Impact of the tangential traction for radial hydraulic fracture D. Peck1 G. Da Fies2 1Department of Mathematics Aberystwyth University

2025-05-08 0 0 1.64MB 31 页 10玖币

侵权投诉

Impact of the tangential traction for radial hydraulic fracture

D. Peck(1,∗)& G. Da Fies(2)

(1)Department of Mathematics, Aberystwyth University,

Aberystwyth, Wales, United Kingdom

(2)Rockﬁeld Ltd, Swansea, UK

(∗)Corresponding author: dtp@aber.ac.uk

Abstract

The radial (penny-shaped) model of hydraulic fracture is considered. The tangential traction

on the fracture walls is incorporated, including an updated evaluation of the energy release rate

(fracture criterion), system asymptotics and the need to account for stagnant zone formation near

the injection point. The impact of incorporating the shear stress on the construction of solvers,

and the eﬀectiveness of approximating system parameters using the ﬁrst term of the crack tip

asymptotics, is discussed. A full quantitative investigation of the impact of tangential traction on

solution is undertaken, utilizing an extremely eﬀective (in-house build) adaptive time-space solver.

1 Introduction

Hydraulic fracture (HF) involves a ﬂuid driven crack propagating in a solid material. This process is

widely studied, due to it’s appearance in nature, for example in subglacial drainage and the ﬂow of

magma in the Earth’s crust, as well as it’s use in energy technologies, most notably geothermal energy,

unconventional hydrocarbon extraction and in the relatively new process of carbon sequestration. While

many advanced models exist of this phenomena, the 1D models of hydraulic fracture developed in

the 1950’s and 1960’s: PKN, KGD and radial (penny-shaped), still maintain their relevance. This is

particularly true when it comes to examining the roles certain physical eﬀects play in determining the

fracture behaviour.

One approach to updating the 1D models is the recent drive to better describe the behaviour of

the ﬂuid which drives the fracture. This has previously been considered as either purely Newtonian or

as following a power-law description (see eg. [28, 32]), however recent works attempt to incorporate a

truncated power-law [20], Herschel-Bulkley law [16], or a Carreau ﬂuid description [42] into HF models.

Other major developments in this area have involved approaches which provide a better description the

inﬂuence of proppant (particles within the ﬂuid) on the apparent viscosity of the ﬂuid [41] and near

front behaviour [2], as well as incorporation of turbulence within the fracture ﬂuid [8, 52], plasticity or

porosity of the fracture walls [36, 47, 48], investigations of the impact of toughness heterogeneity [5, 11],

amongst others. Of crucial importance for this paper however, is the recent incorporation of shear stress

induced by the ﬂuid into the 1D models of HF [38, 45].

The incorporation of hydraulically induced tangential traction on the fracture walls into the PKN and

KGD models was provided in [45]. One crucial result was that, when the shear stress was accounted for,

there was no longer a diﬀerence in aperture asymptotics between the viscosity and toughness dominated

regimes. Given the high dependence of most modern algorithms for modeling hydraulic fracture on

these asymptotic terms (see eg. [29,30,32]), this suggested that signiﬁcant simpliﬁcations could be made

to the numerical modeling of hydraulic fracture. In addition, incorporating the hydraulically induced

arXiv:2210.00046v1 [physics.geo-ph] 30 Sep 2022

tangential traction can also have a noticeable eﬀect on fracture redirection, as outlined in [31, 49], and

unstable crack propagation [37].

It should also be noted however that the original paper on the incorporation of tangential traction

into hydraulic fracture models [45] was not without controversy, sparking signiﬁcant discussion about

whether the tangential traction on the fracture walls needs to be accounted for when modeling hydraulic

fracture [24, 25, 46]. To ensure the presented paper addresses the key aspects of this discussion, here a

full quantitative analysis of the time-dependent case is provided in Sect. 4.

The paper is arranged as follows. The problem formulation of the radial model incorporating the

tangential traction is outlined in Sect. 2, including the updated elasticity equation, fracture criterion and

system asymptotics for the viscosity dominated regime, as well as modifying the shear stress formulation

at the injection point. Next, in Sect. 3 the self-similar formulation is used to examine the eﬀect of the

updated formulation on the construction of the algorithm, most notably the eﬀect of the changed system

asymptotics. Finally, in Sect. 4 a full quantitative investigation of the impact of the shear stress for the

time dependent formulation is conducted, and the applications for which it may play a role are discussed.

A summary of the most important results is given in the concluding Sect. 5.

2 Problem formulation

Sect:ProbForm

2.1 Governing equations

We consider the case of a radial hydraulic fracture, driven by a Newtonian ﬂuid. The system is considered

in cylindrical coordinates {r, θ, z}. The crack dimensions are given by l(t), w(r, t), describing the fracture

radius and aperture respectively. The fracture is driven by a point source located at the origin, with

known pumping rate: Q0(t). Due to the axisymmetric nature of the problem, the solution will be

independent of θ, and only 0 ≤r≤l(t) needs to be considered.

