. 2, along with the Ar-component of the fluid velocity profile in a representative xy plane; the y and z components being zero. Further, as indicated in the figure, the crack tip shape near x = L(t) is assumed such that the stress just outside the tip, ay (x, t) at x = L{t), tending to open new fracture is finite. Such a crack is known as an equilibrium crack [8, 9] and differs from the classical or Griffith notion of a crack in an elastic solid [10], in which the slope of the crack opening at the tip is infinite and hence the tip stress is infinite. Both notions of the crack tip lead to useful characterizations of the near crack stress field. For a discussion of the Griffith crack stress field and crack shapes in infinite media see [11-13]. Note that the equilibrium crack tip condition of smooth closure is consistent with experimental observations and theoretical arguments which demonstrate that fracturing fluid does not penetrate completely to the crack tip [14-16, 41, 42]. The most basic version of the model, therefore, consists of a one-dimensional problem in the x direction of fluid flow in a rectangular opening and a two-dimensional plane strain crack problem in the xy plane. The two problems

w(x,z,1)

Fig. 3 Geometry of the PKN rectangular fracture of constant height confined between bounding layers

are coupled through a fluid volume balance condition to find L(t), which states that the volume of fluid pumped less the volume of fluid leaked to the reservoir must equal the volume of the fracture. This condition is based on the assumption that the fracture resistance or toughness at the tip x = L(t) is negligible compared to the necessary pressure in the fracture to oppose the applied stress, ayao, and therefore new fluid volume immediately creates additional length, regardless of fluid pressure near the tip.1 For materials with significant fracture toughness, the actual process is more likely one of short time periods of constant length and increasing width and pressure followed by sudden length propagation.

Later Geertsma and De Klerk [17] and Le Tirant and Dupuy [18] included the effects of leak-off of fracturing fluid into the formation. Daneshy [19] later added the effects of certain non-Newtonian fluids and a sand transport algorithm [20] to create a useful numerical simulator, although many details are not included in the only reference [19]. McLeod [21], however, presented a short programmable calculator program based on Geertsma and De Klerk [17] and Daneshy [19, 20] which includes leak-off and a detailed high-viscosity proppant transport model. All of these models, referred to as CGDD models, named after their principal developers, specify a priori, the spatial form of the pressure p(x, t) in order to obtain special case closed form solutions (no leak-off, con-stant pressure) or tractable numerical iterative schemes for more general cases.

Also commonly used today are simulators based on the model of Perkins and Kern [22], which is illustrated in Fig. 3. In this model plane strain is assumed in vertical planes, and the plane strain response in each vertical plane of constant x is assumed independent of its neighboring vertical planes. Although not dictated at all by the plane strain assumptions, but just as in CGDD models, gx(x, t) is assumed to be the only nonzero fluid velocity component. Thus, as the fracture propagates, its vertical cross sections are elliptical, as determined by the plane-strain elastic crack problem in the zy plane [12, 13]. The ellipse size varies in time and with horizontal position due to the pressure gradient in x associated with one-dimensional fluid flow through an elliptical channel. The fluid flow and crack problems are coupled through a fluid volume balance condition as in CGDD models. Here, too, even though the elastic response is ignored in xy planes, zero fracture toughness is assumed

See [87] for a discussion of crack propagation criteria, including the fracture toughness criterion.

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through the condition of zero fluid overpressure at the tip x=L, i.e.,p(L, t) = aya>. Nordgren [23] extended the model to include leak-off and the effect upon flow rate due to the change in fracture width in time. This type of model will be referred to as a PKN model, named after its developers.

The treatment of the leak-off problem was first considered by Carter [24] and is based on the experimental observation that leak-off at a point on the fracture face is a strong func-tion of exposure time to fluid; the highest leak-off rates occurring immediately on exposure (known as spurt loss) followed by the rapid buildup of filtercake causing greatly reduced flow velocities. Carter's model for adjusting created fracture length is limited to constant height and width fractures and quasi-static flow, and ignores the effect of reservoir fluid and pressure; however it is attractive because it only requires fluid loss data obtainable from simple laboratory tests. Williams [25] later refined this so-called classical model to account for dynamic effects due to the high flow velocity in the fracture parallel to the crack face. In an excellent review, Geertsma and Haafkens [26] detailed these corrections for fluid leak-off as well as those for non-Newtonian fluids in both the PKN and CGDD models. They also compare the two models making the natural observation that for large length/height ratios the PKN model should be used, and for values of length/height near unity or less, the CGDD model is better suited.

