Examination of the Cavity Expansion Model:

Predicting Hydrofracture During Horizontal

Directional Drilling

Mary Asperger

ECI 284: Theoretical Geomechanics

Term Project

Professor Boris Jeremi

March 23, 2012

Table of Contents

Introduction .................................................................................................................................................. 1

Development and Application of the Cavity Expansion Model to HDD ........................................................ 3

Comparison of the Delft Equation and Finite Element Models .................................................................... 7

The Tensile Failure Fracture Mechanism and Suggested Revisions to the Delft Equation ........................... 9

Comparison of the Delft Equation and Experimental Results .................................................................... 11

Comparison of the Delft Equation and Field Results .................................................................................. 12

Summary and Conclusions .......................................................................................................................... 14

References .................................................................................................................................................. 15

1

Introduction

Modeling the expansion and contraction of spherical and cylindrical cavities is an important part

of theoretical soil mechanics because it can be widely applied to geotechnical problems. Cavity

expansion theory has been used to model situations in both soil and rockincluding interpreting and

modeling the results of pressuremeters and cone penetrometers, as well as determining the breakout

resistance of anchors, the capacity of driven piles, the stability of tunnels, the behavior of geomaterials

during compaction grouting, and the likelihood of hydraulic fractures occurring during oil well and

horizontal directional drilling. This paper will focus on the application of cavity expansion theory to

horizontal directional drilling (HDD).

HDD is a trenchless construction method that was first developed in the 1970s from oil well

drilling technology. Trenchless refers to the fact that the method requires minimal surface excavation,

and therefore has minimal impact on the surrounding area. Trenchless methods are commonly used

where open cut trenching is either impractical or undesirable because of factors such as environmental

impacts, social impacts (e.g. noise or traffic), or obstacles than cannot be disturbed. In its infancy HDD

was primarily used to install small oil and gas pipes. The technology grew in popularity during the 1980s

and experienced a boom in the 90s when it became a very common method of installing ducts for fiber-

optic cable. (Arends, 2003) In addition to small pipes and ducts, in its current state the technology can

be used to install pipelines up to 54 inches in diameter over distances exceeding 8,200 feet for a variety

of uses. (Bennett and Ariaratnam, 2008) Figure 1 shows a typical HDD rig.

Figure 1: A typical medium sized HDD rig.

Installation of a product pipe by HDD is a three step process: a pilot bore is drilled, the hole is

enlarged to the required diameter, and the product is pulled in to the prepared bore. HDD is a surface

launched method; unlike a tunnel boring machine, the drilling rig remains in one location on the ground

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surface during drilling. The operator steers the drill bit over a pre-determined alignment which typically

consists of a straight section, followed by a vertical curve to reach the desired depth, another straight

section to extend the bore to the required length, followed by a second vertical curve and straight

section to rise back to the ground surface. Figure 2 shows a typical pilot bore.

Figure 2: Pilot bore of an HDD operation (Bennett and Ariaratnam, 2008)

After the pilot bore is completed, it is enlarged to the diameter required to install the product

pipe using one or more reaming passes. Once the hole is large enough, the drill pipe is connected to the

product pipe on the side of the bore opposite the rig and the product pipe is pulled into place. The

product pipe must be welded, joined, or fused into a single string in advance of this step so that it can

take place in one continuous operation, as shown in Figure 3.

Figure 3: Pullback of product pipe (Bennett and Ariaratnam, 2008)

Drilling fluid, which is constantly pumped into the borehole through the drill pipe, is a critical

part of the HDD process and serves many purposes. Spoils are removed by suspending them in the

drilling fluid, allowing the cuttings to be pumped to the ground surface through the annulus between

the drill pipe and the bore. It also helps to clean and cool the drill bit during drilling, as well as lubricate

the drilling tools and product pipe to reduce friction during pullback. Finally, it performs the critical

function of stabilizing the bore by creating a filter cake (which prevents inflow from groundwater and

loss of drilling fluid into the formation) and providing positive hydrostatic pressure to prevent collapse.

(Bennett and Ariaratnam, 2008) Drilling fluid at its most basic is a mixture of water and bentonite clay.