The ﬂuid mass balance equation is as follows:

∂w

∂t +1

∂

∂r (rq) + ql= 0,0< r < l(t).(2.1) Nobelfluidmass

where ql(r, t) is the ﬂuid leak-oﬀ function, representing the volumetric ﬂuid loss to the rock formation in

the direction perpendicular to the crack surface per unit length of the fracture. Throughout this paper

we will assume it to be predeﬁned and bounded at the fracture tip.

Meanwhile q(r, t) is the ﬂuid ﬂow rate inside the crack, for a Newtonian ﬂuid, is given by the Poiseuille

law:

q=−w3

∂p

∂r ,(2.2) NobelPoiseville

where the constant M= 12µis the ﬂuid consistency index.

The elasticity relation deﬁning the deformation of the rock needs to be updated to incorporate

the eﬀect of tangential traction on the crack faces, with the derivation provided in the supplementary

material (ﬁrst provided by the authors in [27], with a similar form also derived independently in [38]).

The elasticity equation takes the form:

p(r, t) = −1

l(t)Z1

0k2

∂w(ρl(t))

∂ρ −k1l(t)τ(ρl(t))Mr

l(t), ρdρ, 0≤r < l(t),(2.3) newPre2

with its inverse:

k2w(r, t)+k1Zl(t)

τ(s, t)ds =

π2l(t)



Z1

∂p(yl(t), t)

∂y Ky, r

l(t)dy

| {z }

w1(r,t)

+s1−r

l(t)2Z1

ηp(ηl(t), t)

p1−η2dη

| {z }

w2(r,t)







(2.4) Nobel_InvElast

where the kernel functions are given by:

M[˜r, ρ] = 





˜rKρ2

˜r2+˜r

ρ2−˜r2Eρ2

˜r2,˜r > ρ

ρ2−˜r2E˜r2

ρ2, ρ > ˜r, (2.5)

K(y, ˜r) = yEarcsin(y)

˜r2

y2−Earcsin(ψ)

˜r2

y2, ψ = min y

˜r,1,(2.6) Almighty_Kernel_K

with E(φ|m) denoting the incomplete elliptic integral of the second kind, while:

k1=1−2ν

π(1 −ν), k2=E

2π(1 −ν2).(2.7)

Note that if we take k1= 0 (ie. ν= 0.5), this is identical to the ‘classical’ elasticity equation.

We can also utilize the elasticity equation to parameterise the fracture regime, as outlined in [11].

Note that in (2.4), the fracture aperture wcan be represented as the sum of the term denoted w2,

which represents the impact of the material toughness KIc, and w1, representing the contribution of

the (viscous) ﬂuid pressure, alongside some ﬁnal shear term. Consequently, we can deﬁne the associate

volumes

Vv(t)=2πZl(t)

rw1(r, t)dr, VT(t)=2πZl(t)

rw2(r, t)dr. (2.8)

The ratio of these two terms

δ(t) = VT(t)

Vv(t),(2.9) defn_delta

will provide a (rough) measure of the extent to which fracture evolution is governed by the ﬂuid viscosity

or the material toughness. This can therefore be used to parameterise whether the fracture is within

the viscosity (0 ≤δ1), transient (δ∼1), or toughness (1 δ) dominated regime, which will

prove useful when conducting the time-dependent investigation. Note that for the radial model this will

change over time, as the fracture transitions from the (initially) viscosity dominated to the toughness

dominated regime as it grows (see e.g. [9, 21, 35] for details of the fracture regimes). For more details of

the parameterisation by δ(t), see [11].

These equations are supplemented by the boundary condition at r= 0, which deﬁnes the intensity

of the ﬂuid source, Q0:

lim

r→0rq(r, t) = Q0(t)

2π,(2.10) Nobel_SourceIntense

alongside the tip boundary conditions:

w(l(t), t) = 0, q(l(t), t)=0.(2.11) Nobel_TipBC

We assume that there is a preexisting fracture, starting with appropriate non-zero initial conditions for

the crack opening and length:

w(r, 0) = w∗(r), l(0) = l0,(2.12) Nobel_InitCond

Finally the global balance equation takes the form:

Zl(t)

r[w(r, t)−w∗(r)] dr +Zt

0Zl(t)

rql(r, τ )dr dτ =1

2πZt

Q0(τ)dτ. (2.13) Nobel_fluidbalance1

In addition to the above, we employ a new dependent variable named the ﬂuid velocity, v, deﬁned

by:

v(r, t) = q(r, t)

w(r, t)=−w2(r, t)

∂p

∂r ,(2.14) NobelPVInit

It has the property that, provided the ﬂuid leak-oﬀ qlis ﬁnite at the crack tip:

lim

r→l(t)v(r, t) = v0(t)<∞,(2.15)

which, given that the fracture apex coincides with the ﬂuid front (no lag), allows for fracture front

tracing through the so-called speed equation [23]:

dt =v0(t).(2.16) Nobel_SpeedEq

Note that this replaces boundary condition (2.11)2, which now immediately follows from (2.11)1, (2.14)-

(2.16). This Stefan-type condition has previously been employed in 1D hydraulic fracture models, the

advantages of which (alongside technical details) are shown in [18, 32, 43–45]. Of crucial importance is

the fact that the fracture tip can now be considered in terms of the ﬁnite variable v, with clearly deﬁned

leading asymptotic coeﬃcient v0, eliminating the singular term qfrom computations entirely. These

singular terms are however closely related to the ﬂuid velocity (2.14), and as such can easily be obtained

in post-processing.

