# Dynamics of fluid-driven fractures across material heterogeneities

Sri Savya Tanikella1, Marie C. Sigallon1,2, Emilie Dressaire1
1\. University of California Santa Barbara
2\. Ecole Polytechnique, CNRS, Institut Polytechnique de Paris

###### Abstract

Fracture propagation is highly sensitive to the conditions at the crack tip.
In heterogeneous materials, microscale obstacles can cause propagation
instabilities. Macroscopic heterogeneities modify the stress field over scales
larger than the tip region. Here, we experimentally investigate the
propagation of fluid-driven fractures through multilayered materials. We focus
on analyzing fracture profiles formed upon injection of a low-viscosity fluid
into a two-layer hydrogel block. Experimental observations highlight the
influence of the originating layer on fracture dynamics. Fractures that form
in the softer layer are confined, with no penetration in the stiffer layer.
Conversely, fractures initiated within the stiffer layer experience rapid
fluid transfer into the softer layer when reaching the interface. We report
the propagation dynamics and show that they are controlled by the toughness
contrast between neighboring layers, which drives fluid flow. We model the
coupling between elastic deformation, material toughness, and volume
conservation. After a short transient regime, scaling arguments capture the
dependence of the fracture geometry on material properties, injection
parameters, and time. These results show that stiffness contrast can
accelerate fracture propagation and demonstrate the importance of macroscopic
scale heterogeneities on fracture dynamics. These results have implications
for climate mitigation strategies involving the storage of heat and carbon
dioxide in stratified underground rock formations.

##  1 Introduction

Fractures propagate when the elastic stress conditions at the crack tip are
favorable to the formation of new surfaces [17]. The advance of the crack
front is, therefore, controlled by local properties, including material
composition, stress, and local heterogeneities. When the energy available
within a material reaches the energy required to create new fracture surfaces,
the crack propagates through an intermittent process that starts with a
localized nucleation, which moves the tip forward [28, 34, 6]. At the
nucleation side, the high curvature of the crack front triggers rapid lateral
growth. The irregular propagation is exacerbated by micro-heterogeneities that
can pin the fracture front [9, 23, 4, 29]. Heterogeneities whose
characteristic length scale is larger than the tip region have also been
reported to affect fracture growth [5, 35, 25]. Such heterogeneities are found
in natural and engineered substrates, including rocks, adhesives, and
composite materials. In heterogeneous substrates, fluid-driven fractures can
be diverted by a plane of lower toughness, such as the interface between two
material layers [31, 26, 19, 30]. As the fracture tip propagates through a
single material at any given time, understanding how macroscopic
heterogeneities affect propagation requires coupling stress distribution and
local material rupture conditions.

This study focuses on the interaction of a penny-shaped fracture with a
material discontinuity in a brittle elastic medium with no stress contrast.
Fluid-driven fractures have previously demonstrated that non-local effects,
such as viscous dissipation [16, 7, 32, 33] and in-situ stress distribution
can affect the fracture propagation [36]. Yet the role of heterogeneous
material properties on a macroscopic scale and the corresponding fracture
dynamics remain to be studied. We investigate fracture propagation when a
fluid is injected into a hydrogel block composed of two layers that exhibit
different stiffness. Since the material toughness governs the propagation, the
stiffness contrast determines the fate of the fracture at the interface
between the two layers. We are particularly interested in the fracture
dynamics when the tip crosses from a stiff into a soft layer, as the
macroscopic heterogeneity influences the propagation rate over length scales
larger than the crack tip. The theoretical model of the propagation dynamics
relies on scaling arguments to couple volume conservation, elastic stress, and
fracture propagation criterion. The model yields a good agreement with
experimental results.

##  2 Experimental methods

The experimental system allows the study of planar fractures growing across a
multilayered substrate under controlled loading conditions. We form the
fracture upon injection of liquid in hydrogel. We use a substrate composed of
two horizontal gelatin layers of comparable height in an acrylic cube of
dimensions equal to 150 mm, as shown in Fig. 1 [11, 13, 16, 32, 33]. The
bottom and top layers are labeled 1 and 2, and their Young’s moduli are
$E_{1}$ and $E_{2}$, respectively. The relative masses of ultra-pure water and
gelatin powder (Gelatin type A; Sigma-Aldrich, USA) to achieve the Young’s
modulus $E_{1}$ [32] are combined and heated at 60oC. The mixture is poured
into the cube around a needle of inner diameter ID = 1 mm [3].

Figure 1: Schematic of the experimental set-up.

We use a small plastic washer at the tip of the blunt injection needle to
ensure the fracture propagates in the vertical plane, in the focal plane of
the camera lens. The gelatin for the second layer is prepared using a
different amount of gelatin powder to obtain Young’s modulus $E_{2}$. To avoid
softening the first layer while ensuring adhesion of the second layer, the
gelatin mixture for layer 2 is cooled from 60oC down to 40 oC before pouring
on top of layer 1. The hydrogel block then sets over 24 hours before
fracturing [12]. To form the fracture, silicone oil is injected into the block
at the tip of a needle at a constant flow rate $Q$ with a syringe pump. As the
volume of fluid increases, the fracture grows and reaches the interface
between the two layers for $V=V_{0}$ and propagates until the injection stops
at $V=V_{T}$. We vary the stiffness of the two layers, fluid viscosity, and
injection parameters (flow rate and total volume), as detailed in Table 1.
These parameters ensure the fracture propagation occurs within the toughness-
dominated regime for both layers [27, 16, 21]: the viscous dissipation
associated with the flow in the fracture is negligible. In the toughness-
dominated regime, the elastic deformation of fracture surfaces leads to its
radial expansion. The material properties are measured in the laboratory as
described in [32] and the measurement errors are estimated to $10\%$ for
Young’s modulus $E$, $10\%$ for the fracture toughness $K$, $0.35\%$ for the
flow rate $Q$ and, $1\%$ for the viscosity $\mu$.

