The rheological behavior of energized fluids and foams with application to hydraulic fracturing: Review

Accepted Manuscript The rheological behavior of energized fluids and foams with application to hydraulic fracturing: Review Salah Aldin Faroughi, Antoine Jean-Claude Jacques Pruvot, James McAndrew PII:

S0920-4105(17)31008-2

DOI:

10.1016/j.petrol.2017.12.051

Reference:

PETROL 4541

To appear in:

Journal of Petroleum Science and Engineering

Received Date: 3 July 2017 Revised Date:

5 December 2017

Accepted Date: 15 December 2017

Please cite this article as: Faroughi, S.A., Jean-Claude Jacques Pruvot, A., McAndrew, J., The rheological behavior of energized fluids and foams with application to hydraulic fracturing: Review, Journal of Petroleum Science and Engineering (2018), doi: 10.1016/j.petrol.2017.12.051. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

ACCEPTED MANUSCRIPT

The Rheological Behavior of Energized Fluids and Foams with Application to Hydraulic Fracturing: Review Salah Aldin Faroughia , Antoine Jean-Claude Jacques Pruvota , James McAndrewa Liquide Delaware Research and Technology Center, Newark, DE, USA

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Abstract

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The use of aqueous energized fluids and foams for hydraulic fracturing reduces water consumption and formation damage, while improving proppant transport and placement. Thus, coupling an improvement in well productivity with a reduction in environmental impact appears possible. However, several advances in scientific understanding of these complex fluids, e.g., fracture-induced shear thinning, elasticity and osmotic effects, have not been fully integrated into the engineering practice of well stimulation. These properties may lead to an optimized fracturing fluid that can be tuned to satisfy both sides of the stimulation design spectrum better fracturing and efficient proppant transport. This review aims to progress in that direction by reviewing recent advancements in combination with earlier and more engineering-oriented works, and to connect the complex rheology and mechanics of energized fluids and foams with practical usage in hydraulic fracturing.

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1. Introduction

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In the last decade, resources such as shale gas and tight oil 36 have become extremely important thanks to the successful de- 37 velopment of hydraulic fracturing [209]. The resulting increase 38 in the use of natural gas has allowed the United States econ- 39 omy to prosper, even while reducing its carbon footprint. De- 40 spite these positive aspects, hydraulic fracturing is controver- 41 sial and widely threatened by regulations, to the point where it 42 has been banned in several areas. The reasons for this include 43 the consumption of large amounts of water and the risks inher- 44 ent in its disposal. The productivity of hydraulically fractured 45 wells also declines very quickly, and is usually much less than 46 would be expected considering the resources in the reservoir. 47 Thus, there is a strong motivation to improve the productivity 48 and lifetime of fractured wells, while reducing the quantity of 49 water and other chemicals consumed in the fracturing process. 50 Foam fracturing fluids are promising in both respects, as a 51 significant fraction of water is replaced by nitrogen or carbon 52 dioxide. The use of foam fracturing fluids is well-established 53 in the small percentage of important reservoirs that are under- 54 pressured, where the quick and easy clean-up offered by foams 55 is sufficient to justify the effort, specialized equipment and ex- 56 pertise required. Their wider use is hampered by logistical chal- 57 lenges, mainly because modern hydraulic fracturing is increas- 58 ingly applied in long horizontal wells with 20-40 stages which 59 require delivery of several hundred truckloads of liquid nitrogen 60 over a few days, or even higher volumes in the case of carbon 61 dioxide (because of the higher density of carbon dioxide under 62 downhole conditions). However, these challenges are manage- 63 able and foams can provide productivity benefits beyond under- 64 pressured reservoirs, mainly due to improved proppant trans- 65 port and reduced water damage, see e.g., [167, 139]. In this 66

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review, we will discuss the basis for these benefits, and particularly how recent scientific insights support their implementation. In hydraulic fracturing, pressurized fluids are used to fracture impermeable rock and transport proppant into the fracture to prevent fracture closure after the pressure release. Proppant placement within the fracture directly impacts productivity, because it controls both short-term and long-term conductivities of the fractured well. To prevent fracture closure and enhance the conductivity of wells, different fracturing fluids and proppant types have been used, see review papers by Al-Muntasheri [3] and Liang et al. [120], respectively. The main types of fluids are: (i) low viscosity fluids (e.g. slickwater, which is water with a drag-reducing additive at a few percent by weight) that are efficient in creating long fractures in the flow direction, but not in proppant transport due to a high rate of proppant segregation [166], (ii) gels and polymer-based fluids that lead to fractures that propagate less in the flow direction, more in the direction perpendicular to the flow, and transport proppant well, but may damage the fracture surface [74], (iii) energized fluids (bubbly liquids) and foams that have a high effective viscosity and therefore behave more like gels, but avoid damage to the fracture surface and reduce water consumption [167, 231], and (iv) a mixture of the aforementioned fluids using methods like hybrid [207], reverse hybrid [123], multi-stage [133], alternateslug [131] fracturing. In under-pressurized geologic plays, energized fluids (see definition in section 2) and foams serve better than other fluids in all steps of the treatment, starting from drilling [127, 107, 52], fracturing [167], proppant transport [226, 41, 76, 241], and finally deep well cleaning [68]. In terms of geography, the use of energized fluids and foams is most prevalent in Canada and in severely water-stressed regions of the U.S. (mainly in New Mexico). Multiple studies, see e.g., Burke et al. [20], Reynolds

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• High bubble deformation in narrow fractures and near the 153 fracture tip, which will reduce the apparent viscosity and 154 facilitate fracture growth [64] . 155

• Osmotic pressure, or the tendency of foams to retain wa- 156 ter driven by interfacial energies. Thanks to this effect, 157 leakoff of water from foams is much lower than would be 158 expected based on the volume fraction of water present. 159 Lower leakoff from foams has been observed in the labo- 160 ratory but its origin in osmotic pressure was not explained 161 162 [183].

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• Elastic behavior of foams which can improve proppant 164 transport without necessarily requiring an increase in ap- 165 parent viscosity [74, 75, 76]. 166 167

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highlight some areas where work is ongoing and more understanding is needed. 2. Regimes of liquid-gas mixture A fluid-fluid mixture is a system where two or more fluids with different physical properties are mixed, but they are not combined chemically (i.e. they are immiscible). In these systems, one fluid is continuous, called the ambient phase, and the rest are suspended, called dispersed phase(s). Depending on the nature of the dispersed phase(s), the fluid is classified either as an emulsion (liquid) or a foam (gas). In the petroleum literature, an additional distinction is often made within the latter category between mixtures with low volume fraction of gas dispersed, called ”energized fluids”, while the label ”foam” is restricted to high gas volume fraction systems. The term ”energized fluid” is somewhat confusing, especially to those not in the petroleum industry. Its origin is in the use of such fluids to provide ready flowback from the well once pressure is released, even in an underpressured reservoir [156] - the fluid is said to have ”energy” because it flows back by itself, without requiring reservoir pressure or a pump. Of course, this benefit applies to foams also, so that strictly speaking ”energized fluid” should be a generic term that includes foams. However, in practice it is reserved for fluids where the gas volume fraction is too low to provide a sharp increase in apparent viscosity, similar to what is usually called a ”bubbly liquid” in the foam physics literature. We will follow the petroleum literature terminology here. In a stress-free state, the dispersed phase typically forms spherical bubbles (energized fluids and foams) or droplets (emulsions) inside the continuous fluid. Hereafter, we refer to the dispersed phase as ”bubbles” in all cases. In our case of interest in this review, a gas (typically nitrogen, N2 or carbon dioxide, CO2 ) is dispersed inside an aqueous phase forming an energized fluid or a foam. Figure 1 shows a schematic two-dimensional representation of a two-phase gas-liquid mixture and the classification based on the bubble volume fraction (i.e. dispersed phase volume fraction, ψ). This parameter is known as ”quality” in the petroleum industry. Different categories based on volume fraction can be identified at this point. Within the energized fluid category, at a very low volume fraction known as dilute regime, 0 ≤ ψ ≤ 0.1, bubbles are few and far between in the ambient fluid (see Fig. 1). Collisions between bubbles are almost nonexistent, and the assumption of no hydrodynamic interactions between bubbles is valid [56, 7, 18, 121]. The effective rheological behavior of dilute energized fluids is a weak linear function of the bubble volume fraction following Einstein [56], Taylor [219], Mackenzie [128], Frankel & Acrivos [66] when ψ → 0, and a weak non-linear function following the work of Saito [200], Mendoza [142], Faroughi & Huber [64] in the entire range of 0 ≤ ψ ≤ 0.1. With the increase of dispersed phase volume fraction, the disturbance of the background flow streamlines due to presence of bubbles increases leading to a weak hydrodynamic interaction between bubbles (see Fig. 1 where these interactions are symbolized with dashed arrows). This regime is known as

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et al. [181], have been published discussing the performance of 121 foam fracturing fluids in the Montney and Cardium (in Canada). 122 Again, these reservoirs are largely under-pressured. A few studies, mainly from the 1980’s- 1990’s, have discussed foam frac123 turing applications in more highly-pressured reservoirs. For example, Harris et al. [81] discussed results from the Red Fork 124 formation, which is at a depth of 3,900 m and highly over- 125 pressured. A larger study of over 85 wells was carried out in 126 strata above the Haynesville [232], with depths to below 4,300 127 m and pressures as high as 910 bar, some of the greatest depths 128 ever recorded for foam fracturing. 129 Polymeric and polymer-free foams (or energized fluids) are 130 viscoelastic materials whose mechanical and rheological respon-131 ses to transmitted stresses fall between elastic solids and vis- 132 cous fluids [242, 75]. The mechanical and rheological responses133 and flow patterns of such fluids are strongly affected by the 134 stress conditions (e.g. applied shear rates), flow geometry (e.g. 135 pipe or fractures size), time-scale (e.g. aging and drainage), mi- 136 crostructures (e.g. bubble size distribution and volume fraction, 137 packing configuration, surfactant molecules, and other chemi- 138 cal additive), and state physical parameters (e.g. operating tem- 139 perature and pressure). 140 During the initial development of foam fracturing (1970s to 141 1990s), multiple studies (e.g. [81, 174]) described their rheol- 142 ogy from a more or less practical point-of-view and with par- 143 ticular application to fracturing. Since that time, fundamental 144 understanding of the complex rheological behavior of foams 145 has made considerable progress through many studies, covering 146 aqueous foams [89, 213, 43, 60], non-aqueous and polymeric 147 foams [12], polymer-free foams [89, 78], and particle-laden sta- 148 bilized foams [58]. In this review we will point out relatively 149 new fundamental concepts that are particularly relevant to the 150 use of foams in fracturing, notably the following: 151

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This review is organized as follows. In Section 2, we ex168 plain the differences between energized fluids and foams, and 169 explain the effect of the dispersed phase volume fraction on 170 the properties of each. In sections 3 and 4, we present deeper 171 reviews of the properties of energized fluids and foams, respec172 tively. Section 5 discusses how these concepts are applied in 173 fracturing. Next, in section 6, we conclude with a discussion of 174 the potential for broader application of foams in fracturing and 2

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portant difference between energized fluids and foams happens while passing the wet limit (see Fig. 1), when the shape of bubbles changes from spherical to polyhedral [227, 65] due to geometrical constraints. Thereafter, the energy at the interface between bubbles increases above the minimum surface energy. The interface energy includes the chemical energy of the substances inside bubbles and thin films (made by ambient fluid), and the energy associated with surface tension [5, 115]. The wet limit is defined as the state where the maximum possible volume of dispersed phase presents in fully spherical shapes that are closely packed whether hexagonally or randomly, thus 0.64 < ψt ≤ 0.74. Note that the mentioned delineation for wet limit (or the threshold packing) is only valid for monomodal dispersion of bubbles, and as modality increases, i.e. broader bubble size distribution, this range also shifts to higher volume fraction values [47]. Any addition of the dispersed phase volume fraction beyond the wet limit results in osmotic pressure (i.e. tendency to take up ambient phase into the bubble configuration) [239, 98] and distortion of the sphericity of bubbles to polyhedra [5, 35]. The change in the surface area of interfaces comes with an energy cost, because interfaces are in a higher energy state (i.e. molecules are in a thermodynamically unfavorable state) due to surface tension. Further increase of the volume of dispersed phase (to create foam) requires further expenditure of energy which is not tolerable for a pure continuous fluid that needs to be transformed to a network of thin films (see Fig. 1) with extremely large surface area. One way to overcome this issue and form stable foams beyond the wet limit is to add an impurity (e.g. additional surfactants) to the continuous phase. Surfactant molecules accumulate at interfaces which is an energetically favorable place for them [5]. The positioning of surfactant then reduces the surface tension (or surface energy) to some extent that improves the stability of films possessing high superficial area. Ultimately, even in the presence of enough surfactant, increasing the volume fraction of bubbles decreases the amount of fluid available to sustain the films network, and the foam finally collapses. The onset of collapsing is referred to as dry limit beyond which the phase inversion occurs, i.e. the dispersed gas phase becomes continuous, and the liquid phase forms suspended droplets. The region beyond the dry limit is known as mist [179, 89].