Since the inception of the basic PKN and CGDD models there has been considerable work done on improving any or all of the treatments of leakoff, fluid flow in the fracture, proppant transport and formation-fracturing fluid heat transfer. While much of this work is in the form of refined PKN or CGDD simulators, the concepts can be applied to the three-dimensional simulators to be discussed in Part II [87].

The problem of modeling fluid loss from the fracture to the reservoir has received much attention since Carter's model [24] which merely specifies the flow velocity out of the fracture (based on empirical data), ignoring entirely the presence of the reservoir fluid, and any non-Newtonian fluid behavior. Settari [27] recently modified the Carter model, eliminating several simplifying assumptions allowing him to obtain variable flow velocities dependent on fluid properties and flow conditions. Both Biot, et al. [28] and Rubin [29] took a more fundamental approach and treated the case of pistonlike, one-dimensional displacement of fracturing fluid into the reservoir in the y direction, governed by Darcy's law, coupled with the one-dimensional flow of reservoir fluid driven by the moving (unknown) boundary between the fracture and the reservoir fluids. Rubin [29] discussed in detail the limitations of the Carter model by incorporating his leak-off model into a CGDD-type simulator using a critical stress intensity factor (fracture toughness) extension criterion rather than a fluid volume balance. Biot, et al. [28] also presented a CGDD-type simulator solved by applying the formal Lagrangian formulation using the crack length L ( 0 and the maximum width w(0, t) as generalized coordinates. A distinct advantage of this method is that the individual effects of leak-off, crack shape, frictional flow, etc., are accounted for separately by analytical methods and then coupled automatically in a global virtual work expression.

Settari, et al. [30-32], Advani [33] and Nghiem [34] all treated, with various levels of simplifications, the more rigorous problem of two-dimensional flow in the reservoir containing both fracturing and reservoir fluid. The models in [30-34] treat rigorously the miscible two-phase flow resulting from mixing of fracturing and reservoir fluids. Models which approximate the situation of very compressible reservoir fluids may assume single-phase flow of the fracturing fluid with a compressibility coefficient representing the resistance to flow due to the reservoir fluid [31]. In all cases [27-34] the coupling of the leak-off flow with the fracture flow is through

boundary conditions specifying continuity of pressure and flow rate at the fracture/reservoir interface and not through a specification of the flow velocity out of the fracture. When included in simulators for the CGDD geometry which each of [27-34] do, these models can show marked effects on the volume of fluid lost and, therefore, on predicted fracture dimensions as compared to Carter's model.

More accurate solutions of the fluid flow problem inside the fracture, governed by the Navier-Stokes equations, without a priori specified pressure distributions, and in some cases including pressure gradients in both the height and the length, may be found in [28, 30-37]. These have been solved by a variety of standard numerical techniques. The addition of proppant transport calculations to CGDD-type simulators, initially by Daneshy [20], and later by Settari, et al. [30, 32] and Biot, et al. [28], has required the development of particle settling velocity correlations with fluid properties, flow velocity, particle diameter and roughness, etc. Clark and Quadir [38] have written an up-to-date summary of work in determining particle settling velocities. It appears that ac-curate prediction of settling velocities in commonly used, highly viscous, cross-linked fluids is still quite difficult. A more recent paper by Shah [39] treats this problem specifically. The addition of heat transfer calculations to fracture simulators has also been accomplished and Settari [30] presents his own model as well as a reveiw of most other previous models. The rheology of the fracturing fluid will not be discussed in this paper. For a brief but complete review see the paper by Veatch [ 1,1983 (Part 2)].

