J. Fluid Mech. (2017), . 821, pp. doi:10.1017/jfm.2017.234...

J. Fluid Mech. (2017), vol. 821, pp. 5984. c Cambridge University Press 2017doi:10.1017/jfm.2017.234

59

Gravity-driven flow of HerschelBulkley fluid ina fracture and in a 2D porous medium

V. Di Federico1,, S. Longo2, S. E. King3, L. Chiapponi2, D. Petrolo2and V. Ciriello1

1Dipartimento di Ingegneria Civile, Chimica, Ambientale e dei Materiali (DICAM),Universit di Bologna, Viale Risorgimento, 2, 40136 Bologna, Italy

2Dipartimento di Ingegneria e Architettura (DIA), Universit di Parma, Parco Area delle Scienze,181/A, 43124 Parma, Italy

3School of Mathematics, University of Edinburgh, Edinburgh, EH9 3FD, UK

(Received 30 January 2017; revised 9 April 2017; accepted 10 April 2017)

New analytical models are introduced to describe the motion of a HerschelBulkleyfluid slumping under gravity in a narrow fracture and in a porous medium. A usefulself-similar solution can be derived for a fluid injection rate that scales as time t; anexpansion technique is adopted for a generic injection rate that is power law in time.Experiments in a Hele-Shaw cell and in a narrow channel filled with glass ballotiniconfirm the theoretical model within the experimental uncertainty.

Key words: gravity currents, Hele-Shaw flows, non-Newtonian flows

1. Introduction

Implications of fluid rheology on flow in fractures and porous media have beenextensively analysed in recent years. Artificial fluids and foams are designed to fulfilspecific requirements related to aquifer remediation, fracking technology and soilreinforcement. In some conditions, carbon dioxide stored in aquifers may behave asa non-Newtonian fluid (Wang & Clarens 2012). Darcys law, valid for Newtonianfluids, has been extended, with various methodologies, to power-law non-Newtonianfluids and experimentally validated; see Cristopher & Middleman (1965), Barletta &de B. Alves (2014) and references therein.

Viscous gravity currents of power-law (Ostwald 1929) fluids in wide channels andin fractures have been extensively investigated, formulating specific models for variousgeometrical configurations and providing experimental verification. Gratton, Minotti &Mahajan (1999) and Perazzo & Gratton (2005) presented a comprehensive theoreticalframework for unidirectional and axisymmetric flow over a horizontal plane and downan incline. Longo et al. (2013a) investigated experimentally horizontal spreading inradial geometry, while Longo, Di Federico & Chiapponi (2015c,d) examined the

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60 V. Di Federico and others

advance in horizontal and inclined channels, taking into account the shape of thecross-section, and longitudinal variations of cross-section and bottom inclination.

Gravity currents of a power-law fluid in porous media have recently been analysedwith a combination of analytical, numerical and experimental techniques (e.g. Longoet al. 2013b; Di Federico et al. 2014; Longo et al. 2015a; Ciriello et al. 2016).

However, even though the power-law approximation provides an accurate interpre-tation of fluid behaviour in several flow conditions, it does not cover other classes offluids exhibiting yield stress. These are better described by models such as HerschelBulkley (three parameters, Herschel & Bulkley 1926), Cross (four parameters, Cross1965) and CarreauYasuda (four or five parameters, Carreau 1972; Yasuda, Armstrong& Cohen 1981).

Gravity currents of HerschelBulkley (HB) fluids on horizontal and inclinedplanes, or in wide channels, have been analysed theoretically and experimentallyby several authors. Hogg & Matson (2009) modelled two-dimensional currents,focusing their attention on the front geometry, its role in the overall dynamics andthe arrested state for a dam-break process. Huang & Garcia (1998), Vola, Babik &Latch (2004) and Balmforth et al. (2006) investigated the propagation numerically.Further experimental contributions were provided by Ancey & Cochard (2009) inthe dam-break configuration and by Chambon, Ghemmour & Naiim (2014) in thesteady uniform regime. The special case of Bingham fluids was analysed by Liu &Mei (1989); the effect of finite-width channels was explored by Mei & Yuhi (2001)and Cantelli (2009). A recent review (Coussot 2014) critically lists the numerouspapers on flows of HB fluids in several geometries and conditions. The effect ofa realistic channel geometry mimicking natural channelized flow was analysed inLongo, Chiapponi & Di Federico (2016), where an HB fluid was injected with aconstant discharge rate in a channel widening and reducing its bottom inclinationdownstream.

Regarding porous flow of yield stress fluids, a key element is the reliability of themodel relating the flow rate and the pressure gradient. According to Chevalier et al.(2013), porous flow of an HB fluid is characterized by multiple length scales, and atleast one of them is not related to the geometry of the pores and connecting channels,but depends on the pressure drop. Hence, the flow starts along specific limited pathsnear the threshold pressure drop. Subsequently, the sequence of converging anddiverging throats encountered by the fluid facilitates a progressive increment of themobilized fluid domain as the pressure drop increases, rather than a sharp increasein the extent of mobilized fluid. Further experiments (Chevalier et al. 2014) havedemonstrated that the domain of fluid at rest is very limited even for very lowvelocity. This complex behaviour increases the difficulties in modelling yield stressfluids and strengthens the need to verify existing formulations of the flow law validat Darcys scale with carefully conducted experiments.

The existing body of knowledge on HB flows, accumulated mostly in recentyears, leaves open several avenues of investigation. To the best of our knowledge,the behaviour of HB fluids flowing in narrow channels (fractures) has not beeninvestigated to the same extent as flows in wide channels, and deserves a morein-depth analysis due to the numerous practical applications of the process, suchas polymer processing, heavy-oil flow, gel cleanup in propped fractures and drillingprocesses. In addition, the flow of HB fluids in a porous medium still requiresexperimental validations to enable extension of the results obtained in viscometricflows to more realistic configurations. The present theoretical approach aims tocontribute to these aspects, with the crucial support of laboratory experiments.


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2D flow of a gravity current of HerschelBulkley fluid 61

Impermeable bottom

Injection

Current fluid

Ambient fluid

x

y

z

FIGURE 1. (Colour online) Diagram showing the set-up of axes and fluid orientation inflow through a narrow fracture (Hele-Shaw cell).