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semi-dilute regime where 0.1 ≤ ψ ≤ 0.2 [31, 71, 118]. When 231 bubbles start to interact, mediated by the ambient fluid, the rate 232 of viscous dissipation may increase depending on the shear con- 233 dition and bubble deformation. These interactions thus invali- 234 date the linear relationship between the relative viscosity and 235 bubble volume fraction [158, 191, 121]. 236 For regimes with bubble volume fraction of 0.2 ≤ ψ ≤ 0.4, 237 known as intermediate regimes, collisions between bubbles be- 238 come possible, and strong hydrodynamic interactions develop 239 between bubbles. The force balance in this regime is still dom- 240 inated by the shear and surface tension stresses at the interface 241 of mixture constituents. Many experimental studies show the 242 presence of slight shear thinning behavior (reduction in relative 243 viscosity with increasing shear rate) due to the shear-induced 244 microstructural rearrangement. At a given shear condition, the 245 effect of bubble volume fraction on the relative viscosity be- 246 comes more pronounced within this regime [49, 134]. 247 Beyond 40% volume fraction of bubbles, the rheological 248 and mechanical behavior of energized fluid becomes very com- 249 plex [57, 162, 163, 164, 125, 40, 64]. This degree of complexity 250 arises due to high frequency of bubble-bubble interactions as 251 well as the complex many-body hydrodynamic interactions oc- 252 curring between bubbles. This regime (0.4 ≤ ψ ≤ ψt where ψt 253 is a threshold bubble volume fraction at which the maximum 254 possible volume of the dispersed phase is made of spherical 255 bubbles) is known as the concentrated regime, where bubbles 256 are placed close and form common boundaries (i.e. made of 257 films of continuous phase) together. For suspensions of solid 258 spherical particles, ψt is known as the amorphous maximally 259 random jammed packing at which the transition between fluid- 260 like and solid-like states occurs, and generally found to vary 261 in 0.58 < ψt ≤ 0.637 for a monomodal particle size distribu- 262 tion [203, 210, 17]. For any mixture, the value of the threshold 263 packing volume fraction depends strongly on the state of stress, 264 bubble size distribution and the packing configuration, i.e. the 265 presence or absence of randomness, and thus in energized flu- 266 ids with spherical monomodal particles, ψt may vary between 267 0.52 (corresponding to cubic lattice packing) to 0.74 (corre- 268 sponding to the face centered cubic or hexagonal close pack- 269 ing). We note that the value of ψt increases as modality (e.g. 270 from monomodal to bimodal) increases. The interactions be- 271 tween bubbles in this regime are both direct and indirect, which 272 changes the rate by which momentum is exchanged between 273 constituents of the mixture. Other phenomena like lubrication and the gradient of surface tension caused by uneven distribu274 tion of surfactant concentration also affect the bulk properties of concentrated energized fluids. The relative viscosity in this 275 regime is a nonlinear function of bubble volume fraction with 276 a sharp increase close to ψt . The shear-induced microstructural 277 rearrangements occurring before the threshold volume fraction 278 affect relative viscosity to a great extent [228, 90, 119, 230]. 279 Figure 1 schematically shows the transition of monomodal bub280 ble configuration that occurs in the concentrated regime. 281 For bubble volume fraction larger than that of the thresh282 old packing ψ > ψt , the terminology to describe the mixture 283 changes from ”energized fluids” to ”foams”. The most im-

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3. Energized fluids 3.1. Physical description The presence of a cloud of gas bubbles in a liquid dramatically changes the transmission of stress by the bulk ambient fluid, and varies the rate at which the constituents of the mixture exchange momentum. According to the response of such fluids to external forces like shear force, their behavior may generally be categorized as Newtonian or non-Newtonian. For Newtonian fluids, the relation between shear stress and shear rate is linear, and the constant of proportionality is the shear dynamic viscosity. For non-Newtonian fluids showing shear

µr =

µy µm

= 1+

1 y 1 + 2.5l 140(l3 + l2 l 1) ( ) · + kCa2 , 2 2 1 + kCa 1 y 1+l 28(2l + 3)(l + 1)

where

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S.A. Faroughi et al.

k=

(2l + 3)(19l + 16) 40(l + 1)

(8

2

,

Energized Fluids

Foam

(9

Mist

that for the case of a very dilute energized fluids ( y ! 0) with non-deformable gas bubbles (Ca ! 0 and l ! 0) reduces to Semi-Dilute the linear viscosity Dilute model proposed by TaylorIntermediate (1932), which Concentrated reads as µr =

µy = 1 + y. µm

(10

10%

µr

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20%

N

40%

~63-74%

◆1 N ✓ ◆ NW 2 2 ( M 1) + M kCa2 (µBubble volume fraction, y r) = 1 W 1 y N + M kCa2

~95-99%

1

,

(11

Figure 1: A 2D schematic to represent the different types of energized fluids and foams and the routinely used classification based on gas volume fraction. Here the dispersed phase is considered to be monomodal.

1 yt 1 + 2.5l 140(l3 + l2 l 1) W = , N = , M = . rates, T is the absolute temthinning, shear thickening, and finite normal stress differences where γ˙ denotes the applied shear 2 y 1 + l 28(2l + 3)(l + 1) t [126, 243], the relation between the shear stress and shear rate perature and K is the Boltzmann constant. At low Re num-

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At higher dispersed phase concentration the viscosity model is no longer a linear (or a weak-linear) function, and be comes a highly non-linear function of capillary number, viscosity ratio and also the volume fraction of particles. Following the work of Faroughi & Huber (2014), the extension of Eq. (8) to concentrated emulsions using the fixed volume DEM theory becomes,

317 318

B

(12

is nonlinear, which makes it difficult to define the shear dy- 319 ber (Re << 1) and high Peclet number, (Pe  103 ), the flow Equation (11) has been successfully validated with experimental data over a wide range of particle volume fraction 288 namic viscosity. In these fluids the4 shear dynamic 320 dynamics of energized fluids is mainly controlled by external 4 viscosity (0 <289y < ytincrease ), capillary number < Ca < 10 andshear viscosity between constituent phases (see Faroughi may or decrease with(10 the applied shear rates) (or 321 bodyratio forces, bubble-bubble forces, and hydrodynamic interac-& Hube (2014) for detailed information on model For energized fluidsbubbles, (l ! 0), (11) reduces 290 stress) showing shear thickening or shearverification). thinning behaviors, 322 tions between andEq. Brownian motion isto vanishing. In291 respectively. In almost all practical applications, energized flu- 323 deed, at low bubble Stokes number (Stk  1) defined as, 1 ✓ ◆ response 292 ids exhibit non-Newtonian behaviors, and mostly to2 ! shear45 12 2tend Bubble W time ρb 1 Ca (µ ) y r = = Rb ςRe, (3) 293 thin. The rheological behavior ofµenergized 5fluids is heteroge- 324= 1 Stk W , (13 r Fluid response time ρ m 12 internal 2 294 neous microscopically, and this is rooted1in their mi1 y Ca 5 295 crostructure (e.g. bubble size, spatial and orientation distri- 325 the phase segregation to study the rheological response can be 296 bution) evolution under different shear conditions. For exam- 326 neglected. In Eq. 3, ρb is bubble density and ς is a shape factor. 297 ple, when flowing through a pipe where the shear stress field 327 This assumption assures that bubbles follow the bulk fluid flow 298 is inhomogeneous, the viscosity varies spatially. It has been 328 streamlines. Non-interacting gas bubbles may deform due to shearing. 299 shown that the macroscopic rheology of such fluids depends 329 330 The spherical shape, which possesses the minimum surface en300 strongly on the dispersed phase volume fraction, shape, state 331 ergy, is deformed by a surface stress proportional to the viscous 301 of dispersion, shear history and finally the rate of deformation 332 stress, µm γ˙ . The stress that acts on the surface of bubble to re302 [31, 191, 212, 40, 64, 218]. Due to the non-Newtonian nature of 333 store its spherical shape is proportional to the surface tension 303 energized fluids, the coefficient of viscosity is measured at spe334 between constituents, σRb , where σ represents the surface ten304 cific values of the shear rate to represent the so-called apparent 335 sion. One can derive a dimensionless number between these 305 or effective (non-Newtonian) viscosity. 336 stresses acting on the surface of bubbles to determine the equi306 To characterize the dynamics of these fluids, several dimen337 librium shape of bubbles, the so-called capillary number which 307 sionless numbers can be defined that carry important informa338 is defined as, 308 tion about the fluid’s microstructures, (e.g. bubble size), shear 309 conditions, thermal effect and the relative density between conViscous forces µm γ˙ Rb Ca = = . (4) 310 stituent phases. One may define the bubble Reynolds number 339 Surface tension forces σ 311 as, 340 As the capillary number increases, bubbles start to deform. Inertial Forces ρm Rb 2 γ˙ 341 At early stage of deformation, i.e. at low and intermediate Re = 312 = , (1) 342 capillary number, the isotropy of emulsions is retained. HowViscous Forces µm 343 ever, at high capillary numbers, all bubbles are deformed and 313 where ρm and µm are the ambient fluid density and shear vis- 344 aligned with the imposed shear direction, which results in grow314 cosity, respectively, and Rb is the bubble radius. The other im- 345 ing anisotropic microstructures in the fluid [30, 134, 40]. De315 portant dimensionless number is the Peclet number, 346 formation of a bubble to the second order in a shear flow is 3 347 characterized by Hydrodynamic Forces 6πµm Rb γ˙ 316 Pe = = , (2) Brownian Force KB T 348 r − Rb (1 + f (˙γ, r)DR + f 0 (˙γ, r)D2R ) = 0 (5)

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where r is the magnitude of the position vector from the center 399 of a bubble to the interface between bubble at the ambient fluid, 400 and f and f 0 are complex functions that depend on the shear 401 rate tensor and position vector from the center of bubbles (see 402 Faroughi & Huber [64] and Greco [72] for more details about 403 these functions). In Eq. 5, DR characterizes the deformation 404 of bubbles which can be obtained as (considering the viscosity 405 ratio between constituents, λ = µd /µm is very small, λ → 0), 406 DR =

407

Rmax − Rmin ' Ca Rmax + Rmin

(6) 408 409

where Rmax and Rmin are respectively the maximum and mini- 410 mum axis lengths of an ellipsoid obtained from a spherical bub- 411 412 ble under deformation. 413

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(τm + Nτb )u · ndA =

Z

429

(τψ )u · ndA

(7)

= 1+

430

where τm , τb and τψ are respectively the ambient fluid, bubble and equivalent fluid stress tensors, n is a outward unit vector 431 normal to the surface of bubble and u is the fluid flow at large distances from the bubble center. The deviatoric stress tensor 432 for the energized fluid (left-hand-side of Eq. (7)) consists of the stress associated with ambient fluid and the stress associated with the disturbance due to the presence of a bubble summed 433 over the number of bubbles, N. The latter part of the stress tensor may include different orders of bubble deformation. The 434 zeroth order of deformation (Ca  1) was calculated by Tay- 435 lor [219], and the first and the second order of deformation was 436 437 estimated in later studies by several authors [30, 201, 72].

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414 3.2. Effective viscosity of energized fluids It is common practice to treat an energized fluid as a ho- 415 mogeneous Newtonian fluid under specific shearing conditions 416 with corresponding effective viscosity. The viscosity of the ho- 417 mogenized or equivalent fluid is generally expected to be higher 418 than that of the ambient fluid as the rate of energy dissipation 419 per unit volume increases. However, if the surface tension be- 420 tween constituents of the energized fluids is not high enough to 421 restore the spherical shape of bubbles under deformation, the 422 effective viscosity may fall below than the viscosity of the am- 423 424 bient fluid [165, 212, 134, 191, 64]. The ”homogenization” process assumes the equivalence of 425 the work performed by the deviatoric component of the stress 426 tensor (symbolized by τ) on the boundary of the actual ener- 427 gized fluids and the homogenized fluid. 428 Thus, the homogenization process reads as,

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found that the relative viscosity becomes a quadratic function of the bubble volume fraction. At higher volume fractions of bubbles, the strong hydrodynamic and inter-bubble (e.g. lubrication, collision and friction) interactions need to be considered within the stress disturbance function. Theoretical approaches are unable to describe these highly complex many-body interactions, and thus, experiments [163, 164, 191, 212, 148] and numerical studies [104, 243, 223, 230] are utilized to constrain the highly non-linear relative viscosity of intermediate and concentrated energized fluids. Phenomenological approaches have also been used widely to characterize concentrated systems. See Table 1 for a non-exhaustive list of well-known relative viscosity models for concentrated energized fluids and their range of applicability based on the volume fraction of dispersed phase as well as deformation parameters. Faroughi & Huber [64] recently proposed a generalized non-linear viscosity model using the differential effective medium (DEM) theory [154]. This model satisfactorily predicts the nonNewtonian relative viscosity from a dilute system to onset of foam formation while considering the effect of capillary number up to the second order of bubble deformation, i.e., Ca2 . Note that the effective non-Newtonian viscosity for fluids with time-dependent microstructure is defined at an equilibrium condition where a dynamic force balance is present. This leads to a state where microstructural rearrangements or evolution occur mutually over time to keep isotropy. Faroughi & Huber [64] showed that for a dilute emulsions with finite viscosity ratio between constituents, the relative non-Newtonian viscosity, µr , becomes, µψ µr = µm

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3.2.1. Dilute Energized fluids The main assumption embedded in Eq. (7) (i.e. the linear summation of the effect of bubbles on the bulk stress field) is 439 strictly valid for dilute regimes where bubbles do not interact at 440 all. The macroscopic relative viscosity of energized fluids pre- 441 dicted using this assumption is linearly proportional to the bub- 442 ble volume fraction, especially when ψ → 0. For semi-dilute 443 regimes, the non-linear effect of bubble perturbation of the bulk stress field was considered by Batchelor & Green [7], who 5

 1 ψ 1 + 2.5λ ( ) × 1 + κCa2 1 − ψ 1+λ  3 2 140(λ + λ − λ − 1) 2 κCa , + 28(2λ + 3)(λ + 1)2

(8)

where κ=



(2λ + 3)(19λ + 16) 40(λ + 1)

2

,

(9)

that for the case of a very dilute energized fluids ( ψ → 0) with non-deformable gas bubbles (Ca → 0 and λ → 0) reduces to the linear viscosity model proposed by Taylor [219], which reads as µψ = 1 + ψ. (10) µr = µm 3.2.2. Concentrated Energized fluids At higher dispersed phase concentration the viscosity model is no longer a linear (or a weak-linear) function, and becomes a highly non-linear function of capillary number, viscosity ratio and also the volume fraction of bubbles. Following the work

n further (e.g.

shear direction. Therefore, the upper bound is a function of particle shape. For spherical 454

451

452

particles it is shown to be around 20%, and it becomes smaller as particles become more elongated.

453

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cles), the particle spacing is such that interparticle forces, friction at contacts, lubrication

455

and viscous forces in fluid films between particles, and microstructure arrangement be-

459

they are designed to describe the same phenomena, highlighting the complexity of suspen-

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sion dynamics. A common feature, for a suspension of rigid particles, is that the e↵ective

461

viscosity should theoretically satisfy the following asymptotic behavior,

!

t

µ ' 1, µm

(20)

Bube↵ective shear modulus of suspenwhich is equivalent to the trivial upper bound for the ble V 0 olum e Fra ctiona transition r 463 sions, see Eq.be(17). In Eq. (20), t is the threshold packing fraction at which m u N y r fluid-like and solid-like states occurs, i.e., where the suspension becomes rheolog464 between illa 0 ap C 465 ically locked. 462

466

0.74 t 3.4.1. Threshold packing,

Relative Viscosity

Relative Viscosity

458

Enerof concentrated suspensions. Various upscaling come dominant and control the rheology gize d Flu ids methods have been proposed to approximate interparticle and hydrodynamic interactions Foam s in concentrated suspensions. The strategy behind the proposed models di↵ers even though

lim

ion at contact and viscous forces in fluid film s between particles, and micr ostructure arra 456 come dominant and control the rheology of concentrated susp ensions. Vario 457 methods have been Petroleum Science and prop Engineering (2017) XXX-XXX osed to apprXXX oximate interparticle and hydr odynamic 458 in concentrated suspensions. The strategy behind the prop osed models di↵ers 459 they are designed to describe the same phenomena, highlight ing the complexit Energized Fluids Foams 460 sion dynamics. A common featu re, for a suspension of rigid part icles, is that 461 viscosity should theoretically satisfy the following asymptot ic behavior, µ lim ' 1, ! t µm

ACCEPTED MANUSCRIPT > 0.2 for sphericalJournal partiof

particle volume fraction increases even further (e.g. S.A. FaroughiAsettheal.