As the fracture forms and grows, we record images with a DSLR camera; the oil
is dyed with Sudan red to improve the contrast and allow accurate area
measurement in the plane of propagation. The fluid pressure is measured with a
membrane sensor (ATO-PRES-P350) placed on the oil tubing connecting the
syringe and the needle. We also record the fracture aperture using the light
attenuation method [2, 32] with Sudan red at 0.025 g.L-1: the light absorbance
at $\lambda=520$ nm varies linearly as a function of the fracture aperture. We
obtain aperture values across the fracture after calibration in a custom-made
acrylic wedge [32].

Figure 2: Times series of the fracture propagation for (a) experiments 1, (b)
experiment 2, and (c) experiment 3 at times t = [0, 120, 180, 330, 590] s,
from left to right with $t=0$ the onset of the fracture in layer 1. The layers
have Young’s moduli $E_{\mbox{soft}}=15$ kPa (clear yellow) and
$E_{\mbox{stiff}}=30$ kPa (dark yellow). In experiment 3, the layers have the
same Young’s modulus $E_{\mbox{stiff}}=30$ kPa.

##  3 Results

###  3.1 Phenomenology

As oil is injected into the gelatin, a fracture forms around the needle tip in
layer 1 and propagates axisymmetrically in the vertical plane. The penny-
shaped fracture expands towards the interface between layers 1 and 2, as shown
in Fig. 2.

Table 1: List of experiments: symbols and parameters

|  | $E_{1}$ | $E_{2}$ | $V_{T}$ | $\mu$ | $Q$ | $V_{0}$ | ${R_{1,f}}$
---|---|---|---|---|---|---|---|---
Exp. |  | (kPa) | (kPa) | (ml) | (Pa.s) | (ml/min) | (ml) | (m)
1† | $\color[rgb]{0.703125,0.19921875,0.25}\definecolor[named]{pgfstrokecolor}{rgb}{0.703125,0.19921875,0.25}\blacksquare\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\square$ | 15 | 30 | 12 | $0.02$ | 1 | - | -
2*† |  $\color[rgb]{0.75390625,0.75390625,0.75390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.75390625,0.75390625,0.75390625}{{\blacklozenge}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}$ $\lozenge$ | 30 | 15 | 12 | $0.02$ | 1 | 2.77 | 0.016
3 | ★ ☆ | 30 | 30 | 10 | $0.02$ | 1 | - | -
4 |  $\color[rgb]{1,0.5,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0.5,1}{{{\blacktriangle}}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}$ $\triangle$ | 15 | 30 | 10 | $0.01$ | 0.5 | - | -
5 |  $\blacktriangle$ $\bigtriangleup$ | 15 | 60 | 10 | $0.02$ | 1 | - | -
6* | $\color[rgb]{0.59765625,0.984375,0.59765625}\definecolor[named]{pgfstrokecolor}{rgb}{0.59765625,0.984375,0.59765625}{{{{\bullet}}}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{{{{\circ}}}}$ | 30 | 60 | 10 | $0.02$ | 0.5 | - | -
7 |  $\color[rgb]{0.5,0.5,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.5,1}{{\blacklozenge}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}$ $\lozenge$ | 30 | 15 | 12 | $0.1$ | 0.5 | 3.85 | 0.018
8 |  $\blacktriangle$ $\bigtriangleup$ | 60 | 15 | 10 | $0.02$ | 1 | 3.35 | 0.016
9 |  $\blacktriangle$ $\bigtriangleup$ | 60 | 15 | 12 | $0.02$ | 0.5 | 3.68 | 0.021
10* | $\color[rgb]{0.82421875,0.70703125,0.55078125}\definecolor[named]{pgfstrokecolor}{rgb}{0.82421875,0.70703125,0.55078125}\blacksquare\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\square$ | 60 | 30 | 10 | $0.02$ | 0.5 | 4.04 | 0.022
11 | $\color[rgb]{0.5,1,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,1,0.5}\blacksquare\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\square$ | 30 | 15 | 10 | $0.01$ | 0.1 | 7.79 | 0.026
12* | $\color[rgb]{1,0.85546875,0.7265625}\definecolor[named]{pgfstrokecolor}{rgb}{1,0.85546875,0.7265625}{{{{\bullet}}}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{{{{\circ}}}}$ | 88 | 60 | 10 | $0.02$ | 0.5 | 2.88 | 0.018
13 | $\color[rgb]{1,0.5,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{1,0.5,0.5}{{{{\bullet}}}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{{{{\circ}}}}$ | 30 | 15 | 10 | $0.01$ | 0.5 | 5.3 | 0.023
2r |  $\color[rgb]{0.19921875,0.19921875,0.80078125}\definecolor[named]{pgfstrokecolor}{rgb}{0.19921875,0.19921875,0.80078125}{{\blacklozenge}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}$ $\lozenge$ | 30 | 15 | 12 | $0.02$ | 1 | 1.7 | 0.012

Pressure measurements for experiments marked with ∗ and aperture measurements
for experiments marked with †. $\bf{V_{0}}$ and $\bf{R_{1,f}}$ are measured.
In experiments $1$ and $4-6$, the fracture is confined in layer 1.