Additives such as polymers can also be mixed into the drilling fluid to obtain desirable qualities such as

minimizing the hydration of a swelling clay formation. Unfortunately, because pressure is required to

3

maintain a stable bore and to return cuttings to the surface, there is a risk that drilling fluid can end up

on the ground surface in undesirable places. These inadvertent returns (often referred to as

hydrofractures or frac-outs) can be a significant issue because of concerns that they will negatively

impact sensitive environments such as vernal pools, salmon spawning beds, etc. In less environmentally

sensitive locations they can also impact man-made features such as roadways, existing utilities, and

foundations. (Stauber et al, 2003) Inadvertent fluid returns are often more of an eyesore than an

environmental catastrophe or serious problem. However, they can be detrimental and are of great

concern to project owners and regulators, therefore it is important to be able to predict and minimize

the risk of them occurring. (Wallin and Bennett, 2008)

At its most basic, predicting the risk of hydrofracture for an HDD bore is a question of two

limiting pressures. The upper limit is the maximum pressure the soil can sustain without failing. The

lower limit is the minimum pressure required to transport the drilling fluid and cuttings back to the

ground surface and prevent collapse of the borehole. The minimum pressure portion of the equation is

usually predicted during design using the Bingham plastic fluid model. This is an interesting problem in

and of itself due to the complicated nature of flow through the bore and the properties of the drilling

fluid. However, it is not addressed in this paper which instead focuses on the prediction of the upper

pressure limit.

Development and Application of the Cavity Expansion Model to HDD

A rational method for predicting hydraulic fracturing, or hydrofracture, during HDD was first

proposed by Luger and Hergarden at the 1988 International No-Dig Conference in Washington D.C.

Their method and resulting equation are based on the theory and general solutions for the expansion of

spherical and cylindrical cavities introduced by Vesi in 1972. Since the radius of the bore is very small in

comparison to its length, the problem can be treated as a two-dimensional cylindrical expansion

problem. (The spherical expansion solutions are useful for modeling the behavior of soil around the drill

bit if the bore behind the bit becomes blocked off. This case does occasionally occur, but rarely does it

occur in a predictable manner so it is not typically analyzed during HDD design.) Luger and Hergarden

reasoned that the drilling fluid in the bore will exert pressure on the surrounding soil, causing elastic

deformations. When the pressure in the bore exceeds a certain value, plastic deformation will occur.

Initially the plastic deformation will be limited to a region adjacent to the borehole, but as the pressure

continues to increase the zone of plastic deformation will also increase. Drilling fluid reaching the

ground surface or other damages occur when the plastic zone reaches a certain radius. Therefore, to

prevent inadvertent returns or damages the pressure in the bore must remain low enough to keep the

plastic zone within a safe radius. (Keulen, 2001)

Figure 4 shows a schematic of the borehole and the plastic zone. The bore starts with an initial

radius of R0 and expands to radius Rg as the pressure p increases. The radius of the plastic zone reaches

a value of Rp, depending on the pressure in the borehole. Outside of radius Rp it is assumed that the soil

is completely elastic.

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Figure 4: Schematic of the borehole (Keulen, 2001)

The Luger and Hergarden maximum allowable pressure is determined by first selecting the

maximum allowable plastic zone radius (Rp max). Once this is selected, using the model developed by

Vesi, the radial stress (r) can be derived as a function of the distance to the borehole (r) (shown as line

B in Figure 5 below). The radius of the borehole is also a function of pressure (line B in Figure 5) and at

the boundary of the borehole and the soil the drilling fluid pressure and radial stresses must be equal.

Thus, the intersection of lines A and B give the maximum allowable drilling fluid pressure in the

borehole. (Keulen, 2001)

Figure 5: Pressure of the drilling fluid in the borehole versus the bore radius (line A) and the radial total

stresses versus the radial distance outside the borehole (line B). The intersection of the two lines gives

the maximum allowable pressure. (Keulen, 2001)

5

Figure 5, in addition to showing the maximum allowable pressure pmax, also shows the limit

pressure plim. This limit pressure is the upper bound pressure that can be sustained by the cavity.

Luger and Hergarden assume that the maximum allowable drilling fluid pressure should be no greater

than 90 percent of this limit pressure.