In this paper, we present a theoretical model and its experimental validation for two-dimensional (2D) flows of a HB fluid in a narrow fracture and in a porous medium.The theoretical model is general, while the computations and experiments refer mainlyto a specific situation (an injected volume quadratic in time) where a simple self-similar solution is available. An expansion method has been applied to handle, withsome restrictions, the general case of an injected volume that is power law over time;the general method has likewise been experimentally validated.

The paper is structured as follows. Section 2 presents the model for HB flow in anarrow fracture. The self-similar solution is illustrated in 3. Flow in a homogeneousporous medium is examined in 4. Section 5 describes the experiments conducted ina Hele-Shaw cell and in an artificial 2D porous medium. The last section containsthe conclusions. Details on the rheometry of the yield stress fluids employed in theexperiments are included in appendix A.

2. Model description for flow in a narrow fractureThe HB model for a shear-thinning/thickening fluid with yield stress is

= (0n1+ p

1) , > p, = 0, < p,

}(2.1)

given in terms of the stress and the strain rate . A slightly more complicateddescription using tensor invariants is required for three-dimensional flows; thisformulation is not reported here as we consider a one-dimensional problem below.The parameter 0, the consistency index, represents a viscosity-like parameter, whilep is the yield stress of the fluid, and n, the fluid behaviour index, controls theextent of shear thinning (n < 1) or shear thickening (n > 1); n = 1 corresponds tothe Bingham case. For flow through a narrow fracture (such as a Hele-Shaw cell) ofwidth Ly, as depicted in figure 1, the primary balance is between cross-gap quantities.The relevant relationship is that the velocity u(x, y) in the x-direction must satisfy

xy =

(0

uyn1 + p uy

1)uy, xy > p,

uy= 0, xy < p,

(2.2)




where xy represents the cross-gap stress and y is the cross-gap direction. Furthermore,the primary balance between the cross-gap stress and the along-fracture pressuregradient is given by y(p/x) = xy. We define the height h(x, t) of fluid above ahorizontal impermeable base. Then, under the relevant shallow water approximation,the pressure is to leading order hydrostatic, and the pressure gradient becomesp/x=1 g (h/x), where 1 is the density difference driving the flow, betweenfluid within and that outside the gravity current, and g is the acceleration due togravity. We assume that the wetting characteristics at the walls of the fracture areunimportant (or alternatively that the flow takes place in a prewetted fracture).Combination of these equations, using the condition that xy = p as the fluidyields, leads to an expression for the location of the yield surface in the gap,|yyield| = p/(1 g|h/x|), where Ly/2 6 yyield 6 Ly/2 is required.

To continue, we need to solve for the cross-gap flow structure in yielded andunyielded regions of the flow. The continuity of mass of the fluid layer may bewritten as

ht=

x(uh), (2.3)

where u(x, t) is the gap-averaged velocity. Assuming a zero slip velocity, and uponintroducing the expression for u(x, t) into (2.3), we can form an evolution equationfor h(x, t) alone, namely

ht=

(Ly2

)(n+1)/nsgn(hx

)(n

2n+ 1

)(1 g0

)1/n

x

h hx1/n(

1 hx

1)(n+1)/n (

1+(

nn+ 1

)

hx1) , (2.4)

where = 2p/(1 gLy) is a non-dimensional number representing the ratio betweenyield stress and gravity related stress, or the ratio between the Bingham and Rambergnumbers. Additionally, setting

=

(Ly2

)(n+1)/n ( n2n+ 1

)(1 g0

)1/n, (2.5)

where is a velocity scale, allows us to write this equation slightly more succinctlyas

ht= sgn

(hx

)

x

h hx1/n(

1 hx

1)(n+1)/n (

1+(

nn+ 1

)

hx1) .(2.6)

To this equation we must add the further condition that when the flow is fully plugged(that is yyield=Ly/2), then the velocity throughout the gap is zero (u= 0), so thereforeh/t = 0 for such regions. This is the limiting version of the equation above inthe limit h/x= , which is in turn the condition for the flow to be fully plugged.Consequently, the equation for the height of the current is continuous through such aplugging transition.

In the model, the slip contribution was neglected also because most experimentswere conducted by roughening the Hele-Shaw cell with commercial transparentantislip tape, which is commonly adopted to make slippery surfaces safe. Independentrheometric measurements were conducted with plates roughened with sandpaper.




3. Self-similar solutionVarious self-similar solutions exist to describe the spreading of gravity currents of

constant or variable volume in cases similar to our problem where = 0, n= 1 (e.g.King & Woods 2003; Lyle et al. 2005) and where = 0 (Pascal & Pascal 1993;Longo et al. 2013b; Longo, Di Federico & Chiapponi 2015b; Ciriello et al. 2016).For the current study where both yield stress and shear-thinning/thickening effects arepresent, a generalized form of such solutions is not available since the presence ofyield stress breaks self-similarity. However, one form of self-similar solution does existif we allow a variable injection rate of fluid (a more general type of solution withself-similarity of the second kind might be possible, although it is not attempted here).To this end, suppose that the volume of fluid in the gravity current varies as

Ly

0h(x, t) dx=Qt, (3.1)

where , Q > 0. Then, the principal dimensions of the three parameters are[] = L/T, [Q/Ly] = L2/T, [] = 1. It is possible to rewrite our principal variablesin dimensionless form as

h= h(

QLy

)1/(2), x= x

(Q

Ly

)1/(2), t= t

(Q

Ly2

)1/(2). (3.2ac)

Introduction of these variables immediately reduces (2.6)(3.1) to the dimensionlessform

h t=

x

h h x

1/n1 h x

1(n+1)/n 1+( n

n+ 1

)

h x1 , (3.3)

0h dx= t , (3.4)

where we have assumed that h/ x< 0. Hereafter, the tilde is dropped. We can seekself-similar solutions of these equations by looking for solutions in the form h= t f (),where = x/t . Substitution into (3.3)(3.4) yields

t1f t1 f =

[t+( )/nf | f |1/n

(1

t

|f |

)(n+1)/n(1+(

nn+ 1

)t

|f |

)]t ,

(3.5) e0

t+ f () d= t, (3.6)

where primes denote differentiation with respect to . Comparison of powers of tgives immediately + = , = 0, 1 = ( )(1 + 1/n). This has oneconsistent solution, namely = 2 and = = 1. A fluid injection rate scaling as tallows a self-similar solution, for which both the height and the length of the currentincrease with time, similarly scaling as t. However, for = 2, the scales in (3.2) breakdown and an additional velocity scale embedded in the integral constraint of massconservation given by (3.1) arises beyond , given by (Q/Ly)1/2. A similar case is