2.2 The shear stress at the fracture inlet

The_wall_jet

The normal and tangential stress on the fracture walls, created by the ﬂuid pressure, follows directly

from lubrication theory (see for example [40]), in this case being given by:

σ0=−p, τ(r, t) = −1

2w(r, t)∂p(r, t)

∂r .(2.17) taudef

It should be noted that this representation of the shear stress is singular at both the crack tip (r=l(t))

and the fracture opening (r= 0). While the former singularity is physically meaningful for deﬁning the

total ﬂux within the fracture, following the same principals as that for the stress at the crack tip in

linear elastic fracture mechanics, the singularity at r= 0 should be properly addressed.

There is a clear explanation for the singularity at the fracture opening. HF models typically treat

the ﬂuid source as a singularity at the fracture inlet (r, θ, z) = (0, θ, 0). Tangential traction is induced by

ﬂuid traveling in a single (turbulence-free) streamline from this source directly to the fracture wall, and

along this wall to the fracture front. However, this behaviour is a clear violation of established rules for

ﬂuids in such situations, where it has been demonstrated that instead stagnant regions will form in the

region where the ﬂuid source makes contact with the fracture wall (r, θ, z) = (0, θ, ±w(0, t), preventing

ﬂuid from the source from reaching these points (see Fig. 1). These secondary streamlines will typically

be stable, even though it arises from turbulent eﬀects acting on the ﬂuid, however its precise form will

depend upon both the problem geometry and ﬂuid properties (Reynold’s number). This can be thought

Figure 1: Exaggerated depiction of the primary streamlines within a quarter-segment of a penny-shaped

hydraulic fracture, which determine the tangential traction on the fracture walls. The red line indicates

the longest streamline within the stagnant zone (wall-jet eﬀect), while the blue line indicates the longest

streamline connecting the ﬂuid source (blue dot at r= 0) to the fracture tip. Jet_effect_fig

of as a form of the ‘wall jet’ eﬀect, analogous to the behaviour of a rocket exhaust hitting the ground

(reviews can be found in [15,19]).

Consequently, while the singularity at the fracture front needs to be maintained to properly model

the radial geometry, the formulation needs to updated to eliminate this non-physical singularity at r= 0.

There are three primary options for doing so:

•Incorporating the wellbore will (artiﬁcially) cut-oﬀ the current left-hand boundary (r= 0),

with the ﬂuid ﬂow instead ending some distance away from the origin (the half-width of the

wellbore), and thus remove the singularity. This has previously been incorporated for the classical

radial model, for example in [21] where it eﬀectively predicted experimental results.

•Fixing the opening height by adding an additional boundary condition such that w(0, t) =

w∗(0), a constant, where w∗(r) is the initial fracture proﬁle (2.12). This could be enforced numer-

ically, and would eliminate the eﬀect of the tangential traction at the crack opening.

•Modifying the tangential traction formulation to eliminate the singularity at r= 0 from

(2.17). Unfortunately, there is no simple formula to describe the eﬀect of these stagnant zones on

the tangential traction induced on the fracture walls. Subsequently, this requires a more general

modiﬁcation, allowing multiple ‘possible’ forms of the shear stress to be considered.

As the aim of this paper is to incorporate the tangential traction into the general radial model, rather

than for some speciﬁc application, we will take the third option and modify the formulation. This has

the added beneﬁt of being the most generalised approach, allowing for a diﬀerent forms of the tangential

traction to be investigated. Note however that the other two approaches could be utilized for speciﬁc

applications, if it were preferable.

In order to control the extent to which the shear stress is changed away from the point r= 0, we

introduce the updated formulation of the tangential stress on the fracture wall:

τ(r, t) = −1

χ(r, t)

l(t)w(r, t)∂p(r, t)

∂r ,(2.18) New_tau

where the particular form of χis not ﬁxed (to allow for various possible formulations to be considered),

but is always a continuous function such that

χ(r, t)∼r, r →0, χ(r, t) = l(t), r →l(t).(2.19) New_rtsar

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

ImpactofthetangentialtractionforradialhydraulicfractureD.Peck(1;)&G.DaFies(2)(1)DepartmentofMathematics,AberystwythUniversity,Aberystwyth,Wales,UnitedKingdom(2)RockeldLtd,Swansea,UK()Correspondingauthor:dtp@aber.ac.ukAbstractTheradial(penny-shaped)modelofhydraulicfractureisconsidered.Thetangentia...

展开>> 收起<<

Impact of the tangential traction for radial hydraulic fracture D. Peck1 G. Da Fies2 1Department of Mathematics Aberystwyth University.pdf

共31页,预览5页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Impact of the tangential traction for radial hydraulic fracture D. Peck1 G. Da Fies2 1Department of Mathematics Aberystwyth University

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: