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Morphology of multiple constant height hydraulic fractures versus propagation regime E.V. Dontsov 1 1 [email protected], ResFrac Corporation, Palo Alto, USA Abstract The purpose of this study is to investigate morphology of simultaneously propagating hydraulic fractures. Simultaneous propagation of hydraulic fractures occurs during stimulation of horizontal wells, and, in particular, several initiation points or perforation intervals along the well are often used to promote the growth of multiple hydraulic fractures at the same time. Numerical simulations demonstrate that there are situations, in which stress interaction between the fractures is minimal and this results in a very similar geometry of each fracture. At the same time, for some other parameters, the stress interaction is very strong, so that the fractures interact with each other and develop complex shapes. By focusing on the constant height hydraulic fractures, it is shown that there is a dimensionless parameter that controls such a behavior. In particular, fractures propagating in the toughness dominated regime lead to complex shapes, while when fluid viscosity dominates the response, then the fractures are more regular and uniform. A series of numerical examples is presented to illustrate the findings. 1 Introduction Multistage hydraulic fracturing is often used to enhance recovery of hydrocarbons from low permeability formations [1]. To optimize the process, multiple hydraulic fractures are initiated simultaneously from a horizontal wellbore. In addition, this is usually supplemented by a limited entry design, which promotes uniform flow distribution between the fractures. It was shown, however, that even in the simplest case of no geological layers, such a problem can lead to the development of instabilities that can cause complex fracture shapes [2]. In addition, similar behavior was observed for field scale simulations of hydraulic frac- turing [3]. This behavior is important from both fundamental and practical points of view. First, it allows to better understand processes happening underground, and second, it permits to control this behavior using operational procedures. As was shown in [2], hydraulic fracture propagation regime is the key factor that determines morphology of simultaneously propagating hydraulic fractures. For simple fracture geometries, such as plane strain, radial, and constant height cracks, there are four limiting cases or propagation regimes [4]. These are determined by the competition between viscous dissipation and fracture toughness, as well as the fluid storage inside the fracture or the surrounding formation. It is important to note that these mathematical conclusions are based on the model that features linear elastic rock deformation, Newtonian fracturing fluid, linear elastic fracture mechanics, Carter’s leak-off, and uniform injection rate. Other important physical processes, such as proppant transport or the effect of hydrostatic pressure are ignored. Nevertheless, it was concluded in [2] that when viscosity dominates, then all the fractures are nearly identical. At the same time, when toughness dominates, then the fractures strongly compete with each other and develop petal-like shapes. This was the case for “radial” geometry, or when there were no geological layers. Qualitatively similar observations were made for field scale hydraulic fracture modeling in [3]. Namely, the increase of fracture toughness leads to more fracture asymmetry, while the increase of viscosity (and/or decrease of toughness) leads to more regular and symmetric fractures. One of the challenges, however, is to quantitatively determine the parameter that governs such a transition in fracture morphology. As a result, this study aims to extend the previous analysis [2] for the case of constant height hydraulic fractures. While such a geometric assumption 1 arXiv:2110.13347v1 [physics.geo-ph] 26 Oct 2021
Transcript of propagation regime
Morphology of multiple constant height hydraulic fractures versus
propagation regime
E.V. Dontsov1
[email protected], ResFrac Corporation, Palo Alto, USA
Abstract
The purpose of this study is to investigate morphology of simultaneously propagating hydraulic fractures.Simultaneous propagation of hydraulic fractures occurs during stimulation of horizontal wells, and, inparticular, several initiation points or perforation intervals along the well are often used to promote thegrowth of multiple hydraulic fractures at the same time. Numerical simulations demonstrate that thereare situations, in which stress interaction between the fractures is minimal and this results in a verysimilar geometry of each fracture. At the same time, for some other parameters, the stress interactionis very strong, so that the fractures interact with each other and develop complex shapes. By focusingon the constant height hydraulic fractures, it is shown that there is a dimensionless parameter thatcontrols such a behavior. In particular, fractures propagating in the toughness dominated regime lead tocomplex shapes, while when fluid viscosity dominates the response, then the fractures are more regularand uniform. A series of numerical examples is presented to illustrate the findings.
