# On viscous fluid flow near the moving crack tip

Huy Duong Bui, Corinne Guyon, Bernadette Thomas

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Huy Duong Bui, Corinne Guyon, Bernadette Thomas. On viscous fluid flow near the moving crack tip. Gérard A. Maugin; Raymonde Drouot; François Sidoroff. Continuum thermodynamics: the Arts and Sciences of modeling material behavior, 76, Springer, pp.63-74, 2000, Solid Mechanics and Its Applications, 978-0-7923-6407-8. 10.1007/0-306-46946-4_4. hal-00112275

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# On viscous fluid flow near a moving crack tip

Huy Duong Bui¹,², Corinne Guyon and Bernadette Thomas²

(1) Laboratoire de Mécanique des Solides, CNRS UMR 7649,
Ecole Polytechnique, 91128 Palaiseau, France

(2) Division Recherche & Développement, Electricité de France, Clamart

**Abstract:** We consider a crack partially filled with a fluid. We show that the presence of a lag avoids the appearance of pressure and velocity singularities. For the static equilibrium, we recall the previous result on the Capillary Stress Intensity Factor which provides a purely mechanical explanation of the Rehbinder effect, according to which the toughness of the material can be lowered by humidity. For the steady state propagation of a crack due to viscous fluid flow, we set the coupled system of non-linear equations.

**Keywords:**

## 1. INTRODUCTION

Most applications of hydraulic fracturing of rocks are found in the petroleum industry. To increase the out-flow of petroleum, one generates a multiple fracture in the rock-oil by injecting a pressurized fluid in the well. This problem involves the analysis of a fluid-crack interaction, for initiation as well as for crack propagation. Most works known in the literature are related to the Boussinesq model of 1D-fluid flow inside the crack [Abe et al., 1976], [Bui et al., 1982], [Balueva et al., 1985]. One-dimensional flow models are justified on almost all the crack length because of the smallness of the crack opening displacement compared to the crack size. However it leads to some difficulties in the vicinity of the crack tip. For example, the pressure in the 1D flow model is unbounded at the tip. Therefore, to avoid a pressure singularity, one cannot ignore the two-dimensional nature of flow near the crack tip, [Bui et al., 1982]. There is another possibility for having a finite pressure, by assuming that a lag exists between the fluid and the crack tip [Advani et al., 1997], [Garash et al., 1999].

The existence of a lag is a necessity for some physical reasons. As discussed in [Bui, 1996], there is some physical and/or geometrical incompatibility near the crack tip zone when the fluid completely fills the crack. This can be illustrated by an example. Let us consider the following constants of a rock : Toughness ($K_{IC}=10^6$ Nm$^{-3/2}$), Poisson's
ratio ($v=0.3$), Young's modulus ($E=5.10^{10}$ Nm$^{-2}$). Typically, the radius of curvature of the crack opening displacement, corresponding to the critical toughness $K_{Ic}$ is given by $R_c=4K_{Ic}^2(1-v^2)^2/\pi E^2$ or $R_c=5.10^{-10}$ m = 0.5nm, i.e the size of two water molecules. Therefore, from the physical point of view, a continuous description of the fluid-solid may be questionable in models without a lag between the fluid and the crack tip.

The second reason is purely geometrical. Suppose that the incompressible viscous fluid completely fills the opening crack and adheres to the solid wall. The no-slip condition for fluid particles in contact with both crack surfaces implies singular velocity and singular resistant hydrodynamic forces on the crack tip region, precluding its propagation.

Some experimental results confirmed the lag assumption near the crack tip. This is schematically shown in Figure 1, adapted from [Van Dam et al., 1999], representing a hydraulic fracture in plaster, after splitting the sample over the fracture surface. One can observe that the fluid front and the crack front are clearly separated by a lag, which is about one tenth of the crack radius.

<figure><img src="image_2.png" /><figcaption>Figure 1. Surface roughness pattern and dry lag at the tip of a hydraulic fracture (after Van Dam et al, 1999).</figcaption></figure>

In this paper we will discuss qualitatively different kinds of incompatibility, physical or geometrical, and inconsistencies which exist near the crack due to simplified assumptions introduced by models. We shall set up general equations for investigating the particular case of Newtonian incompressible non-wetting fluid flow near the moving crack tip, in the presence of surface forces.
## 2. WHAT MODELS OF FLUID-FILLED CRACKS?