Some recent field measurements and laboratory ex-periments have focused on obtaining the true character of length and width in time for a constant height propagating fracture. Smith, et al. [40] have made down-hole closed-circuit TV studies of fracture width during the creation of the fracture. Smith, et al. observed the fracture to grow vertically and stably during pumping, and were still able to obtain an approximation from the constant height PKN model to match fairly well with width versus pressure data. Warpinski [41] has made in-situ measurements by mineback of fracture width and fracture pressure versus pump time by instrumenting pressure ports placed in the fracture's path. Among his ob-servations were: approximately rectangular fractures due to stress barriers, a Barenblatt-type smooth crack tip [8, 9], very large pressure drops along the fracture due to large scale roughness, offsets at natural fractures, and pressure and length behavior in time which were qualitatively similar to the PKN model. Daneshy [14] created constant height hydraulic fractures in rock plates and found that if stiff barrier plates are bonded to the rock the borehole pressure increases with time and length as the PKN models predict, and if soft barriers are used, the pressure decreases with increasing time and length as the CGDD model predicts. Medlin and Masse' [15] and Rubin [42] have also created constant height propagating hydraulic fractures in the laboratory and measured width and pressure at several fixed positions, and length versus time. Both report pressure and width data qualitatively similar to Geertsma and De Klerk [17] and Rubin obtained reasonable quantitative agreement with a modified CGDD model with a fracture toughness criterion, and corrections for finite laboratory sample size. A rough estimate of viscous effects is also presented in [42]. Papadopoulous, et al. [43] presented preliminary results and methodology for a repeatable simulation of hydraulic fracturing by interface separation. By creating bands of low confining stress the separation area may be restricted to a rectangular shape with increasing length.

4 Poro-Elastic Effects The diffusion of fracture and reservoir fluids in the

reservoir near the crack edge has a significant effect on the

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Propagating Vv yr Plane of Weakness V / or Material Interface

Fig. 4 Fracture arrest, shift and reinitiation at a plane of weakness or material interface stress felt by the rock matrix, and hence potentially upon fracture propagation behavior. Due to the complexity of the poro-elastic equations (see Biot [44-46] for the general for-mulation), little progress has been made in including such effects in simulators. However, the poro-elastic environment and its effect on hydraulically driven crack propagation has been studied quite a bit. An extremely brief review of such studies follows. For more detail consult the references in the cited papers.

There has been a significant effort to calculate the stresses set up surrounding a wellbore due to flow from the wellbore to the porous reservoir or vice versa [47-53]. All but two of these studies [49, 50] apply the analysis to finding the initiation pressure for a hydraulic fracture at a wellbore, obtaining an inverse dependence on Biot's constant (one minus the ratio of matrix to bulk compressibilities). The application of the poro-elastic model to the analysis of the extension of hydraulically created cracks has also received some attention recently [52-59]. The single overriding mechanism appears to be the stabilization or retardation of crack extension due to increased closure stress, which is due to reduced pore pressures or even totally undrained regions near the crack edge. Salz [60] has presented field data from 75 fracturing treatments which showed significant effects on extension pressures due to variations in initial pore pressure. The inclusion of poro-elastic effects in simulators should be a goal for future development.

5 Experimental Motivation for Three-Dimensional Models

Quite often hydraulic fractures are created in more or less horizontal layers of reservoir rock different than the layers above and below. Also, quite often, due to the mismatch in material properties, a discontinuity or steep gradient in the minimum horizontal tectonic stress aya develops around or at the material interface. The previous two-dimensional models assumed these interfaces to be sufficient barriers to fracture growth through one mechanism or another. Unpublished field data have shown, though, that fracture heights at the wellbore significantly greater than the reservoir or payzone thickness can occur. In an attempt to understand and demonstrate the possibility of vertical migration of fractures out of the originating payzone a number of experimental studies in the field and in the laboratory have been performed. They deal with the behavior of fractures near interfaces and planes of weakness as well as fracture response to stress gradients.

Lamont and Jessen [61], Daneshy [62], Hanson [63] and Zoback, et al. [64] have all studied the interaction of propagating fractures with planes of weakness or faults in otherwise homogeneous rock. All found that, although a crack may be temporarily arrested or even shifted from its original path (Fig. 4), it eventually will cross the plane of weakness with its orientation preserved. Hanson, et al. [65-67, 69] and Anderson [68, 70] further studied the crack propagation characteristics across totally unbonded interfaces between blocks of the same rock. They found the existence of a critical frictional interface shear strength, expressed as a critical applied stress normal to the interface, below which the

Bounding Layer

7 1

\

"