0.5

0

1.0

1.5

2.0

0 0.5 1.0 1.5 2.0 2.5 0 0.5 1.0 1.5 2.0 2.5

0.5

0

0.5

(a) (b)

FIGURE 2. (Colour online) (a) Shape of the similarity solution in a Hele-Shaw cell (=2)for different values of n; (b) plug regions.

treated in Di Federico, Archetti & Longo (2012a,b). We define an arbitrary time scalet, the velocity scale u = (Q/Ly)1/2 and the ratio between the two velocity scales = (Ly2/Q)1/2, with x = ut. Equations (3.3)(3.4) become

ht=

x

h hx1/n(

1 hx

1)(n+1)/n (

1+(

nn+ 1

)

hx1) , (3.7)

0h(x, t) dx= t2. (3.8)

We seek a self-similar solution of the form h = tf (), where = x/t; substitutioninto (3.7)(3.8) yields

f f =

[f | f |1/n

(1

|f |

)(n+1)/n (1+

(n

n+ 1

)

|f |

)], (3.9) e

0f () d= 1. (3.10)

This system admits a simple solution, namely a linear profile for f () (Di Federicoet al. 2012a). Supposing a solution in the form f ()= A(e ), for some constantsA> 0 and e > 0, we substitute into (3.9)(3.10) to obtain

Ae = A(n+1)/n(

1

A

)(n+1)/n [1+

(n

n+ 1

)

A

], e =

2A. (3.11)

Elimination of e gives one nonlinear equation to solve for A.Solutions for several values of the parameter n are shown in figure 2, with all other

parameters kept constant. The self-similar solution retains similarity by constrainingthe yield surface to be at a constant location along its length as the current spreads.Furthermore, the thickness of the plugged region simply grows as the fluid becomesmore shear thinning.




3.1. Asymptotic analysis for 6= 2For 6= 2, no self-similar solution is predicted, but it is possible to find anapproximate result starting from the self-similar solution for power-law fluids ( 0).We briefly recall that for 0 equation (3.3) becomes

hx=

x

(hhx

1/n), (3.12)

while the integral mass balance given by (3.4) is unmodified. Equations (3.4)(3.12)admit the similarity solution

h= n+2N tF2 f ( ), = xtF1, N =

( 10

f d)1/(n+2)

, = /N, (3.13ad)

whereF1 =

+ n2+ n

, F2 = F1, (3.14a,b)

and where the shape function f satisfies the following nonlinear ordinary differentialequation:

( f |f |1/n) + F2 f F1 f = 0. (3.15)

The numerical integration of (3.15) for 6= 0 and 6= 2 requires two boundaryconditions at 1. By assuming f a0(1 )b, substituting in (3.15) and balancingthe lower-order terms, b= 1 and a0 = Fn2 are yielded. Hence, it follows that

f |1 = Fn2, f

1 =F

n2, (3.16a,b)

with a small quantity. The two cases = 0, 2 admit an analytical solution with frepresented by a parabola and a straight line respectively.

It is possible to extend the self-similar solution to > 0 with the followingexpansion in the term /|h/x| (see, e.g., Hogg, Ungarish & Huppert 2000; Sachdev2000).

Upon assuming that /|h/x| is a small quantity, (3.3) becomes

ht=

x

[hhx

1/n(

12n+ 1

n(n+ 1)

hx1 + n+ 1 2n22n2 2

hx2 +O(3)

)].

(3.17)We propose the following expansion in the regime tn(2)/(n+2) 1:

h= n+1N tF2[ f0( )+ f1( )+ 2f2( )+ ], (3.18)

x= tF1(1+ X1 + 2X2 + ), (3.19)

where f0( ) and N are given by the similarity solution for power-law fluids (3.13),and X1, X2, . . . are constants to be evaluated.

The variable is selected in order to guarantee that at the zero order O( 0) theyield stress contribution is null and the solution is represented by (3.13). At the firstorder O( ) there is a balance between the terms due to the yield stress and all otherterms. The condition 1 requires that t tc (n+2)/[n(2)] if < 2 and t tc if > 2; tc is defined as a critical time. Figure 3 shows the critical time tc(, n, )




1050

1030

1010

1010

1030

1050

0 0.5 1.0 1.5 2.0 2.5 3.0

FIGURE 3. (Colour online) Dimensionless critical time for = 0.01 as a function of thefluid behaviour index n and of . The vertical dashed line at = 2 is the asymptote; thehatched area indicates the domain where the condition < 1 is satisfied for a fluid withn= 1.2.

for = 0.01 and for different n and . The critical time becomes infinite for = 2;notably, this coincides with the case that permits a self-similar solution without thenecessity of an expansion. It is seen that tc/ > 0 for any n. In addition, tc/n> 0if > 2 and tc/n< 0 if < 2; hence, the domain of validity of the expansion isextended as the fluid becomes more shear thinning.

The previous analysis implies that for a gravity current of a power-law fluid, allterms in the evolution equations evolve at a common rate (or, equivalently, a uniquevelocity scale exists). In contrast, for an HB fluid, the term arising due to the yieldstress evolves at a different rate from other terms (i.e. it introduces a second velocityscale, for a given common length scale). In order to obtain an expansion to solve thisequation, it should be noted that the correction is achieved at first order by computinga second function that evolves with a rate equal to the new one imposed by the yieldstress term. The nonlinearity of the problem requires an increasing number of suchterms in the series to improve the accuracy in the balance when extended to higherorders.

However, since appears always in powers of |h/x|1, we expect a reductionin the accuracy of the asymptotic solution for increasing time, if |h/x| t(2)n/(n+2)decreases in time, which happens for < 2. Conversely, for > 2, the asymptoticexpansion becomes more accurate, but the current evolves to an increasing steepnesswhich renders the thin-current assumption asymptotically invalid. The expansion willbe uniform in x as long as |h/x| does not approach zero anywhere. In particular,the expansion will be non-uniform if = 0, since in this case the current becomesflat at x= 0.

By inserting (3.18)(3.19) in (3.17) and balancing the terms of equal power in , atO( 0) we recover the fundamental balance, with f0 f , and f represented by (3.13).

