ISSN 10283358, Doklady Physics, 2010, Vol. 55, No. 4, pp. 199–202. © Pleiades Publishing, Ltd., 2010. Published in Russian in Doklady Akademii Nauk, 2010, Vol. 431, No. 6, pp. 758–761.
MECHANICS
TwoContinua Model of Suspension Flow in a Hydraulic Fracture1 S. A. Boronin and A. A. Osiptsov Presented by Academician A.S. Koroteev November 11, 2009 Received November 11, 2009
DOI: 10.1134/S1028335810040117
FORMULATION OF THE PROBLEM
velocity vp. The carrier phase is chacaterized by the 0
The aim of the present work is to construct an asymptotic twocontinua model of a suspension flow in a vertical hydraulic fracture with account for a finite particle volume fraction and the particle slip. The study of this phenomenon is important as there is a need to construct mathematical models for the imple mentation into commercial program packages, which are used in numerical modeling of suspension flows in a fracture with application to the technology of hydraulic fracturing of an oil reservoir [1]. Existing models of particle transport in a fracture [2–4] were formulated within the effectivefluid approximation, where the suspension is assumed to be viscous incom pressible fluid with the density and viscosity depend ing on the particle volume concentration. We consider a threedimensional unsteady laminar suspension flow in a vertical hydraulic fracture subject to gravity acceleration g. The carrier phase is a viscous 0 incompressible Newtonian fluid with the density ρ f and the viscosity μ0. The dispersed phase consists of identical noncolloidal spherical particles of the radius σ 0 with the substance density ρ p and the mass of a single particle m. The fracture is assumed to be a vertical channel with smooth walls and a variable width w. We introduce the Cartesian coordinate system Oxyz, with the axes x and y, directed horizontally and vertically, and the axis z directed normally to the fracture mid dleplane. Unit vectors of the coordinate system are denoted e1, e2, e3. The suspension is considered as a combination of two interacting and interpenetrating continua: the medium of particles and the carrier phase. The medium of particles is characterized by the volume concentration C, the number concentra 0 tion np, the density ρp = C ρ p and the massaveraged
1The
article was translated by the authors. Schlumberger Moscow Research Institute of Mechanics, Lomonosov Moscow State University
averaged density ρf = (1 – C) ρ f and the massaver aged velocity vf. Within the twocontinua approach [5], the differ ential equations of the mass and momentum conserva tion laws for each phase are written in the dimensional form: ∂ρ ∂ρ (1) f + ∇ ( ρ f v f ) = 0, p + ∇ ( ρ p v p ) = 0, ∂t ∂t df vf ij ρ f = – ∇p f + ∇ j τ f e i + ρ f g – n p F p , dt dp vp ij ρ p = – ∇p p + ∇ j τ p e i + ρ p g + n p F p , dt
(2)
(3)
dv ∂v ∂v df vf = f + ( v f ∇ )v f , pp = p + ( v p ∇ )v p . dt ∂t dt ∂t Boundary conditions in the dimensional form are written as: x = 0: v f = v f0 ( t, y, z ), C = C 0 ( t, y, z ), w z = ± : u f = v f = 0, 2
1 ∂w w f = ± ± v l , 2 ∂t
(4)
w p = 0. At the fracture inlet, the fluid velocity vf0 and the particle concentration C0 are specified. On the fracture walls, we specify the normal fluid velocity vl and the nooutflow condition for particles. It is assumed that the chaotic velocity of settling particles is small, hence the stress tensor in the particulate medium can be ij neglected: pp = 0, τ p = 0 [6]. Due to a finite particle volume concentration and the nonzero particle slip, the densities of the particulate medium, the carrier phase, and the suspension are not constant, and hence the massaveraged velocities of the phases and the mixture are divergent. We will assume that the stress tensor of the carrier phase can be specified as for a vis cous compressible fluid with the viscosity depending on the particle volume concentration [6]:
199
200
BORONIN, OSIPTSOV ij ij 1 ij τ f = 2μ 0 μ ( C ) ⎛ e – δ ∇v f⎞ , ⎝ ⎠ 3
C ⎞ –1.89 , μ ( C ) = ⎛ 1 – ⎝ C max⎠
3
∂w w ( = ∇ ∇P + Bu [ 1 + C ( η – 1 ) ]e 2 ) ∂t 12μ ( C )
μ ( 0 ) = 1,
C max = 0.65.