Payzone

/ Bounding Layer

Fig. 5 Fracture containment by interface slippage at the material interfaces

crack did not cross the interface. Typical critical normal stresses were 3.5-7.0 MPa (500-1000 psi), and these values showed little dependence on surface roughness or rock type. This phenomenon is discussed and predicted by Daneshy in a paper which discusses in depth the entire vertical growth problem [14]. Hanson, et al. [69] also observed that a region of reduced shear strength (created by lubrication) caused the crack path to shift (as in Fig. 4) and reinitiation to occur at the point of shear stress discontinuity. Teufel, et al. [71, 72] also investigated the frictional interface with similar results, but they found that the critical normal stress was appropriately reduced by increased interface roughness. They also found an increase in fracture-induced interface shear stress with in-creasing tensile strength of the originating rock type. Daneshy [14] postulated that the fracture induced interface shear stress also decreases with increasing fracture area. It is therefore concluded that if containment is to occur by interface slippage or crack blunting, Fig 5, it will occur in thin, strong reservoir layers with weak interfaces. It is, however, generally felt that the overburden azoa at typical modern fracture treatment depths is sufficient to prevent weak interfaces, and con-tainment by this mechanism is therefore unlikely. Frictional effects may still cause temporary arrest, shifting and reinitiation at great depths, and must not be ignored.

The effects of material property contrasts across an in-terface on both crack penetration behavior and relative growth rates following penetration (if it does occur) have also been studied. Theocaris and Milios [73, 74] observed dynamically propagating cracks in metals as they approached, arrested and then penetrated an interface. They found drastic differences in arrest time and fracture path when going from a less to more brittle material as compared to the reverse situation, and in each case penetration occurred. Daneshy [75] also found that a contrast in material properties across bonded interfaces did not prevent penetration in rock sam-ples. Therefore, some attention has focused on relative growth rates for cracks in each material with an attempt to compare vertical to horizontal growth tendencies once growth out of zone has started. Daneshy [14, 75] studied the relative ease of fracturing different rock types by devising several bi-material rock sample configurations which compare relative crack extension pressures or lengths in each rock under comparable conditions. One configuration [14] also allows comparison of necessary critical interface penetration pressures. Results from these tests vary greatly depending on rock type and are fairly independent of whether the rock is a typical reservoir rock (sandstone or limestone) or a typical bounding layer rock such as shale. Hodulak and Peters [76] investigated tensile fracture front propagation in each layer of non-rock composite plates and found a distinct reduction in fracture extent with increasing fracture toughness, the loading and elastic modulus being kept equal in each layer.

In later experiments using rocksamples, Hanson, et al. [67] did find several instances of elastic modulus contrasts preventing cracks from crossing epoxy-bonded interfaces;



however, Biot, et al. [78] demonstrated that if the fluid pressure at the tip of the arrested crack is allowed to rise, causing critical tensile stresses on the other side of the in-terface, the crack will eventually penetrate. In the field, Warpinski, et al. [78, 79] have found by post-fracturing mineback investigation that a bounding rock with an elastic modulus 20 times that of the originating rock did not stop vertical crack growth. Therefore, contrasts in elastic modulus or fracture toughness alone may not guarantee containment, since apparently, the additional crack tip pressures needed for penetration into higher modulus or toughness rock are quite often present at the wellbore. This is due to the close proximity of the crack edge at the interface to the fluid source, and could cause preferential vertical growth in bounding layers near the wellbore rather than lateral ex-tension in the originating layer.

Despite the results noted in the foregoing, there is con-siderable field evidence that hydraulic fractures created under various pressure and flow-rate conditions are often contained in the original reservoir layer. Therefore, favorable discontinuities or gradients in the minimum horizontal in-situ stress rjvco (z) are most likely responsible for the observed containment. Several laboratory and one field observation on fractures propagating in changing stress fields have been made recently. In experiments in homogeneous metallic plates, Pisarenko, et al. [80] found that a region of applied stress normal to the crack plane could completely stop the mechanically driven crack from propagating. Warpinski, et al. [81], Hanson, et al. [82], Teufel, et al. [71, 72], Thiercelin and Lemanczyk [83] and Ahmed, et al. [84] have all in-vestigated the effects of layered stress fields or high stress gradients on the propagation of a hydraulic fracture in a homogeneous material ([71, 72, 81, 82] all use rock and [83, 84] use manmade cast materials such as PMMA). All found qualitatively similar effects; an increase in horizontal stress always retards or entirely prevents growth in that region. It appears from these results that the necessary jump in stress for preventing growth depends on sample size, sample material and the fluid used. The only attempt at quantifying vertical growth as a function of driving pressure was made in [83] in which reasonable agreement for some cases was found with an approximate two-dimensional crack solution. Warpinski, et al. [85] have observed by mine-back that thin regions of horizontal stress gradients with increases of 2.1 to 3.5 MPa (300 to 500 psi) stopped fractures from crossing from above or below. Nolte and Smith [86] discuss the issue of vertical containment from the point of view of using treatment pressures to diagnose height growth versus length growth. The effects upon treating pressures due to several of the mechanisms discussed in the foregoing are postulated and several field pressure records are analyzed, some indicating significant vertical growth.