Coupled problems of fluid-filled cracks can be found either in elasticity or in poro-elasticity. In a poro-elastic cracked medium, the system of equations are provided by Biot's consolidation theory for the porous solid. The crack is treated as a cavity, with a sink or a source located inside the crack. The fluid flow in the crack is then analysed by one-dimensional Boussinesq's approximation. The whole crack faces are permeable and the fluid can be assumed to occupy the whole crack and to be subjected to the difference between the pore pressure of the crack and the fluid pressure in the medium. No particular difficulties arise in the formulations and modelings of coupled hydro-poro-elasticity systems even if the equations are more complicate than those in hydro-elasticity, [Balueva et al., 1985].

When the crack surfaces are impermeable, the viscous fluid flow is parallel to the crack. Some specific difficulties arise in the modelling of the coupled elastic-hydrodynamic problems because there exists a moving interface S between the fluid and the vacuum near the crack tip, where surface forces are present. If the flow is almost one-dimensional in the main part of the crack, it becomes two-dimensional near the crack tip zone. With a lag, in both cases of the Griffith crack and the Barenblatt crack, the fluid rolls on the crack surface and can adhere to the solid at points B, B' far away from the fluid stagnation point and the opening crack tip. Very high velocity gradient and high hydrodynamic stresses are then expected at the interface region S, of high curvature, where surface force can not be neglected. Another difficulty arises in the modeling of the common line between the fluid interface and the solid (the triple point in 2D).

### 2.1 The equilibrium Griffith crack

A fluid-filled Griffith crack in an infinite elastic medium, subjected to a remote tension $\sigma_{22}^\infty$ and a pore pressure p, has been investigated in [Bui, 1996], with a lag and a surface tension force. The case of a non-wetting fluid-solid system, defined by a surface tension coefficient $\alpha$ and the contact angle $\theta$ at the triple point FSV (fluid-solid-vapor or fluid-solid-vacuum) less than $\pi$ has been worked out. The contact angle $\theta$ at the triple point, Figure 2, the slope of the crack opening displacement $-\partial u_2/\partial x_1$ and the angle $\psi$ of the convex meniscus are related by

$$ (1) \qquad \theta = -\partial u_2/\partial x_1 + \psi + \pi/2. $$

Neglecting the slope $-\partial u_2/\partial x_1$ of the crack opening displacement, we take
<figure><img src="image_3.png" /><figcaption>Fig. 2. The equilibrium of a fluid-filled crack of size 2a, with the meniscus BK</figcaption></figure>

$$ (2) \qquad \theta \approx \psi + \pi/2. $$

This approximate equation makes it possible to derive a closed form solution of the coupled systems of equations, given in [Bui, 1996]. The wet zone size 2c (< 2a the crack length) is found to be a function of p. It is found that for vanishing remote tension, $\sigma_{22}^\infty=0$, a wetting crack $c(p) \neq 0$ exists for a well-defined pore pressure p which gives rise to an initial stress-intensity factor,

$$ K_I^0(c) = \frac{2p(c)}{\pi} \arcsin\left(\frac{c}{a}\right) (\pi a)^{1/2}. $$

This stress-intensity factor is due solely to the existence of a wet zone in the crack and is refered to as the Capillary Stress Intensity Factor (CSIF). The plot of CSIF versus the wet zone size c is shown in Figure 3. It increases as the wet zone size increases up to a maximum value $c_{max} < a$.

This simple model explains why humidity can lower the toughness of materials significantly, $K_{Ic}(wet) \le K_{Ic}(dry)$. This is nothing but the well-known phenomenon called Rehbinder's effect. The explanation provided here by the CSIF, which lowers the apparent critical stress intensity factor of wet materials is purely mechanical. It differs from classical arguments based upon Van der Vaals's interaction forces and chemical in interactions between water molecules and atoms. In the classical interpretation of Rehbin-
<figure><img src="image_4.png" /><figcaption>Figure 3. Capillary stress-intensity factor versus the wet zone size, after [Bui, 1996]</figcaption></figure>

der's effect, a water molecule can penetrate the crack tip cavity in glass because of its size smaller than $R_c$. A link is broken between the atoms Si and O of ..Si-OH; another one is broken in H-OH of water molecule. Then the two hydroxyls OH make new chemical bonds with the silicium atoms giving two separate -Si-OH and resulting in a new cavity or an additional crack length, [Michalske et al., 1988]. This mechanism requires less energy than a direct separation of the bond of --Si-O-. The result is a separation of atoms in glass. Water molecule is small compared to methanol or aniline. As a matter of fact, the chemical influence of bigger molecules in the toughness of materials is negligible since there is no room for interactions at the atomic level. In this case, the lowering of toughness by the CSIF is one of the possibilities.