The dependence of the viscosity on the particle vol ume concentration is determined by the Scott formula [7]. In the expression for the interphase force, we include the Stokes force with a correction due to a finite particle volume concentration f(C) and the Archimedes force: 1 – 4 πσ 3 ρ 0 g. F p = 6πσμ 0 ( v f – v p ) f f(C) 3 DERIVATION OF ASYMPTOTIC EQUATIONS Using the characteristic scales of the velocity U, the length (height) L and width d of the fracture, from Eqs. (1)–(3) we obtain the following equations in a nondimensional form: ∇ [ ( 1 – C )v f + Cv p ] = 0, ∂C + ∇ ( Cv p ) = 0, (5) ∂t df vf dp vp –1 ε Re ( 1 – C ) + ηC dt dt ij
= – ∇p f + ∇ j τ f e i – Bu 0 [ 1 + C ( η – 1 ) ]e 2 ,
(6)
dv 1 – St ⎛ η – 1⎞ , εSt pp = ( v f – v p ) e3 f ( C ) Fr 2 ⎝ η ⎠ dt
(7)
d ε = , L
mU St = , 6πσμ 0 d
0
U Fr = , gd
0
ρ f Ud Re = , μ0
ρ Re η = p0 , Bu 0 = . 2 2 ρf ε Fr In the first Eq. (5), the expression in parentheses is the volumeaveraged suspension velocity. We use the lubrication approximation and consider the following asymptotic limit: ε 1, ∇w 1, St ∼ 1, Fr ∼ 1, (8) Re ∼ 1, η ∼ 1. We introduce the new functions averaged across the fracture: w/2
H ( t, x, y ) = 1 2
∫
h ( t, x, y, z ) dz.
– w/2
As in the existing models, it is assumed that the crossfracture particle concentration profile is uni form. From Eq. (5)–(7), in the asymptotic limit Eq. (8) we obtain the following twodimensional aver aged equations: ∂wC (9) + ∇ ( wCV p ) = 0, ∂t
– wCV s – 2v l ,
(10)
w V f = – ( ∇P + Bu [ 1 + C ( η – 1 ) ]e 2 ), 12μ ( C ) Vp = Vf + Vs ,
(11)
2
St η – 1 V s = – 2 ⎛ ⎞ f ( C )e 2 , Fr ⎝ η ⎠ C ⎞ α , f ( C ) = ⎛ 1 – ⎝ C max⎠
Re . Bu = 2 Fr For solving Eq. (9), we prescribe the following initial and boundary conditions for concentration (4): t = 0: C = 0,
L ( x, y ) ∈ 0, × [ 0, 1 ]; H
x = 0: C = C 0 , y ∈ [ y 1, y 2 ]. Boundary conditions for Eq. (10) follow from (4): ∂p = – 12μ ( C) , y ∈ [ y , y ]; x = 0: 1 2 2 ∂x w ∂p = 0, y ∈ [ 0, y 1 ], [ y 2, 1 ], ∂x L ∂p x = : = – Bu; H ∂y ∂p y = 0, 1: = – Bu ( 1 + C y = 0, 1 ( η, –1 ) ). ∂y Particle settling velocity is determined by an empirical formula, which accounts for the decrease in the rate of settling with the increase in the particle volume con centration. Existing effectivefluid models contain an assumption that the volumeaveraged suspension velocity is governed by the Poiseuille law, whereas in the present work within the twocontinua approach on the basis of the conservation laws it is shown that the Poiseuille law governs the massaveraged velocity of the carrier phase Eq. (11). In addition, in earlier mod els the expression for the particle velocity Eq. (11) contained the volumeaveraged suspension velocity, instead of the massaveraged fluid velocity. As a result, in contrast to the existing effectivefluid models, the two continua model contains an additional term –∇(wCVs) on the righthand side of the equation for pressure Eq. (10). This term accounts for twospeed effects. NUMERICAL RESULTS In order to estimate the twospeed effects on the transport and settling of particles in a hydraulic frac ture, the system of Eqs. (9), (10) is solved numerically on a uniform rectangular mesh. The solution of elliptic DOKLADY PHYSICS
Vol. 55
No. 4
2010
TWOCONTINUA MODEL OF SUSPENSION FLOW y 1.0
(a) C
201
y 1.0
(a)
0.6 0.5 0.4
0.5
0.5
0.3
0.2 0.1 0
(b) 1.0
1.0
0.5
0.5
0
0.5
1.0 x
Fig. 1. Particle concentration distributions in a fracture of elliptic vertical crosssection at t = 0.5, obtained within the effectivefluid (a) or twocontinua (b) models for Bu = 3260.