Luger and Hergarden made a series of assumptions to derive equations from the theoretical

framework discussed above. First, it is assumed that the cylindrical cavity is expanding symmetrically

about the axis in an infinite space (there are no boundary effects and gravity is not taken into

consideration). Next, the forces are in equilibrium. The material itself is assumed to be isotropic and

homogenous. The elastic behavior of the soil is described by Hookes law for increments of elastic

deformation (the soil behaves as a linear elastic material prior to yielding). Failure of the soil is defined

by Mohr-Coulombs criterion and once the soil yields it is perfectly plastic.

The following derivation is based on Keulens summary of the derivation used by Luger and

Hergarden. According to the Mohr-Coulomb failure criteria plasticity first occurs when the drilling fluid

pressure p reaches a value pf (shown in Figure 5) equal to:

[1]

Where: 0 initial effective stress friction angle c cohesion u initial in-situ pore pressure The radius of the borehole for effective pressures that have not exceeded pf is described by:

[2]

Where: p = p u effective drilling fluid pressure Rg radius of the borehole R0 initial radius of the borehole G shear modulus

Equation [2] describes Line A (Figure 5) for drilling fluid pressures that are below pf and is

derived from Hookes law.

For the next step of the derivation, the transition zone where position s = Rp must be

considered. The position of a soil particle at this point can be determined using equations [1] and [2]

and is given by:

[3]

Where:

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s0 initial position of the particle s actual position of the particle Since the soil in the plastic zone is assumed to be perfectly plastic, no volume change occurs.

Therefore the volume between r = s0 and r = s (= Rp) is equal to the volume r = R0 and r = Rg. The current

radius of the borehole can be expressed as a function of the initial radius and the radius of the plastic

zone:

[4]

This equation [4] describes the geometry of the borehole and the plastic zone. From this

equation, the value of radial stress at the transition from elastic to plastic behavior, and the assumptions

of equilibrium and Mohr-Coulomb yield function the radial stress can be determined in terms of the

radial coordinate:

[5]

Where:

r' radial effective stress

Equation [5] is the equation of Line B in Figure 5 where Rp max is substituted for Rp. Therefore,

using equations [4] and [5] the relationship between the drilling fluid pressure in the bore and the actual

radius of the borehole can be determined. This uses the relationship that the effective radial stress at

the borehole wall is equal to the drilling fluid pressure (r = p):

[6]

Where:

Equation [6] describes the section of Line A in Figure 5 for drilling fluid pressures greater than pf.

The maximum allowable pressure pmax is given by the intersection of lines A and B:

[7]

Equation [7] is the Delft Equation, which is still the equation commonly used in analyzing the risk

of hydrofracture occurring during a horizontal directional drill.

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The final equation of this derivation is the equation for the limit pressure that was discussed

previously. From equation [7] the limit pressure occurs when Rp,max approaches infinity:

[8]

Comparison of the Delft Equation and Finite Element Models

There have been several studies conducted which examined the accuracy of the Delft Equation

by comparing it to finite element models. The two that will be discussed in this paper are a study by

Matthew Kennedy, Graeme Skinner, and Ian Moore that was presented in 2004, and a study by Ian

Moore that was presented in 2005.

Kennedy et al (2004) compared a simplified version of the Delft equation to an elastic finite

element analysis. They assumed that the soil was purely cohesive and had a friction angle of zero. This

assumption causes the Delft equation to reduce to:

[9]

This does not take any potential tensile strength due to cementation or chemical processes into

account. It is a conservative assumption, but a reasonable one for the purposes of this study as it is

intended as a comparison between the finite element model and the Delft equation.

The finite element model used in this study makes a number of assumptions similar to the Delft

model. Gravity is neglected; therefore there is no gradient in earth pressures across the bore cavity and

no gradient in drilling fluid pressure across the bore cavity. In addition, the finite element model did not

consider the potential for shear failure in the soil around the bore cavity. This assumption is generally

considered reasonable for deep bores. In shallow bores, shear failure (where a wedge of soil is pushed

out of the ground) is considered the more likely failure mechanism. (Keulen, 2001) The finite element

model used for this study also assumes two-dimensional plane-strain conditions. While many of these

assumptions are similar to the assumptions for the Delft equation, one primary difference is that the

finite element model could account for variability in K0, which it is not possible using the Delft equation.

