The effects of proppant on the near-front behavior of a hydraulic fracture

https://doi.org/10.1016/j.engfracmech.2020.107110Get rights and content

Highlights

  • Effects of proppant slurry near the tip of a hydraulic fracture are quantified.

  • Proppant effectively increases the apparent fracture toughness.

  • Larger proppant volume and smaller particle radius amplify the effect.

  • Fluid filtration plays an important role for large leak-off.

  • Bridging criterion does not always significantly influence the result.

Abstract

The qualitative and quantitative effects of proppant on the near-front region of a hydraulic fracture are investigated by solving the problem of a semi-infinite fracture driven by a slurry of Newtonian fluid with proppant. The results demonstrate that proppant bridging and slurry dehydration cause proppant accumulation near the crack front and, in turn, a pressure difference over the proppant pack that leads to significant fracture widening. Specific attention is paid to the sensitivity of the solution to proppant volume, particle radius, the permeability of the proppant pack, and bridging criterion.

Introduction

Unconventional reservoirs are a considerable source of oil and gas. However, before the advent of hydraulic fracturing technology, it was difficult to extract the hydrocarbons from within them because they are comprised of low permeability rocks. Unconventional oil and gas production has significantly improved over the last decades due to a combination of hydraulic fracturing and horizontal drilling techniques [21], [3]. The introduction of multiple cracks from a wellbore creates paths along which the hydrocarbons can move from within the matrix to the wellbore and, in turn, can be extracted from the underground. The injected fluid usually contains proppants, granular materials such as sands and ceramics, which prevent complete closure of the cracks after the pumping pressure is released, thus leaving conductive paths for hydrocarbons to flow to the surface. The review of fracturing fluids and their rheology in application to hydraulic fracturing can be found in [2].

The global behavior of a hydraulic fracture (HF) is dominated by the processes occurring near its front. This region can be modeled as a semi-infinite crack in a linear elastic medium propagating steadily under plane strain conditions [40]. Because the problem is two-dimensional, the crack front will henceforth be referred to as the crack tip. A wealth of research has shown that the extension of an HF is dominated by three distinct processes that occur within three regions in the wake of the crack. These processes are characterized as being dominated by either the rock fracture toughness, the fluid viscosity, or fluid leak-off [14]. The solution for the toughness dominated regime originates from linear elastic fracture mechanics (LEFM), a one-parameter theory that dictates crack extension occurs when the stress intensity factor reaches a value equal to the fracture toughness of the matrix material [41]. The solution for the viscosity dominated regime was obtained in [13] and for the leak-off dominated regime in [32]. Both solutions were derived for power-law fluid rheology. A numerical solution that captured all three phenomena for a Newtonian fluid was obtained in [23]. A more efficient computational procedure based on an accurate approximate solution for the general problem was developed in [16]. The approach was based on the removal of the fluid pressure singularity along the crack front, which allowed the use of standard numerical techniques. The same methodology was subsequently applied to solve the general problem that accounts for fluid viscosity, fluid leak-off, and fracture toughness for power-law fluid rheology in [18] and for Herschel-Bulkley fluid rheology in [6]. However, the aforementioned studies of the HF’s near-tip behavior did not include the effects of proppant. An attempt to account for proppant was made in [5] under the assumption that the region in the wake of the crack tip that extends to the fracture is filled with proppant at the maximum concentration, and that the fluid filtrates through the packed proppant according to Darcy’s equation. In this paper, we generalize the formulation by representing the slurry flow within the HF as a multi-phase flow of Newtonian fluid containing a proppant whose concentration is treated as an unknown variable.

Strategies for modeling slurry flows, otherwise called granular suspension flows, include the diffusive flux method [31] and the suspension balance model [35], [39], [34]. The suspension balance model consists of the balance equations for fluid and particles and accounts for the shear and normal stresses exerted by the particles. Proppant transport has also been modeled using the kinetic theory of granular flows [25], [22], in which a statistical particle velocity distribution is assumed and the concept of granular temperature is used to describe slurry flow using statistics (similar to the kinetic theory of gases). Models of proppant transport inside an HF can be characterized using either single-phase or multi-phase approaches, see e.g., review [36] and references therein. The single-phase approach uses the fact that the addition of proppant to the fracturing fluid effectively increases the viscosity of the mixture, and that apart from settling, particles move together with the carrier fluid. This scenario can be accounted for through constitutive relations for viscosity that depend on proppant concentration [1]. In the multi-phase approach, the granular phase and the carrier fluid are allowed to move separately [22], [7]. Both single-phase and multi-phase approaches require the use of phenomenological constitutive equations to model the effect of particles in the slurry [12], [17]. Generally, a constitutive model relates suspension stress to kinematics depending on particle concentration to take into account (1) suspension thickening and (2) particle contribution to suspension stress tensor [26]. To obtain constitutive relations, a combination of experimental work and computational modeling is necessary. The broad review of the constitutive modeling of dense suspensions was performed in [45].