To explore the role of the relative stiffness of the layers, we conduct three
experiments with layers of Young’s moduli $E=[15,30]$ kPa (experiments 1 to 3
in Table 1). In experiment 1, layer 1 is the softer layer, and the fracture
growth is presented in Fig. 2(a) and Movie S1. For experiment 2, layer 1 is
the stiffer layer, and the results are in Fig. 2(b) and Movie S2. Despite
identical injection parameters ($Q=1$ ml min-1 and $\mu=20$ mPa s), the
fractures produced exhibit different geometry and propagation dynamics once
they reach the interface. In experiment 1, the fracture continues to propagate
in the soft layer, adopting an elongated geometry [37]. In experiment 2, the
fracture crosses over to the second and softer layer upon reaching the
interface. The fracture initially adopts a half-disk shape in the soft layer.
In comparison, when the two layers have the same Young’s modulus, the fracture
adopts a circular geometry (see Fig. 2(c)) and Movie S3). This observation
indicates that the contrast in stiffness between the layers is responsible for
the different shape and dynamics and not the presence of an interface between
two gelatin layers.

To rationalize these results, we consider the pressure required to propagate
the fracture in the different layers. The ratio depends on the material
stiffness and can be approximated as
$P_{2}/P_{1}\propto\sqrt{\gamma_{1}G_{1}/\gamma_{2}G_{2}}$, where $\gamma$ is
the fracture energy and $G$ the shear modulus [1]. Since both layers are made
of gelatin and have the same fracture energy $\gamma\approx$ 1 J m-2, and
Poisson’s ratio $\nu\approx 0.5$ [20], the relation simplifies to
$P_{2}/P_{1}\propto\sqrt{E_{1}/E_{2}}$. Therefore, when the penny-shaped
fracture reaches the interface between the two layers, it preferentially
propagates in the layer with the lowest Young’s modulus at the lowest
propagation pressure [19, 37]. This is consistent with our observations: a
fracture that originates in the softer layer remains confined in the softer
layer, and a fracture that forms in a stiffer layer crosses the interface and
propagates in the softer neighboring layer.

###  3.2 Propagation in a homogeneous substrate

We first consider the propagation of the fracture in the injection layer. When
a penny-shaped fracture forms and grows in a homogeneous infinite material,
the propagation dynamics are driven by elastic deformation and limited by the
fracture opening or toughness of the material. In all experiments presented
here, the viscous dissipation associated with the flow in the fracture is
negligible. The fluid injection starts at $t_{1}=0$. At time $t_{1}$, the
fracture volume equals $V=Qt_{1}$. We can model the propagation of the
fracture using scaling arguments [27, 15, 21]. The average radius of the
fracture is noted $R_{1}$. The aperture of the axisymmetric fracture is
maximum at the injection point where $w(r=0)=w_{1}$. The elastic pressure that
drives the fracture propagation is $P_{1}$. Using those parameters, the
fracture volume scales as:

Figure 3: Fracture dynamics in layer 1. (a) The time dependence of the radius
is reported for all experiments and (b) rescaled following Eq. 4. The black
line represents the best-fit line with a slope of 1. (c) The radial dependence
of the fracture aperture is measured at t = [100, 200, 300, 400, 500, 600,
700] s for experiment 1 and (d) the fracture profiles are rescaled using Eq.
5. (e) The pressure is measured over time for experiments 2, 6, 10, and 12.
(f) The pressure data are rescaled with Eq. 6. The black line represents the
best-fit line with a slope of -1/5.

$V\approx R_{1}^{2}w_{1}$ (1)

Assuming there is no fluid lag at the low injection flow rates considered here
[10], the theory of Linear Elastic Fracture Mechanics (LEFM) holds. The
fracture propagates in mobile equilibrium such that the elastic pressure is

$P_{1}\approx E^{\prime}_{1}w_{1}/R_{1}$ (2)

where $E_{1}^{\prime}=E_{1}/(1-\nu^{2})$ and $\nu\approx 1/2$ is the Poisson’s
ratio of gelatin. The stress intensity factor at the fracture tip should equal
the material toughness for the fracture to grow. The fracture propagation
criterion yields

$K_{1}^{\prime}\approx P_{1}\sqrt{R_{1}}=8\sqrt{E_{1}^{\prime}\gamma/\pi}$ (3)

where $\gamma=1$ J m-2 is the fracture energy for gelatin [24, 22, 20]. These
three equations yield scaling expressions governing $R_{1}$, $w_{1}$, and
$P_{1}$ [27, 8],

$\displaystyle R_{1}$ $\displaystyle\approx$
$\displaystyle\left(\frac{E_{1}^{\prime}}{K_{1}^{\prime}}\right)^{2/5}V^{2/5}$
(4) $\displaystyle w_{1}$ $\displaystyle\approx$
$\displaystyle\left(\frac{K_{1}^{\prime}}{E_{1}^{\prime}}\right)^{4/5}V^{1/5}$
(5) $\displaystyle P_{1}$ $\displaystyle\approx$
$\displaystyle\left(\frac{K_{1}^{\prime
6}}{E_{1}^{\prime}}\right)^{1/5}V^{-1/5}$ (6)

In the toughness-dominated regime, the propagation dynamics depend on the
material properties and flow rate at which the liquid is injected $Q$, as
$V=Q\,t_{1}$. The radius measurements shown in Fig. 3(a-b) demonstrate that
the fracture growth in layer 1 is well-captured by Eq. 4. The best-fitting
power law for the experimental data is described by
$R_{1}=\alpha_{1}({E_{1}^{\prime}}/{K_{1}^{\prime}})^{2/5}(Qt)^{2/5}$, with
$\alpha_{1}=0.65$, a value which is in agreement with previous studies [27,
15, 21]. The fracture aperture is measured using light absorbance and plotted
as a function of the radial position in Fig. 3(c): the fracture increases in
diameter and aperture during the injection. Rescaling both parameters using
Eqs. 4 and 5, the fracture profiles collapse on a single curve, which is
consistent with the self-similar geometry of the penny-shaped fracture (see
Fig. 3(d)). The fluid pressure can also be rescaled using Eq. 6 as shown in
Fig. 3(e-f). It is worth noting that the results presented in Fig. 3 include
data collected when after the fracture has reached the interface between a
soft and a stiff layer, and its growth is confined to the soft injection
layer. The shape of the fracture becomes elongated. Yet, the time dependence
of all three parameters follows the scaling behavior derived for an infinite
substrate and an axisymmetric fracture, illustrating the robust growth
dynamics of the self-similar penny-shaped fractures.