There are approximations inherent to classical models of surface tension. Due to surface tension force on the meniscus S, the solid deforms as a dimple at the FSV point. In 2D elasticity, the displacement due to point force should be logarithmically singular. However the ratio of surface force acting on S to the pressure acting on the wet zone is of the order $O(v(c)/a) \ll 1$. Hence the effect on the stress intensity factor due to surface force acting on the dimples is *negligible*, compared to the effect due to pressure. Perhaps
the model is inconsistent regarding the appearance of a *local* singularity at the triple points, but it gives a good idea of *global* physical quantities such as the stress-intensity factors etc. The deficiency of the theory about the behavior near the triple point is simply due to the approximations used in the model, such as linear elasticity, simple theory of surface force etc. A better description of the triple point zone may be possible with more elaborate models (non linear elasticity, second gradient theory of fluid/solid interaction). But it is difficult to know a priori what kind of model to use for a better modeling of specific quantities encountered in the coupled systems fluid-solid.

## 2.1 The moving triple point

A more elaborate model of the triple point has been provided in [Seppecher, 1989] where the second gradient theory is used. The theory introduces a third order stress tensor, symmetric in its first two indices, generalizing the power of internal forces in a fluid domain $\Omega$ as follows

$$ (3) \qquad \mathbf{P}_{\text{int}} = - \int_{\Omega} [\sigma : \nabla \mathbf{v} + s : \nabla \nabla \mathbf{v}] \, dx $$

By relaxing the incompressibility condition, he succeeded in analysing the structure of the density field $\rho(\mathbf{x})$ in the vicinity of the stationary triple point. The iso-density lines have a constant contact angle, outside the triple point zone, Figure 4.

Such a refined model, when extended to the moving interface, is expected to provide better informations about the structure of the triple point.

<figure><img src="image_5.png" /><figcaption>Figure 4. Iso-density curves near the triple point, after [Seppecher, 1989].</figcaption></figure>
<figure><img src="image_6.png" /><figcaption>Figure 5 Structure of flow near the moving interface Fluid-Fluid-Solid. (after [Dussan et al., 1974].)</figcaption></figure>

Analyses of the moving interface fluid 1-fluid 2 along a solid wall have been worked out theoretically and experimentally by many investigators, [Moffath, 1964], [Dussan et al., 1974], [Durbin, 1989]. For example fluid 1 is the water injected in the well and fluid 2 is the oil extracted from the crack tip. This system is very intricate even outside the common line, as is shown in Figure 5, adapted from [Dussan et al., 1974]. Fluid 1 is rolling on the rigid wall while fluid 2 is rolling off at the triple point. Since the fluid points of the moving interface $F_1F_2$ map to the triple point P at certain angle, an ejected line of fluid 2 is predicted (and observed experimentally) when rolling off the solid wall, going inside the domain $F_2$. There exists a multi-valued velocity singularity at the triple point due to model assumptions: incompressibility, Newtonian fluid and no-slip condition. It is observed experimentally that the no-slip condition does not mean that fluid particles can never leave the solid at some instant. The adherence condition is stronger than the no-slip condition because it states that fluid particles can never leave the solid wall.

In [Durbin, 1989] the no-slip condition is relaxed by introducing a slip zone to be determined, with a constant shear. Durbin's boundary condition avoids the appearance of singularity in the fluid, just as the Barenblatt assumption of constant yield stress eliminates the stress singularity in linear fracture mechanics.

For large surface tension or small capillary number $c_a=V\mu/\alpha$, the constant contact angle $\theta$ may be a good approximation. Some arguments are given in [Dussan et al., 1974] for a rapid change of the tangent to the interface $F_1F_2$ near the wall, suggesting that the fluid rolls on the solid wall smoothly and that the contact angle is constant just outside
the contact line zone. Thus the structure of the fluid-fluid interface line near the contact line seems to be very complex, since it depends very much on the boundary condition used. In this discussion, we do not consider the deformation of the solid at the triple point which can modify the flow.