1 Introduction
Multistage hydraulic fracturing is often used to enhance recovery of hydrocarbons from low permeabilityformations [1]. To optimize the process, multiple hydraulic fractures are initiated simultaneously from ahorizontal wellbore. In addition, this is usually supplemented by a limited entry design, which promotesuniform flow distribution between the fractures. It was shown, however, that even in the simplest case ofno geological layers, such a problem can lead to the development of instabilities that can cause complexfracture shapes [2]. In addition, similar behavior was observed for field scale simulations of hydraulic frac-turing [3]. This behavior is important from both fundamental and practical points of view. First, it allows tobetter understand processes happening underground, and second, it permits to control this behavior usingoperational procedures.
As was shown in [2], hydraulic fracture propagation regime is the key factor that determines morphology ofsimultaneously propagating hydraulic fractures. For simple fracture geometries, such as plane strain, radial,and constant height cracks, there are four limiting cases or propagation regimes [4]. These are determinedby the competition between viscous dissipation and fracture toughness, as well as the fluid storage insidethe fracture or the surrounding formation. It is important to note that these mathematical conclusions arebased on the model that features linear elastic rock deformation, Newtonian fracturing fluid, linear elasticfracture mechanics, Carter’s leak-off, and uniform injection rate. Other important physical processes, suchas proppant transport or the effect of hydrostatic pressure are ignored. Nevertheless, it was concluded in [2]that when viscosity dominates, then all the fractures are nearly identical. At the same time, when toughnessdominates, then the fractures strongly compete with each other and develop petal-like shapes. This wasthe case for “radial” geometry, or when there were no geological layers. Qualitatively similar observationswere made for field scale hydraulic fracture modeling in [3]. Namely, the increase of fracture toughnessleads to more fracture asymmetry, while the increase of viscosity (and/or decrease of toughness) leads tomore regular and symmetric fractures. One of the challenges, however, is to quantitatively determine theparameter that governs such a transition in fracture morphology. As a result, this study aims to extend theprevious analysis [2] for the case of constant height hydraulic fractures. While such a geometric assumption
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is still far from a general field scale simulation, it is arguably a better approximation relative to the radialcase, especially when strong barriers limit fracture propagation in the vertical direction. Finally, and mostimportantly, it is possible to perform a relatively simple analysis for the constant height hydraulic fractures,which is an essential component that is needed to quantify the fracture morphology.
Constant height hydraulic fractures are often referred to as Perkins–Kern–Nordgren (PKN) [5, 6] frac-tures. Many researchers have investigated propagation of PKN fractures from various perspectives. A fewpapers that are relevant to this work are [7, 8, 9]. The classical PKN model does not account for the effectof fracture toughness and therefore is unsuitable for analyzing the problem under consideration since thetransition from toughness domination to viscosity domination presents the most interest. The first attemptto include the effect of toughness was done in [7], where the propagation condition in terms of pressurewas taken from the uniformly pressurized radial fracture with the diameter equal to the reservoir height.This methodology was then revised in [8], in which the authors proposed a more accurate procedure thatutilizes energy considerations. Finally, it was shown in [9] that the second energetic approach is indeed moreaccurate (albeit quite marginally) by comparing the results to the solution with non-local elasticity as wellas to the simulations performed with a fully planar model.
One prerequisite for conducting the study on simultaneous propagation of multiple hydraulic fractures isthe knowledge of the parametric space for the problem. For instance, for the case of radial geometry [2], theprerequisite paper is [10], which builds on top of many previous publications and outlines boundaries in theparametric space that quantify dominance of various physical processes. In the context of constant heightor PKN fractures, such a parametric space was recently constructed in [11]. In particular, the model thataccounts for toughness based on the results in [8] is used due to its simplicity and still admissible accuracy.Section 2 summarizes mathematical formulation for the model and describes the aforementioned parametricspace. Equipped with this parametric space, several cases are constructed for multiple fractures that liewithin different regions of the parametric space. Numerical simulations are performed using commercialcode ResFrac1 [12] and results of simulations for simultaneous propagation of multiple hydraulic fracturesare presented in section 3 for different regimes. Finally, conclusions are summarized in section 4.
2 Parametric analysis of a single constant height hydraulic frac-ture
This section outlines the mathematical model for a single PKN fracture and also presents the parametricspace for the problem. With the reference to Fig. 1, let the fracture be vertical and occupy the (x, y) plane,where y is the vertical coordinate and x is the horizontal coordinate. The fracture height is denoted by H,while the half-length is l(t). The model assumes that H l, which ensures that the flow is predominantlyhorizontal and also confirms validity of the local elasticity assumption.
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Figure 1: Schematics of one wing of a constant height hydraulic fracture.