At O( ), equation (3.17) becomes(f1|f 0|

1/n

1n

f0|f 0|1/n1f 1

)+ F2f1 F1 f 1 =

1nN

2n+ 1n(n+ 1)

( f0|f 0|1/n1), (3.20)




which is an inhomogeneous linear ordinary differential equation for the unknownfunction f1, with a forcing term modulated by the fundamental solution f0. Thenumerical integration of equation (3.20) requires two boundary conditions for 1,obtained again by expanding in series near the front of the current. Assuming thatf1 a1(1 )b (with f0 Fn2(1 ), see (3.16)), substituting in (3.20) and equatingthe lower-order terms (corresponding to b= 1) yields

a1 =F2

(F2 F1n+ F2n)2n+ 1nN(n+ 1)

, b= 1. (3.21a,b)

Hence, the function can be approximated by f1 a1(1 ) for 1, and theboundary conditions for equation (3.20) are

f1|1 = a1, f

1|1 =a1, (3.22a,b)

where is a small quantity.Even though the two functions f0 and f1 are defined in the domain 0 6 6 1, the

nose of the current is N = 1+ X1+ , which is expected to be smaller than unitysince the additional resistance supplied by the yield stress reduces the propagation ratecompared with the case = 0. The integral constraint given by (3.4) becomes

n+2N (1+ X1 + ) 1+X1+

0[ f0( )+ f1( )] d = 1, (3.23)

which at O( ) yields

X1 =

10

f1( ) d 10

f0( ) d. (3.24)

Figure 4 shows the correction to the front position for waning inflow rate ( = 0.5),constant inflow rate ( = 1), waxing inflow rate ( = 1.5) and very waxing inflowrate (= 2.5). The case = 2 is not shown since the correction is null. The smallestcorrection is for waxing inflow rates, with minimum effects for shear-thickening fluids,while for low values of , the corrections are minor for shear-thinning fluids. Thefirst-order correction term may be valid for a limited time, as shown in figure 4(a) forshear-thickening and Newtonian fluids with = 0.05; a divergence from the first-orderapproximation solution appears for t < 10 and t < 20 respectively. Since the criticaltimes are tc 100 and tc 400 for the two cases, we conclude that the limiting factoris the number of terms in the expansion. An extension of the range of validity canbe achieved by increasing the number of these terms (see, e.g., Hogg et al. 2000).

Figure 5 shows the profiles of the current at t= 5 for constant volume (= 0) andconstant inflow rate (= 1). The case = 0, = 0 has a closed-form solution (Cirielloet al. 2016) and is a parabola for a Newtonian fluid (n = 1). The presence of yieldstress reduces the front position and increases the average steepness of the profile,without other significant variations as long as is a small quantity.




5

10

15

20

25

0 20 40 60 80 100 0 20 40 60 80 100

0 20 40 60 80 100 0 20 40 60 80 100

10

20

30

40

20

40

60

80

100

200

300

400

t t

(a) (b)

(c) (d )

FIGURE 4. (Colour online) The effects of the first-order correction on the front position,for a Hele-Shaw cell: (a) = 0.5, (b) = 1, (c) = 1.5, (d) = 2.5. The thick, midand thin curves refer to n= 1.5, n= 1 and n= 0.5 respectively. The continuous, dashedand dot-dashed curves refer to = 0 (power-law fluid), = 0.01 and = 0.05 respectively.The time decreasing curves, in grey, are unphysical. Variables are non-dimensional.

4. Two-dimensional flow in a porous medium

The case of flow through a porous medium requires the formulation of theequivalent Darcys law for an HB fluid, which may be written as (Chevalier et al.2013)

dp= p + 0

(ud

)n, (4.1)

where d is the diameter of grains, p is the pressure gradient, p is the yieldstress, 0 is the consistency index, n is the flow behaviour index, u is the Darcianvelocity, and and are coefficients. The coefficient is governed by the maximumwidth of the widest path of the flowing current while the coefficient depends onpore size distribution and structure. Their values should, in general, be determined




0.2

0.4

0.6

0.8

1

2

3

10 2 3 4 5

10 2 3 4 5

H

H

x

(a)

(b)

FIGURE 5. (Colour online) The effects of the first-order correction on the current profilesat time t= 5, for a Hele-Shaw cell: (a) = 0, (b) = 1. The thick, mid and thin curvesrefer to n= 1.5, n= 1 and n= 0.5 respectively. The continuous and dashed curves referto = 0 (power-law fluid) and = 0.01 respectively. Variables are non-dimensional.

experimentally, and theoretically they are related to the distribution of the secondinvariant of the strain tensor (Chevalier et al. 2014). Here, a pragmatic approach isadopted, and the values reported in Chevalier et al. (2013), = 5.5 and = 85, areused; the diameter of the glass beads employed in our experiments falls in the rangeadopted in their experiments (from 0.26 to 2 mm). Equation (4.1) indicates that theflow is possible only if |p/x| > p/d; otherwise a plug is formed. Indeed, theexperiments indicate that percolation takes place even at a very low pressure gradient,with a progressive increment of the flow rate for increasing pressure drop. Chevalieret al. (2014) have shown that even at very low Darcian velocity values, the regionof fluid at rest is negligible and the velocity density distribution is similar to thatobtained for a Newtonian fluid.

Under the relevant shallow water approximation, the pressure gradient becomesp/x = 1 g (h/x); inverting (4.1), and under the constraint |h/x| > p, theaverage velocity is equal to

u=sgn(hx

)d(n+1)/n

(1 g850

)1/n (1 p

hx1)1/n hx

1/n , (4.2)the Cambridge Core terms of use, available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/jfm.2017.234Downloaded from https:/www.cambridge.org/core. Universita' di Parma. Dipartimento di Filologia Classica e Medievale, on 17 May 2017 at 13:58:03, subject to



where p = 5.5p/(d1 g). Insertion of the average velocity in the local massconservation yields for h/x< 0

ht=

d(n+1)/n

(1 g850

)1/n

x

h hx1/n(

1 p

hx1)1/n , (4.3)

where is the porosity. Assuming again a current with time variable volume, theintegral mass conservation reads

Ly

0h(x, t) dx=Qt. (4.4)

Upon introducing the velocity and length scales

u =d(n+1)/n

(1 g850

)1/n, (4.5)

x =d(n+1)/n

(Q

Lyd(2n+2)/n

)1/(2) (8501 g

)/[n(2)], (4.6)

with the resulting time scale

t =x

u=

(Q

Lyd(2n+2)/n

)1/(2) (8501 g

)2/[n(2)], (4.7)

equations (4.3)(4.4) may be written in non-dimensional form as

ht=

x

h hx1/n(

1 p

hx1)1/n , (4.8)