457

462

463

0

464

465

0

466

which is equivalent to the trivi al upper bound for the e↵ective shear modulus sions, see Eq. (17). In Eq. (20) ? , t is the threshold packing fraction at which a between fluid-like and solid-like states occurs, i.e., where the suspension becom ically locked. ? 3.4.1. Threshold packing , 0.74 t

1

Bubble Volume Fraction

SC

1

(a)

(b)

448

449

450

451 452 453 454 455 456

457

458

459

TE D

447

EP

446

of Faroughi & Huber [64], the extension of Eq. (8) to concen- 460 for the case of Ca → 0 (zero deformation), and trated emulsions using the fixed volume DEM theory becomes,   5 Ω−1 3 ψ N 461 µ = 1 − Ω . (15) 1 r   ( −1) 1−ψ N + M κCa2 (µr )2 2 M µr N + M κCa2 462 for flow conditions where Ca  1 (bubbles are highly deformed  −N Ω−1 463 and oriented towards the shear direction). ψ = 1−Ω , (11) 464 Recently, Gu & Mohanty [78] conducted an experimental 1−ψ 465 study on the rheology of aqueous energized fluids and foams 466 and compared their experimental data with the model proposed where 467 by Brouwers [19] which is a simplified version of Eq. (13) in 1 − ψt 1 + 2.5λ 468 the limit of Ca → 0 and λ → 0 (is basically equivalent to Eq. Ω= , N = , ψt 1+λ 469 (14). The model by Brouwers [19] is essentially derived for 140(λ3 + λ2 − λ − 1) 470 solid particle suspensions (λ → ∞). This successful comparison . (12) M= 28(2λ + 3)(λ + 1)2 471 between the measured data and Eq. (13) provides additional 472 support to the use of the generalized model represented in Eq. Equation (11) has been successfully validated with experi473 (13) to characterize the rheology of energized fluids. mental data over a wide range of bubble volume fraction (0 < 474 A schematic representation of Eq. (13) in 3D (relative visψ < ψt ), capillary number (10−4 < Ca < 104 ) and viscosity ra475 cosity as function of space variables such as capillary numtio between constituent phases (see Faroughi & Huber [64] for 476 ber and bubble volume fraction) and 2D (relative viscosity as detailed information on model validation). For energized fluids 477 function of bubble volume fraction at different values of capilwhere λ → 0, Eqs. (11) and (12) reduces to, 478 lary number) to predict the relative non-Newtonian viscosity is !− 4  479 shown in Fig. 2(a,b). These figures also show the presence of a −1  5 −Ω 2 2 1 − 12 ψ 5 Ca (µr ) 480 critical capillary number, Cacr , on which the relative viscosity = 1−Ω , (13) µr 2 1 − ψ 1 − 12 Ca 481 is a null function of particle volume fraction, i.e., particles are 5 482 deformed such the equivalent viscosity of the medium is equal and further reduces to 483 to that of the ambient fluid, µr = 1 (see the red curve in Fig. 2).  −Ω−1 484 Crtical cappilary number can be obtained using Eq. (13), which ψ µr = 1 − Ω , (14) 485 is found to be Cacr = 0.645 for energized fluids (λ → 0). Note 1−ψ 486 that Eq. (13) is valid when 0 < ψ < ψt , and the curve beyond

AC C

445

M AN U

Figure 2: Schematic viscosity ofD Renergized fluids and solid lines show the behavior D R foams. D R A F T representation March of 9, the 2016,relative 6:26pm A F T A F T In panel (a), theMarc h 9, 2016, 6:26pm of energized fluids (ψ < ψt ) at several different capillary numbers as predicted by Eq. (13)). The dashed line shows that of foams (ψ > ψt ) at very small capillary numbers only. The dot-dashed line follows ψt as a function of capillary number. Panel (b) shows the projection of the same curves onto the 2D plane of relative viscosity versus bubble volume fraction. Insets show the state of bubble configuration and deformation associated with each curve (i.e. capillary number). One observes that at higher capillary number bubbles are severely deformed, and the shear viscosity of the mixture could be even less than that of the ambient phase due to high shear localization within bubbles. Question marks in panel (b) show the lack of well-defined models and datasets (to the best of authors’ knowledge) for mixtures with ψ > ψt , i.e. foams sheared at intermediate to high capillary numbers.

444

> 0.2 for sp

that interparticle forces, frict

455

3.4. Rheology of concentrated suspensions

456

cles), the particle spacing is such

RI PT

450

6

D

ACCEPTED MANUSCRIPT Journal of Petroleum Science and Engineering XXX (2017) XXX-XXX

488

489 490 491 492

that (the dashed blue curve in Fig. 2(a,b)) represents the foam 540 relative viscosity that will be discussed in section 4. 541 542 3.3. General rheological behavior of energized fluids 543 The general rheological behavior of energized fluids (λ → 544 0), according to Eq. (13) and Fig. 2(a,b), can be summarized 545 for the following scenarios: 546

493 494 495 496 497 498 499 500 501 502 503

• At very small capillary number: When the capillary num- 547 ber is Ca < 0.1, surface tension and capillary stress are 548 sufficient to resist deformation. The bubbles thus remain 549 spherical due to high surface resistance, and the relative 550 viscosity increases with volume fraction of bubbles (i.e. 551 the overall rate of energy dissipation due to the presence 552 of bubbles increases). The increment is linear at very 553 small volume fraction and then becomes non-linear at 554 higher volume fraction up to ψt , see the black curve in 555 Fig. 2(a,b). Equation (14) best describes this regime for 556 557 energized fluids. 558

508 509 510 511 512 513 514 515

516 517 518 519 520 521 522 523 524 525 526 527

528 529 530 531 532 533 534 535 536 537 538 539

M AN U

507

TE D

506

EP

505

• At high capillary number: At a very large capillary number (Ca > 1), bubbles become severely deformed and act like shear banding zones where most of the strain localizes (see the schematic inset in Fig. 2(b) on the purple curve). In this region, the relative viscosity, depends mainly on the viscosity ratio, λ, between constituent phases in emulsions, because the resisting force restoring the deformation is mostly controlled by shear stresses within the dispersed phase. At small viscosity ratios (λ → 0, like energized fluids where gas bubbles are dispersed in a viscous liquid), the restoring force is small, i.e. less energy is dissipated, and thus energized fluids show a smaller effective viscosity than that of the ambient fluid, µr (ψ) < 1, see the cyan to purple curves in Fig. 2(a,b) where an increase in bubble volume fraction cause a larger reduction in the relative viscosity. However, at high viscosity ratios (λ > 1, like droplets of oil dispersed in water) the effective viscosity of the emulsion is greater than the viscosity of the ambient fluids for the entire range of bubble volume fraction, µr (ψ) < 1 and capillary number. Equation (15) best describes the regime where Ca  1, and for a very dilute energized fluids reduces to the well-known linear viscosity model proposed by Mackenzie [128].

• At small to critical capillary number: When capillary 559 number falls between 0.1 < Ca < Cacr , due to an in- 560 crease in the applied shear rate, the network of bubbles 561 tends to rearrange and resist deformation to maintain the 562 minimal surface energy. The overall rate of energy dis- 563 sipation in this regime again increases, but the increment 564 in the relative viscosity is less than that of a very low cap5 illary number (Ca < 0.1), due to a slight deformation and 565 (16) µr = 1 − ψ. rearrangement of bubbles. The shift from black to green 3 curves in Fig. 2(a,b) is an indication of the shear thinning behavior in energized fluids (Ca ∝ γ˙ ), mainly because of 566 An S-shaped curve is observable when looking at the relamicrostructural rearrangement in this regime. 567 tive viscosity - capillary number plane in Fig 2(a) (the schematic 568 dot-dashed line). It includes the upper and lower plateaus where • At critical capillary number: When the capillary number 569 energized fluids behave like Newtonian fluids (i.e. the relative comes close to the critical capillary number (Ca = Cacr ), 570 viscosity does not vary with shear rate or capillary number). an equilibrium force balance between the imposed shear 571 The upper plateau indicates the relative viscosity in the presforce and forces associated with capillary stresses devel- 572 ence of spherical bubbles, and the lower plateau indicates the ops at interfaces such that the overall rate of energy dis- 573 relative viscosity when bubbles are severely deformed. The gap sipation due to the presence of bubbles is the same as 574 between these two plateaus parametrizes the amount of shear the absence of bubbles. Therefore, the relative viscos- 575 thinning, i.e. places where non-Newtonian behaviors become ity becomes independent of bubble volume fraction, but 576 dominant because of microstructural rearrangement and evoonly up to intermediate regimes. See the red curve in 577 lution. The gap size increases as volume fraction increases, Fig. 2(a,b). The critical capillary number can be obtained 578 i.e. the S-shaped curves becomes a straight line at the limit from Eq. (13), and has a value of 0.645 for energized flu- 579 of ψ → 0. In concentrated energized fluids, ψ → ψ , the upt ids λ → 0. 580 per Newtonian plateau may disappear due to the formation of • At intermediate capillary number: In this range of capil- 581 microstructures possessing a considerable strength. This phelary number (Cacr < Ca < 1), the applied shear force on 582 nomenon is called yielding, and when applying a shear stress the surface of bubble is greater than that exerted by the 583 below the yield stress (which is a function of the microstrucsurface tension, and causes a considerable distortion in 584 tural formation) no deformation occurs. the spherical shape of bubbles. This results in a reduction in the overall surface resistances and by extension a 585 4. Foams reduction in the relative viscosity. In this regime, the relWith transition to foams, ψ > ψt , the liquid-gas mixture ative viscosity falls below unity meaning that the energy 586 dissipation in energized fluids is less than that of pure am- 587 becomes more complex. When the volume fraction increases bient fluids. The shear thinning, in this regime, appears 588 beyond the wet limit, see Fig. 1, bubbles lose their spherical due mainly to the bubble deformation that enhances shear 589 shape due to the geometrical constraints and become polyhelocalization. The sensitivity to shear thinning behavior 590 dra. This increment in the surface area between constituents

AC C

504

increases strongly with bubble volume fraction, see the change from red to cyan curves in Fig. 2(a,b).

RI PT

487

SC

S.A. Faroughi et al.

7

ACCEPTED MANUSCRIPT Journal of Petroleum Science and Engineering XXX (2017) XXX-XXX

S.A. Faroughi et al.

Table 1: A non-exhaustive list of well-known viscosity models to predict the relative viscosity of energized fluids. The range of bubble volume fraction, ψ, and capillary number, Ca, over which they intend to be applied is also reported [64].

Equation

Taylor [219] Mackenzie [128]

µψ µm µψ µm

Oldroyd [158]

µψ µm

Frankel & Acrivos [66]

µψ µm µψ µm µψ µm

Llewellin & Manga [125]

µψ µm µψ µm

Faroughi & Huber [64]



1+K1 ( 5 )

2 1− 12 5 Ca 1+( 65 Ca)2 ψ = exp(−k 1−ψ )

= 1+ψ

2 2 1− 12 5 Ca (µr ) 2 Ca 1− 12 5

− 4

µr −Ω−1  ψ 1 − Ω 1−ψ

5

=

)

4.1. Foam structure and morphology

597 598 599 600 601

604 605 606 607 608 609 610 611 612 613 614 615 616 617 618

EP

596

631 632

AC C

594 595

TE D

603

593

Up to 20%

All range

1+ 32 ψ 1− 25 ψ 1−ψ 1+ 35 ψ

Up to 10%

All range

2.5 ≤ k ≤ 4 ( −16 K1 = (1 − ψψt ) 15 ψt

Up to 45%

Ca > 1

Up to ψt

All range

Minimum model

Up to 7%

8

K2 = (1 − ψψt ) 5 ψt

Ω=

602

592

Ca  1 Ca > 1

Maximum model

leads to an increase in the interfacial energy, and the foams be- 619 come more thermodynamically unstable [185, 26, 193]. Be- 620 yond the wet limit, the bubble size distribution naturally be- 621 comes broader, and the mean bubble size becomes larger [91, 622 238]. The broader bubble size distribution alters ψt , shifts wet 623 limit to higher volume fraction, and causes more contacts (small 624 bubbles fill the voids formed by larger bubbles and form com- 625 mon contacts). With this transition, the mixture also experi- 626 ences a higher rate of phase segregation, called gravity-driven 627 drainage, due to large density differences [47]. This section 628 discusses foam structures, stability and responses to different 629 external forces in more detail. 630

591

ψ→0 ψ→0

—

2 1+K1 K2 ( 6Ca 5 ) 2 1+K12 ( 6Ca ) 5



— —   K1 =  K2 =



= (1 − ψψt )−ψt (  (1 − ψ)−1 = 5/3  (1 − ψ) 1 + 9ψ = (1 + 22.4ψ)−1

Deformation

Up to 50% Up to ψt

Ca ≤ 1 Ca > 1 Ca ≤ 1 Ca > 1

Up to ψt

All range

SC

Pal [165]

1− 5 ψ

Concentration

M AN U

Ducamp & Raj [49]

= 1+ψ = 1 − 35 ψ   2 1+ 3 ψ 1+K1 K2 ( 6Ca 5 ) = 52 2 6Ca 2

Specific Parameter

RI PT

Reference

The structure of a foam has a significant effect on the bulk 633 rheological behavior, and thus, controlling bubble size distribu- 634 tion is an important process in any foam-involved treatments. 635 There are a wide range of length scales in a time-dependent 636 foam structure; from macro-scale (bulk skeleton), micro-scale 637 (bubbles size) to nano-scale (thin liquid film and stabilizing 638 agents at interfaces) [61]. In an equilibrium condition and at 639 a given bubble volume fraction, the equilibrium structures of 640 foam have a minimal surface energy. A small perturbation from 641 the equilibrium structure may lead to other equilibrium struc- 642 tures. In general, the known equilibrium structures of foams 643 are (i) wet monodisperse ordered, (ii) dry monodisperse or- 644 dered, (iii) monodisperse disordered or (iv) polydisperse dis- 645 ordered [46, 98, 236]. Wet monodisperse ordered foams are 646 made of bubbles of perfectly sorted size, and possess either 8

1−ψt ψt

face-centered cubic or hexagonal close-packed structure, ψt ≈ 0.74 [233]. Dry monodisperse ordered packs based on bodycentered cubic (Kelvin) structure, and wet monodisperse disordered foams possess a random structures with ψt ≈ 0.64 [233, 35]. In the case of ordered foams, the packing structure of the bubbles is determined by capillary pressure and surface energy minimization considerations [14, 227, 46, 96, 98]. Additionally, in equilibrium, the packing structure of foams follows the equilibrium law described by Plateau in 1873. This law states continuous liquid films forming the common contacts of neighboring bubbles meet three-by-three (forming curved triangular channel) at equally divided angles of 120◦ . The junctions at which three films join are called Plateau borders that are connected four-by-four at equally divided angles of 109.47◦ degrees [35]. Note that the cross-sectional area (and by extension the volume) of these concave channels increases as we move from dry limit to wet limit (decrease bubble volume fraction), see Fig. 1. It must be noted that bubble size distribution and packing structures strongly affect the isotropy of foam properties. Ordering foams can result in developing strong anisotropy in foams’ mechanical and rheological properties [235]. In practical industrial systems including hydraulic fracturing, because of the complexity of the phenomena, foams are mostly disordered and polydisperse with bubbles of a broad (but usually monomodal) size distribution packed randomly[46, 168, 47]. Such foams typically possess more than 36 different bubble topologies [138, 112].