1www.resfrac.com
2
By following the classical PKN model [5, 6], fracture width in each vertical fracture cross-section isassumed to be elliptical, and the fracturing fluid pressure is determined based on the local plane strainelasticity assumption:
w(x, y) =4
πw(x)
√1−
(2y
H
)2, p(x) =
2E′w(x)
πH, w(x) =
1
H
∫ H/2
−H/2w(x, y) dy. (1)
Here w(x, y) is the fracture opening, w(x) is the effective or averaged width, p(x) is the fluid pressure, andE′ = E/(1−ν2) is the plane strain Young’s modulus. The primary governing equation is the vertically-averaged volume balance, which reads:
∂w
∂t+∂qx∂x
+C ′√
t−t0(x)=
Q0
Hδ(x), qx = − 1
12Hµ
∂p
∂x
∫ H/2
−H/2w3 dy = − w3
π2µ
∂p
∂x. (2)
In the above equation µ is the fluid viscosity, qx denotes the horizontal flux, C ′ = 2Cl is the scaled Carter’sleak-off coefficient, and t0(x) corresponds to time when the fracture front was located at the point x.
Propagation condition at the lateral fracture tip is based on the results from [8] and can be written interms of the toughness dependent pressure condition as
p(l) =2KIc√πH
, (3)
where KIc is fracture toughness. This propagation condition is supplemented by the zero flux at the tip,which translates into the following equation for fracture length:
dl
dt=qx(l)
w(l)= − 2E′
3π3µH
∂w3
∂x
∣∣∣∣x=l
. (4)
By following the results in [11], one possibility to reduce the number of parameters in the problem is tointroduce the following normalization:
Ω =w
w∗, λ =
l
l∗, τ =
t
t∗, ξ =
x
l. (5)
Here the width, length, and time scales are given by
w∗ =(πH)1/2KIc
E′, l∗ =
H2K4Ic
2πE′3µQ0, t∗ =
H7/2K5Ic
2π1/2E′4µQ20
. (6)
By adopting such a normalization, the governing equations (1)–(4) become
∂Ω
∂τ− ξλ
λ
∂Ω
∂ξ− 1
λ2∂2Ω4
∂ξ2+
φ√τ−τ0(ξ)
= δ(ξ), Ω(1) = 1,dλ
dτ= − 4
3λ
∂Ω3
∂ξ
∣∣∣∣ξ=1
, (7)
where the dimensionless parameter that quantifies leak-off is
φ =( H5K6
IcC′4
4π3E′4µ2Q40
)1/4. (8)
Thus, the solution to the problem in terms of length and effective width depends on two dimensionlessparameters, namely dimensionless time τ and dimensionless leak-off φ. Conceptually, these parameters aresimilar to those for a radial fracture [10], but quantitatively they are very different.
Parametric space for this problem was constructed in [11] and is shown in Fig. 2. The blue, red, green,and magenta regions indicate zones of applicability of storage viscosity or M , storage toughness or K, leak-off viscosity or M , and leak-off toughness or K solutions, respectively. These are the zones, in which thedifference between the “true” solution and the limiting self-similar solution falls below a small threshold,
3
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log10(
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Figure 2: Parametric space for a single constant height hydraulic fracture.
see [11] for more details. Here the storage viscosity limit corresponds to dominance of viscosity over toughnessand small leak-off, i.e. when most of the injected fluid stays inside the fracture. In contrast, storage toughnesslimit occurs when toughness dominates over the viscosity, while most of the injected fluid still stays insidethe fracture. The leak-off regimes are defined in a similar fashion, but most of the injected fluid leaks intothe formation in these limits. Equations that define boundaries of the transition regions are given by
τMK = τ, τMK,1 = 0.11, τMK,2 = 2.3× 103,
τKK = τφ2, τKK,1 = 5.7× 10−5, τKK,2 = 3.1× 103,
τKM = τφ−2, τKM,1 = 0.18, τKM,2 = 6.5× 104,
τMM = τφ10/3, τMM,1 = 2.0× 10−7, τMM,2 = 2.9× 103. (9)
In the context of the analysis of propagation of multiple hydraulic fractures, this parametric space allowsto select problem parameters that correspond to dominance of either toughness, viscosity, or fall into thesituation when both are important.
