Propagation regimes, transition times, and approximate universality in 2D hydraulic fracture propagation with fluid lag

Highlights

• Transition from early to intermediate time regime is strongly toughness-dependent.

• Transition from intermediate to late time regime is mildly toughness-dependent.

• An approximately universal behavior covers a large part of the MKO triangle.

• Setting of initial conditions for complex simulations can be made easier.

• A “symmetrized” parametrizationon of the MKO triangle is presented.

Abstract

During its lifetime, a hydraulic fracture is known to traverse a trajectory in a region of a parametric space of non-dimensional evolutionary parameters. The topology of this diagram depends upon the phenomena considered. For the specific case of a 2D-plane strain fracture propagating in an elastic solid on a straight path normal to the minimum compressive stress, with a constant rate of injection of an incompressible newtonian fluid, and without leak-off, the diagram is a triangle whose vertices are typically called O, M, and K. The non-dimensional parameters are the toughness $\mathcal{K}$ and remote stress $\mathcal{T}$ (monotonically increasing with time). At each point in the trajectory $\mathcal{P}\left(t\right)=(\mathcal{K},\mathcal{T})\left(t\right)$, the configuration of the fracture is essentially described by several non-dimensional variables, in this case the opening ${\Omega }_0$ and pressure ${\Pi }_0$ at the inlet, and the length $\gamma$. When fluid lag is considered, as in this case, a fourth variable (e.g., the fluid fraction ${\xi }_f$) can be appended to build the descriptive set ${\mathcal{F}}_0=\{{\Omega }_0,{\Pi }_0,\gamma ,{\xi }_f\}$. Various propagation regimes are observed across the MKO triangle.

As the main results, we: (1) provide specific, $\mathcal{K}$-dependent transition times among the propagation regimes; and (2) found that the transient evolutions of all propagating cracks with moderate values of the non-dimensional toughness ($\mathcal{K}\gtrsim 0.3$), from the OK edge to the MK edge, are contained in a thin bundle about a universal curve in the ${\mathcal{F}}_0$-space. This result can be applied, e.g., to readily setup approximate initial conditions for more detailed hydraulic fracture propagation simulations. In addition, we developed a four-parameter family of parametrizations of the MKO triangle suitable for plotting trajectories and other loci on the triangle.

Introduction

Fluid-driven fracturing is a phenomenon involved in a variety of geoscience and civil engineering problems and applications. These include magma-driven cracks [1], [2], soil remediation [3], [4], carbon sequestration [5] and cave preconditioning [6]. It has even been linked recently to biological processes [7]. But perhaps the most relevant application is in the stimulation of hydrocarbon reservoirs (fracking), mostly unconventional, to enhance productivity to economical levels [8]. Reservoir stimulation may be even associated with carbon sequestration [9]. Hydraulic fracturing has also been proposed for enhanced geothermal systems (EGSs) [10], [11], the Habanero Project in Australia probably being the most prominent as of today.

From the standpoint of modeling and simulation, a variety of approaches is found in the literature. Analytical and semi-analytical solutions were developed for simple cases like straight cracks propagating in 2D [12], [13], [14]. These solutions provide significant qualitative and quantitative insight into the process, set a reliable benchmark for comparison, and guide the development of other techniques like Finite Elements [15], [16], Extended Finite Elements [17], Displacement Discontinuity Method [18], etc., that can deal with more complex cases, including simultaneous and/or interacting fractures. When using the analytical solutions as a benchmark for other calculations, it is essential to ensure that the hypotheses involved correspond with each other. Similarly, analyses were performed for penny-shaped cracks propagating in 3D [19], [20], which also served as benchmarks for more complex simulations [21].

Two dimensional models typically provide an easier ground to look for analytical or numerical solutions than the three dimensional counterparts, capturing many of the common essential features but also at the expense of a poorer or different description of other phenomena. For instance, when dealing with the interaction between an incident hydraulic fracture and a preexisting fracture, 2D models can account for the effect of remote stresses and interaction angle [22], [23], but they cannot provide for the mechanism of bypassing the fracture by surrounding it.

Another case in point, which relates specifically to the topic of the present work, is the analysis of the complete lifetime of a straight propagating crack, when a region about the crack tip not invaded by the fluid is allowed for. This so-called fluid lag was the subject of some debate in the literature since the proposal by Khristianovic and Zheltov [13], [24]. Later on, it was directly observed in the laboratory [25], [26], where a low confining stress that promotes the existence of a sizeable fluid lag was applied. This was also observed in field experiments [27]. The evolution of a plane strain fracture with fluid lag is known to be controlled by two non-dimensional parameters, only one of which evolves in time [28], [29]. In connection with this parameter, a time scale can derived. On the other hand, the evolution of a penny-shaped crack with fluid lag is controlled by two evolutionary parameters, which in turn give rise to two different time scales [30]. This phenomenology is well understood at this point. Nevertheless, few models endeavored inclusion of the fluid lag, either under rather controlled circumstances as straight propagation [28], [29], [30], [31] or under more general conditions [32], [33], [34], [35]. Overall, results for the 3D case are scarcer than those for 2D.

Several works have provided before solutions that can be used as practical methods to evaluate the evolution of some variables [13], including their usage as a starting point for more elaborate solutions [29]. In particular, Garagash [29] focused on the early-time of the fracture life. In the same spirit, we perform here an analysis of the evolution of the fundamental variables across the whole life of a fracture, covering the range from large to small fluid lag with various propagation regimes. We explore the validity of an approximate universal behavior, which was described before exclusively for the early and late time regimes [29]. We extend here the result to a larger region in the domain of system parameters. In addition, we provide quantitative time bounds for the applicability of each of the propagation regimes, and a method for the quantitative mapping of related quantities.

In Section 2 we describe the basic equations of the model, which are in line with standard literature. In Section 3 we present some essential results on the structure of the solution that motivates the analysis of this paper, shown in Section 4. In Section 5 we summarize the conclusions.