The finite element model was developed using the assumption that when the circumferential

stress equals zero, failure occurs. Therefore, the critical location where failure was expected was at the

crown of the bore. The equation for the maximum allowable mud pressure used in the analysis (derived

from elastic theory) is:

[10]

Results of the Kennedy et al (2004) study are presented in Figure 6. The authors concluded that

for cohesive soils in the elastic range the Delft equation generates results that are very unconservative

for values of K0 less than one. They hypothesized that this is due to the Delft equation assumption of K0

= 1 and that the soil will fail in shear. However, they stated that additional analysis of hydraulic fracture

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in elastic-plastic soils was needed to both examine maximum allowable pressures beyond the elastic

range and provide more reliable design equations.

Figure 6: Maximum allowable drilling fluid pressures versus K0; comparison with the maximum allowable

pressure from the Delft equation (equation [7] in this paper. The stars on the blue line showing the

results of the elastic equation represent the point past which the equation is not valid because shear

failure has occurred. (Kennedy et al, 2004)

In 2005, Ian Moore reported on additional finite element comparisons with the Delft equation.

Finite element calculations in this study were performed using a nonlinear finite element program called

AFENA developed by J.P. Carter at the University of Sydney in 1992. The study examined the growth of

the plastic zone as drilling fluid pressures were increased for four different conditions. The first solution

assumed that the soil was purely cohesive and that K0 was equal to zero. The second solution examined

a purely cohesive soil with a K0 of 0.85. The final two solutions looked at purely friction soils with K0

values of one and 0.4, respectively.

The results of this study indicate that for both frictional and cohesive soils for values of K0 equal

or close to one (K0 > 0.85) the Delft equation estimates a radius of the plastic zone that is very similar to

the radius calculated by the finite element analysis. The results of these comparisons are presented in

Figure 7.

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Figure 7: Comparison of the extent of the plastic radius from the Delft equation (shown as Arends (2003)

in the figure) and the finite element analyses performed by Moore. (Moore, 2005)

In the same paper, Moore presented results of further examination of the work by Kennedy et

al. (2004) discussed earlier in this section. Kennedy et al. hypothesized that rather than the radius of the

plastic zone being the critical criteria for anticipating hydrofracture, the analysis should instead be based

on the initiation of tensile fracture at the crown or springline (midpoint on the vertical axis) of the

borehole. This hypothesis will be discussed further in the following section.

The Tensile Fracture Failure Mechanism and Suggested Revisions to the Delft

Equation

According to Moore (2005), the tensile failure mechanism can be separated into two processes

which can be considered separately: fracture initiation, caused by the circumferential stresses being

reduced from a state of compression to zero; and fracture propagation, which can be modeled using

classical fracture mechanics. In his thesis work, Hong Chang reports that laboratory experiments by Lo

and Kaniaru (1990) have shown that a lower pressure is required to propagate a fracture than to initiate

it (see Figure 8, below). Therefore to prevent hydrofracture, the critical portion of this two-phase

process is the fracture initiation.

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Figure 8: Injection pressure of a hydraulic fracturing test. Note that the peak pressure occurs at the

initiation of the fracture. (Lo and Kaniaru, 1990 as reported in Chang, 2004)

After observing both small and large-scale experiments in the lab, Chang concludes that

hydraulic fracturing in sands is a three step process: cavity expansion before the injection pressure

reaches its peak; fracture front initiation from the expanding cavity near the pressure peak; and

propagation of the developed fracture after peak.(Chang, pg 229) He concludes that a relatively large

plastic zone is formed before a fracture initiates and the pressure reaches its peak. This suggests that

the Delft equation could be applied to the fracture initiation problem.

Keulen in his 2001 thesis suggests just such a revision to the Delft equation. Rather than basing

the maximum allowable drilling fluid pressure on the radius of the plastic zone, Keulen presents a failure

criteria based on allowable maximum strain. He theorizes that as the drilling fluid expands the borehole

the soil particles are pushed further and further apart until they reach a distance where the mud can

form a kind of wedge (Keulen, pg 25) that could initiate a fracture (see Figure 9). The equation he

derives based on a maximum allowable strain is as follows:

[11]

Where:

t,max maximum allowable strain

11

dilation angle

When no volume change is assumed (as is the case in the original Delft equation) and therefore

the dilation angle is zero, equation [11] becomes:

[12]

Figure 9: The influence that the expansion of the borehole has on strain at the outer radius, suggesting

that failure could be initiated by a kind of wedge of drilling fluid. (Keulen, 2001)

Comparison of the Delft Equation and Experimental Results

In their 1988 paper which first applied the cavity expansion model to HDD, Luger and Hergarden

reported good correlations between the Delft equation and a full scale experiment.