Figure 4: Fracture dynamics in experiment 2: (a) Effective sub-fracture radius
in layers 1
$\color[rgb]{0.49609375,0.09765625,0.8984375}\definecolor[named]{pgfstrokecolor}{rgb}{0.49609375,0.09765625,0.8984375}{{{{\bullet}}}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{{{{\circ}}}}$
and 2
$\color[rgb]{0.8984375,0.09765625,0.49609375}\definecolor[named]{pgfstrokecolor}{rgb}{0.8984375,0.09765625,0.49609375}{{{{\bullet}}}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{{{{\circ}}}}$.
(b) Fluid pressure. (c) Fracture aperture. Fracture profiles are measured
along the white dashed line. We note (1) the start of the injection, (2) the
initiation of the fracture, (3) the crossing of the interface between layers 1
and 2, and (4) the end of the injection.

###  3.3 Propagation through a layered substrate

####  3.3.1 Experimental results

We now focus on the dynamics of a fracture that grows from a stiff into a soft
layer, as observed in experiment 2. After the fracture crosses the interface,
it comprises two sub-fractures: sub-fracture 1 in layer 1 and sub-fracture 2
in layer 2. We measure the areas of the sub-fractures and estimate the
effective radii plotted in Fig. 4(a). Before the fracture reaches the
interface (at point 3), its radius increases in layer 1. Upon crossing the
interface, sub-fracture 1 stops expanding, while the radius of sub-fracture 2
increases rapidly. We record the fluid pressure values, starting when the
syringe pump is turned on, as shown in Fig. 4(b). At the start of the
injection (point 1), the fluid pressure increases until it reaches a critical
value necessary to initiate the crack [18]. Once the fracture forms (point 2),
there is a rapid pressure release, and the propagation becomes quasi-steady in
layer 1, following the scaling in Eq. 6. The pressure drops rapidly when the
fracture reaches the interface (point 3). After this transient regime, the
pressure evolution becomes quasi-steady again until the end of the injection
(point 4). Using light absorbance, we measure the aperture of the fracture,
which is represented using a heat map in Fig. 4(c). As the fracture propagates
through layer 1, its aperture increases. When the fracture crosses the
interface, the aperture of sub-fracture 1 decreases, whereas the aperture of
sub-fracture 2 increases.

These observations show that the coupling between the two sub-fractures
controls the propagation of the fracture when it crosses an interface. Indeed,
the collapse of sub-fracture 1 fuels the rapid growth of sub-fracture 2. This
fluid transfer is driven by the larger elastic stresses in the stiffer layer
1. Based on our measurements, the radius of sub-fracture 1 remains constant,
equal to $R_{1,f}$, while the maximum aperture $w_{1}(t)$ decreases over time.
As the fracture crosses the interface, the pressure decreases from the value
set by the toughness of layer 1, necessary to grow the fracture in layer 1, to
the value set by the toughness of layer 2, as the fracture now only grows in
layer 2. After a short transient regime, the pressure can be considered
uniform in the two sub-fractures, with $P\approx P_{1}\approx P_{2}$.

Figure 5: Fracture radius in layer 2. (a) Effective radius for experiments 2
and 7-13 and (b) corresponding scaled effective radius using Eq. 12. The black
line represents the best-fit line with a slope of 1.

####  3.3.2 Theoretical model

To understand how the elastic relaxation of sub-fracture 1 is coupled with the
propagation of sub-fracture 2, we use scaling arguments and derive expressions
for the fracture parameters, i.e., the effective radius $R_{2}$ and maximum
aperture $w_{2}$ in layer 2, maximum aperture $w_{1}$ in layer 1, and pressure
$P$. We note $V_{0}$ the fracture volume when it reaches the interface. After
crossing, the fracture volume is $V=V_{0}+Qt_{2}$ where $t_{2}=0$ when the
fracture reaches the interface and

$V\approx R_{1,f}^{2}\,w_{1}+R_{2}^{2}\,w_{2}.$ (7)

The elastic pressure scales as $P_{1}\approx E^{\prime}_{1}w_{1}/R_{1,f}$ and
$P_{2}\approx E^{\prime}_{2}w_{2}/R_{2}$ with $P\approx P_{1}\approx P_{2}$
[27, 8]. We can approximate the fracture aperture in both layers as follows:

$\displaystyle w_{1}$ $\displaystyle\approx\frac{PR_{1,f}}{E^{\prime}_{1}},$
(8) $\displaystyle\mbox{and}\;\;w_{2}$
$\displaystyle\approx\frac{PR_{2}}{E^{\prime}_{2}}$ (9)

Assuming there is no fluid lag and the theory of Linear Elastic Fracture
Mechanics (LEFM) holds, the material properties of layer 2 set the propagation
criterion [10]:

$K^{\prime}_{2}=8\sqrt{E_{2}^{\prime}\gamma/\pi}\approx P\sqrt{R_{2}}.$ (10)

Figure 6: Fluid pressure for experiments 2, 6, 10, and 12 in layer 2: (a)
pressure as a function of time, and (b) rescaled pressure data using Eqs. (1)
and (4).