ACCEPTED MANUSCRIPT Journal of Petroleum Science and Engineering XXX (2017) XXX-XXX

650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672

720

674 675 676 677 678 679

Eb =

hSb i

hVb i

(17)

2 3

721 722 723

which intuitively determines the average sphericity of bubbles. 724 hSb i and hVb i indicate the average surface area and volume of a 725 bubble, respectively. For monodisperse foams, it is found that 726 4.836 < Eb < 6 [47]. For disordered monodisperse foams Eb ≈ 727 5.32 [112], and the range of packing efficiency parameter is 728 much narrower in real foams [94]. 729

TE D

673

730

683 684 685 686 687 688 689 690 691 692

693

694 695 696 697

EP

681 682

4.1.2. Foam osmotic pressure 731 Unlike energized fluids, a foam at a given structure between 732 wet and dry limit in contact with a reservoir of the ambient 733 phase compound will draw the liquid into its structures. This 734 phenomenon occurs based on osmotic pressure driven by the 735 energetic contributions of the interfaces [173]. The added liquid 736 allows bubbles to retrieve their original spherical shape, which 737 results in a lower overall interfacial energy stored in the foam 738 (i.e. this increases the stability of foam). At the jamming con- 739 dition or wet limit, where all bubbles are spherical, the osmotic pressure is zero, because the change in total surface area of the 740 bubbles over the total volume of foam vanishes. The osmotic 741 pressure, Π(ψ), mathematically can be represented by [173, 48] 742   σ 2 d S(ψ) 743 Π(ψ) = −3 ψ , (18) 744 Rb32 d(1 − ψ) S0

AC C

680

the osmotic pressure can be well approximated up to the dry limit using, Π(ψ) = κ

σ (ψ − ψt )2 √ , Rb32 1−ψ

(19)

where κ ≈ 7.3 and ψt ≈ 0.74 for monodisperse ordered foams, and κ ≈ 3.2 and ψt ≈ 0.64 for monodisperse disordered as well as polydisperse disordered foams [97, 129]. The impact of osmotic pressure effects on hydraulic fracturing is further discussed in Section 5.

RI PT

648 649

4.1.1. Foam limits 698 The wet limit of a foam depend on the experimental condi- 699 tions, foam structural type (bubble size distribution and pack√ ing) and its composition. For a monodisperse foam, ψt = π/3 2700' 0.74 (with 12 neighbours) is a well-accepted value for wet limit [233, 47]. However, the wet limit can go down to ψt = 0.64 701 (with 6 neighbours) for monodispersed disordered foams [33, 702 48]. For a polydisperse disordered foam with very narrow bub- 703 ble size distribution, the wet limit is typically suggested to be 704 0.71 < ψt ≤ 0.74, and it shifts to higher values as bubble size 705 distribution becomes broader, i.e. small spherical bubbles fill the voids formed by larger spherical bubbles [174, 40]. 706 The dry limit of a foam, defined as the onset of foam col707 lapsing and phase inversion, is also a strong function of exper708 imental conditions and foam structures. In practice, monodis709 perse and polydisperse disordered foams can reach up to ψ = 710 0.96 [40, 105, 132, 238] and even up to ψ → 0.99 [174, 35] 711 at their dry limit. For a monodisperse ordered foam, typical 712 packing structure at the dry limit is body-centered cubic (or 713 Kelvin structure), where bubbles are forming a tetrakaidecahe714 dron which is a truncated octahedron with 14 faces (or neigh715 bours) [202]. For dry monodispersed disordered foams, bub716 bles are shaped as various polyhedra with an average of 13.7 717 faces [138, 112, 48]. It is common practice to compare differ718 ent structures of foams using the normalized energy density (or 719 packing efficiency parameter)

4.2. Foam stability and aging

Foams are intrinsically and thermodynamically unstable due to the large interfacial energy and surface tension between the constituents [196, 193]. For this reason, the structure of foams continuously evolves in time, and ultimately tends to collapse. In a real foam, e.g. polydisperse disordered foam, the Laplace pressure is different from one bubble to another. Smaller bubbles possess a higher pressure, whereas larger bubbles possess lower pressures (pressure is proportional to 1/Rb ). The presence of pressure gradients between neighboring bubbles leads to the diffusion of gas through the liquid film from small to large bubbles. This diffusion or gas exchange causes coarsening, which is fostered by gas volume fraction and solubility as well as diffusion of the gas within the liquid [33, 193]. As large bubbles expand, the difference in size increases, and tiny bubbles accumulate around the large ones. Ultimately, the lack of enough gas in the small bubbles to handle the pressure difference cause film rupture; this phenomenon is called coalescence [168, 47]. In addition to these processes, the liquid phase contained in a foam column naturally tends to flow downward due to gravity, which creates concentration gradients. The bubble volume fraction increases at the top, and decreases at the bottom yielding a mechanism which is called drainage [14, 109, 152]. These three phenomena, coarsening, coalescence and drainage are known as time-evolution, self destruction processes that result in aging the foam, and as foam ages, the rates of these processes decrease [196, 35, 193]. In many practical applications, foam aging needs to be controlled, retarded or suppressed to a large extent. Owing to the fact that foam stability is strongly tied to film stability, many promising surface active agents and methods are proposed to keep the foams’ coarsening, coalescence and drainage under control by sufficiently coating the bubbles, see the review by Rio et al. [184]. Some of these methods are listed below:

M AN U

647

SC

S.A. Faroughi et al.

where S0 and S(ψ) are the total surface area of spherical (which 745 are associated with Π(ψ) = 0) and deformed (polyhedra) bub- 746 bles, respectively. In Eq. 18, Rb32 is the Sauter mean radius, and σ is the surface tension. For most practical applications, 9

• Adding surfactant molecules to the liquid (ambient) phase to reduce the interfacial energy [16, 211, 62, 55]. • Adding a small amount of non-Brownian (solid) particles to the liquid phase to enhance its viscoelasticity due to capillary attraction [34, 93, 61, 80] • Using a concentrated suspension of Brownian or nanoparticles as the foam liquid phase that strongly retards

ACCEPTED MANUSCRIPT Journal of Petroleum Science and Engineering XXX (2017) XXX-XXX

S.A. Faroughi et al.

753 754

755 756 757 758

759 760

761 762 763 764

765 766 767 768 769 770 771 772 773 774 775 776 777 778 779 780 781 782 783 784 785 786 787

788 789 790 791 792 793 794 795 796 797 798

RI PT

752

in Plateau borders, and decrease the rate of drainage [84, 810 4.3. Mechanical and rheological response of foams 85, 108, 79, 242, 238] 811 The mechanical and rheological behavior of foams is con• Adding oil droplets to the films liquid along with the 812 trolled by processes that occur within the microstructures, e.g. presence of surfactant [114, 229, 208] 813 films, Plateau borders, and bubbles rearrangement. The defor814 mation of foams due to compression and shearing is of interest • Injecting CO2 with a suspension of pH-sensitive micro815 in many applications. When used as a fracturing fluid, foam scale particles to the films liquid which makes extraordi816 undergoes strong compression as pressure builds up, and then narily stable bubbles and greatly reduces the rate coales817 expands as fracture grows. This expansion, along with its low cence [168, 47] 818 density, ensures efficient well cleanup which has made foam 819 a popular fluid in underpressured production areas where the 4.2.1. The key role of surfactant 820 reservoir pressure is insufficient to flow-back fracturing fluids Increasing surfactant concentration is the most common way 821 [36, 68]. However, it is shown that the bulk modulus of most to extend the foam lifetime. Surfactants are used to reduce the 822 foams exceed the shear modulus by several orders of magniinherent tension between an aqueous phase and a gaseous phase 823 tude, meaning that foams can be well approximated as incomand create a more stable foam, slowing the temporal evolution 824 pressible fluids in many practical treatments [234, 39]. In other of the structure and delaying collapsing [86]. Surface tension 825 words, when stimulated, bubbles tend to deform and slide along between the two phases decreases with concentration up to the 826 each other than to compress or expand. critical micelle concentration (CMC). At the CMC, the inter827 Upon shearing, foams generally show viscoelastic behavface is fully occupied by surfactant molecules. Above this limit, 828 ior. When applying a shear stress to foams, the average orientathe surfactant surplus stays free in the aqueous phase, and sur829 tion of films tends to align towards the applied shear direction. face tension remains constant [1, 50, 62]. Additionally, com830 For small applied shear stresses (below a certain value so-called monly used surfactants have a hydrocarbon tail. Fluorocarbon831 yield stress), the deformation is reversible and the imposed entailed surfactant can reduce surface tension to a lower level [21] 832 ergy is stored at interfaces (i.e. an increase in surface energy). but, due to environmental concerns, their use is restricted [145]. 833 In this regime, the force balance is such that the Plateau criteMoreover, increasing the surfactant concentration enhances vis834 rion (i.e. films meet at angles of 120◦ ) is always satisfied by cosity, especially when surface area is large, typically at high 835 films. The stored surface energy is released upon removing the bubble volume fraction [78]. For a surfactant solution to form 836 stress. This is a solid-like behavior by which a foam restores micelles, a minimum temperature is required at a given con837 its original shape and stabilized structure. Indeed, the shear centration, called Krafft temperature. Below this value, surfac838 modulus G remains proportional to the surface tension, bubbles tant molecules are inactive. When the solubility is equal to the 839 size distribution and configuration, and the composition of the CMC, this temperature is called Krafft point, and is a tabulated 840 continuous phase (e.g. the inclusion of surfactant molecules). property of the surfactant [38]. Table 2 shows different types of 841 The surface tension, σ, provides the resistance against the small surfactant, their properties and their typical uses. 842 strains in the solid-like regime, thus, the elastic shear modulus 843 is linearly proportional to σ, and is approximated as G = ασ/Rb 4.2.2. Operating conditions and composition effects 844 where Rb is the bubble radius [47, 35]. α is a coefficient of proFoam lifetime or stability can be strongly affected by oper845 portionality which is a function of the ordering of the foam. For ating conditions such as temperature and pressure as well as the 846 ordered Kelvin structure α = 0.51 [111], and for monodisperse composition of constituents phases. It is shown that, at higher 847 disordered foams α = 0.55 [112]. More generally, for polydisoperating temperatures, the rate of coarsening and coalescence 848 perse foams, Saint-Jalmes & Durian [195] derived, increases, and thus, foam decays faster [130, 29]. Coalescence 1.4 increases due to the reduction in surface tension with tempera849 σψ(ψ − ψt ), if ψ > ψt (20) G= ture [105]. Drainage is also affected by temperature. The rate Rb32 of drainage increases with temperature, which can be attributed to a reduction in the effective viscosity of film liquid, which in 850 where Rb32 is Sauter mean radius. Equation (20) indicates that 851 G increases with bubble volume fraction, i.e. wet foams are turn also increases foam coarsening. [206, 63, 238].

SC

751

M AN U

750

TE D

749

drainage due to their yielding (non-Newtonian) behav- 799 Unlike the temperature, the effect of pressure on foam staior [9, 178, 10, 22, 198], especially in treatments con- 800 bility has received less attention. At a given temperature and ducted in high salinity environments or at high tempera- 801 concentration of stabilizing agents, one may generally find that tures where surfactants tend to degrade [58] 802 the foam stability increases with increasing pressure. However, 803 the overall effect of pressure depends greatly on the surfactant • Adding hydrophobic [124, 214, 190] or partially hydropho804 type and concentration [122]. For example, Holt et al. [92] bic/amphiphillic [77] colloidal or nano-particles as foam 805 showed that the stability of foams including C16 alpha olefin stabilizing agents to strongly increase the films dilational 806 sulphonate (AOS) and fluorinated betaine increases with inviscoelasticity 807 creasing operating pressure, while the foam with fluorinated 808 betaine in the presence of oil displays a reduction in stability • Using starch particles (or guar, other polymers or gelling 809 with increasing pressure. agents) which thicken the liquid films by accumulating

EP

748

AC C

747

10

ACCEPTED MANUSCRIPT Journal of Petroleum Science and Engineering XXX (2017) XXX-XXX

S.A. Faroughi et al.

Table 2: A non-exhaustive list summarizing the types and main properties of commonly used surfactants.

857 858 859 860 861 862 863 864 865 866 867 868 869 870 871 872 873 874 875 876 877 878 879 880 881 882 883 884 885 886 887 888

889

890

Anionic

Hydrocarbon

No

Perfluorooctane-Sulfonic Acid (PFOS)

Anionic

Fluorocarbon

No

Quaternary Amonium Salt

Cationic

Hydrocarbon

Yes

Ethylene Oxide Polymer

Nonionic

Hydrocarbon

No

Amine Phosphate

Amphoteric

Hydrocarbon

No

τ(t) = G0 (ω)γ0 Sin(ωt) + G00 (ω)γ0Cos(ωt).

928

(21) 929

This range is non-destructive with no microstructural al- 930 11

Domestic products Hydraulic fracturing Firefighting foams Water and stain repellent *Banned for polluting concerns Polymer-free viscoelastic fluid Fabric softeners Non-emulsifiers Foaming agents Anti-oxidant Anti-wear

teration, and is generally called the linear-viscoelastic range. Equation (21) shows that the shear modulus of the foam considered as a linear-viscoelastic material has two degrees of freedom usually captured and studied through the complex form of G∗ (ω) = G0 (ω) + iG00 (ω). The real part G0 is called the storage modulus (associated with energy stored in elastic form and yields the in-phase component of the viscosity, µ0 = G0 /ω) and the imaginary part G00 is called the loss modulus (associated with viscous dissipation and yields the out-of-phase component of the viscosity, µ00 = G00 /ω). The phase angle, which is a measure of damping within the viscoelastic material, is defined as ξ = tan−1 (G00 /G0 ). For a purely solid material ξ = 0◦ and for a purely viscous fluid ξ = 90◦ which indicates the steady stress and strain deformation are perfectly out-of-phase. The balance between G0 and G00 under specific shearing conditions specifies the dominant rheological behavior of the foam (solid-like or fluid-like). To differentiate G0 and G00 in different regimes, three measurements are performed; (i) frequencysweep (at fixed amplitude of the exciting sinusoidal signal), (ii) strain-sweep (at fixed frequency of the exciting sinusoidal signal), (iii) constant shear rate (i.e. simultaneously varying frequency and strain such that their product which yields the shear rate remains constant). Generally, the strain-sweep test is first performed to differentiate the linear and non-linear viscoelastic regimes of materials. Then, the frequency-sweep test is performed at a fixed strain located within the linear viscoelastic regime to determine the extension of fluid-like and solid-like (and others such as gel-like or cross-linked) behaviors of materials. The reverse approach is also practical, where the frequency-sweep test is performed first, and then, in a region where G0 and G00 do not vary with frequency, the strain-sweep test is conducted. It is shown that at low frequencies (long timescale) the loss modulus, G00 , is bigger than the storage modulus G0 . However, this part of the behavior is rather less understood because of the irreproducibility associated with the experiment timescale, i.e. the effect of aging and coarsening on the foam rheology [137]. At intermediate frequencies and in linear-viscoelastic regime, both moduli are almost constant, neither a function of amplitude nor of frequency, but G0 > G00 rendering a more solid-like