In summary, experimental work to date has shown that below a critical and relatively shallow depth corresponding to a critical interface shear strength (dependent on the cracked layer's thickness and modulus) containment due to interfacial slip and crack blunting may not be guaranteed. While not entirely understood, containment by elastic modulus and/or fracture toughness contrast may also not be counted on, at least until critical pressures are better defined and can be avoided during pumping. This leaves in-situ stress gradients as the most probable mechanisms responsible for much of the observed containment in the field. Additional field and laboratory data are needed to better quantify all of these effects in order to obtain rational fracture propagation criteria which govern growth in both vertical and horizontal directions.

6 Conclusion Since the inception of hydraulic fracture modeling research

around 1960 the work has focused on the interaction of a crack of growing dimensions and the flow of the pressurized fluid inside the crack which is causing it to grow. The com-plicating effects of leak-off of the fluid into the permeable rock being fractured, heat transfer between rock and fluid, proppant transport in the fluid, and fluid flow in the rock as it effects the stress around the crack have all been added to the analysis with a fair amount of success (except possibly in the latter instance). However, due to the inherent difficulty in coupling the two major components of the analysis, crack growth and fluid flow, models have been limited until recently (c.f., 1979), to vertical planar cracks of rectangular shape confined to their geological layer of origin. While this is a highly desirable result in many situations, recent observations indicate that it does not always occur by any means. The variation of the fracture from a rectangular shape has been the subject of an intense modeling effort which is aimed at characterizing and quantifying the mechanisms which control vertical growth in the hydraulic fracture environment as well as the inclusion of these results in integrated fracture simulators. Only then will it be possible to define with con-fidence treatment parameters which will avoid vertical growth, if it is possible. The study of vertical or three-dimensional hydraulic fracture growth is the subject of Part 2 of this paper, [87].

Acknowledgment The author would like to thank the management of

Halliburton Services for permission to publish this work, a portion of which was completed while the author was a Senior Engineer in the Chemical Research and Development Department. The author is also grateful to Professor S. H. Advani for many enlightening discussions.

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27 Settari, A., "A New General Model of Fluid Loss in Hydraulic Frac-tures," Paper SPE 11625, SPE/DOE Symposium on Low Permeability Gas Reservoirs, Denver, 1983.

28 Biot, M. A., et al., "A Two-Dimensional Theory of Fracture Propagation," Paper SPE 11067, 57th SPE Annual Fall Technical Conference, New Orleans, 1982.

29 Rubin, M. B., "A Quantitative Evaluation of Two Classical Ap-proximations Used to Predict the Extent of Vertical Hydraulic Fractures," ASME JOURNAL OF ENERGY RESOURCES TECHNOLOGY, Vol. 105, Dec. 1983, pp. 512-527.

30 Settari, A., "Simulation of the Hydraulic Fracturing Process," Society of Petroleum Engineers Journal, Vol. 20, 1980, pp. 487-500.

31 Hagoort, J., Weatherill, B. D., and Settari, A., "Modeling the Propagation of Waterflood-Induced Hydraulic Fractures," Society of Petroleum Engineers Journal, Vol. 20, Aug. 1980, pp. 293-303.

32 Settari, A., and Price, H. S., "Simulation of Hydraulic Fracturing in Low Permeability Reservoirs," Society of Petroleum Engineers Journal, Vol. 24, Apr. 1984, pp. 141-152.