The most recent and accurate semi-empirical constitutive relations for slurry were obtained in [8] and were used to study slurry flow in a channel using multi-phase methods [30], [19]. Also, the accounting for the slip velocity between the phases in [19] allowed the transition between Poiseuille flow for clear fluid (or fluid with vanishingly small particle volume fractions) and Darcy filtration through the packed proppant (once the particle concentration reaches the packing limit).

Another relevant phenomenon in proppant transport is referred to as proppant bridging or tip screen-out [24], [11], [20], [2]. Proppant bridging occurs when a proppant carried by the fracturing fluid creates a bridge of several particles across the fracture by blocking the flow of the slurry. This condition near the fracture tip results in the formation of a proppant pack that prevents the fracture from propagating. If this happens, an increase in pressure is required to continue the crack propagation – tip screen-out. Here we use a classical approach to account for bridging based on the assumption that the proppant bridges near the fracture tip when the fracture width divided by the proppant diameter becomes smaller than a prescribed amount, which is varied to simulate the effect of different bridging criteria and their influence on the solution.

The focus of this paper is to elucidate the effects of proppant on the near-front region of an HF propagating with a constant velocity without fluid lag. The problem is formulated for a slurry flow of proppant particles and a Newtonian fluid inside the fracture. We use a proppant transport model that accounts for the transition between Poiseuille flow for clear fluid and Darcy flow for highly concentrated proppant suspensions [19] and is based on the experimental work of [8]. The paper is organized as follows. Section 2 outlines the problem statement. Section 3 formulates the governing equations for the semi-infinite fracture with proppant slurry and the model for proppant transport. To calculate the numerical solution for the problem with proppant, we employ a non-singular formulation [16], which is adapted to the current problem and summarized in Section 4. Section 5 specifies limiting solutions for the HF without proppant to compare with the numerical results for the problem with proppant provided in Section 6. A summary and conclusions are discussed in Section 7.

Section snippets

Problem statement

The near-tip region of a hydraulic fracture is represented by a semi-infinite crack in a state of plane strain [40], as shown schematically in Fig. 1. Fig. 1a shows the fracture in the coordinate system (x̂,y) centered at the tip. Crack propagation is accounted for by allowing the front to move with a constant velocity V. The problem is solved in the moving coordinate system (x,y), in which the x coordinate is defined by: x=Vt-x̂ (Fig. 1b). Note that the positive values of x lie inside the

Governing equations

The equations governing the crack propagation include an elasticity equation, mass balance equations for proppant and slurry, and a fracture propagation criterion. To shorten mathematical expressions, all equations are cast in terms of the following scaled material parameters,E'=E1-ν2,K'=32πKIc,μ'=12μ,C'=2CL,where E and ν are Young’s modulus and Poisson’s ratio, KIc is the (rock) fracture toughness, μ is the intrinsic viscosity of the carrier Newtonian fluid, and CL is Carter’s leak-off

Scaled form of governing equations

In this section, the governing equations are written using a non-singular formulation similar to [16], [18], [6]. To decrease the number of unknowns and to be consistent with previous works, the equations are formulated in terms of the following dimensionless quantities,w~=wE'K'x1/2,x~=xl1/2,s~=sl1/2,x~0=x0l1/2,x~1=x1l1/2,l=K'3μ'VE'22,χ=2C'E'V1/2K',a~=aE'K'l1/2=aE'3μ'VK'4,V~p=VpE'2K'l3/2=VpE'7μ'3V32K'10,where the scaled parameters are referred to as follows: w̃ is fracture width, x̃ is the

Asymptotic solutions for fracture opening

It is useful to compare the effects of proppant on the different regimes of fracture propagation with those associated with the limiting solutions for the semi-infinite hydraulic fracture problem without proppant. For a Newtonian fluid, there are three such limiting (asymptotic) solutions,wk=K'E'x1/2,wm~=βm~4μ'2VC'2E'21/8x5/8,wm=βmμ'VE'1/3x2/3,where βm̃=4/(150.25(2-1)0.25) and βm=21/335/6 are constants. Here wk denotes the fracture opening solution for the “dry” or toughness dominated fracture

Numerical results

Integral Eq. (29), the concentration Eq. (33) and the proppant volume Eq. (37) are solved simultaneously for three unknowns: w~(x~),ϕ¯(x~),x~1. First, integral in (29) is discretized using Simpson’s rule. Then, the discretized Eq. (29) is solved using Newton’s method with an initial guess for x̃1 and with an initial guess for w̃ corresponding to a solution without proppant. At each iteration, the proppant concentration ϕ¯(x̃) and the proppant boundary x̃1 are recalculated using the updated w̃.

Summary

The paper presented an extensive study of the effects of proppant of varied volume and particle radius on the near-tip behavior of a hydraulic fracture. The near-tip region of the crack is modeled as a semi-infinite fracture that propagates with a constant velocity.

The main conclusions are summarized as follows:

  • It was determined that, when plotted on a logarithmic scale, the effects of proppant on the fracture opening appear to be localized, i.e., the solution with proppant is similar to the

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

The support of the University of Houston in the form of a research assistantship to A.B. is acknowledged.

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