SC

softer than dry foams. The elastic shear modulus decreases 891 when fluid (the ambient phase) is drawn into the foam’s film, 892 and ultimately vanishes when bubbles all become spherical. For 893 disordered and ordered monomodal foams, the rigidity loss oc- 894 curs at ψt = 0.64 and ψt = 0.74, respectively (i.e. right at wet 895 limit which indicates G ≈ 0 for energized fluids). Equation (20) 896 assumes that all chemical formulations and chemistry of the in- 897 terfaces is included into the elastic modules using the corrected 898 value of surface tension (a macroscopic approach). 899 At applied shear stresses larger than the yield stress, bub- 900 bles start to deform, and films separating bubbles either shrink 901 or thicken. At some point when bubbles slide along each other, 902 films are in contact such that the Plateau criterion is not satisfied 903 and the surface area between constituents and consequently sur- 904 face energy increases. The resulting energetically unfavorable 905 structure quickly transforms itself to another static equilibrium 906 configuration with lower surface energetic level. Therefore, at 907 this stress, foams show a transition from solid-like behavior to 908 fluid-like behavior, i.e. foam flows irreversibly and a plastic 909 deformation occurs [195, 46, 234, 39]. It is experimentally ob- 910 served that the yield stress is also linearly proportional to σ/Rb 911 and increases as the volume fraction of the dispersed phase in- 912 creases [97]. The reduction in the volume fraction of the ambi- 913 ent phase (moving towards the dry limit of foams, see Fig. 1) 914 makes the bubble interfaces more rigid [152]. The experimental 915 timescale, due to phenomena like coarsening and drainage, also 916 affects the yield stress; the longer the time is, the smaller the 917 yield stress, and thus the transition from solid-like to fluid-like 918 behaviors occurs at lower stresses [213]. Foams in fluid-like 919 regimes behave as strongly shear thinning fluids. 920 To differentiate the behaviors of foam deforming under shear921 conditions, oscillatory rheometry is applied [137, 213]. In this 922 context, the foam is subjected to a sinusoidal strain deformation 923 such as γ(t) = γ0 Sin(ωt), where ω is the angular frequency, or a 924 sinusoidal shear stress. When foams are excited with sinusoidal 925 strain deformation at small shear amplitude γ0  1, the steady 926 state response of the shear stress is characterized by 927

Typical Use

RI PT

Alpha Olefin Sulfonate (AOS)

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855 856

Viscoelasticity

TE D

854

Tail

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853

Charge

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852

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4.4. Effective shear viscosity of Foams

932 933 934 935 936 937 938 939 940 941 942 943

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950 951 952 953 954 955 956 957 958 959 960 961 962 963 964 965 966 967 968 969 970 971 972 973 974 975 976 977 978 979 980 981 982 983 984 985

4.4.1. Herschel-Bulkley constitutive model The model proposed by Herschel & Bulkley [88] is a structural rheological model derived from considerations of the nonNewtonian fluid structure. These types of structural models (see Rao [177] for a discussion of different structural models), introduce a parametric study of non-linear flow behaviors with the help of experimental data. Parameters within these models vary significantly under different flow and shear conditions and integrate the structural rearrangements and evolution. The Herschel-Bulkley model relates the shear stress, τ, and shear rate, γ˙ , as, τ = τy (ψ, γ˙ ) + K(ψ, γ˙ )˙γn(ψ,˙γ) ,

12

(22)

where τy (ψ, γ˙ ) represents the yield stress with the unit of [Pa], K(ψ, γ˙ ) is the consistency with the unit of [Pa.sn ], and n(ψ, γ˙ ) is a dimensionless parameter, the so-called flow index. For Newtonian fluids τy = 0 and n(ψ, γ˙ ) = 1, while for foams it is commonly shown that τy > 0 (indicating yielding behavior) and n(ψ, γ˙ ) < 1 (indicating shear thinning behavior). For Newtonian fluids, the consistency, K(ψ, γ˙ ), is identical to the shear dynamic viscosity, µ. The consistency physically represents the shear stress when τy = 0, and γ˙ = 1(s−1 ), otherwise, it does not refer to any physical quantities, i.e. it acts as a fitting parameter only. In this model, Eq. (22), the coefficient and the exponent of the shear rate are strong functions of the shear rate itself. If the value of the yield stress is known a priori from independent experiments, a linear (or non-linear) regression of the plot of log(τ − τy ) versus log˙γ provides the best fitting value for the other two parameters: n from the slope of the obtained line, and logK from the intercept of the line to the vertical axis. It is also possible to consider all three parameters as tuning parameters, however, the result of even a perfect non-linear regression may totally conceal the underlying physics, and may not indicate the corresponding structures and the flow of the foam under deformation. Other techniques like Golden Section (GS) [107], Genetic Algorithms [187] and General Regression Neural Network (GRNN) [188] might be applied, in fitting procedures, to more realistically constrain the three parameters of the Herschel-Bulkley model. The continuum description of the bulk flow behavior of foams characterized by Herschel-Bulkley model within different geometries and rheometers have been discussed in many studies, e.g. see Calvert & Nezhati [23], H¨ohler & Cohen-Addad [97], Sexton et al. [205] and Kahara et al. [103]. Here we review the method of measurement of the Herschel-Bulkley parameters within a (recirculating or straight) pipe rheometry which is commonly used for high pressure foams with application to hydraulic fracturing [84, 15, 199, 215, 108, 91, 100, 13, 67, 78].

M AN U

949

TE D

948

EP

947

At the onset of transition from energized fluids to foams, a1001 sudden increase in the effective viscosity has been widely ob1002 served, see the dashed line in Fig. 2 that represents the relative viscosity of foams and mists [179, 183, 53]. The sudden incre-1003 ment in the effective viscosity with this transition is attributed1004 to several factors; (i) the transition of random close packing to1005 an ordered close packing and the appearance of yield stress, (ii)1006 local distortions of bubble shape from spherical to polyhedral1007 that increases the total surface energy and resistance to flow1008 due to the development of surfactant concentration gradients,1009 (iii) surface rigidity due to other stabilizing agents, (iv) vis-1010 coelastic properties and applied shearing conditions, (v) the vis-1011 cous dissipation associated with structural rearrangement and1012 evolution, e.g. bubble sliding, films flowing and restructuring1013 [108, 97, 35, 136, 43]. 1014 To determine the effective shear viscosity of foams, the ef-1015 fect of all these phenomena (occurring on different length-scales) 1016 should be taken into account. Due to the limitation of theo-1017 retical approaches developed for foam rheology over different1018 length-scales, it is very difficult to generalize and link all these1019 phenomena to quantify the bulk rheological behavior. To cope1020 with this, a large number of experimental studies have been de-1021 voted to parametrizing the rheology of foams at macro-scale1022 using continuum approaches, see e.g. Khade & Shah [108],1023 Gu & Mohanty [78] and references therein. It is commonly1024 shown that, at imposed stresses beyond the yield stress, the1025 relative viscosity (ratio of effective foam viscosity to ambient1026 fluid viscosity) decreases as shear rate or shear stress increases,1027 which is an indication for the shear thinning behavior of foams1028 [35]. These studies provided several empirical correlations de-1029 veloped for different types of foams. The Herschel-Bulkley1030 model [88] is widely used for this aim, see for example Khade1031 & Shah [108], H¨ohler & Cohen-Addad [97] and Gu & Mo-1032 hanty [78]. 1033 There are, however, serious complexities and limitations1034 within empirical studies that prevent development of a gener-1035 alized (or constitutive) model valid under a wide range condi-1036 tions. In many cases, published experimental datasets by dif-1037 ferent authors do not follow the same trend. These difficulties, as pointed out by Hutchins & Miller [100], are mainly associ-

AC C

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ated with differences in instruments to generate foams, operating conditions, applied measurement methodology, and last but not least, data analysis. We discuss the relevant approaches to characterizes the rheological behavior of foams in the following sections.

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behavior. Beyond the yield strain, the non-linear viscoelastic 986 behavior becomes apparent showing the foam starts to flow, i.e. 987 bubbles slide along each other. In this regime, the storage mod- 988 ulus, G0 , decreases, and the loss modulus, G00 , first increases, 989 meets a peak where it overtakes the storage modulus, and ul- 990 timately decreases at large strains, but still G00 > G0 rendering a more fluid-like behavior. Moving towards the dry limit, i.e. 991 increasing bubble volume fraction much beyond ψt , strongly 992 affects the shear modulus and yield strain and by extension the 993 yield stress [137]. Therefore, dryer foams possess higher G0 994 and G00 . Indeed, as shown by Drenckhan & Langevin [47] the 995 packing configuration and bubble size distribution (e.g. ordered 996 monodisperse, disordered monodisperse and polydisperse) sub- 997 tly vary the shear modulus and the onset of foam flowing. 998

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∆PD . τw = 4L

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(29)

and imposing a large enough shear rate by which the foam starts to flow, i.e. the effect of the yield stress, τy , can be neglected [78], a power law model for the rheological flow curve can be obtained, n

τw = K˙γw .

(30)

Substituting Eq. (26) in Eq. (30), and equating the resultant formula with Eq. (28),   0 0 3n + 1 n n (31) γ˙ w = K 0 γ˙ nw , K 4n0

From the measurement of the volumetric flow rate, q, or the1091 flow average velocity, U, corresponding to the equivalent Newtonian fluid, one can determine the magnitude of the apparent1092 provides a way to obtain the flow index which is a fluid intrinsic 1093 property as n = n0 and the true consistency parameter, K, as wall shear rate, which reads as,  n 4n 32q 8U 0 K = K 1094 . (32) γ˙ w = = . (24) 3n + 1 πD3 D

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n

τw = τy + K˙γw ,

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This shear rate is also known as ”Newtonian wall shear1095 rate”. To derive the wall shear rate for a Non-Newtonian fluid1096 flowing through the same pipe with the same volume flow rate,1097 q, the correction proposed by Rabinowitsch [175], 1098 ! 1099 1 ∆P dq 1100 γ˙ w = 24q + 8( ) πD3 L d ∆P 1101 L ! 8U 3 1 d ln( D ) = γ˙ w + (25)1102 4 4 d ln(τw )

TE D

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n0 and K 0 vary along the flow curve [153]. By considering the Herschel-Bulkley model to constrain the true rheological flow curve (τw versus γ˙ w ) of foams as,

As already discussed in above sections, with the help of different fitting techniques (linear and non-linear regression, or general regression neural network) and with the assumption that either τy is known, or is negligible, one can find n = n0 and K from Eq. (32). Knowing this information, the non-Newtonian effective viscosity of the foam characterized as a power law fluid is calculated by,   3n + 1 n−1 n−1 µψ = K γ˙ w . (33) 4n

can be applied that yields the non-Newtonian wall shear rate as,1103 Note that in Eq. (33), n and K are intended to be constant  0  1104 along the flow curve, but are functions of the bubble volume 3n + 1 γ˙ w = γ˙ w . (26)1105 fraction and imposed shear rate (or stress). Several experimen4n0 1106 tal studies on the rheology of foamed N2 and CO2 made efforts 0 1107 to constrain self-consistent models for n, K and τy under speIn Eq. (26), n is the slope of the log-log plot of the flow 1108 cific conditions. A non-exhaustive summary of these studies is curve (wall shear stress, τw , versus the magnitude of the New1109 reported in Table 3. It should be noted that an alteration in op8U tonian shear rate, D ), which is defined as, 1110 erating temperature and pressure and the presence of possible 1111 shear localization (e.g. caused by wall slip) during flow might d ln(τw ) n0 = . (27) 1112 severely change the value of flow parameters. These effects are d ln(˙γw ) 1113 discussed in details in the following sections. The value of n0 varies along the plot of the flow curve; for Newtonian fluids, n0 = 1 meaning the slope of the log-log plot1114 4.4.3. Wall-slip and shear localization of the flow curve is unity everywhere, while for shear thinning1115 Rheology models for foams and energized fluids do not exfluids, n0 < 1 that indicates the correction factor for wall shear1116 plicitly account for the effect of confining walls in their develrate in Eq. (26) is greater than one. Therefore, the shear thin-1117 opment. At best, it is assumed that the wall acts as a stress ning fluids experience larger rate of deformation than Newto-1118 or kinematic boundary condition (imposed strain-rate). In fact, nian fluids in a fixed setup [153]. Based on Eq. (27), one can1119 most theoretical models assume either an infinite domain or, define the flow curve equation for a general fluid as 1120 in limited cases, a finite domain with fixed boundary condi-

EP

1039 1040

4.4.2. Flow index, n, and consistency, K, in pipe rheometry 1080 For pipe rheometry, we consider a laminar flow of an in-1081 compressible foam flowing through a pipe with length L and1082 diameter D. To ensure the validity of a continuum descrip1083 tion of the bulk foam flow within a pipe, the bubble diameter, Db should be much smaller than the characteristic length scale1084 (e.g. the diameter of the pipe) meaning Db /D  1. The incom-1085 pressible assumption is supported because the shear modulus of1086 foams is much smaller than their bulk modulus. At the steady1087 state condition, the flow in the pipe is resisted by a force at the boundary of the system yielding the relationship between the1088 pressure drop, ∆P, and wall shear stress, τw [147],

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(28)1121 tions. Under natural or laboratory conditions, these fluids are 1122 generally bounded either by solid confining boundaries or in where n0 and K 0 are measured at different points and their value1123 some particular cases with a free surface. Over the last three is geometry-dependent. Note that Eq. (28) is different from that1124 decades or so, the effect of solid boundaries on the flow of of a power law fluid (with constant n and K), because, here, both1125 these fluids has received more attention. More specifically, the 0

τw = K 0 γ˙ wn ,

13

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Table 3: A non-exhaustive list of well-known models to determine the relative viscosity of foams.