3 Numerical results for multiple fractures
A series of numerical simulations for simultaneous propagation of multiple hydraulic fractures is performedin ResFrac, which is a fully coupled 3D hydraulic fracturing and reservoir simulator [12]. Three basic casesare considered: toughness domination (denoted by K), transition (denoted by MK), and viscosity (denotedby M) domination. The input parameters as well as the corresponding dimensionless parameters τ and φare summarized in Tab. 1, see (5), (6), and (8) for the definitions of τ and φ. Note that leak-off cases arenot considered because previous study [2] showed that leak-off does not qualitatively affect the behavior.Uniform flux distribution is achieved by substantially increasing perforation friction, the effect of gravity isneglected, and the height constraint is introduced by having strong barriers above and below the primarylayer of interest. All the injection points are located in the middle of the layer. Only 5 fractures are initiated,all of which are planar and perpendicular to the well in the base case. Also, the base simulations have 10 muniform spacing between the fractures, while other values are used to investigate the effect of spacing, namely20, 40, 80, and 160 m.
Fig. 3 shows results of numerical simulations for 10 m spacing. First of all, the parametric map (similarto the one shown in Fig. 2) is shown and three circular markers indicate relative location of the three casesunder consideration. As can be seen, one point lies within the toughness domination zone, the second oneis in between the toughness and viscosity domination zones, while the third point lies within the viscositydomination zone. Hydraulic fractures for all these three cases are shown in two views: original, in whichphysical proportions are preserved, and stretched, in which the result is stretched 10 times along the wellborefor better fracture visualization.
4
Property K MK M
KIc [MPa·m1/2] 5 3 3H [m] 80 80 80E [GPa] 10 10 25
ν 0.25 0.25 0.25µ [Pa·s] 0.003 0.01 0.05
Cl [m/s1/2] 0 0 0Q0 [m3/s] 0.1 0.1 0.2t [hrs] 1 1 1τ 0.35 15 1.2× 104
φ 0 0 0
Table 1: Input parameters for numerical simulations of propagation of multiple hydraulic fractures in thetoughness K, transition MK, and viscosity M regimes.
As can be seen from the fracture geometries in Fig. 3, toughness domination (case 1) leads to unstablefracture growth, whereby the fractures tend to avoid each other and form irregular shapes. Some fracturespropagate predominantly to the left from the well and some propagate to the right. Moreover, the “local”fracture height of each individual fracture is not equal to the reservoir height (or height of the primary layer).Once the viscosity is increased and toughness is reduced (case 2), the fractures become more symmetric andhave nearly identical lengths, even though the “local” fracture height is still variable along each individualfracture. Finally, when viscosity dominates (case 3), all the fractures become nearly identical and thefracture height of each fracture is equal to the reservoir height in most regions of the fracture. The exceptionis the near tip region, whose size is on the order of fracture height O(H), where the non-uniform behaviorstill persists. Thus, these results demonstrate that the behavior is qualitatively similar to that observedfor “radial” geometry in [2], i.e. the case without the barriers restricting vertical growth. One very bigdifference, however, is the parameter that quantifies the transition from one limiting case to another. Forthe radial case, the transition from the viscosity to the toughness regime is determined by the parameter
4.5× 10−2 <(32
π
)9/2 K9Ict
(125µ5E′13Q30)1/2
< 2.6× 106, (10)
where the lower bound corresponds to the onset of the viscosity regime, while the upper bound correspondsto the onset of the toughness regime. For constant height fractures, on the other hand, the result is
0.11 <2π1/2E′4µQ2
0t
H7/2K5Ic
< 2.3× 103, (11)
where the lower bound corresponds to the onset of the toughness regime, while the upper bound correspondsto the onset of the viscosity regime. Clearly, these equations (10) and (11) are very different and lead tovery different results for these geometries.
Fig. 4 illustrates sensitivity of the results to fracture spacing. Numerical simulations are performed forthe base simulations (see Tab. 1), but for various spacing ranging from 10 m to 160 m. Given that thereservoir height is 80 m, then the ratios between the spacing and height are from 1/8 to 2/1. The samenumber of fractures, i.e. 5, is used in all simulations. The vertical axis corresponds to the approximatevalue of log10(τ), but essentially shows the results for toughness (lowest τ), transition (intermediate τ), andviscosity (highest τ) cases. As can be seen from the results, the normalized spacing significantly affects thefracture morphology, as expected. Once the spacing is twice as much as the fracture height, there is almost nointeraction between the fractures, which is consistent with the fact that the stress shadow decays quickly atdistances commensurate with fracture height (for constant height fractures). Once the fractures are broughtcloser together, stress interaction starts to distort the fracture shapes if toughness dominates or sufficientlylarge. Viscosity dominated fractures, on the other hand, remain very similar. These results illustrate howfracture height and spacing, as well as the fracture regime affects the resulting fracture morphology.