Can the structure of flow near the contact line be predicted by a refined theory ? Is such a theory a necessity or not, regarding the question of determining the macroscopic behaviour of the fluid-crack system ? These are still open problems.

### 3. A SIMPLIFIED MODEL OF FLUID-FILLED PROPAGATING CRACK

We consider a steady state propagation of a crack partially filled by an incompressible viscous fluid. The absolute crack velocity is $V$. The geometry of the coupled system in the moving frame is shown in Figure 6. Since the crack is propagating (inertial force neglected), the geometry of the crack opening displacement in the vicinity of the crack tip is well-defined by the toughness $K_{IC}$ of the solid.

<figure><img src="image_7.png" /><figcaption>Figure 6. The fluid-filled moving crack with a convex meniscus</figcaption></figure>

The fluid is separated from the crack tip by a lag (a vacuum), with the surface tension $\alpha$, the contact angle $\theta$. The boundary condition at the section AA' is given by a parabolic profile velocity, with a vanishing total flow rate and the velocity at the wall equal to $-V$. The equations of the fluid-solid system are summarized below:

* Navier-Stokes equations :

$$ (4) \qquad \operatorname{div} \mathbf{v} = 0, \quad \text{in } \Omega, $$

$$ (5) \qquad (\mathbf{v} \cdot \operatorname{grad}) \mathbf{v} + \frac{1}{\rho} \operatorname{grad} p - \eta \Delta \mathbf{v} = 0, \quad \text{in } \Omega, $$
(where η is the kinematic viscosity, p the pressure, ρ the constant density)

• Capillary force:

(6) $p = \alpha \operatorname{div} n$ on S (n is the unit outward normal to S),

(7) $\mathbf{v} \cdot \mathbf{n} = 0$ on S,

• In-flow velocity (2h is the crack opening at A'A)

(8) $$v_1 = \frac{3V}{2} \left(1 - \frac{x_2^2}{h^2}\right) - V, \quad v_2 = 0 \quad \text{on } A'A,$$

• Boundary conditions at the wall:

(9) $v_1 = -V, \quad v_2 = 0$ on AB and A'B$,

• Parabolic crack profile:

(10) $$x_1 = -x_2^2 / 2R_0, \quad \text{with } R_0 = 4K_{Ic}^2(1-v^2)^2 / \pi E^2,$$

• Global equilibrium equation :

(11) $$K_{Ic} = f(p, \sigma_{22}^{\infty}), \quad (\sigma_{22}^{\infty} \text{ : remote applied stress}).$$

Equation (11) expresses the global relation between the stress-intensity factor in the symmetric mode I, the applied loads on the cracked body and the pressure p on the crack, without the point force at FSV. It corresponds to the matching between the outer expansion solution with the asymptotic inner solution defined by Eqs (4) - (10). Such a matching can only be made when the solution of (4)-(10) is obtained for any $K_{Ic}$ and p.

For low velocity, small crack opening and high viscosity, the Reynolds number $\mathcal{R}$ is small and the Stokes approximation may be justified. The above equations are still non-linear, even if the linearized fluid equations are used, because the contact points B, B' are yet unknown and the pressure is coupled to the curvature of the meniscus.
4. NUMERICAL RESULTS

In numerical computations, the value θ=2π of the contact angle has been used, in order to avoid either the appearance of velocity singularity at the triple point (no-slip condition) or the consideration of a slip zone between the fluid and the solid (Durbin's boundary condition). This restriction is not essential since the magnitude of the point force at FSV is small in comparison with the total force due to pressure acting on the wet zone. The point force and the velocity singularity should have a little effect on the overall behavior of the fluid-solid system. Instead of solving the complete nonlinear system of equations, we are searching an approximate solution of Equations (4), (5), (7), (8), (9), (10), for a fluid domain having an elliptical geometry and for given triple point positions. Since the pressure and the radius of the meniscus are practically constant near the stagnation point, the equation (6) is satisfied approximately, a posteriori, by adjusting some constant α. This inverse fitting applies also to (11) where the toughness is assumed to be related to the meniscus radius R_o and to the loading.

The non-linear problem, without the Stokes approximation, is solved by an
iterative procedure. The computations are carried out, using the finite element method
provided by the non-linear computer code N3S of Electricité de France (3D Navier-
Stokes).