0h(x, t) dx= t. (4.9)

As for the flow in the fracture, equations (4.8)(4.9) admit a self-similar solution onlyfor = 2, which breaks down the scales in (4.6)(4.7). By defining an arbitrary timescale t, the new velocity scale u = (Q/Ly)1/2 and the coefficient

p =

(Lyd(2n+2)/n

Q

)1/2 (1 g850

)1/n, (4.10)

which is the ratio between the velocity scale (4.5) and the new velocity scale, (4.3)may be written in non-dimensional form (the tilde is dropped) as

ht=p

x

h hx1/n(

1 p

hx1)1/n , (4.11)

while (4.4) becomes

0h(x, t) dx= t2. (4.12)




The self-similar solution is again h= tf (), = x/t, and substitution into (4.11) gives

f f =p

[f |f |1/n

(1

p

|f |

)1/n]. (4.13)

The system of equations (4.11)(4.12) admits a solution f () = Ap(ep ), whichupon substitution yields

2Ap= 2p(Ap p)

2/n, Ap2ep = 2. (4.14a,b)

For given values of p and p, it is possible to solve the first equation numerically inthe unknown Ap and then to compute ep. The condition for flow requires that AP>p.The general case 6= 2 can be treated with an expansion similar to that adopted forthe flow in a Hele-Shaw cell.

There are many conceptual and formal similarities between the equations arisingin the description of Hele-Shaw and 2D porous flows of an HB fluid, while the mainpoint of difference is the treatment of the plug region. While in a Hele-Shaw flow thepresence of a plug region is an explicit part of the model (possibly with wall slip), fora porous flow the model predicts the cessation of the flow below a threshold withinthe entire body of the granular medium. During the experiments described in thefollowing, it was noted that below a threshold value of the pressure gradient (h/x inour approximation), a percolation develops and the flow never completely stops. Thisbehaviour suggests that a biviscous model, able to smooth the transition from pre- topostyield behaviour, could better interpret the experiments.

5. The experimentsIn order to validate the theoretical model, two series of experiments were conducted

(i) in a Hele-Shaw cell with a small gap, simulating a fracture, and (ii) in the samecell with a larger gap and filled with glass beads of uniform size, reproducing a 2Dporous medium. Figure 6 shows the experimental device and two different snapshots.The 75 cm long cell was made of two parallel plates of transparent plastic, the gapwidth between which could be varied as necessary. In order to limit slip for theexperiments without glass beads, a commercial transparent antislip tape was used toline the inside of the plates. Fluid was injected with a syringe pump for the fracturetests, and with a vane pump controlled by an inverter for the porous flow tests,requiring higher flow rates.

The HB fluids were obtained by mixing deionized water, Carbopol 980 (0.05 %0.14 %) and ink, with subsequent neutralization by adding NaOH. The mass densitywas measured with a hydrometer (STV3500/23, Salmoiraghi) or a pycnometer, withan accuracy of 1 %.

The rheologic behaviour of the fluid was analysed in a parallel-plate rheometer(dynamic shear rheometer, Anton Paar Physica MCR 101), conducting severaldifferent tests, both static and dynamic, to evaluate the fluid behaviour index, theconsistency index and the yield stress. When dealing with non-Newtonian fluids, eitherOstwaldde Waele (power law) or HB, the characterization of the correct parametersrepresenting the rheological behaviour of the fluid can be challenging. However, thefinal aim of the experiments is clear, and that is to verify the proposed model by usingindependent measurements of both the flow field characteristics (front position and




Front view Side view

Photocamera

18 c

m

75 cm

Levellingsupport

Injection

Lock

To the pump

(a) (b)

(c) (d )

FIGURE 6. (Colour online) A sketch of the experimental rectangular channel. (a) Frontview, (b) side view, (c) a snapshot of the channel during experiment B1 (the shaded areais the advancing current) and (d) a snapshot of the channel filled with glass beads duringexperiment A1 (the dark area is the advancing current and the grey area is the porousmedium not yet reached by the current).

current thickness over time) and the fluid rheometric parameters. It is noteworthy thatthe models used to describe the fluid rheology and obtain the rheometric parametersdo not have a general validity but are an approximation of the fluid behaviour in alimited shear range. Furthermore, none of the flow fields generated in the measuringinstruments (mainly rheometers) are perfectly viscometric. The problem of correctlyestimating the rheometric parameters is particularly significant for HB fluids, forwhich yield stress estimation (and definition) is far from trivial. See the review paperby Nguyen & Boger (1992) for a discussion on the topic and a description of themethodology adopted in our laboratory experiments. Therefore, in order to evaluatethe accuracy of the measurements of the yield stress, we carried out numerousadditional experiments with different methods, broadly classified as direct andindirect methods; see the further description in appendix A.

The glass beads forming the porous medium had nominal diameters of 3, 4 and 5mm 0.1 mm, depending on the test. The profile of the current was detected witheither a stills camera (Canon EOS 3D, 3456 2304 pixels) operating at a rate of0.5 frames s1 or a high-resolution video camera (Canon Legria, 1920 1080 pixels)operating at 25 frames s1. The images were then postprocessed with a proprietarysoftware package in order to be referenced to a laboratory coordinate system andfor the boundary between the (dark) intruding current and the empty cell or the(light) porous medium to be extracted and parameterized. With this set-up, the overallaccuracy in detecting the profile of the current was approximately 1 mm, while theuncertainty in measuring timings was negligible. During all experiments, the naturalpacking prevented any movement of the beads, as demonstrated by the images usedfor extracting the fluid interface.