5
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1 2 3
3.2.
1.Original
Stretched 10x
Original
Stretched 10x
Original
Stretched 10x
Figure 3: Results of numerical simulations for 5 simultaneously propagating hydraulic fractures with 10 mspacing. Parametric space is shown in the top left corner, in which circular markers indicate locations ofthe problem parameters summarized in Tab. 1. Actual fracture geometries are shown in two views: originaland stretched. In the latter, the result is stretched 10 times along the wellbore to better visualize individualfractures.
Finally, Fig. 5 presents sensitivity of the results to stress orientation. In particular, the change of thestress orientation angle allows to effectively adjust the angle between the well and the direction of fracturepropagation. The zero angle corresponds to fractures propagating perpendicular to the well, while othercases correspond to the rotation (measured in degrees) relative to this baseline position. Note that thefracture spacing along the well is kept fixed, so that the physical spacing between the fractures, measuredperpendicularly to the fracture surface, is effectively multiplied by the cosine of the rotation angle. Theinput parameters for the “base” simulation, i.e. the one corresponding to zero degree stress rotation, aregiven in Tab. 1. Results shown in Fig. 5 demonstrate that changing the stress orientation angle noticeablyaffects the results. First of all, once the symmetry is broken, the fractures that have an initial lateral shifttend to propagate predominantly in the direction of this shift. This applies to all cases, albeit to a differentdegree. The viscosity dominated cases still have significant fracture overlap, and this overlap drasticallyreduces towards lower values of τ or toughness domination. Another interesting observation is that theheight of each individual fracture tends to be equal to the reservoir height for larger stress angles, even inthe toughness domination regime. That is, the complexity of fracture morphology is reduced once the stressorientation angle increases. This is caused perhaps by the initial bias that fractures have, which significantlyaffects the stress interaction at early times of propagation. Note that such a behavior persists even if thefluid viscosity is further reduced, i.e. when the system is brought further inside the toughness domination
6
Spacing/Height 1/8 1/4 1/2 1/1 2/1
log10(
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4
1
-1
Figure 4: Variation of the fracture morphology versus normalized spacing and log10(τ) that determines thefracture regime.
zone.
4 Summary
In this work, hydraulic fracture morphology for the case of simultaneous propagation of several fracturesis investigated in view of dominance of either fluid viscosity or toughness. In particular, the results areobtained for the case of height contained fractures. First, the dimensionless parameters that define the para-metric space are outlined and a single parameter that determines the transition from toughness to viscositydomination is defined. Numerical results for 3 base cases (toughness domination, viscosity domination, andtransition) are performed for the case of 5 uniformly spaced fractures. It is observed that toughness domina-tion leads to strong stress interaction between the fractures, which leads to the creation of complex fracturegeometries. At the same time, relatively large viscosity tends to regularize the fracture morphology andleads to the formation of nearly identical fractures. Numerical simulations with different fracture spacingsindicate that the ratio between the fracture spacing to the reservoir height plays a crucial role, since thestress interaction decays quickly as the spacing to height ratio is increased. Also, the influence of stressorientation on the results is investigated, which showed that it also affects the fracture morphology, but toa smaller degree. Since the results are presented with the reference to the dimensionless parametric space,these findings can help to quickly identify the fracture regime and morphology for any input parameters andcan be used as a guideline to reach a desired fracture morphology configuration.
Acknowledgments
The author would like to acknowledge Mark McClure for useful recommendations regarding this work.
7
Stress angle0 15 30 45
log10(
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Figure 5: Variation of the fracture morphology versus stress orientation angle (in degrees) and log10(τ) thatdetermines the fracture regime.
References
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[9] E.V. Dontsov and A. P. Peirce. Comparison of toughness propagation criteria for blade-like and pseudo-3d hydraulic fractures. Eng. Frac. Mech., 160:238–247, 2016.
[10] E.V. Dontsov. An approximate solution for a penny-shaped hydraulic fracture that accounts for fracturetoughness, fluid viscosity, and leak-off. R. Soc. open sci., 3:160737, 2016.
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[11] E. Dontsov. Analysis of a constant height hydraulic fracture. arXiv:2110.13088, 2022.
[12] M. McClure, C. Kang, C. Hewson, and S. Medam. Resfrac technical writeup. arXiv:1804.02092, 2018.
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