To predict the evolution of $R_{2}$, $w_{2}$, $w_{1}$, and $P$, the system of
equations 7-10 needs to be solved. The coupled equations yield a 6th-order
polynomial equation whose exact roots are not conducive to capturing the
physics of the fracture propagation. Consequently, we use the scaling
expressions derived for an infinite uniform material to get approximate
closed-form expressions for the fracture parameters.

We hypothesize that the expressions for the radius, aperture, and pressure of
sub-fracture 2 have the same form as the scaling relations for a fracture
growing in a uniform infinite material. In an infinite material, the volume of
the fracture is equal to the injected volume. Here, the volume of sub-fracture
2 is the sum of the volume transferred from sub-fracture 1, noted
$V_{0}^{\prime}$ and the injected volume $Qt_{2}$, which is assumed to flow
entirely into sub-fracture 2 since sub-fracture 1 collapses:

$V_{2}=V_{0}^{\prime}+Q\,t_{2}.$ (11)

The scaling relations 4, 5 and 6 become:

$R_{2}\approx\frac{E_{2}^{\prime 2/5}}{K_{2}^{\prime
2/5}}(V_{0}^{\prime}+Qt_{2})^{2/5},$ (12)

$w_{2}\approx\frac{K_{2}^{\prime 4/5}}{E_{2}^{\prime
4/5}}(V_{0}^{\prime}+Qt_{2})^{1/5},$ (13)

$P\approx\frac{K_{2}^{\prime 6/5}}{E_{2}^{\prime
1/5}}(V_{0}^{\prime}+Qt_{2})^{-1/5}.$ (14)

Figure 7: Fracture profiles measured during experiment 2: (a) Aperture values
$w$ at $t_{2}=[21,111,201,291,381,471]$ s, with time increasing from light to
dark purple, (b) rescaled aperture profiles using Eqs. (2) and (3), for the
propagation in layer 2, and (c) rescaled aperture profiles using $R_{1,f}$ and
Eq. (5), for the collapse in layer 1.

Substituting Eq. 14 in Eq. 8, we solve for $w_{1}$:

$w_{1}\approx\frac{K_{2}^{\prime 6/5}}{E_{2}^{\prime
1/5}}(V_{0}^{\prime}+Qt_{2})^{-1/5}\frac{R_{1,f}}{E_{1}^{\prime}}.$ (15)

The volume of the fluid $V_{0}^{\prime}$ from layer 1 in layer 2 at time
$t_{2}$ is

$V_{0}^{\prime}\approx V_{0}-R_{1,f}^{2}\,w_{1}.\\\ $ (16)

Substituting the expression for $w_{1}$ in Eq. 15, we define the transferred
volume:

$V_{0}^{\prime}\approx V_{0}-\frac{K_{2}^{\prime 6/5}}{E_{2}^{\prime
1/5}}\frac{R_{1,f}^{3}}{E^{\prime}_{1}}(V_{0}+Qt_{2})^{-1/5}.$ (17)

The volume transferred $V_{0}^{\prime}$ is an increasing function of time that
converges toward $V_{0}$ at large times when the fluid initially in sub-
fracture 1 is completely transferred into sub-fracture 2. In the limit of
$t_{2}=0$, the transferred volume has a finite value
$V_{0}^{\prime}(t_{2}=0)=V_{0}\left[1-(E_{2}/E_{1})^{2/5}\right]$. This
initial transferred volume equals 0 when the two layers have the same Young’s
modulus, and no fluid flow between the two layers is expected. The value of
the initial transferred volume is finite for different layers as the model
does not capture the rapid transient regime during which the pressure in the
two layers becomes equal to $P_{2}$. The initial transferred volume increases
as the ratio of Young’s moduli increases, and more fluid flows into sub-
fracture 2 during the transient regime. For small values of $t_{2}$, the
transferred volume increases linearly with time. This results in the faster
propagation of sub-fracture 2, with an increase in effective flow rate. The
contribution of sub-fracture 1 to the growth of sub-fracture 2 decreases over
time.

####  3.3.3 Discussion

To compare the expressions derived with experimental results, we focus on the
fracture dynamics once the tip crosses the interface between the two layers.
For the range of parameters in Table 1, we measure the effective radius of
sub-fracture 2; the results are presented in Fig. 5(a). Initially, we observe
a rapid increase in radius, characteristic of the swift transfer occurring
between the two layers. Using Eqs. 12 and 17 to define the dimensionless
radius of sub-fracture 2 and the transferred volume, all datasets collapse on
a master curve as shown in Fig. 5(b). The best-fitting power law for this
unified curve is
$R_{2}({K_{2}^{\prime}}/{E_{2}^{\prime}\,V_{0}})^{2/5}=\alpha_{2}(V_{0}^{\prime}/V_{0}+Qt_{2}/V_{0})^{2/5}$,
with $\alpha_{2}=0.57$. This prefactor is consistent with the values obtained
in the toughness regime for an infinite homogeneous material [21, 15, 32].
Figure 6(a) reports the liquid pressure once the fracture has advanced into
layer 2. When rescaled, the pressure data converge to a constant, as depicted
in Fig. 6(b). The best-fit line is given by $P=\alpha_{4}({K_{2}^{\prime
6}}/{E_{2}^{\prime}})^{1/5}(V_{0}^{\prime}+Qt_{2})^{-1/5}$, where
$\alpha_{4}=0.65$. The results validate the hypothesis that the fluid pressure
is set by the toughness of layer 2. The shape of sub-fracture 2 is initially
ellipsoidal as the fluid transfer from sub-fracture 1 occurs over a
distributed interface region. The fracture apertures $w_{1}$ and $w_{2}$ are
measured along a line normal to the interface and going through the injection
point, as shown in Fig. 4(c). The results presented in Fig. 7(a) show that the
aperture of sub-fracture 1 decreases as the aperture of sub-fracture 2
increases over time. The aperture of fracture 2 reaches values up to 7 mm. We
rescale the fracture profiles with the maximum radius $R_{2}$ and aperture
$w_{2}$, using Eqs. 12 and 13. We present the results in Fig. 7(b): the
profiles of sub-fracture 2 collapse, indicating that the fracture geometry is
self-similar and the scaling for $R_{2}$ and $w_{2}$ agree with the
measurements, despite the elongate shape of sub-fracture 2. If we rescale the
fracture aperture using Eq. 15 for $w_{1}$ and a fixed radius, $R_{1,f}$, the
profiles of sub-fracture 1 collapse onto each other, as shown in Fig. 7(c). We
conclude that the sub-fractures exhibit self-similar profiles, which are
robustly observed for planar or penny-shaped hydraulic fractures [14, 22].

