A model of non-Newtonian slurry flow in a fracture
Graphical abstract
An accurate calculation of a non-Newtonian slurry flow in a fracture is an important issue for fracture design. A model of such a flow based on the kinetic theory of granular media has been developed. The model allows reliable estimates of such important parameters as the pressure gradient determining a fracture width and the solids concentration distribution across a fracture that strongly affects the proppant settling rate.
Introduction
Hydraulic fracturing is an important technology for stimulating wells to increase the rate of production of oil or gas. The principle is simple. A slurry consisting of a viscous fluid (usually an aqueous solution of a natural polymer such as guar) and near-spherical particles (proppant or sand) is pumped into a cased wellbore at high pressure. It flows into the reservoir through perforations in the casing creating a large planar fracture, provided that the fluid pressure at the perforations is sufficiently high to overcome the least principal earth stress (the closure pressure) in the reservoir. As slurry flows along the fracture the fracture widens and propagates; frictional (net) pressure drop in the flowing slurry means that the fluid pressure decreases from perforations to the fracture tip. The geometry of a growing fracture is dependent on this pressure distribution. When pumping ceases, most of the slurry fluid left in the fracture leaks off into the reservoir rock bordering the fracture walls, and the fracture closes onto the proppant left in the fracture. This leaves a narrow permeable channel (typically about 10 mm wide) between the walls of the fracture (which can be several hundred meters long and is typically 50 to 100 m high). Fig. 1 shows a horizontal section of a typical fracture. The propped fracture is much more permeable than the reservoir rock around it because the proppant particles are much larger than the grains of the rock. This large increase in effective reservoir conductivity near the wellbore leads to a large increase in well productivity.
Details about the modeling and design of hydraulic fractures can be found in [1]. Various computational codes have been developed for the necessary calculations. The predictions made are only as good as the models they use. A disappointingly high proportion (30%) of fracturing jobs fail in one sense or another. This paper provides an improved model for calculating the transport of proppant and associated pressure gradients within the fracture, consistent with the framework presented in [2].
Section snippets
Model
Slurry flows moving in fractures during fracturing procedures are characterized by high mean shear rates (up to γ˙m = um / w = 200 1/s), where um is the mean (superficial) slurry velocity and w is the fracture width (see Fig. 2). In our analysis we will consider a steady-state slurry flow in a channel of a constant width. We do not consider the leak-off impact on slurry dynamics. This assumption corresponds to the real situation when the formation permeability is relatively low. It is also important
Numerical examples and discussion
The model equations were integrated numerically. Note that Eqs. (6), (22), (24), (27) are the governing equations while the other equations composing the model represent the boundary conditions and the constitutive relations. The shooting (iterations) method was used to satisfy both the boundary condition in the channel center for Eq. (6) expressed by Eq. (8) (zero granular temperature gradient), and the boundary condition for Eq. (22) presented by Eq. (23) (the given solids mass flow rate per
Model validity and practical applications
The numerical results obtained by the model developed look plausible. Nevertheless, there is an important issue. Our calculations showed that the condition of applicability of the kinetic theory formulated above as the ratio of the particle relaxation time to the time of the mean particle free path between collisions, which should be equal to or bigger than 2 [3], was satisfied only in the near-wall regions where the granular temperature was relatively high (see Fig. 4, Fig. 7, Fig. 9, Fig. 11
Conclusions
A model of slurry flow in a fracture taking into account particle dynamics on the micro-level has been developed. The computations have shown that to a significant extent the slurry dynamics is governed by particle fluctuations generated in a high shear-rate flow. Particles migrate from high shear-rate zones at the fracture walls towards the fracture center therefore a slurry flow is characterized by non-uniform concentration distribution across a fracture. One of the important conclusions is
Nomenclature
- c
Volume particle concentration
- cm
Mean solids volume concentration
- ds
Particle diameter, m
- F
Hydrodynamic force acting on a fluctuating particle in a fluid, N
- f(c)
Ratio of the slurry viscosity to the viscosity of a carrying liquid
- g0
Radial distribution function
- JL
Liquid mass flow rate per unit of a fracture height, kg/(m s)
- Js
Solids mass flow rate per unit of a fracture height, kg/(m s)
- kn
Particle–particle restitution coefficient
- K(c)
Coefficient
- kΘ
Granular conductivity, W s2/m3
- L
Fracture half-length, m
- ms
Symbol used.
Acknowledgment
The authors are grateful to Prof. J.R.A. Pearson (Schlumberger Cambridge Research Center) for the constructive advices and discussion of this research.
References (15)
On suspension transport in a fracture: framework for a global model
J. Non-Newton. Fluid Mech.
(1994)Modeling dilute gas-particle flows in horizontal channels with different wall roughness
Chem. Eng. Sci.
(2005)Settling and slumping in a newtonian slurry, and implications for proppant placement during hydraulic fracturing of gas wells
Chem. Eng. Sci.
(1995)- et al.
Reservoir Stimulation
(2000) - et al.
Rheology of suspensions with high particle inertia and moderate fluid inertia
J. Fluid Mech.
(2003) Multiphase Flow and Fluidization: Continuum and Kinetic Theory Descriptions
(1994)- et al.
Pressure-driven flow of suspensions: simulation and theory
J. Fluid Mech.
(1994)
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2019, Journal of Petroleum Science and EngineeringCitation Excerpt :These effective viscosity relationships are thus excluded from further study here. The relationships postulated by Eskin and Miller (2008), Eilers (1941), and Krieger and Dougherty (1959) incorporate the expected behaviours in the limit and will be further studied. Numerical methods don't constrain the solution from exceeding the physically admissible limits, so an additional algorithm is implemented in Section 3.6 to explicitly constrain the problem.