Valko & Economides (1992)

H-B model

Power law

Bingham

1129 1130 1131 1132 1133 1134 1135 1136 1137 1138 1139 1140 1141 1142 1143 1144 1145 1146 1147 1148 1149 1150 1151 1152

Power law

Gu & Mohanty [78]

H-B model

report of unexpectedly low effective viscosity in concentrated1153 energized fluids and foams supports, in many cases, the oc-1154 currence of wall-slip [90, 113], shear localization and banding1155 [37, 186, 234, 237, 132]. For example, in three-dimensional1156 foams, there is an apparent tendency to localize shear by which1157 the fluid is separated into regions of smaller and larger bub-1158 bles [90]; however, in contrast, it has been shown that three-1159 dimensional foams do not display an observable sign for the1160 presence of shear banding [160]. 1161 Wall-slip is actually a misnomer (it is often confused with1162 slip boundary condition at wall) and refers to a thin lubricating1163 layer with thickness δ depleted in bubbles that develops near1164 smooth solid boundaries. See Fig. 3 for the development of the1165 wall-slip phenomenon and its effect on the velocity profile (or1166 stress partitioning) for a foam (or concentrated energized fluid)1167 flowing in a tube. In Fig. 3, the thickness of the layer is over-1168 sized for the visualization purposes. It has been shown that the1169 slip thickness may vary from the order of a film to a bubble1170 [221, 234]. The slip layer eases the flow (i.e. leads to a smaller1171 overall rate of viscous dissipation and a non-realistic effective1172 viscosity). Its effect, however, is different for various types of1173 viscometers [6, 90]. 1174 Different experimental datasets obtained without consider-1175 ing wall-slip should not be directly compared, and the effective1176 viscosity obtained from experiments in which the wall-slip is1177 overlooked are most likely unreliable. 1178 Within pipe rheometry, there are two possible causes of1179

TE D

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Khade & Shah [108]

EP

1127

H-B model

AC C

1126

Bonilla & Shah [15]

Polymer-free CO2 foams T = 24◦C P = 62 bar Polymer-free N2 foams T = 24◦C P = 62 bar Guar N2 foams (Concentration not stated) T = 16 − 24◦C P = 21 bar Guar N2 foams (Concentration not stated) T = 16 − 24◦C P = 21 bar 2.4 kg.L−1 guar N2 foams T = 38◦C P = 69 bar 2.4 kg.L−1 guar N2 foams T = 38◦C P = up to 83 bar Polymer free N2 foams T = 35 − 65◦C P = 69 bar

14

n[], K[Pa.sn ] and τy [Pa] n=1 2 K = 0.00002e3.6ψ+0.75ψ τy = 0.0002e9ψ n=1 2 K = 0.00002e4ψ+0.75ψ τy = 0.00012e8.9ψ

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Materials & conditions

n = 0.47 K = 1.4(1 − ψ)−0.53 τy = 0 n=1 K = 0.048 τy = 7(1 − ψ)−1

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n = 0.77 2 K = 0.0007e0.5168ψ+4.1224ψ 9.425ψ τy = 0.0002265e n = 0.73(1 − 2.1006ψ7.3003 ) 2 K = 0.0009e−1.9913ψ+8.9722ψ τy = 0 n = 1.54 − 1.64ψ2 2 K = 105.89ψ +0.43ψ−4 τy = 0

the depletion layer at the wall. The first is purely geometrical (static), i.e. the presence of a straight wall breaks the homogeneity in the distribution of the dispersed phase, because bubbles cannot intersect the boundary. The second is a dynamical effect and relates to the propensity of bubbles to migrate from high to low shear rate regions when shear-rate gradients (or stress inhomogeneities) are present. Wall-slip strongly affects the pressure gradient, i.e. the correct pressure gradient used in Eq. (23) to obtain n and K might be far from that of being read from experiment depending the bubbles and tube diameters, dispersity, and depletion layer thickness. When using a pipe rheometer, corrections can be applied to the measured pressure drop to cope with wall-slip effects [70]. Alternatively, at a given pressure, temperature, bubble volume fraction and size, one may remove the wall-slip effects by measuring and analyzing data within different pipe diameters. The latter is plausible using the fact that the flow curve obtained from pipe rheometry shifts to lower stresses with decreasing diameter of the tube, i.e. a significant dependence on the diameter of the pipe [59], or, more generally, the ratio of slip thickness layer to pipe diameter. To take this route and account for the effects of wall-slip, it is necessary to quantify the thickness of the depleted layer as well as the slip velocity. Different formulas have been suggested to estimate δ and express the velocity of the slip layer accordingly. Table 4 reports a non-exhaustive list of well-known methods and their key features. As can be inferred from Table 4, these methods try to find

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Figure 3: A schematic representation of the wall-slip effect. Panel (a) shows the velocity profile for a general fluid without internal microstructure (e.g. Newtonian fluids (n = 1) or an equivalent fluid considered for a foam or energized fluid with n < 1). Panel (b) shows the actual velocity profile and the state of bubbles for a foam or energized fluids near a smooth solid surface that causes a depletion layer with thickness δ close to wall and a slip velocity of uslip . Table 4: A non-exhaustive list of well-known models to determine the slip layer thickness, slip velocity and real Newtonian wall shear rate (˙γw )real from observed wall shear rate (˙γw )obs = −8U/D.

1185 1186 1187 1188 1189 1190 1191 1192 1193 1194 1195 1196

1197

Oldroyd [157], Jastrzebski [102]

δ=

βµ D

Enzendorfer et al. [59]

δ=

(a+bτw )µ D

uslip = βτw

the real Newtonian shear rate, (˙γw )real , by an extrapolation to1204 1/D → 0 or 1/D2 → 0 (i.e. an infinite pipe diameter) where1205 the effect of wall-slip is minimal. In the methods proposed by1206 Mooney [146] and Oldroyd [157], β is called the slip coeffi-1207 cient which is determined from the slope of the linear plot of1208 observed Newtonian shear rate, (˙γw )obs , versus 1/D and ver-1209 sus 1/D2 , respectively, at a given wall shear stress, τw . Note1210 that (˙γw )real is the intercept of the linear fits with the vertical1211 axis in both cases. Enzendorfer et al. [59] proposed a gen-1212 eralized model in which the slip coefficient is assumed to vary1213 linearly with the wall shear stress yielding β = a + bτw with1214 a = 8.6 × 10−6 and b = 7 × 10−7 for foamed polymer solutions.1215 The corrected flow curve, τw versus (˙γw )real has no dependence1216 on tube diameters, and thus, different datasets for different di-1217 ameters should collapse on one single curve. The inferred ef-1218 fective viscosity with ignoring the wall-slip is then constrained1219 by 1220 1221  n−1 3n + 1 n−1 µψ = K ((˙γw )real ) . (34)1222 4n 1223 1224

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Reference

In this section, we discuss how aforementioned concepts1227 from more fundamental studies can be applied to fracturing,1228 from the preparation of the fluid to the stimulation of the reser-1229 voir. Unlike those of water, the physical properties of ener-1230 gized fluids and foams can be greatly modified through multiple1231 15

Wall shear rate correction w (˙γw )real = (˙γw )obs − 8βτ D

uslip =

βτw D

w (˙γw )real = (˙γw )obs − 8βτ D2

uslip =

(a+bτw )τw D

w )τw (˙γw )real = (˙γw )obs − 8(a+bτ D2

parameters: quality, temperature, pressure, surfactant choice, phase composition, etc. Many of these properties, described in section 3 and 4, can be calibrated to ensure the efficiency of fracturing. In the case of foams, we will show how some additional specific features introduced in section 4 are beneficial to production. In fracturing, the fluid is selected following its ability to perform two main functions: (1) transmission of pressure generated at the surface to the fracture tip, preferably with a low pressure drop, and (2) transport of proppant particles from the surface to the fracture, preferably distributing them more or less uniformly throughout the length and height of the fracture with minimal settling. The apparent viscosity requirements of these two purposes are in obvious conflict, as low pressure-drop requires a low viscosity while proppant transport requires high viscosity. However, the flexibility of energized fluids and foams allows a fluid to be optimized for individual cases. In addition, the elastic nature of foams is expected to provide some advantage in proppant transport without imposing a pressuredrop penalty. In practice, characterization of foam fracturing fluids has been dominated by pipe rheometry measurements, as described for example by Harris [84] and Reidenbach et al. [179]. As a consequence, within the well-bore, fluid flow and pressure drop are well-predicted and proppant transport is not an issue (due to high flow rate). However, in the fracture, the situation is much less understood, not only due to the uncertainty in the fracture geometry, but also due to uncertainties in proppant transport,

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Effective viscosity (cP)

(b)

Cawwiezel & Niles (1987): ψ=0.55 aqueous energized fluid at 1200 s−1 Cawwiezel & Niles (1987): ψ=0.55 guar energized fluid at 1200 s−1 Gu & Mohanty (2015): ψ=0.59 aqueous energized fluid at 500 s−1 Gu & Mohanty (2015): ψ=0.59 aqueous energized fluid at 1200 s−1 Gu & Mohanty (2015): ψ=0.80 aqueous foam at 1200 s−1

1

Cawwiezel & Niles (1987): ψ=0.70 aqueous foam at 1000 s−1 Cawwiezel & Niles (1987): ψ=0.70 aqueous foam at 1500 s−1 Bonilla & Shah (2000): ψ=0.63 guar energized fluid at 500 s−1 Gu & Mohanty (2015): ψ=0.62 aqueous energized fluid at 500 s−1 Gu & Mohanty (2015): ψ=0.83 aqueous foam at 500 s−1 Gu & Mohanty (2015): ψ=0.63 VES energized fluid at 500 s−1

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the importance of suspended transport vs. translation along the1253 bottom of the fracture, and etc. In a recent work (McAndrew1254 et al. [141]) we have attempted to reduce this uncertainty by1255 proppant transport measurements in high pressure foams. 1256

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5.1. Fluid preparation for fracturing 1258 When used as a fracturing fluid, energized fluids and foams1259 are generated at the wellhead by flowing gas turbulently into the1260 liquid phase. The path followed by the fracturing fluid in the1261 well and down to the fracture reduces the impact of the forma-1262 tion process, as the bubbles rearrange while flowing, especially1263 in the case of deep reservoir. This method creates a chaotic1264 structure governed exclusively by energy minimization [192].1265 Thus, following section 4.1, it is rather a reasonable assumption1266 to consider that the downhole foam is polydisperse and disor-1267 dered. Readers may refer to thorough reviews by Drenckhan1268 & Langevin [47], Cohen-Addad et al. [35] and Drenckhan &1269 Hutzler [48] and studies of Hofmann et al. [95] and Murtagh et1270 al. [149] for detailed information regarding different structures1271 1272 of foams. The choice of the surfactant is particularly difficult for reser-1273 voir stimulation applications, and many formulations have been1274

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Figure 4: Evolution of the effective viscosity of different energized fluids and foams with (a) operating pressure and (b) operating temperature. Panel (c) and (d) show the evolution flow index and consistency of different energized fluids and foams with operating temperature.

16

tested. As we discussed in section 4.2.1, several aspects need to be considered when looking at surfactant properties, e.g. interfacial tension reduction, CMC, foaming capacity, electrical charge, wettability, and acidity [16]. Additionally, surfactants lose their ability to reduce interfacial tension at high temperature [55, 122, 155, 176]. This issue can be addressed up to a certain limit by increasing the surfactant concentration [106, 143]. The main factor to select a surfactant for petroleum industry is the charge. Most surfactants are anionic, but they can also be cationic, nonionic, amphoteric or zwitterionic [4, 8]. Anionic surfactants are preferred for petroleum applications to reduce reservoir interactions, as clay particles generally possess a negative charge [51]. Some surfactants, typically mixtures including a quaternary ammonium salt, give viscoelastic properties to the ambient fluid. These are of particular interest in the case of fracturing, as they allow the production of polymer-free water based viscoelastic fluids [159]. In static conditions, micelles of viscoelastic surfactant (VES) entangle and enhance elasticity. When sheared, they adopt a rod-shape structure and flow easily, decreasing viscosity and elasticity [106]. The importance of elasticity for proppant transport is discussed in section 5.3.

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As discussed in section 4.2, foam stability, interface mobility and rheological properties strongly depend on the experimental and operating conditions, including temperature and pressure. As foam is compressed at the wellhead, it acquires a thinner texture [83]. In general, using Eq. (33) to fit viscosity measurement data, it is shown that the effective viscosity of foams increases with pressure - accordingly the flow index n decreases and the consistency K increases with pressure, enhancing the proppant transport [28, 78]. This effect can be attributed to (i) an increase in the gas densities, (ii) a slight increase in gas viscosity, and, most importantly, (iii) a change in the bubble configuration, size distribution and mean diameter [91]. Figure 4(a) shows a subset of experimental data published for different foams indicating an increase in effective viscosity by increasing pressure. Note that with this wellhead compression the dispersed gas reaches supercritical condition. In practice, to deliver the required pressure for fracture propagation, it is of obvious importance to predict pressure drop as a function of flow rate and travel distance in the well-bore. Pipe rheometry measurements, combined with Herschel-Bulkley models from section 4.4.1, enable hydraulic fracturing service companies to quantify wellhead requirements for each reservoir. At this point, within the smooth surface of the well, foam wallslip remains significant and has to be considered to determine downhole conditions, either by estimating the slip layer thickness following models from table 4 and analysis by Gardiner et al. [69, 70] and Tisne et al. [222], or by roughening or hydrophobizing the rheometer/tube wall surface [113]. The rheological parameters and predicted pressures have been verified by many comparisons with downhole pressure measurements. The same parameters are also used to predict pressure drops in fractures and to estimate the effective viscosity for the purpose of predicting proppant transport. Of course it must be realized that proppant transport in a fracture cannot be verified in the same way as downhole pressures, so the validity of the predictions is much less certain. In particular, proppant transport models neglect several factors that are probably of importance in reality, including elastic effects (further discussed later) and translational transport of proppant along the bottom of the fracture. On the other hand, the roughness of the rock surface reduces the thickness of the depletion layer, and consequently, wall-slip may not occur within the fracture [25]. The strong increase of temperature at downhole conditions affects the rheology and stability of foams more significantly than pressure variation. We previously discussed that increasing the operating temperature increases the drainage and coalescence phenomena, and thus reduces the foam life time. Simultaneously, surfactant molecules become inoperative, i.e., when bubbles are sheared, the concentration of free surfactant molecules in the liquid phase is insufficient to nullify the Marangoni effect [135]. With the resulting excess of tension, the interfacial energy becomes very large, and thus the liquid film mobility increases. As a result, foam deformation is accelerated, and the effective viscosity is reduced [28, 171, 105]. Figure 4(b) summarizes a non-exhaustive collection of ex-

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As discussed earlier in section 4.2.2, the composition of the1330 aqueous (ambient) phase flowing in the films between bubbles1331 has a strong impact on foam stability. The effect of the compo-1332 sition of the gaseous dispersed phase, including gas permeabil-1333 ity and solubility in the aqueous phase, may also affect the foam1334 stability to a great extent [172, 84, 151]. These potential effects1335 are a nontrivial function of operating temperature [150]. Fara-1336 jzadeh et al. [63] performed a series of static-stability measure-1337 ments using gas permeability and solubility index to analyze the1338 effect of gas type (e.g. N2 , CO2 He, and CH4 ) and composition1339 on foam stability. They showed that the rate of coarsening, co-1340 alescence and drainage correlates with the gas solubility, and1341 thus, foams with gases that possess lower solubility constant1342 (e.g. N2 ) last longer than those with higher constant (e.g. CO2 )1343 [151, 197]. These results, which were obtained at atmospheric1344 pressure in an apparatus open to the air, are in agreement with1345 common experience, for example in beer [11]. However, high1346 pressure static-stability analyses in closed system carried out1347 by Harris [84], on foams with 70% volume fraction of bubbles,1348 found foams with CO2 to be more stable than those with N21349 (by a factor of 5), which is in sharp disagreement with obser-1350 vations at atmospheric pressure. At the conditions where they1351 prepared their foams (136 bar and 24◦ C), CO2 is liquid (close1352 to the two-phase region) with density about 0.86 kg/m3 i.e. ap-1353 proaching that of water, while N2 is supercritical, with density1354 about 0.15 kg/m3 . Presumably, the slower drainage of CO21355 foams observed by Harris results from the smaller difference1356 in density between CO2 and water under that conditions, while1357 at atmospheric pressure, the density difference between CO21358 and water is about the same as between N2 and water. More1359 measurements of foam stability at high pressure are desirable1360 to clarify the behavior in this regime. 1361 The use of specific equipment for foam generation at the1362 wellhead is not widely reported, based on the above logic that1363 chaotic processes during fluid delivery will govern the foam1364 structure. However, downhole foam generation close to the1365 fracture offers significant advantages over surface generation1366 such as reduced pressure drop and improved control [204]. If1367 this technology were to be implemented in practice, it would1368 likely require careful design of the foam generator.. 1369 Another approach to dynamic optimization of fluid proper-1370 ties is to combine more than one fluid in a hybrid frac. While1371 this approach is typically associated with combining slickwater1372 and gel fluids, it can be applied to foams also. Combining a CO21373 pad with a foam for proppant transport was discussed recently1374 by Ribeiro et al. [182], and indeed was previously proposed1375 by Layne & Siriwardane [117] and Yost et al. [240]. Hybrid1376 fracs can be considered a way to reduce the overall pressure1377 drop due to the fracturing fluid while maintaining the ability to1378 transport proppant, e.g. by using a more viscous (and elastic)1379 fluid when proppant is injected. Somewhat counter-intuitively,1380 reverse-hybrid fracs using less viscous fluids for proppant injec-1381 tion have also been shown to be effective by Liu et al. [123].1382 These approaches need to be further explored for foam fractur-1383 ing. 1384