##  4 Conclusion

Our results demonstrate the long-range interactions between macroscopic
material heterogeneities and fracture dynamics: the fracture can undergo
enhanced or confined propagation because of differences in stiffness between
two neighboring layers. Our results show that a fracture growing from a stiff
into a soft layer experiences faster expansion dynamics due to fluid transfer
from the stiff to the soft layer. The fracture geometry also depends on the
growth history and relative stiffness of the layers encountered. The proposed
model uses scaling arguments to describe fluid transfer from high to low
elastic stress regions and fracture propagation. After a short transient
regime, the model captures the geometric parameters of the fracture and the
fluid pressure. The role of the fluid is a notable difference between the
influence of micro-scale and macro-scale heterogeneities on fracture
properties. For micro-heterogeneities, the deformation of the crack front
around the obstacle is responsible for the micro-instabilities in the
propagation. Macro-heterogeneities are responsible for inhomogeneous stress
fields, which drive flow in a fluid-filled fracture. For a low-viscosity
Newtonian liquid, the fluid transfer is controlled by the differences in
elastic stresses between the two sub-fractures. More generally, the
compressibility and rheology of the fluid are expected to influence the
fracture growth. A description of fracture dynamics combining the effects of
micro and macro-heterogeneities will result in a better understanding of the
propagation in disordered media.

The authors acknowledge the support of the UCSB Academic Senate and thank Dr.
Emre Turkoz for helpful conversations.

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# Definition of tolerances and corrector strengths for the orbit control of
the High-Energy Booster ring of the future electron-positron collider††thanks:
This work was supported by the European Union’s Horizon 2020 Research and
Innovation programme under grant no. 951754 (FCCIS).

B. Dalena<EMAIL_ADDRESS>T. Da Silva  A. Chancé  A. Ghribi1
Paris-Saclay University and CEA Irfu  91191 Gif-sur-Yvette  France
1also at CNRS GANIL  14118 Caen  France

###### Abstract

After the discovery of the Higgs boson at the LHC, particle physics community
is exploring and proposing next accelerators, to address the remaining open
questions on the underlying mechanisms and constituents of the present
universe. One of the studied possibilities is FCC (Future Circular Collider),
a 100-km-long collider at CERN [1]. The feasibility study of this future
proposed accelerator implies the definition of tolerances on magnets
imperfections and of the strategies of correction in order to guarantee the
target performances of the High Energy Booster ring. The efficiency of the
correction scheme, used to control the orbit, directly bounds the corrector
needs and magnet tolerances. Analytic formulae give a first estimation of the
average RMS values of the required dipole correctors’ strengths and of the
allowed magnets misalignments and field quality along the entire ring. The
distribution of the correctors along the ring is simulated, in order to verify
the quality of the residual orbit after the proposed correction strategy and
to compare it with the analytical predictions. First specifications of the
orbit correctors strength and tolerances for the alignment of the main
elements of the ring are presented. The limits of the studied correction
scheme and method are also discussed.

##  1 INTRODUCTION

The aim of the present study is to first, determine the tolerances for the
misalignment of the elements of the High Energy Booster (HEB) of the lepton
version of the Future Circular Collider (FCC-ee). Then, to estimate the
correctors’ strength to correct for closed orbit perturbations caused by
machine errors. The type of errors considered are:

random dipole field error and random dipole roll ;

quadrupole alignment errors ;

beam profile monitors (BPM) alignment and reading errors ;

sextupole alignment errors. The expected Root Mean Square (RMS) values of the
residual orbit and corrector strengths in presence of the considered errors
have been calculated according to the following formulae [2, 3].

$\displaystyle x_{\text{rms}}$ $\displaystyle=$
$\displaystyle\frac{\pi}{\sqrt{2}\sin{\pi
Q_{x}}}\frac{\bar{\beta}}{\sqrt{N_{d}}}\left(\frac{\Delta
B}{B}\right)_{\text{rms}}+$ (1)
$\displaystyle\frac{\sqrt{N_{q}}}{\sqrt{2}\sin{\pi Q_{x}}\cos{\mu/2}}(\Delta
q_{x})_{\text{rms}}+$
$\displaystyle\frac{\sqrt{1/2}}{\left[1+\sin(\mu/2)\right]}(\Delta\sigma_{x})_{\text{rms}}$
$\displaystyle y_{\text{rms}}$ $\displaystyle=$
$\displaystyle\frac{\pi}{\sqrt{2}\sin{\pi
Q_{y}}}\frac{\bar{\beta}}{\sqrt{N_{d}}}(\Delta\theta)_{\text{rms}}+$ (2)
$\displaystyle\frac{\sqrt{N_{q}}}{\sqrt{2}\sin{\pi Q_{y}}\cos{\mu/2}}(\Delta
q_{y})_{\text{rms}}+$
$\displaystyle\frac{\sqrt{1/2}}{\left[1+\sin(\mu/2)\right]}(\Delta\sigma_{y})_{\text{rms}}$