Numerical results are shown in Figure 7 for the stream lines and the pressure, which decreases while approaching the crack tip.

<figure><img src="image_8.png" /><figcaption>Figure 7. Pressure and stream lines near the crack tip</figcaption></figure>
The stream lines patterns show clearly an interfacial motion similar to the moving
tractor tread described in [de Gennes, 1985].

5. CONCLUSIONS

1. In studying the coupling between a crack and a fluid, it seems reasonable to consider only *microscopic* surface force phenomena having a significant effect on the *macroscopic* response of the cracked solid. One can neglect microscopic details with short range effects such as the dimple, the singular velocity at the triple point.

2. The pressure which is related to the curvature of the meniscus has long range effects because it applies to the crack surface.

3. Simplified theory of surface force is used for investigating the main features of the solution.

6. REFERENCES

[Abe et al., 1976] Abe, T.; Mura, T.; Keer, L.M.; Growth rate of a renny-shaped crack in hydraulic fracturing of rock; In: *J. Geophys. Res.*, p. 5325; **81**, n° 29.

[Advani et al., 1997] Advani, S.; Lee, T.; Dean, R.; Pak, C.; Avasthi, J.; Similarity solution of a semi-infinite fluid-driven fracture in a linear elastic solid ; In: *Int. J. Numer. Anal. Meth. Geomech*, p. 229-240; **21**.

[Balueva et al., 1985] A.V. Baluaeva, A.V.; Zazovskii, A.F.; Elastic-hydrodynamic problem of inflow of fluid in a crack in a porous medium ; In: *Izv. AN SSSR. Mekhanika Tverdogo Tela*, pp. 157-166; **20**, n° 5.

[Bui et al., 1982] Bui, H.D.; Parnès, R.; A reexamination of the pressure at the tip of a fluid-filled crack; In: *Int. J Engng. Sci.*, p. 1215-1220; **20**, n° 11.

[Bui, 1996] Bui, H.D.; Interaction between the Griffith crack and a fluid: theory of Rehbinder's effect; In: *FRACTURE: A Topical Encyclopedia of Current Knowledge*; Ed. Cherepanov, G.P.; p. 99-105; Chap. 6.

[Conca et al., 1994] Conca, C.; Planchard, J.; Thomas, B.; Vaninathan, M.; Problèmes mathématiques en couplage fluide-structure; *Collection des Etudes & Recherches*; **85**, Eyrolles, Paris, 1994.
[De Gennes, 1985] De Gennes, P.G.; Wetting: statics and dynamics. In: *Reviews of Modern Physics*, p. 827; **57**, 3, I.

[Durbin, 1989] Durbin, P.A.; Stokes flow near a moving contact line with yield-stress boundary condition; In: *Quart. J. Mech. & Math.*, p. 99-113; **42**.

[Dussan et al., 1974] Dussan, E.B.; Davis, S.H.; On the motion of a fluid-fluid interface along a solid surface; In: *J. Fluid. Mech.*, p. 71; **65**, I.

[Garash et al., 1999] Garash, D.; Detournay, E.; Similarity solution of a semi-infinite fluid-driven fracture in a linear elastic solid; Private Comm. (to appear).

[Michalske et al., 1988] Michalske, T.; Bunker, B.; La fracture du verre ; In: *Pour la Science*, p.52-59; Février 1988.

[Moffatt, 1964] Moffatt, H.K.; Viscous and resistive eddies near a sharp corner; In: *J. Fluid. Mech.*, 1-18; **18**.

[Monla, 1990] Monla, M. ; Sur la détermination de la forme d'une bulle en ascension dans un liquide visqueux; *Thèse de l'Ecole Centrale de Lyon*; n° 90-08.

[Rehbinder, et al. 1957] Rehbinder, P.; Lichtman, V.; Effect of surface active media on strain and rupture of solid ; In: *Proc. 2nd Int. Cong. Surf. Act.*, p. 563; **3**.

[Seppecher, 1989] Seppecher, P.; Etudes des conditions aux limites en théorie du second gradient: cas de la capillarité; In: *Comptes Rendus Acad. Sci. Paris*, p. 497-502; **II 309**.

[Van Dam et al., 1999] Van Dam, C.J.; De Pater, C.J.; Romijn, R.; Reopening of dynamic fractures in laboratory experiments; In: *Cong. Int. Mech. Rocks*, Paris August 1999, p. 792-794; **II**.