Tables 1 and 2 list the parameters for the two sets of experiments (fracture and 2Dporous medium) and four series with increasing concentration of Carbopol (labelledfrom A to D, corresponding to 0.05 %, 0.08 %, 0.10 % and 0.14 % concentrationrespectively). Figure 7(a) shows the dimensionless front position of the currents versus




0 200 400 600 800 1000

1000

200

400

600

800

1

2

3

4

0 0.2 0.4 0.6 0.8 1.0

Exp.B1B2B3B4

C1C2C3C4C5

D1D2D3D4

t

(a) (b)

FIGURE 7. (Colour online) A comparison of theory with experiments for the fracture. (a)The front position xN/e as a function of dimensionless time t for all tests. The three boldlines represent the perfect agreement with theory for the three different fluids used in theexperiments. (b) The dimensionless profile of the current at different times for experimentB3. The error bars refer to the profile at t = 66 s and correspond to one STD, thebold straight line indicates the theoretical profiles and the dashed lines are the confidencelimits of the model. For clarity, in both (a) and (b), one point of every three is plotted.

time for 13 tests conducted in the fracture with three different HB fluids. The plottingvariables have been chosen to collapse all of the experimental data into a single line.In order to separate the results for the three different fluids, the experimental frontposition was multiplied by 0.5 and 1.5 for experiments A and C respectively. Theuncertainty in the model and experimental data was computed following the sameprocedure as reported in Di Federico et al. (2014), and is represented by the dashedlines and error bars corresponding to plus or minus one standard deviation (STD) forthe data. The front propagation is generally linear for all tests, with some discrepancyat early times due to the effects of the injection geometry and the poor adherence ofthe experiments to model assumptions. We recall that the solution is an intermediateasymptotic to the general solution, and is valid for times and distances from theboundaries large enough to forget the details of the boundary (or initial) conditionsbut far enough from the ultimate asymptotic state of the system. Hence, we expectthat also for longer time the experimental results will deviate from the self-similarsolution.

Figure 7(b) shows the shape of the current at different times for experiment B3.The normalized profiles show a fairly good collapse to a single curve, even thoughthe discrepancy between experiments and theory becomes evident for /e< 0.30 andnear the front. This is due to the disturbances at the inlet (the inflow is located nearthe bottom in the experiments, while it is assumed to be evenly distributed along thevertical in the theory), to non-negligible vertical velocities near the inlet (see Longo& Di Federico 2014) and to the bottom stress, which becomes relevant at the tip ofthe current. The experimental profiles are similar for all other tests.

Figure 8(a) shows the front position for the six experiments conducted in a porousmedium with three different HB fluids. The profile was corrected for capillaryeffects following the procedure outlined in Longo et al. (2013b). The front velocity




Exp

erim

ent

QL y

n

0 p

x N

exp

x N

theo

rFl

uid

(cm

3s

2 )(c

m)

(Pa

sn)

(Pa)

(kg

m

3 )(

10

3 )(c

ms

1 )(c

ms

1 )

B1

0.01

20.

320.

680.

90.

310

0019

0.55

0.54

Car

bopo

l0.

08%

B2

0.01

20.

480.

680.

90.

310

0013

0.53

0.54

B3

0.02

40.

480.

680.

90.

310

0013

0.71

0.71

B4

0.02

40.

480.

680.

90.

310

0013

0.73

0.71

C1

0.00

60.

480.

662.

50.

610

0030

0.29

0.28

Car

bopo

l0.

10%

C2

0.01

20.

480.

662.

50.

610

0030

0.37

0.37

C3

0.01

20.

480.

662.

50.

610

0030

0.38

0.36

C4

0.02

40.

480.

662.

50.

610

0030

0.46

0.48

C5

0.02

40.

480.

662.

50.

610

0030

0.53

0.48

D1

0.00

60.

480.

453.

115

.010

0063

70.

160.

18C

arbo

pol

0.14

%D

20.

012

0.48

0.45

3.1

15.0

1000

637

0.22

0.22

D3

0.01

20.

480.

453.

115

.010

0063

70.

230.

23D

40.

024

0.48

0.45

3.1

15.0

1000

637

0.31

0.30

TAB

LE

1.Pa

ram

eter

sfo

rth

eex

peri

men

tsin

the

frac

ture

.T

hein

ject

edvo

lum

esc

ales

with

t2.




Exp

erim

ent

Q

dL y

n

0 p

px N

exp

x N

theo

rFl

uid

(cm

3s

2 )(c

m)

(cm

)(P

asn)

(Pa)

(kg

m

3 )(

10

3 )(c

ms

1 )(c

ms

1 )

A1

0.10

02

0.3

30.

700.

20.

110

0019

0.76

0.72

Car

bopo

l0.

05%

A2

0.20

02

0.4

40.

700.

20.

110

0014

1.06

1.00

B5

0.02

52

0.5

50.

680.

90.

310

0034

0.23

0.23

Car

bopo

l0.

08%

C6

0.02

02

0.4

40.

602.

50.

610

0084

0.17

0.16

Car

bopo

l0.

10%

C7

0.03

02

0.5

50.

602.

50.

610

0067

0.17

0.17

C8

0.03

02

0.4

30.

602.

50.

610

0084

0.17

0.17

A3

16.0

1.0

0.3

30.

700.

20.

110

0018

Car

bopo

l0.

05%

A4

26.0

0.6

0.4

40.

700.

20.

110

0014

B6

4.0

1.0

0.5

50.

680.

90.

310

0030

Car

bopo

l0.

08%

B7

30.0

0.6

0.5

50.

680.

90.

310

0030

TAB

LE

2.Pa

ram

eter

sof

the

expe

rim

ents

ina

poro

usm

ediu

m,

with

volu

me

t.

The

expe

rim

enta

lan

dth

eore

tical

fron

tsp

eed

isno

tco

nsta

ntfo

rth

ela

stfo

urex

peri

men

ts.




0 200 400 600 800 1000

1000

200

400

600

800

0 0.2 0.4 0.6 0.8 1.0

1

2

3

4Exp.A1A2

B5

C6C7C8

t

(a) (b)

FIGURE 8. (Colour online) A comparison of theory with experiments for 2D porousflow (rectangular channel filled with glass beads). For caption, see figure 7. (b) Refersto experiment A2.

shows a fairly good agreement with the theoretical constant velocity, even thoughasymptotically there are larger discrepancies due to the geometry of the current, witha very thin nose affected by the bottom boundary effects. The experimental profilesof the current at different times are shown in figure 8(b) for experiment A2. Theprofiles collapse to a single curve, which is significantly affected by the disturbancesat the inlet. The experimental points are within the confidence limits of the theoreticalmodels and show a clear linearity for > 0.5ep. In order to also check the modelfor the asymptotic solution with 6= 2, some experiments were designed for constant( = 1) and waning ( = 0.6) inflow rate, with two different HB fluids and threedifferent bead diameters (see the last four experiments in table 2). Figure 9 shows theexperimental front position (symbols) and the first-order expansion of the self-similarsolution (solid curves). The overlap in each case is satisfactory, in particular forexperiments at constant inflow rate.