$\displaystyle(\delta_{x})_{\text{rms}}=$
$\displaystyle\biggl{\\{}\frac{\bar{\beta}}{\beta_{\text{max}}}\left(n\left[\frac{2\pi}{N_{d}}\left(\frac{\Delta
B}{B}\right)_{\text{rms}}\right]^{2}+2(\Delta
q_{x})_{\text{rms}}^{2}K_{q}^{2}L_{q}^{2}\right)+$ (3)
$\displaystyle\frac{1+2\cos^{2}(\mu)}{2(L_{\text{cell}}/2)^{2}\left[1+\sin(\mu/2)\right]^{2}}(\Delta\sigma_{x})_{\text{rms}}^{2}\biggr{\\}}^{1/2}$
$\displaystyle(\delta_{y})_{\text{rms}}=$
$\displaystyle\biggl{\\{}\frac{\bar{\beta}}{\beta_{\text{max}}}\left(n\left[\frac{L_{d}}{\rho}\left(\Delta\theta\right)_{\text{rms}}\right]^{2}+2(\Delta
q_{y})_{\text{rms}}^{2}K_{q}^{2}L_{q}^{2}\right)+$ (4)
$\displaystyle\frac{1+2\cos^{2}(\mu)}{2(L_{\text{cell}}/2)^{2}\left[1+\sin(\mu/2)\right]^{2}}(\Delta\sigma_{\text{y}})_{\text{rms}}^{2}\biggr{\\}}^{1/2}$

where $N_{d}$ is the number of dipoles, $Q_{x,y}$ the horizontal/vertical
tunes, $\bar{\beta}=\frac{L_{\text{cell}}}{\sin(\mu)}$ the mean betatron
function, $L_{\text{cell}}$ the cell length, $\mu$ the phase advance per cell,
and $N_{q}$ the number of quadrupoles. For each of 100 different error
configurations of the lattice model we compute the corresponding analytical
value. We compare the maximum among the analytical RMS estimates with the
distribution of 100 numerical results, after applying the correction strategy
described in the next section.

##  2 Correction Strategy

Before correcting the orbit, we first need to define and choose the correctors
and BPMs to use. If a quadrupole is focussing in the horizontal plane the
neighbouring BPM will read in horizontal and the following corrector will
correct in horizontal. Reciprocally, if the quadrupole is focussing in the
vertical plane, the BPM and corrector will act in vertical. The range of
correctors and BPMs selected for each arc is defined in order to have the same
number of correctors and BPMs used in each plane. In the $x$ plane: from the
first BPM and the two correctors before the beginning of the arc to the three
BPMs and correctors after the end of the arc. In the $y$ plane: from the first
BPM and the two correctors before the beginning of the arc to the two BPMs and
correctors after the end of the arc.
The same strategy has been used for the LHC during commissioning [5]: a first
correction, Segment-by-Segment (SbS) i.e. in our case, arc by arc. Each
segment of the machine is handled as a line, using the optical parameters at
the first segment entry. The correction procedure is divided in two sections:
one with the sextupoles turned off and the other with the sextupoles on. This
decomposition sextupoles off/on of the orbit study is commonly done in
accelerators such as SuperKEKB during commissioning [4]. We assume that all
the correctors of the Booster will be individually powered.
Procedure part 1: sextupoles off. After turning off the sextupoles, we first
do the SbS. This gives us a first orbit corrected to inject in a singular
value decomposition (SVD) algorithm over the entire machine. The SVD is done
with the MadX command CORRECT [6]. By doing an SbS before a SVD on all arcs,
we can make more seeds converge than starting directly with all arcs together.
After the SbS, two iterations of SVD are made on all arcs and in line, in
order to reduce the residual orbit. This is sufficient to get an orbit reduced
enough to find the closed orbit. Finally, multiple iterations of SVD in ring
are made and the number of iterations vary for each seed. Iterations are made
until the RMS of the orbit in each plane is below the analytical RMS, while
the number of iterations is under a limit of maximum iterations (fixed to 15).
Procedure part 2: sextupoles on. The sextupoles are turned on and we do one
iteration of SVD in ring. More iterations are not useful because the command
CORRECT does not give a better correction of the residual orbit.
To illustrate the different steps of the correction procedure, the Fig. 1
represents the orbit for the seed 3, with an RMS random error on the
quadrupole offset of $150\text{\,}\mathrm{\SIUnitSymbolMicro m}$, dipole
relative field error of $10^{-4}$, main dipole roll of
$300\text{\,}\mathrm{mrad}$, BPM offset of
$150\text{\,}\mathrm{\SIUnitSymbolMicro m}$ and sextupole offset of
$150\text{\,}\mathrm{\SIUnitSymbolMicro m}$, after each important step.

Figure 1:  Example of the evolution of the orbit during three different steps
of the correction procedure (seed=3, MQ offset
($150\text{\,}\mathrm{\SIUnitSymbolMicro m}$), MB field error ($10^{-4}$),
main MB roll ($300\text{\,}\mathrm{mrad}$), BPM offset
($150\text{\,}\mathrm{\SIUnitSymbolMicro m}$), and MB offset
($150\text{\,}\mathrm{\SIUnitSymbolMicro m}$)). Top panel shows residual orbit
with errors and no correction. Middle panel shows residual orbit after last
iteration of the SVD correction with sextupoles OFF. Bottom panel shows
residual orbit after the SVD correction with sextupoles ON.