Eq. (10) is obtained using a finitedifference method with the second order approximation. The system of linear algebraic equations with the fivediagonal matrix obtained as a result of discretization is solved using the conjugated gradient method. For solving transport Eq. (9), we use a TVD scheme with the first order approximation in time and the second order approximation in spatial coordinates [8]. Calculations were conducted on the computational domain with the aspect ratio L/H = 5 for two types of hydraulic fractures: with a rectangular or elliptic vertical cross section for vl = 0, α = 5, y1 = 0, y2 = 1, σ = 1 , η = 3, d 6 C0= 0.2. In order to verify the accuracy of the results, a number of calculations was conducted on a series of refined meshes of the initial mesh. The final results were obtained on a mesh providing the required accu racy of calculations. A parametric study is conducted. In Fig. 1, we present typical particle concentration distributions at the end of a certain time period since the injection DOKLADY PHYSICS
Vol. 55
No. 4
2010
0
(b)
0.5
1.0
x
Fig. 2. Position of the particle concentration front in a fracture of elliptic vertical crosssection at t = 0.5 for Bu = 326 (a) and Bu = 3260 (b). Results are obtained using the effectivefluid (dashed line) and twocontinua (solid line) models.
started. These results are obtained within the effective fluid model and within the twocontinua model pro posed in the present work. For calculations within the effectivefluid model we used Eqs. (9), (10) without additional terms due to twocontinua effects. It is shown that the ratio of the additional term to the buoyancy number Bu in (10) is proportional to ξ = 2 ⎛σ ⎞ (η – 1). For the fixed ξ, the increase in Bu leads ⎝ d⎠ to the increase in the twospeed effect on the concen tration distribution in a fracture (Fig. 2). For Bu > 103, the difference between the calculations using the effectivefluid model and the twocontinua model reaches the value of 15% or higher (Fig. 2b). We also conducted a preliminary numerical study of twocontinua effects of suspension transport on hydraulic fracture propagation using a commercial software package. It is established that twocontinua effects have only a slight impact on the law of fracture propagation in the range of parameters considered.
202
BORONIN, OSIPTSOV
However, it is recommended to conduct a more detailed investigation of this coupled problem in the future. CONCLUSIONS A twocontinua model of a suspension flow in a vertical hydraulic fracture is formulated. As compared with existing models formulated within the effective fluid approximation, the twocontinua equations include the additional differential term, which con tains the particle volume fraction and the settling velocity. A series of numerical calculations is con ducted in order to estimate the effect of the additional term on particle transport and settling. It is shown that the difference between the suspension flow model, constructed earlier within the heuristic singlespeed approach, and the twocontinua model, proposed in the present work based on the conservation laws of the mechanics of multiphase media, is significant, how ever it has only a slight effect on the law of fracture propagation, as it follows from the preliminary numer ical study of a coupled problem. In the future, it would be of interest to take into account polydispersity of suspension and interactions between particles, to con duct a comparison of the model proposed with exper iments, and also to carry out a more detailed study of the coupled problem of hydraulic fracture propagation and of particle transport in a fracture within the two continua model of suspension flow.
REFERENCES 1. M. J. Economides and K. G. Nolte, Reservoir Stimula tion (John Wiley and Sons, New York, 2000). 2. J. R. A. Pearson, “On Suspension Transport in a Frac ture: Framework for a Global Model,” J. NonNewto nian Fluid Mech. 54, 503–513 (1994). 3. P. S. Hammond, “Settling and Slumping in a Newto nian Slurry, and Implications for Proppant Placement During Hydraulic Fracturing of Gas Wells,” Chem. Eng. Sci. 50 (20), 3247–3260 (1995). 4. S. J. McCaffery, L. Elliott, and D. B. Ingham, “Enhanced Sedimentation in Inclined Fracture Channels, Topics in Engineering,” (CDROM ISBN 1853125466, 1997), Vol. 32. 5. R. I. Nigmatulin, Dynamics of Multiphase Media (Hemi sphere Publ., 1991), Vols. 1, 2. 6. S. A. Boronin, “Investigation of the Stability of a PlaneChannel Suspension Flow with Account for Finite Particle Volume Fraction,” Fluid Dyn. 43 (6), 873–884 (2009). 7. D. G. Thomas, “Transport Characteristics of Suspen sions: VIII. A Note on Viscosity of Newtonian Suspen sions of Uniform Spherical Particles,” J. Colloid. Sci. 20, 267–277 (1965). 8. C. Hirsch, Numerical Computation of External and Internal Flows, 2nd ed. The Fundamentals of Compu tational Fluid Dynamics, Elsevier (ISBN: 97807506 65940, 2007).
DOKLADY PHYSICS
Vol. 55
No. 4
2010