6. Conclusions

A general model was developed for gravity-driven flows of HB fluids in a narrowfracture or 2D porous medium, extending existing formulations for Newtonian andpower-law fluids. For the special case of an inflow rate linearly time-increasing, aself-similar solution was derived for both cases: the front of the current advances atconstant speed and with a linear profile. For the general case of a power-law inflowrate, an expansion of the self-similar solution valid for an Ostwaldde Waele fluidwas developed, in the limit 1 or p 1, with these two parameters markingthe deviation from pure power-law behaviour. The expansion has a validity controlledby the value of ; hence, it is valid within a time interval limited by a critical timevalue beyond which either the approximation in deriving the differential equation orthe adopted expansion become invalid.

The rheological parameters of the HB fluids have been measured with independenttests, with two different rheometers and with both direct and indirect methods for theyield stress. The results for the yield stress vary according to the different methods




0 20 40 60 80 100 120

12

B6 1.0

0.6B7A3

A44

8

16

20

Test

t

FIGURE 9. (Colour online) A comparison of theory (solid curves) with experiments(symbols) for the 2D porous medium, showing xN as a function of dimensionless timet for all tests. The injected volume scales with t . The experimental parameters are listedin table 2.

adopted. However, a single value of the yield stress (the lowest in the list of measuredvalues) has proven to correctly characterize all of the experiments performed with thesame fluid in the Hele-Shaw cell and in the porous medium.

The experiments were mostly conducted with volume scaling in time as = 2, withsome additional experiments conducted with = 0.6, 1. Each test shows reasonableagreement with the theory for flow in fractures and porous media, especially so forthe former. Deviations from the linear profile forecast for = 2 occur near the inlet.In the 2D porous medium, a dome develops near the inlet, followed by a profile ingood agreement with the theory. In all cases, the theoretical prediction lies withinthe confidence limits of the experimental data. These experimental results provide averification of the flow model for HB fluids in fractures and also a further verificationof the extended Darcy law proposed by Chevalier et al. (2013) for HB fluid flow inporous media. However, the existence of non-zero Darcy flow below the thresholdpressure gradient requires further fundamental experimental investigation.

An additional promising area of research suggested by our work is the interactionof rheologically complex fluids (described, e.g., by the HB model) with spatialheterogeneity in key problem parameters, using either a deterministic or a stochasticapproach. For fracture flow, heterogeneity may be represented by a spatially variableaperture (Lavrov 2013), or by obstructions, local contractions and expansions (Hewittet al. 2016). For porous medium flow, spatial variability of permeability may bemodelled using deterministic trends or directly as a random field, using approachesand methods typical of stochastic hydrology. We are currently working on theseproblems and plan to report on our efforts in the near future.

Appendix A. Rheometry of the fluidsMost measurements were conducted with a parallel-plate rheometer (dynamic

shear rheometer, Anton Paar Physica MCR 101, equipped with a moving planeplate of diameter 25 mm, with the gap equal to 1 mm in most cases) and with astrain-controlled rheometer (coaxial cylinders, Haake RT 10 RotoVisco, equipped withcup and rotor according to DIN 53019, with internal radius equal to 19.36 mm andexternal radius equal to 21 mm). In order to limit the slip, the surfaces of the cupand the rotor were roughened with strips of Sellotape, and the surfaces of the plates




100101102103104 102101 100101102 101106 105

12

0

4

8

16

20

0

2

4

6(a) (b)

FIGURE 10. (Colour online) (a) Experimental shear-stress shear-rate curves for the threefluids Carbopol 980 0.08, 0.10 and 0.14 %, measured with the coaxial-cylinder rheometer.(b) Experimental shear-stress shear-rate curves for Carbopol 980 0.10 %, measured withthe parallel-plate rheometer. The curves are the HB model interpolation for the reducedseries of experimental points, represented by the filled symbols.

were roughened with sandpaper P-60 glued onto the smooth surfaces (see Carotenuto& Minale 2013, for an in-depth analysis of the effects of sandpaper on rheologicalmeasurements of Newtonian fluids). Additionally, in order to prevent absorption, thesandpaper was painted with transparent acrylic.

Figure 10(a) shows the classical stress strain-rate experimental results obtained withthe coaxial-cylinder rheometer; figure 10(b) shows similar results for the 0.10 % fluidobtained with the parallel-plate rheometer. The continuous curves represent the HBmodel interpolation for the reduced series of experimental points, visualized with filledsymbols. The reduction of the sample is required partly due to the limits of accuracyof the instruments at very low shear rate and partly as a consequence of the pooradaptation of the model to the real rheological behaviour of the fluid. This is thesimplest and most common procedure to estimate the yield stress, but the resultsdepend on the cutoff value of the shear rate used to select the reduced sample. Forthe present data, p= 1.1, 2.9, 7 Pa for measurements in the coaxial-cylinder rheometerand for fluids with increasing concentration of Carbopol. For measurements taken withthe parallel-plate rheometer, the yield stress for the mixture with 0.10 % of Carbopol980 is equal to 1.3 Pa. This method of estimation of the yield stress is an indirectone.

A direct method is based on stress relaxation. The fluid is sheared, reaching aspecific value of strain, then shearing stops and the shear stress required to guaranteethe reached strain is recorded. Its asymptotic value is taken to be equal to the yieldstress.

Figure 11(a) shows the results for various strains in the 0.10 % Carbopol 980fluid. The asymptotic value lies in the range 0.61.0 Pa and is a function of theimposed strain; it decays for increasing strain since the degree of disturbances in thefluid controls the late behaviour of the system. Figure 11(b) shows the asymptoticshear stress for imposed shear rate. After an initial growth, the shear stress reaches




0.4

0

0.8

1.2

1.6

2.0(a)

0.4

0

0.8

1.2

1.6

2.0(b)

100 10310210150 100 150 200 250

t (s)

t (s)

t (s)

FIGURE 11. (Colour online) Carbopol 980 0.10 %. (a) Asymptotic stress at constant strain;(b) asymptotic stress at constant strain rate. The insets show the procedure adopted intesting.