33 Advani, S. H., et al., "Fluid Flow, Structural, and Fracture Mechanics Modeling Associated With Hydraulic Stimulation Operations," Paper SPE/DOE 10846, SPE/DOE Unconventional Gas Recovery Symposium, Pittsburgh, 1982.

34 Nghiem, L. X., et al., "A Fully Implicit Hydraulic Fracture Model," Journal of Petroleum Technology, Vol. 36, July 1984, pp. 1191-1198.

35 Advani, S. H., Lee, J. K., and Komar, C. A., "The Roles of Rock Mechanics in Optimizing Devonian Gas Production from the Appalachian Basin," Paper SPE 10370, SPE Eastern Regional Meeting, Columbus, 1981.

36 Hanson, M. E., Anderson, G. D., Shaffer, R. J., Lin, W., Towse, D. F., Cleary, M. P., and Haimson, B. C , "LLL Gas Stimulation Program, Quar-terly Project Report," Lawrence Livermore Laboratory, Apr.-June 1979.

37 Hanson, M. E., Anderson, G. D., Shaffer, R. J., Towse, D. F., Hearst, J. R., Lin, W., and Cleary, M. P., "LLL Gas Stimulation Program, Quarterly Project Report," Lawrence Livermore Laboratory, July-Sept. 1979.

38 Clark, P. E., and Quadir, J. A., "Proppant Transport in Hydraulic Fractures: A Critical Review of Particle Settling Velocity Equations," Paper SPE 9866, SPE/DOE Low Permeability Gas Reservoirs Symposium, Denver, 1981, accepted for publication in Society of Petroleum Engineers Journal.

39 Shah, S. N., "Proppant Settling Correlations for Non-Newtonian Fluids under Static and Dynamic Conditions," Society of Petroleum Engineers Journal, Vol. 22, 1982, pp. 164-170.

40 Smith, M. B., Rosenberg, R. J., and Bowen, J. F., "Fracture Width-Design vs. Measurement," Paper SPE 10965, 57th Annual Fall Technical Conference, New Orleans, 1982.

41 Warpinski, N. R., "Measurement of Width and Pressure in a Propagating Hydraulic Fracture," Paper SPE 11648, SPE/DOE Symposium on Low Permeability Reservoirs, Denver, 1983.

42 Rubin, M. B., "Experimental Study of Hydraulic Fracturing in an Im-permeable Material," ASME JOURNAL OF ENERGY RESOURCES TECHNOLOGY, Vol. 105, June 1983, pp. 116-124.

43 Papadopoulos, J. M., Narendran, V. M., and Cleary, M. P., "Laboratory Simulations of Hydraulic Fracturing," Paper SEP/DOE 11618, SPE/DOE Symposium on Low Permeability Reservoirs, Denver, 1983.

44 Biot, M. A., "General Theory of Three-Dimensional Consolidation," Journal of Applied Physics, Vol. 12, Feb. 1941, pp. 155-164.

45 Biot, M. A., "Theory of Elasticity and Consolidation for a Porous and

Anisotropic Solid," Journal of Applied Physics, Vol. 26, Feb. 1955, pp. 182-185.

46 Biot, M. A., "General Solutions of the Equation of Elasticity and Consolidation for a Porous Materials," ASME Journal of Applied Mechanics, Vol.22, 1955, pp. 91-96.

47 Scheidegger, A. E., "Stresses in the Earth's Crust as Determined from Hydraulic Fracturing Data," Geology Bauweseu, Vol. 27, 1962, pp. 41-46.

48 Haimson, B., and Fairhurst, C , "Initiation and Extension of Hydraulic Fractures in Rocks," Society of Petroleum Engineers Journal, Vol. 3, Sept. 1967.

49 Seth, M. S., and Gray, K. E., "Transient Stresses and Displacement Around a Wellbore Due to Fluid Flow in Transversely Isotropic, Porous Media: I. Infinite Reservoirs," Society of Petroleum Engineers Journal, Vol. 8, 1968, pp.63-78.

50 Seth, M. S., and Gray, K. E., "Transient Stresses and Displacement Around a Wellbore Due to Fluid Flow in Transversely Isotropic, Porous Media: II. Finite Reservoirs," Society of Petroleum Engineers Journal, Vol. 8, Feb. 1968, pp.79-86.

51 Medlin, W. L., and Masse, L., "Laboratory Investigation of Fracture Initiation Pressure and Orientation," Society of Petroleum Engineers Journal, Vol. 19, April 1979, pp. 129-144.

52 Cleary, M. P. , "Fundamental Solutions for Fluid-Saturated Porous Media and Application to Localized Rupture Phenomenon," Ph.D. thesis, Brown University, 1975.

53 Rice, J. R., and Cleary, M. P., "Some Basic Stress Diffusion Solutions for Fluid Saturated Porous Media with Compressible Constituents," Review of Geophysical and Space Physics, Vol. 14, 1976, pp. 227-241.

54 Wong, H. Y., and Farmer, I. W., "Hydrofracture Mechanisms in Rock During Pressure Grouting," Rock Mechanics, Vol. 5, 1973, pp. 21-41.

55 Peslyak, Y. A., "Equilibrium of a Crack in a Porous Medium with In-jection of the Filtering Liquid," Prikladnor Mekhaniki Technkheskoi Fiziki, Vol.3, 1974, (English Translation), pp. 155-161.

56 Cleary, M. P. , "Moving Singularities in Elasto-Diffusive Solids with Applications to Fracture Propagation," International Journal of Solids Struct., Vol. 14, 1978, pp. 81-97.

57 Hagoort, J., "Hydraulic Fracturing Pressures in Permeable Sub-Surface Layers," Paper SPE 7110, May 1978.

58 Ruina, A., "Influence of Coupled Deformation-Diffusion Effects on the Retardation of Hydraulic Fracture," Proceedings of the 19th Rock Mechanics Symp., 1979, pp. 274-282.

59 Cleary, M. P., et al., "Surface Integral Schemes for Fluid Flow and Induced Stresses Around Fractures in Underground Reservoirs," Paper SPE/DOE 11632, SPE/DOE Symposium on Low Permeability Gas Reservoirs, Denver, 1983.

60 Salz, L. B., "Relationship Between Fracture Propagation Pressure and Pore Pressure," Paper SPE 6870, 1977.

61 Lamont, N., and Jessen, F. W., "The Effects of Existing Fractures in Rocks on the Extension of Hydraulic Fractures," Journal of Petroleum Technology, Feb. 1963, pp. 203-209.

62 Daneshy, A. A., "Hydraulic Fracture Propagation in the Presence of Planes of Weakness," Paper SPE 4852, SPE European Spring Meeting, 1974, Amsterdam.

63 Hanson, M. E., "Some Effects of Macroscopic Flaws on Dynamic Fracture Patterns Near a Pressure-Driven Fracture," International Journal of Rock Mechanics and Mineral Sciences, Vol. 12, 1975, pp. 311-323.

64 Zoback, M. D., Rummel, F., Jung, R., and Raleigh, C. B., "Laboratory Hydraulic Fracturing Experiments in Intact and Pre-Fractured Rock," In-ternational Journal of Rock Mechanics and Mineral Sciences, Vol. 14, 1977, pp.49-58.

65 Hanson, M. E., Anderson, G. D., Shaffer, R. J., Haimson, B., Montan, R. J., and Cleary, M. P., "LLL Gas Stimulation Report, Quarterly Progress Report," Lawrence Livermore Laboratories, Jan.-Mar. 1978.

66 Hanson, M. E., Anderson, G. D., Shaffer, R. J., Hearst, J. R., Haim-son, B., and Cleary, M. P., "LLL Gas Stimulation Program, Quarterly Progress Report," Lawrence Livermore Laboratory, Apr.-June 1978.

67 Hanson, M. E., Anderson, G. D., and Shaffer, R. J., "Theoretical and Experimental Research on Hydraulic Fracturing," ASME Paper No. 78-Pet-49, ETCE, Houston, Tex., Nov. 1978.

68 Anderson, G. D., "Laboratory Experiments on Hydraulic Fracture Growth Near an Interface," Proceedings of the 19th Rock Mechanics Sym-posium, 1979.

69 Hanson, M. E., Anderson, G. D., and Shaffer, R. J., "Effects of Various Parameters on Hydraulic Fracturing Geometry," Society of Petroleum Engineers Journal, Vol. 21, 1981, pp. 435-443.