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perimental results for the foam effective viscosity as function of1440 temperature at different compositions, bubble volume fraction1441 and shear rates. One finds that the effective viscosity decreases1442 with operating temperature. The analysis developed in section1443 4.4.1 helps us to capture this evolution. In fact, using struc-1444 tural models to fit data, it is generally expected that the flow1445 index, n, increases and the consistency, K, decreases with tem-1446 perature for energized fluids, and the opposite occurs for foams1447 [78]. Figures 4(c) and (d) partly endorse this conclusion, and1448 the subtle inconsistency between data reported in literature may1449 be the direct result of employing different fitting procedures -1450 section 4.4.2 summarizes a methodology to accurately fit the1451 data assuring the obtained n and K are fluid intrinsic properties1452 1453 and not geometry and flow type dependent. The temperature effect on the effective viscosity of foam de-1454 pends strongly on the type and concentration of surfactant used,1455 especially near the dry limit. For example, viscoelastic surfac-1456 tant foams are affected more significantly than foams based on1457 AOS [28, 15, 78]. This fact explains why foams are currently1458 being used in reservoir of limited temperature, generally less1459 than 80◦C. However, the effective viscosity of foams is gener-1460 ally preserved against temperature more than that of the corre-1461 sponding base fluid [82, 78]. Also, Hutchins & Miller [100]1462 experimentally observed that, at high temperature, the viscosity1463 of foamed N2 remains higher than that of the equivalent CO21464 foams, which is in agreement with the calculations of Harris1465 [84]. 1466

1413

5.3. Proppant transport and fracture propagation

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The geometry of the fracture is often assumed to be simi-1469 lar to a narrow, tall and long slot [225]. As the width of the1470 fracture is reduced from the entrance to the tip, shearing is in-1471 creased. Thus, unlike water, energized fluid and foam proper-1472 ties are evolving within the fracture. For the case of energized 1473 fluids, bubble deformation at high capillary number has a strong effect on the effective viscosity. For foams, the analysis is more1474 complex and has not been completed. We can reasonably spec-1475 ulate that similar phenomena occur up to a limit at very high1476 capillary numbers, where strong shear forces break the foam1477 into liquid and gas or supercritical streams. 1478 As shown schematically in Fig. 5, capillary number varies1479 along the length of the fracture. Close to the wellbore, capillary1480 number is relatively low, and bubble size is much less than the1481 fracture width, while at the tip of the fracture, capillary number becomes very high. Given a distribution in bubble sizes,1482 we can expect that larger bubbles will be destroyed as they en-1483 ter narrow regions, resulting in smaller bubbles at the fracture1484 tip. Nevertheless, as the fracture becomes very thin, capillary1485 number will still be large even for smaller bubbles. 1486 In order to develop an example for discussion, we take a1487 simple case with a uniform bubble size distribution. We also1488 limit our consideration to energized fluids, using equations de1489 fined for that case in section 3. To this end, we re-define the1490 capillary number as 1491

AC C

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Ca =

6µmURb , wσ

for a 3D slot flow as a geometrical proxy for the fracture, where U and w are the average inlet velocity and the width of the fracture, respectively. Here, we assume that the ambient phase is pure water µm = 0.001(Pa.s), the average radius of gas bubbles (suspended bubbles) is Rb = 1(mm), and the surface tension is reduced to σ = 20 × 10−3 (N.m−1 ), which is a typical value obtained using commonly available surfactants (e.g. Alpha Olefin Sulfonate [62]). Finally, we assume that the energized fluid is injected with the velocity U = 0.3(m/s) at the inlet. Substituting above variables into Eq. (35), and the resulting equation into Eq. (13) leads to a non-linear description for the effective viscosity as function of the width of the fracture, w, as well as the volume fraction of dispersed phase, ψ. We use NewtonRaphson method to solve the resulting non-linear model at two different concentrations, 40% and 65%, assuming ψt = 0.74. The results for the effective viscosity as function of the width of the fracture for intermediate, 40%, and concentrated, 65%, energized fluids are depicted in Fig. 6. The shaded area in this figure shows the range of fracture width which is practically achievable using energized fluids. One readily observes that the effective viscosity of the energized fluids is not constant over the fracture length and changes by more than an order of magnitude (as its width becomes narrower towards the tip of fracture, see the inset in Fig. 6). Thus energized fluids exhibit a beneficial rheological behavior because:

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• close to the inlet where the width is maximum, the effective viscosity is the greatest. This phenomenon is favorable, because it decreases the rate of proppant segregation close to the inlet, when proppant is injected along with the energized fluids. Referring to Fig. 6, at the fracture inlet, fracture width is of order 1 cm, which is 5 times of the bubble radius, sufficiently large to maintain a viscosity several times that of the ambient phase. • towards the tip of fracture the viscosity is reduced, and thus, the rate of energy dissipation becomes smaller, which in turn causes longer fractures. At the fracture tip, where the fracture width reaches around 0.1 mm, substantially less than the average bubble radius of most energized fluids , the effective fluid viscosity is equal to (or even less than) that of the ambient phase (due to the large bubble deformation accommodating most of the shear). • as the fracture grows in length, it also becomes wider along more of its length, allowing viscosity to be maintained for more of the fracture, which will carry proppant deeper into the fracture.

The viscosity of foams is likely to follow a similar behavior as described above. In addition, the foams’ elasticity will greatly enhance the particle residence time (i.e. keeps proppant suspended for a longer time) that ultimately leads to a better proppant positioning in the fractures. Taking advantage of elastic, ”solid-like”, behavior of foams described in section 4.3 is a promising approach to avoid the conflicting requirements of low viscosity for fracture growth

Abstract

S.A. Faroughi et al.

to improve well productivity. However, the behavior of f by reviewing the scientific literature have been mechanics fully integrated into the engineering of energized fluids and fos The use of aqueous foamsfoams enables thenot reduction in water consumption in reservoir by reviewing literature combination with capillary and impact of w to improve well productivity. However,thethescientific behavior ofnumber, foamsinis notthe fully underst Abstract mechanics ofinto energized fluids and foams, The use of aqueous enables fully the reduction in water consumption in reservoir by hydraulic fracturing, and promises elasticity and the importance of the elastp foams have notfoamsbeen integrated the stimulation engineering practice ofincluding well stimulatio ACCEPTED MANUSCRIPT to improve well productivity. However, the of behavior of foams isScience not fully understood, and advances in scientific of Journal Petroleum and Engineering XXX understanding (2017) XXX-XXX capillary number, and the impact of wall effects on labora foams have not beenthe fully integrated into the engineering practice ofin wellcombination stimulation. This review aims to progress in that direction by reviewing scientific literature with earlier, more engineerin by reviewing the scientific literature in combination with earlier, more engineering-oriented works. We discuss the rheology and elasticity and theofimportance ofpossible elastic in the case mechanics of energized fluids and foams,fluids including the possible variation rheology with fracture length and width, theeffects effect of mechanics of energized and foams, including the variation of rheol capillary number, and the impact of wall effects on laboratory measurements. We also review recent discussions of fracturing fluid capillary number, the impact wall effects on laboratory measurements. We a elasticity and the importanceand of elastic effects in the case ofof foams. elasticity and the importance of elastic effects the case of foams. 1 1.inIntroduction a Air

Liquide Delaware Research and Technology Center, Newark, DE, USA

this review, we will discuss the basis for these benefits, and particularly how recent scientific insights support their impleDb 36 mentation. 2 2 Db 1 37 In hydraulic fracturing, w pressurized fluids are used to frac3 . 38 ture impermeable rock and transport proppant into the fracture 3 4 In the last decade, resources such as shale gas and tight oil 39 to prevent fracture closure after the pressure release.34Proppant Db Db directly impacts productivity, be5 have become extremely important thanks to the successful de- 40 placement within the fracture 2 4 35 6 velopment of hydraulic fracturing [200]. The resulting w increase 41 cause it controls both short-term and long-term conductivities 1 w 7 in the use of natural gas has allowed the United States econ- 42 of the fractured well. To prevent fracture closure and enhance 3 5 8 omy to prosper, even while reducing its carbon footprint. De- 43 the conductivity of wells, different fracturing fluids 36 and propDbsee review papers by Al-Muntasheri D 9 spite fracturingD isbcontrover- 44 pant types have been used, b these positive aspects, hydraulic 4 6 2 37 10 sial w where it 45 [4] and Liang et al. [116], w respectively. The main types of fluw and widely threatened by regulations, to the point 11 has been banned in several areas. The reasons for this include 46 ids are: (i) low viscosity fluids (e.g. slickwater, which is water 5 7 3 38 12 the consumption of large amounts of water and the risks inher- 47 with a drag-reducing additive at a few percent by weight) Db fractured 48 are efficient in creating long fractures in the flow direction,that D 13 entbin its disposal. The productivity of hydraulically but 6 8 4 39 14 wells w less than 49 not in proppant transport due to a high rate of proppant segw also declines very quickly, and is usually much 15 would be expected considering the7 resources in the reservoir. 50 regation [158], (ii) gels and polymer-based fluids that lead to 9 ˙ = 6U/w, Figure 5: A schematic top view of5 a hydraulic fracture developing underthe the injection of energized fluids. The shear rate, 40 γ 16 Thus, there is a strong motivation to improve productivity 51 fractures that propagate less in the flow direction, more in the D 17 andb lifetime of fracturedthe wells, while reducing the quantity of and consequently the capillary number increases towards tip of the fracture, which leads to a higher bubble deformation and shear 52 direction perpendicular to the flow, and transport proppant well, 8 10 6 41 18 water consumed in the fracturing process. 53 but may damage the fracture surface [73], (iii) energized w and other chemicals fluids thinning behavior due to the severe shear localization. Different regimes show different orders of bubble deformation. The larger the 19 Foam fracturing fluids are promising in both respects, as a 54 (bubbly liquids) and foams that have a high effective viscosity 9 11 42 bubble deformation, the lower the 7effective viscosity. Note thatishere Dbby= 2Rb or carbon 55 and therefore 20 significant fraction of water replaced nitrogen behave more like gels, but avoid damage to the 21 dioxide. The use of foam fracturing surface and reduce water consumption [159, 221], and 10 fluids is well-established 56 fracture12 8 43 22 in the small percentage of important reservoirs that are under- 57 (iv) a mixture of the aforementioned fluids using methods like 23 pressured, where the quick and easy reverse hybrid [119], multi-stage [129], alternate11 clean-up offered by foams 58 hybrid [198], 13 9 44 24 is to justify the effort, specialized equipment and ex- 59 slug [127] fracturing. 1531 particle-particle friction may contribute to the force balance. vs. high viscosity for proppant transport onsufficient several occasions. 25 pertise required. Their wider use In under-pressurized geologic plays, energized fluids and 12 is hampered by logistical 60 14 10 45 loss shear Generally, forces increase and Gomaa and co-workers, in a series of papers, [74, 75,because 76] modern ex-1532 26 challenges, mainly hydraulic fracturing is these in- 61 foams serve better than otherboth fluids instorage all steps of the treatment, 27 creasingly applied in long horizontal wells with 20-40 stages 62 starting from drillingflowing [123, 106, 51], fracturing [159], proppant 13 15 1533 moduli [34]. Thus, proppant and depositing perimentally studied this hypothesis to28 delineate the reliable 11 46 in the film which require delivery of several hundred truckloads of liquid 63 transport [216, 40, 75, 230], and finally deep well cleaning [67]. rheological properties (other than effective viscosity) design 1534 between of antheaqueous foam may reach to a 29 nitrogen over a fewto days, or even volumes in the case 64 In bubbles terms16of geography, use of energized fluids and foams 14highernetwork 12 47 of carbon dioxide [? ]. However, these challenges are man- 65 is most prevalent in Canada and in severely water-stressed re1535 jamming condition (which is much lower than the usual value efficient proppant transport using both 3031polymer and surfactant ageable and foams can provide 15 productivity benefits beyond 66 gions of17the U.S. (mainly in New Mexico). Multiple studies, 13 48 32 under-pressured mainly improved proppantsolid 1536due to for spherical ψt[19], = 0.64, confinement efbased fluids. They concluded, as expected, that the reservoirs, fluid shear 67 see particles, e.g., Burke et al. Reynoldsdue et al. to [173], have been 33 transport and reduced water damage, fluids ∗ 16 see e.g., [159, 135]. In 68 published 18 discussing the performance of foam fracturing 14 49 34

1. Introduction

1

35

1. Introduction

.