We studied and tested different case scenarios by adding one by one error
types. All tests were run on 100 seeds. Our starting point is the combination
of the offsets of the quadrupoles (designated as MQ), the dipole (designated
as MB) relative field error and main dipole roll error. The statistical study
of this first error configuration revealed that all seeds converged until we
reached a MQ offset of $150\text{\,}\mathrm{\SIUnitSymbolMicro m}$. By
following this method, we added and fixed the other elements errors as
reported in Table 1.

Table 1: Summary of the different error types and their values. Error type (Gaussian RMS ) | Value [Unit]
---|---
MQ offset |  80, 90, 100, 120, 150, 200 $\mathrm{\SIUnitSymbolMicro m}$
MB relative field error |  10-4,10-3
MB main dipole roll error | $300\text{\,}\mathrm{mrad}$
BPM offset |  60, 80, 100, 150, 200 $\mathrm{\SIUnitSymbolMicro m}$
MQ offset $150\text{\,}\mathrm{\SIUnitSymbolMicro m}$ |
MS offset |  60, 80, 100, 120, 150, 200 $\mathrm{\SIUnitSymbolMicro m}$
MQ, BPM offset $150\text{\,}\mathrm{\SIUnitSymbolMicro m}$ |
BPM resolution |  10 and 50 $\mathrm{\SIUnitSymbolMicro m}$
MQ, BPM, MS offset $150\text{\,}\mathrm{\SIUnitSymbolMicro m}$ |

It is worth noticing that all the errors applied on the elements are randomly
Gaussian distributed within $\pm$3 RMS.

##  3 Results and discussion

Figure 2: Distribution of the residual orbits and the corresponding RMS values
for the 99 successful machine configurations and errors described in the text.

Figures 2 and 3 show the orbit and correctors strength distribution with their
respective RMS values for the 99 successful seeds at the end of the correction
procedure. The following errors are included: relative dipole field error of
10-3, main dipole roll of $300\text{\,}\mathrm{mrad}$, BPMs offset of
$150\text{\,}\mathrm{\SIUnitSymbolMicro m}$, sextupole offset of
$150\text{\,}\mathrm{\SIUnitSymbolMicro m}$ and BPM reading error of
$50\text{\,}\mathrm{\SIUnitSymbolMicro m}$. The dashed red lines on the
distributions represents $\pm$3 times the RMS calculated analytically. For
both the orbit and the correctors’ strength, we can see that our analytical
predictions fit well the data from the simulations, for most all of the seeds.
The deviations can be explained by the combination of the different errors and
$\beta$-function, for which the procedure is less effective. The residual
orbit amplitude in both planes is in the order of magnitude of the MQ offsets
(which are the dominant errors as expected) and the pattern of the succession
of the arcs and insertions is visible (because we only applied the errors to
the arcs, the residual orbit after correction in the insertions is expected to
be almost zero). The RMS values for each of the successful seeds are
distributed around the dashed red line representing one analytical RMS
estimate, the blue dots correspond to the RMS after turning on the sextupoles
and correcting the orbit once, always using the SVD algorithm. Apart from the
QP offset, the other main contributors to the residual orbit are the MS offset
and the BPM resolution.

Figure 3: Distribution of the correctors strength and the corresponding RMS
values for the 99 successful seeds.

Concerning the correctors’ strength distribution (Fig. 3), we can notice that
for almost all the correctors, their strength is within the limit of the three
analytical RMS estimates (red dashed line). The RMS values for the 99
successful seeds are the same between the last iteration of the first part of
the correction procedure (sextupoles off) and its end (see sec. Correction
Strategy). The analytical RMS (green dashed line) is well above the numerical
results from the simulations. The values of the corresponding residual orbit
and corrector specifications is reported in Table 2.

Table 2: Residual Orbit and Corrector strengths for $150\text{\,}\mathrm{\SIUnitSymbolMicro m}$ RMS Gaussian distributed random offset values of quadrupoles, sextupoles, and BPMs, $50\text{\,}\mathrm{\SIUnitSymbolMicro m}$ RMS Gaussian distributed random precision of BPM reading, 10-3 RMS Gaussian distributed random relative dipole magnetic field error and $300\text{\,}\mathrm{mrad}$ RMS Gaussian distributed roll of dipoles. |  | Plane |  3$\times$RMS
---|---|---|---
|  |  | Analytic |  Seeds111Mean of RMS of all seeds.
Residual orbit |  [$\mathrm{\SIUnitSymbolMicro m}$] | x | 188 | 174
|  | y | 192 | 180
Corrector strength |  [$\mathrm{m}\mathrm{T}$⋅$\mathrm{m}$] | x | 16 | 12
|  | y | 16 | 12

##  4 Conclusions

We have computed the first specification for the tolerance of mis-alignment of
the main high energy booster elements (of about
$150\text{\,}\mathrm{\SIUnitSymbolMicro m}$) and dipole corrector strengths
(of about ), considering only the orbit corrections. These specifications do
not consider the additional strength that will be required at the extraction
energy to compensate the orbit deviation due to the energy loss in one turn
because of synchrotron radiation (tapering). Moreover, these values need to be
confirmed by the full emittance tuning (i.e. adding the $\beta$-beating, the
dispersion and the coupling correction). Finally, a further optimization of
these specifications and of the correction strategy can be attempted, for
example by mean of machine learning techniques.

##  5 ACKNOWLEDGEMENTS

The Authors would like to thanks I. Agapov, R. Tomas and M. Hostettler for
useful discussions.

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