Keulen (2001) reported the results of an experiment conducted in the Netherlands as part of the

research performed under the heading of Boren van Tunnels en Leidingen (BTL). Mori and Tamura

(1987) looked at hydraulic fracturing by inflating a balloon inside a borehole (thus eliminating loss of

fluid into the formation). They found that the limit pressures seen in the experiment almost perfectly

matched the predictions of the cavity expansion theory.

Elwood and Moore in 2009 reported on five experiments that were conducted on frictional soil

materials (well graded sand and gravel) in a concrete test pit. A borehole was drilled into the

compacted soils in the test pit and then the end of the bore was packed off. After packing the borehole

off it was slowly pressurized using bentonite drilling fluid until failure occurred and drilling fluid returned

to the ground surface. Drilling fluid pressures were measured throughout the process using a resistance

pressure transducer in the borehole. These experimental results were then compared to finite element

and Delft equation estimates. The results are shown in Table 1, below. It is shown that the finite

element model and the Delft equation estimated the maximum drilling fluid pressures very well,

provided reasonably accurate soil input parameters were used.

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Table 1: Experimental results and analytical solutions (Elwood and Moore, 2009)

Comparison of the Delft Equation and Field Results

Keulen briefly examined possible values of maximum strain of 2% and 5% and compared the

predictions obtained to experiments and field tests performed in the Netherlands under the heading of

Boren van Tunnels en Leidingen (BTL). Unfortunately, the comparison was overall inconclusive because

hydrofracture did not occur on any of the field test bores. He did however state that setting the

maximum allowable strain at 2% was likely over-conservative based on the results of the BTL tests.

Arends (2003) further reported that the strain based model better approached the values measured by

the BTL tests.

In 2010 Wallin, Wallin, and Bennett reported the results of a bore drilled in Northern California

where the downhole drilling fluid pressures were measured and compared to the maximum allowable

pressures estimated by the Delft equation. The geotechnical conditions of the crossing were modeled

using two distinct soil layers: an upper layer of very soft organic clay and silt and a lower layer of stiff to

very stiff silt and medium dense to dense silty sand. The results of the Delft equation estimate are

shown in Figure 10 below, where the blue dashed line represents the maximum allowable pressure and

the pink dashed line represents the minimum required pressure to return cuttings to the ground

surface. The results of the estimated pressures show that there is an elevated risk of hydrofracture for

the last approximately 250 feet of the bore.

Figure 11 shows the results of the downhole drilling fluid pressure monitoring system during the

actual drilling of the pilot bore. Yellow stars show where hydrofractures occurred during drilling. As the

graph shows, in three locations where the downhole drilling fluid pressures exceeded the estimated

maximum allowable pressure, hydrofracture occurred. There is one point around approximate Station

6+10 where the downhole pressure exceeded the estimated maximum allowable pressure but no

hydrofracture was observed. It is possible the drilling fluid found a preferential seepage path to a

location other than the ground surface, or that the inadvertent returns simply were not observed as this

location was within the active slough channel. As a side note, the brown blobs indicate approximate

13

locations where wood was encountered during drilling. It is possible that sunken logs contributed or

caused the significant pressure spikes observed.

Figure 10: Graph of the minimum required and maximum allowable drilling fluid pressures for the HDD

crossing of Freshwater Slough in Northern California (Wallin et al., 2010)

Figure 11: Comparison of actual (shown in orange) versus theoretical drilling fluid pressures as estimated

by the Delft cavity expansion equation. (Wallin et al., 2010)

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Summary and Conclusions

The drawbacks of the current model sadly include several of its key assumptions. First, real soil

is never perfectly homogeneous and isotropic, nor is it generally perfectly plastic. Unfortunately, the

Delft equation as proposed by Luger and Hergarden (1988) is not well suited to incorporating changes to

K0, including possible effects from dilation or contraction, or modeling complex soil profiles. However,

revisions to the equation were proposed by Keulen (2001) that can address at least the issue of

including possible dilative effects for particulate materials, as well as including a failure criteria based on

maximum allowable strain as opposed to the radius of the plastic zone. This equation still requires more

testing to check its validity and to determine reasonable values of maximum allowable strain. The

maximum allowable strain criterion also appears to more closely approximate the tensile failure

mechanism suggested by Kennedy (2004), Moore (2005), and Chang (2004).

Going forward, the cavity expansion model should be examined, updated, and refined to

incorporate additional theoretical components and experimental results. For example, Xia and Moore

(2006) suggest a method to incorporate varying K0 values; while Nie (2011) discusses modeling large

strain cavity expansion, which ties in with Keulens maximum allowable strain failure criteria. The cavity

expansion model is so widely used in geotechnical modeling that there is a great deal of information

available that has not yet been examined by the HDD industry.

Although there are some questions as to whether the Delft cavity expansion equation correctly

models the actual mechanism of what happens when hydrofracture occurs, it appears to correlate well

with finite element analyses, and experimental and field results. Thus, it is a useful tool for designers in

the HDD industry to predict and mitigate the risk of hydraulic fracturingleading to more successful

projects with lower environmental and social impacts.

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References

Arends, G. (2003). Need and possibilities for a quality push within the technique of horizontal

directional drilling (HDD), Proceedings of 2003 No-Dig Conference, Las Vegas, Nevada, March 31-April

2, 2003.

Bennett, R.D. and Ariaratnam, S.T. (2008) Horizontal Directional Drilling Good Practices Guidelines, Third

Edition. HDD Consortium, 2008.

Bennett, R.D. and Wallin, K. (2008). Step by Step Evaluation of Hydrofracture Risks for HDD Projects,

Proceedings of 2008 No-Dig Conference, Dallas, Texas, April 27-May 2, 2008.

Chang, H. (2004). Hydraulic Fracturing in Particulate Materials, Georgia Institute of Technology Thesis

Paper.

Elwood, D. and Moore, I.D. (2009). Hydraulic Fracture Experiments in Sand and Gravel and

Approximations for Maximum Allowable Mud Pressure Proceedings of 2009 International No-Dig

Conference, Toronto, Ontario Canada, March 29-April 3, 2009.

Kennedy, M.J., Skinner, G.D., Moore, I.D. (2004). Elastic Calculations of Limiting Mud Pressures to

Control Hydrofracture During HDD, Proceedings of 2004 No-Dig Conference, New Orleans, Louisiana,

March 22-24, 2004.

Keulen, B., Arends, G., Mastbergen, D.R. (2001). Maximum allowable pressures during directional

drilling focused on sand, Delft Geotechnics Thesis Paper: 23-32, Appendices 39-74.

Lo, K.Y. and Kaniare, K. (1990). Hydraulic fracture in earth and rock-fill dams, Canada Geotechniques,

27, 296-506, 1990.

Luger, H.J., Hergarden, H.J.A.M. (1988). Directional drilling in soft soil; influence of mud pressures,

Proceedings of the International No-Dig Conference, Washington D.C.

Moore, I.D. (2005). Analysis of Ground Fracture Due to Excessive Mud Pressure, No-Dig 2005, Orlando,

Florida.

Nie, C. (2011). Unified Analytical Solution for Cylindrical Cavity Expansion of Saturated Soil Based on

Large Deformation and Non-drainage Condition, Advanced Materials Research Vols. 168-170, pp 2406-

2415.

Stauber, R.M., Bell, J., Bennett, R.D. (2003). A Rational Method for Evaluating the Risk of Hydraulic

Fracturing in Soil During Horizontal Directional Drilling (HDD) Proceedings of 2003 No-Dig Conference,

Las Vegas, Nevada, March 31-April 2, 2003.

Vesi, A.S. (1972). Expansion of cavities in infinite soil mass ASCE Journal of the Soil Mechanics and

Foundations Division, Vol. 98: 265-290.

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Wallin, K., Wallin, M., and Bennett, R.D. (2010). HDD Crossing Under Environmentally Sensitive Slough:

Mitigation of Hydrofracture Risk Proceedings of 2010 No-Dig Conference, Chicago, Illinois, May 2-May

7, 2010.

Xia, H.W. and Moore, I.D. (2006). Estimation of maximum mud pressure in purely cohesive material

during directional drilling, Geomechanics and Geoengineering, Vol. 1, No. 1, March 2006, 3-11.

Yu, H.S. (2000). Cavity Expansion Methods in Geomechanics. Kluwer Academic Publishers, 65-73.