70 Anderson, G. D., "Effects of Friction on Hydraulic Fracture Growth Near Unbonded Interfaces in Rocks," Society of Petroleum Engineers Journal, Vol.21, 1981, pp. 21-30.

71 Teufel, L. W., "An Experimental Study of Hydraulic Fracture Propagation in Layered Rock," Ph.D. thesis, Texas A&M University, 1979.

72 Teufel, L. W., and Clark, J. A., "Hydraulic Fracture Propagation in Layered Rock: Experimental Studies of Fracture Containment," Society of Petroleum Engineers Journal, Vol. 24, Feb. 1984, pp. 19-32.

73 Theocaris, P. S., and Milios, J., "Dynamic Crack Propagation in Composites," International Journal of Fracture, Vol. 16, No. 1, 1980, pp. 31-51.

74 Theocaris, P. S., and Milios, J., "Crack-Arrest at a Bimaterial In-



terface," International Journal of Solids and Structures, Vo l . 17, 1981, pp. 217-230.

75 Daneshy, A. A., "Rock Properties Controlling Hydraulic Fracture Propagation," Paper SPE 5752, SPE-European Spring Meeting, Amsterdam, 1974.

76 Hodulak, L., and Peters, W. H., Ill, "Crack Propagation in Bonded Dissimilar Materials," International Journal of Fracture, Vol. 16, 1980, pp. 117-120.

77 Biot, M. A., Medlin, W. L., and Masse", L., "Fracture Penetration Through an Interface," Society of Petroleum Engineers Journal, Vol. 23, Dec. 1983, pp. 857-869.

78 Warpinski, N. R., Northrop, D. A., and Schmidt, R. A., "Direct Ob-servation of Hydraulic Fractures: Behavior at a Formation Interface," Sandia Laboratories Report, SAND78-1935, 1978.

79 Northrop, D. A., Warpinski, N. R., and Schmidt, R. A., "EGR Stimulation Research Project-Direct Observation of Hydraulic and Dynamic Fracturing," 5th Annual DOE Symposium, Aug. 1979

80 Pisarenko, G. S., Naumenko, V. P., and Koval, V. I., "Phenomenon of Crack Arrest in a Plate with Transverse Compression Zones," Problemy Prochnosti, No. 1, 1978, pp. 8-12.

81 Warpinski, N. R., Clark, J. A., Schmidt, R. A., and Huddle, C. W., "Laboratory Investigation on the Effect of In-Situ Stresses on Hydraulic

Fracture Containment," Paper SPE 9834, Society of Petroleum Engineers Journal, Vol. 22, 1982, pp. 333-340.

82 Hanson, M. E., Anderson, G. D., Shaffer, R. J., and Thorson, L. D., "Some Effects of Stress, Friction and Fluid Flow on Hydraulic Fracturing," Paper SPE 9831, Society of Petroleum Engineers Journal, Vol. 22, 1982, pp. 321-332.

83 Thiercelin, M. J., and Lemanczyk, Z. R., "The Effect of Stress Gradient on the Height of Vertical Hydraulic Fractures," Paper SPE/DOE 11626, SPE/DOE Symposium on Low Permeability Gas Reservoirs, Denver, 1983.

84 Ahmed, U., Strawn, J., Wilson, M., and Schatz, J., "Effect of Stress Distribution on Hydraulic Fracture Geometry: A Laboratory Simulation Study in One Meter Cubic Blocks," Paper SPE/DOE 11637, SPE/DOE Symposium on Low Permeability Gas Reservoirs, Denver, 1983.

85 Warpinski, N. R., Schmidt, R. A., and Northrop, D. A., "In-Situ Stresses:The Predominant Influence on Hydraulic Fracture Containment," Paper SPE 8932, SPE/DOE Symposium on Unconventional Gas Recovery, Pittsburgh, 1980.

86 Nolte, K. G., and Smith, M. B., "Interpretation of Fracturing Pressures," Paper SPE 8297, 54th Annual Fall Conference of the SPE of AIME, Las Vegas, 1979.

87 Mendelsohn, D. A., "A Review of Hydraulic Fracture Modeling, Part II: 3D Modeling and Vertical Growth in Layered Rock," to appear in Dec. 1984 JOURNAL OF ENERGY RESOURCES TECHNOLOGY.

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