<< 1

of carbon diox ageable and fo 1. Introduction . . under-pressure <1 ⇡1 << 1 transport and r . . . this review, w ⇡ 1 In the last decade, resources such <1 particularly ho . have become extremely important . mentation. th In the last decade, resourcesofsuch as shale gashydrauli and [20 tig velopment hydraulic fracturing ⇡1 In have become extremely important thanks gas to the successf in the use of natural has allowe . ture impermea velopment of hydraulic fracturing [200]. The resulting inc prosper, reducing In the last decade, resources such as shaleomy gas to and tight oileven while to prevent frac the use thanks of natural gassuccessful has Unitedhydraul States spite theseallowed positive have become extremely in important to the de- theaspects, placement with omy to prosper, evensial while its carbon footprint and reducing widely threatened by regulat velopment of hydraulic fracturing [200]. The resulting increase cause it contro positive aspects, hydraulic is contr has been banned several Th in the use of natural gasspite has these allowed the United States econ-infracturing of theareas. fractured sial reducing and widely by regulations, to point wh the footprint. consumption amounts of omy to prosper, even while its threatened carbon De-of large thethe conductivit has been banned in several areas. The reasons for this in ent inisits disposal. The productivity spite these positive aspects, hydraulic fracturing controverpant types have the consumption of amounts ofit[80]. water and theLiang risks wells also declines very quickly, ande andofwidely by regulations, to large the stop point [4] and fects) that assures a complete of where drainage This phemodulus, G , plays a key role in thesial design efficientthreatened propnomenon highly stable foams (possessing lifetimes pant transport. They showed that higher stor17 ent in itsresults disposal. productivity of50hydraulically fracr 19 The would be expected considering 15 fluids haspossessing been banned in several areas. Theinreasons for this include ids are: (i)the low 0 > G00 ) are a better choice to of years) [10, 99] with the formation of solid-like surface layage modulus than loss modulus (G 18 wells also of declines very quickly, is51usually less 20 and Thus, there isand a strong motivation to 16 the consumption of large amounts water the risks inherwith amuch drag-red suspend the proppant (decrease the rate of settlement). Indeed, ers that basically creates a firm network enhancing proppant 19 would be expected considering the resources in thewhil rese 21 lifetime fractured wells, disposal. The productivity of hydraulically fractured 52 are efficient in transport [184, 161]. Thisand hypothesis and of its practical applicawhen G00 > G0 proppant settles,17 andent thisinis its where the rate of 20 Thus, there is a strong motivation to improve the produc 22 water and other chemicals consumed 18 wells also declines very tion quickly, and elaboration is usually much less than needs more through performing experiments atproppan proppant segregation from the fractuirng fluid strongly depends 53 not in down-hole condition. on the fluid effective viscosity. In proppant is intendedconsidering 21 and lifetime of fractured wells, while reducing the quant 23 fracturing fluids are 19 practice, would be expected the resources inFoam the reservoir. 54 regation promi [158], to be transported far from the fracture opening (towards the tip, 22 water and other chemicals consumed in55the fracturing prop 24 significant fraction of fractures water is replac 20 Thus, there is a strong motivation to improve theOsmotic productivity that 5.4. Foam-Rock Interface and effects see Fig. 5), and retained suspended up until the pressure reFoam fracturing fluids promising indirection bothfracturing dioxide. The lifetime ofopen fractured wells, while reducing the are quantity of of The tendency of 25the aqueous foams to use pull water into itsrespects 56 foam perpe lease to keep a larger volume of21 theand fracture as much as23 structures (osmotic pressure) is a useful intrinsic behavior of 24 significant of isprocess. replaced nitrogen or c 26 fracturing in water the small percentage ofmay important 22 water and other chemicals consumedfraction in the possible. 57 by but damag foams in hydraulic fracturing treatments. Osmotic pressure creBased on the fracture geometry, towards thefracturing end of frac-25 dioxide. The use foam fracturing is well-establ 27 pressured, where the quick and easy 23 Foam fluids are promising inofboth respects, as afluids 58 (bubbly liquids ates a driving force for the foam to retain water and prevent ture, the width decreases and wall shear rate increases, there26 water in the small percentage of important reservoirs that are 28 nitrogen is sufficient to justify the specub 24 significant fraction of is replaced by or carbon 59 clearly andeffort, therefore its leakoff into the reservoir. This effect has been obfore, a foam element experiences a gradually increasing shearpressured, where quick and easy clean-up offered by fi 29the required. Their wider use 25 dioxide.of foams The use of 27foam fracturing fluids ispertise well-established served in laboratory experiments [183], although its origin was surface 60 fracture ing. In other words, the rheological behavior possibly not explained. Reducing water leakoff is critical in water senis sufficient justify the effort, specialized equipment moves from the solid-like (at the fluid-like28 of 30 challenges, mainly modernan 26 injection in the plane) smalltopercentage importanttoreservoirs that are under61 because (iv) a mixture sitive formations where clay particles (e.g. smectite) tend to (at the end of fracture) which is highly favorable for particle 29 pertise required. Their wider use is hampered by logi 31 creasingly longcan horizonta 27 pressured, where the quick and easy clean-up offered by applied foams in hybrid [198], r swell up to 30 times in volume [144, 87, 42, 220]. 62Swelling transport. Indeed, the magnitude of shear moduli can be altered 30 the challenges, mainly because modern hydraulic fracturing 32 which require delivery of several hu 28 is sufficient to justify effort, specialized equipment and ex63 slug [127] frac both block fractures directly and lead to the creation of fines using different additives and surfactants to accommodate differ31 creasingly applied in long horizontal wells with 20-40 that severely impact conductivity of the proppant pack [32]. 33 nitrogen over a few days, even hs 29 pertise required. Their wider use is hampered by logistical 64 Inorunder-p ent down-hole conditions [74]. Foam elasticity also enhances Of more general consequence, water leakoff leads to capillary with quality (gas volume fraction), which motivatesmainly the use of 32 which require deliveryfracturing of severalishundred truckloads 30 challenges, because modern hydraulic in- 65 foams serveof be blocking effects that reduce matrix permeability, even if the fordry foam to enhance proppant transport. Therefore, a careful nitrogen over wells a few with days,20-40 or even higher volumes in the 31 creasingly applied in33 long horizontal stages 66 starting mation is not especially water sensitive [140]. Finally, water from d consideration of foam behaviors under shearing may provide a which require delivery ofleakoff several hundred liquid 67 transport [216, increases water truckloads consumption. of This has a compounding powerful tool to design efficient32proppant transport scenarios. effect on water savings using foams, as not only the fraction of of ge 33 nitrogen over a few days, or even higher volumes in the case It is noteworthy that the addition of proppant itself to foams 68 In terms <1

1498 1499 1500 1501 1502 1503 1504 1505 1506 1507 1508 1509 1510 1511 1512 1513 1514 1515 1516 1517 1518 1519 1520 1521 1522 1523 1524 1525 1526 1527 1528 1529 1530

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may significantly alter their viscoelastic properties concluding1561 from the fact that particles (Brownian or non-Brownian) have1562 been used as stabilizing agent ([189, 61, 58]). The viscoelastic-1563 ity of aqueous foams itself arises from the bubble shape and1564 surface rigidity due to the surface tension, while by adding1565 particles, other forces like capillary attraction, repulsion and1566 1567

19

water present in the fracturing fluid is smaller, but also the total fluid injection rate can also be smaller when leakoff is reduced. Note that the leakoff rate of gas (N2 or CO2 ) from foams is observed to be less than that of the ambient phase [183]. This can be understood by the fact that ambient phase is, generally speaking, in direct contact with the rock surface (at least for water-wet rocks).

SPE SPExxxxxx xxxxxx ACCEPTED MANUSCRIPT Journal of Petroleum Science and Engineering XXX (2017) XXX-XXX

S.A. Faroughi et al.

(a)(a) Subcaption Subcaption 11

(b)(b

Figure Figure 9:9:Example Example ofof mu m

Conclusions Conclusions We Wepresented presentedananapplication applicationof... of... (a) (a)Subcaption Subcaption11 856856 yy==0.65% 0.65% 854854 855855

Figure Figure9:9:Example Exampleof ofmul mu

Acknowledgements Acknowledgements 854 854 Nomenclature Conclusions Conclusions 858 858 Nomenclature 857857

855855

We Wepresented presentedananapplication applicationof... of... y= y = state statevector vector yy==0.40% 0.40%

RI PT

856856

857857 858858

(b) (b)S

Acknowledgements Acknowledgements Nomenclature Nomenclature

mm== vector vectorofofmodel modelpara pa

p p== vector vectorofofprimary primaryvav

d d== vector vectorofofpredicted predictedd

y== state statevector vector dobs dyobs == vector vector ofofobserved observedd

mm== vector vectorofofmodel modelpara para

duc duc== vector vectorofofperturbed perturbedo

pp== vector vectorofofprimary primaryvar va

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g(m) g(m)== vector vectorofofpredicted predictedd

Fracture Width (m)

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1573 1574 1575 1576 1577 1578 1579 1580 1581 1582 1583 1584 1585 1586 1587 1588 1589 1590

dducuc== vector vectorofofperturbed perturbedoo Cg(m) CMD === vector cross-covariance cross-covariance bet MD g(m) = vector ofofpredicted predicted dbd

CCPD C == cross-covariance cross-covariance bet b CYPD state covariance covariancematr mat Y== state

CCYD CYD === cross-covariance auto-covariance auto-covarianceof ofp DD DD C = cross-covariance betw bet CMD C= === cross-covariance covariance covariancematrix matrix of CC cross-covariance betw bet D D MD

H H === cross-covariance augmented augmentedmatrix matrix fo CC = cross-covariance betw bet PD PD

CC = auto-covariance of ofpr p KK === auto-covariance Kalman Kalmangain gainmatrix matri DD DD

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6. Conclusions

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CYCY== state statecovariance covariancemat m

ddobs vectorofofobserved observedda d obs== vector

CYD CYD== cross-covariance cross-covariancebet b

CCDrD= covariance matrix of of r=== covariance correlation correlation matrix matrix fo Figure 6: The effect of bubble deformation on the effective viscosity of energized fluid used to create fractures. Here, the effective H H = = augmented augmented matrix matrix fo fo NdN= totalnumber numberofofobse ob d = total viscosity versus the width of the fracture at two different concentrations, 40% and 65% assuming ψt = 0.74 is plotted using Eq. (13).gain K K = = Kalman Kalman gain matrix matrix NnN= numberofofobserved observe n = number The effective viscosity decreases as width decreases (shear thinning). The width is inversely proportional to capillary number is an matrix rNr= ==that correlation correlation matrix fop N = number number of of model modelfor pa m m indicator showing bubble deformation. The inset shows a schematic 3D representation of a fracture where the width, w, is decreasing NNd d== total totalnumber numberof ofobser obse NN numberofofreservoir reservo p= p = number along its length. NNn n== number numberofofobserved observedd NyN= = size size of of the the state state vect ve y NNmm== number numberofofmodel modelpar pa NeN= numberofofensemble ensemb e = number N N = = number number of of reservoir reservoir s p p There is significant scope for future work on this topic, as1591 in the literature, but is of fundamental importance. is playing NgNIt = numberofofactive activegrig g = number N N = = size size of of the the state state vecto vect y y osmotic effects will depend on factors that influence surface1592 out in the marketplace at present, as we note increasing propOO == number likelihood likelihoodobjective objectiv d (m) dN(m) Ne e== proppant numberofofensemble ensemble tension, mainly surfactant choice, but also the presence and na-1593 pant loadings being used to compensate for the limited OO (m) (m) = = normalized normalized likelihoo likelih N NNN number numberofofactive active grid gri g g== however, ture of thickneners, dissolved salts, pH, etc. 1594 transport capacity of slickwater fracs. This approach, O O (m) (m) = = likelihood likelihood objective objective f d d 1595 has led to extremely high proppant and fluid consumption. It 859859 References References O O (m) (m) = = normalized normalized likelihoo likelihoo N N 1596 is also likely that the fractures created are wider than optimum, 1597

While energized fluid and foam fracturing are well estab-1598 lished in niche applications such as underpressured reservoirs,1599 their scope of application could be significantly enlarged fol-1600 lowing a better understanding of factors presented in this re-1601 view. As we discussed, the achievable reduction in water de-1602 mand is significantly higher than what is calculated simply from1603 the volume fraction of gas replacing water. Productivity im-1604 provements are expected due to enhanced proppant transport,1605 reduced water damage, and easier flowback. These all bene-1606 fit both the operators, by reducing their cost per unit produced,1607 and the environment by allowing the same level of production1608 with fewer wells, less traffic, and less consumption of resources1609 1610 (energy, proppant, water, etc.). In fracturing, there is a clear trade-off between the require-1611 ments of proppant transport and fracture growth. Successful1612 fracturing requires the simultaneous optimization of the length1613 and complexity of the fracture network and the delivery of prop-1614 pant within the network. This trade-off is not often mentioned1615

AC C

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dd== vector vectorofofpredicted predicteddd

20

860860 Aarra, Aarra,M.M.G., G.,Skauge, Skauge, A., A.,Solbakken, Solbakken,J.,and J.,&&Ormehaug, Ormehau and that fractures remain incompletely propped. Energized 861861 pressure. pressure.Journal JournalofofPetroleum PetroleumScience Scienceand andEngineerin Engineer References References foam fracturing fluids can overcome this difficulty due to their Aarra, Aarra, M. M.G., G.,that Skauge, Skauge, A., A.,Solbakken, Solbakken, J., J.,&&Ormehaug, Ormehaug, shear thinning rheological behavior ensures both fracture pressure. pressure.Journal JournalofofPetroleum PetroleumScience Scienceand andEngineering Engineerin propagation and proppant transport. In the case of foams, elasticity enhances suspension and proppant placement without energy compromise. Proper surfactant and gas selection enables foams to enhance elasticity while strongly reducing formation damage via osmotic effects. Foams can be better optimized than other fluids as their rheological properties are tuned by adjusting the volume fraction of ”gas” (actually, supercritical fluid), which can be controlled far more dynamically than, for example, the composition of a gel. This dynamic optimization of properties would require better control of volume fraction than typically practiced in the field, but appears feasible with proper instrumentation. Experimental studies of proppant transport by foams at high pressure are in progress in our laboratory [141]. This work is supporting development of more refined models of proppant transport. The logical next step is to combine this work with 859859 860860 861861

ACCEPTED MANUSCRIPT Journal of Petroleum Science and Engineering XXX (2017) XXX-XXX

S.A. Faroughi et al.

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rock mechanics models to support field trials in which proppant transport and fracture growth are both optimized . This path forward will require a collaborative approach (including both service companies and operators), which we are pursuing.

1620

Acknowledgements

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We would like to acknowledge the support of Air Liquide, the permission to publish this work and the input of our colleagues who commented on the draft.

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S.A. Faroughi et al.

Nomenclature βfoam

Isothermal foam compressibility (Pa−1 )

τm

Ambient fluid stress tensor

τw

Wall shear stress (Pa)

Isothermal gas compressibility

∆P

Pressure drop (Pa)

τy

Yield stress (Pa)

δ

Slip layer thickness (m)

ς

γ˙

Shape factor (m−1 )

Shear rate (s−1 )

ξ

Phase angle (rad)

Ca

Capillary number

Cacr

Critical capillary number

D

Pipe diameter (m)

Db

Bubble diameter (m)

DR

Bubble deformation factor

G

Shear modulus (Pa)

Apparent wall shear rate

γ

Shear strain

γ0 Eˆ

Shear strain amplitude

λ

Viscosity ratio of the dispersed fluid to the ambient fluid

hSb i

Average bubble surface area (m2 )

Packing efficiency parameter

Average bubble volume (m3 )

M AN U

hVb i

(s−1 )

SC

γ˙ w

RI PT

βgas

(Pa−1 )

G

00

Loss modulus (Pa)

G

0

Storage modulus (Pa)

Shear dynamic viscosity (Pa.s)

µ0

In-phase component of the shear viscosity (Pa.s)

G∗

Complex modulus

µ00

Out-of-phase component of the shear viscosity (Pa.s)

K

Consistency (Pa.sn )

µψ

Effective non-Newtonian viscosity (Pa.s)

KB

Boltzmann constant (J.K −1 )

µd

Dispersed fluid shear viscosity (Pa.s)

L

Pipe length (m)

µm

Ambient fluid shear viscosity (Pa.s)

N

Number of bubbles

µr

Relative non-Newtonian viscosity

n

Flow index

ω

Shear strain angular frequency (rad.s−1 )

P

Pressure (Pa)

γ˙ w

Non-Newtonian wall shear rate (s−1 )

Pe

P´eclet number

U

Fracture average inlet velocity (m.s−1 )

q

Flow rate (m3 .s−1 )

Π

Osmotic pressure (Pa)

r

Magnitude of the position vector (m)

ψ

Bubble volume fraction

Rb32

Sauter mean bubble radius (m)

ψt

Threshold packing limit

Rb

Bubble radius (m)

ρb

Bubble density (kg.m−3 )

Re

Reynolds number

ρm

Ambient fluid density (kg.m−3 )

S(ψ)

Total surface area of polyhedral bubbles (m2 )

σ

Surface tension (mN.m−1 )

S0

Total surface area of spherical bubbles (m2 )

τ

Shear stress (Pa)

Stk

Stokes number

τψ

Equivalent fluid stress tensor

T

Temperature (oC)

τb

Bubble stress tensor

w

Fracture width (m)

AC C

EP

TE D

µ

22

ACCEPTED MANUSCRIPT Journal of Petroleum Science and Engineering XXX (2017) XXX-XXX

References

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