102

103

104

101

100

101

100

0 20 40 60 80 100 0 4 8 12 16 20

(a) (b)

t (s)

t (s) t (s)

FIGURE 12. (Colour online) Creeprecovery method for the Carbopol 980 0.10 %. (a)Straintime curves and (b) corresponding apparent viscosity. The yield stress lies between = 1 Pa and = 1.5 Pa. Measurements were conducted with the parallel-plate rheometer;the inset shows the procedure adopted for testing.

a plateau, up to a limiting value that makes the transient state extremely long. Thestress corresponding to the minimum plateau reached is assumed as an upper limitof the yield stress. For the 0.10 % Carbopol 980 fluid, min p = 1.15 Pa.

Another method is called creep and recovery. A constant shear stress is applied insteps and the creep is observed. If a more or less complete recovery is reached, thenthe applied stress was below the yield stress and the continuum behaved as an elasticsolid. If a limited or null recovery is reached, then the applied stress was above theyield stress.

Figure 12(a) shows straintime curves for Carbopol 980 0.10 %, with stressimposed for 20 s (except for the curve relative to = 1 Pa) and a recovery for30 s. Figure 12(b) shows the corresponding time evolution of apparent viscosity. The




0

0.02

0.02

0.04

0.06

0.08

0.10

0

0.002

0.004

0.006

0.008

(a)

(d )

0 20 40 60 80 100 120 140 0 50 100 150 200 300250

0

0.02

0.02

0.04

0.06

0.08

0.10

0.02

0.01

0

0.03(c)

0

0.02

0.02

0.04

0.06

0.08

0.10

0.02

0.01

0

0.03(b)

0 20 40 60 80 100 120

t (s) t (s)

FIGURE 13. (Colour online) (a) A photo of the inclined plane and of the accessories usedfor a direct measurement of the yield stress. The electronic level and the video camera arein the same frame of the fluid layer. (bd) The plots of the free surface velocity versustime for (b) a mixture of Carbopol 980 (0.08 % neutralized), (c) a mixture of Carbopol980 (0.10 % neutralized) and (d) a mixture of Carbopol 980 (0.14 % neutralized). Thesolid lines indicate the fitted free surface velocity; the vertical solid line indicates theassumed start of flow motion. The secondary horizontal axis indicates the angle withrespect to the horizontal; the secondary vertical axis indicates the average shear rateobtained by dividing the free surface velocity and the starting thickness of the layer,neglecting its reduction in time. The thickness of the layer was set to 0.3, 0.3 and 1 cmfor Carbopol 980 0.08, 0.10 and 0.14 % respectively.

residual strain is equal to 28, 32, 54, 85 % for = 0.2, 0.5, 1, 1.5 Pa respectively,and the yield stress is assumed to be in the range [11.5] Pa. The estimation isaffected by a significant uncertainty, because in the present experiments the residualdeformation is monotonic with the imposed stress and no abrupt increase is observed.For other fluids, the behaviour is much sharper (Magnin & Piau 1987).

The yield stress was also measured with a direct method based on the static stabilityof a layer of fluid on an inclined plane (Uhlherr et al. 1984), by adopting the samedevice and experimental technique as detailed in Longo et al. (2016). The yield stresscan be evaluated by assuming that in the incipient motion the following balance holds:

p = gh sin c, (A 1)

where h is the thickness of the layer and c is the critical angle.The incipient motion of the free surface, as detected with a particle image velocity

algorithm applied to the images recorded by a video camera, is assumed as anindicator of the sliding. Figure 13 shows, for three different yield stress fluids,the experimental apparatus and the plots of the free surface for increasing bottom




101

100

101

102 0

1

2

3

10 2 3 2 4 6 8 10

(a) (b)

FIGURE 14. (Colour online) (a) Evolution of the storage modulus G (red dashed lines)and the loss modulus G (grey continuous lines) for the 0.10 % Carbopol 980 mixture asa function of stress for different frequencies. The symbols are the intersection betweenthe moduli (cross-over), conventionally representing the flow point. (b) Cross-over stressas a function of the frequency. Measurements with parallel-plate rheometer, gap 1 mm,T = 298 K.

inclination. The dots represent the average velocity of the free surface, with dispersiondue to vibrations and to noise in the images. It is possible to detect a kink, enhancedby separately interpolating the premotion and the postmotion experimental data withtwo different straight lines. However, a creep motion is detected near the criticalangle, presumably due to the elastic deformation of the layer of material beforeflowing.

The overall uncertainty, computed as

dpp=

d+

dhh+

dctan c

, (A 2)

has a significant contribution due to the uncertainty in the thickness of the layer h.By assuming that the critical angle is detected within the uncertainty of the electroniclevel (0.1), and assuming an uncertainty in the thickness measurements equal to0.3 cm, the estimated yield stress values are p = 1.0 0.1 Pa, p = 1.2 0.2 Pa,p = 26 1 Pa for Carbopol 980 0.08, 0.10, 0.14 % respectively. However, the trueuncertainty is larger than our estimate, due to the intrinsic uncertainty in the definitionof the critical angle.

Another method to estimate the yield stress is based on the dynamic behaviour atsmall deformation, as detected with oscillatory shear tests, with the measurement ofthe storage modulus G() and the loss modulus G(), representative of the elasticand the viscous behaviour of the continuum respectively. Applying a sinusoidal strain = 0 sin t with small amplitude 0 and frequency , the time varying stress = 0 sin( + t) is measured, where is a phase shift. The complex modulusG() / = G() + iG() has the two components G and G, representativeof elasticity (perfect if G = 0, = 0) and viscosity (perfect if G = 0, = 90)respectively. The yield stress can be estimated as the intercept of the tangent of thestorage modulus considering the domains before and after the cross-over (De Graefet al. 2011). Figure 14(a) shows the moduli measured at different frequencies for




101

100

101

102

103100101

FIGURE 15. (Colour online) Sweep experiments with Carbopol 980 0.10 % at a frequencyof = 0.1 Hz. The yield stress is evaluated as the abscissa of the intersection of the twostraight lines interpolating the storage modulus G on the left and right sides of the cross-over.

Method See: p(Pa)

Extrapolation rheom. 1 Figure 10(a) 2.9Extrapolation rheom. 2 Figure 10(b) 1.3Stress relaxation Figure 11(a) 0.61(constant strain)Asymptotic stress Figure 11(b)


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Gravity-driven flow of HerschelBulkley fluid in a fracture and in a 2D porous mediumIntroductionModel description for flow in a narrow fractureSelf-similar solutionAsymptotic analysis for =2

Two-dimensional flow in a porous mediumThe experimentsConclusionsAppendix A. Rheometry of the fluidsReferences

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