Numerical modeling of hydraulic fracturing is complicated when a fracture reaches a stress barrier. For high barriers, it may require changing of a computational scheme. Despite there are examples of modeling propagation through barriers, there is no general theory clarifying when and why conventional schemes may become inefficient, and how to overcome computational difficulties. The paper presents the theory and practical recommendations following from it. We start from the definition of the barrier intensity, which exposes that the barrier strength may change from zero for contrast-free propagation to infinity for channelized propagation. The analysis reveals two types of computational difficulties caused by spatial discretization: (i) general arising for fine grids and aggravated by a barrier; and (ii) specific, caused entirely by a strong barrier. The asymptotic approach which avoids spatial discretization is suggested. It is illustrated by solving bench-mark problems for barriers of arbitrary intensity. The analysis distinguishes three typical stages of the fracture penetration into a barrier, and provides theoretical values of the Nolte-Smith slope parameter and arrest time as functions of the barrier intensity. Special analysis establishes the accuracy and bounds of the asymptotic approach. It appears that the approach provides physically significant and accurate results for fracture penetration into high, intermediate and even weak stress barriers. On this basis, simple practical recommendations are given for modeling hydraulic fractures in rocks with stress barriers. The recommendations may be promptly implemented in any program using spatial discretization to model fracture propagation.
In this paper we demonstrate the influence of the pore pressure to the development of a hydraulically-driven fracture in a poroelastic medium. We present a novel numerical model for propagation of a planar hydraulic fracture and prove its correctness by demonstration of the numerical convergence and by comparison with known solutions. The advantage of the algorithm is that it does not require the distinguishing of the fracture's tips and reconstruction of the numerical mesh according to the fracture propagation. Next, we perform a thorough analysis of the interplay of fluid filtration and redi…
It is usually assumed that hydraulic fracture has two symmetrical wings with respect to the fluid injection point. Hence, authors limit themselves to modelling of one half of the fracture. In our work we demonstrate that the case of a symmetrical fracture occurs only in a homogeneous reservoir with constant physical parameters and confining {\it in situ} stress. Otherwise, inhomogeneity in the stress or the rock permeability can significantly change the dynamics of the fracture propagation. The mathematical model of the hydraulic fracturing used in the paper is adopted from our earlier work \c…
Understanding the behavior of fluid-driven hydraulic fracture in naturally fractured reservoirs is crucial in the development of geothermal energy and unconventional reservoirs. This study presents a fully-coupled numerical model to investigate hydraulic propagation in naturally fractured reservoirs, along with a comprehensive discussion on field observations. The results indicate that, contrary to our common beliefs, "Complex Fracture" rather than "Fracture Networks" is the norm in these naturally fractured reservoirs
The input of the hydraulically induced shear traction into the elasticity equation is estimated for the entire fracture surface. It is established that, except for negligibly small vicinities of a pointed source and of the fluid front, the relative input of the viscous shear does not exceed 10 - 4 of the input of the conventionally accounted term. The estimation is true for Newtonian, as well as non-Newtonian thinning fluids. This implies that there is no need to account for viscous shear not only in the form of conventional near-front asymptotics and fracture conditions, but also in factors e…
The paper presents a method for solving hydraulic fracture problems accounting for the lag. The method consists in matching the outer (basic) solution neglecting the lag, with the inner (auxiliary) solution of the derived 1D integral equation with conditions, accounting for the lag and asymptotic behavior of the opening and the net-pressure. The method refers to practically important cases, when the influence of the local perturbation, caused by the lag, becomes insignificant at a distance, where the leading plane-state asymptotics near the fracture front is still applicable. The universal asy…
The paper presents a solution for axisymmetric propagation of a penny-shaped crack driven by a thinning fluid. The solution to the accuracy of four significant digits, at least, is obtained on the basis of the modified formulation of hydraulic fracture problem by employing the particle velocity, rather than conventionally used flux. This serves to properly organize iterations in the opening after reducing the problem to the self-similar form. Numerical results obtained show relatively small dependence of self-similar quantities (fracture radius, propagation speed, opening, particle velocity, p…
Fluid driven fractures propagate in the upper earth crust either naturally or in response to engineered fluid injections. The quantitative prediction of their evolution is critical in order to better understand their dynamics as well as to optimize their creation. We present a Python implementation of an open-source hydraulic fracture propagation simulator based on the implicit level set algorithm originally developed by Peirce & Detournay (2008) -- "An implicit level set method for modeling hydraulically driven fractures". Comp. Meth. Appl. Mech. Engng, (33-40):2858--2885. This algorithm coup…
Strong interaction of closely located, nearly parallel hydraulic fractures and its influence on their propagation are studied. Both computational and physical aspects of the problem are considered. It is shown that from the computational point of view, when a distance between cracks is small as compared with their sizes, the system becomes ill-conditioned and numerical results deteriorate. The physical consequence of the interaction consists in decreasing of the crack opening and even greater decrease of conductivity. Then the resistance to fluid flow grows what results in the propagation of o…
The paper presents universal asymptotic solution needed for efficient modeling of hydraulic fractures. We show that when neglecting the lag, there is universal asymptotic equation for the near-front opening. It appears that apart from the mechanical properties of fluid and rock, the asymptotic opening depends merely on the local speed of fracture propagation. This implies that, on one hand, the global problem is ill-posed, when trying to solve it as a boundary value problem under a fixed position of the front. On the other hand, when properly used, the universal asymptotics drastically facilit…
In this paper we analyse a problem of a hydraulic fracture driven by a non-Newtonian shear-thinning fluid. For the PKN fracture geometry we consider three different rheological models of fluid: i) the Carreau fluid, ii) the truncated power-law fluid, iii) the power-law fluid. For each of these models a number of simulations are performed. The results are post-processed and compared with each other in order to find decisive factors for similarities/dissimilarities. It is shown that under certain conditions even the basic power-law rheology can be a good substitute for the Carreau characteristic…
We analyse a problem of a hydraulic fracture driven by a non-Newtonian shear-thinning fluid. Fluid viscosity is described by the four-parameter truncated power-law model. By varying the parameters of the rheological model we investigate spatial and temporal evolution of fluid flow inside the crack and the resulting fracture geometry. A detailed quantitative and qualitative analysis of the underlying physical phenomena is delivered. The results demonstrate that rheological properties of fluids significantly affect the process of hydraulic fracture not only by the values of viscosity, but also b…
The paper aims to answer the question: if and how non-Newtonian fluids may be compared in their mechanical action when used for hydraulic fracturing? By employing the modified formulation of the PKN problem we obtain its simple analytical solutions in the cases of perfectly plastic and Newtonian fluids. Since the results for shear thinning fluids are intermediate between those for these cases, the obtained equation for the fracture length suggests a criterion of the equivalence of various shear thinning fluids for the problem of hydraulic fractures. We assume fluids equivalent in their hydrofr…
Natural open joints in rocks commonly present multi-scale self-affine apertures. This geometrical complexity affects fluid transport and heat exchange between the flow- ing fluid and the surrounding rock. In particular, long range correlations of self-affine apertures induce strong channeling of the flow which influences both mass and heat advection. A key question is to find a geometrical model of the complex aperture that describes at best the macroscopic properties (hydraulic conductivity, heat exchange) with the smallest number of parameters. Solving numerically the Stokes and heat equa- t…
These notes address the mechanics of propagating hydraulic fractures (HF). In the first part, we focus on how different physical mechanisms (dissipation in fluid and solid, fluid storage in fracture and its exchange with permeable rock, crack elasticity) manifest near the fracture tip and lead to a `zoo' of fracture propagation regimes, as correspond to the different coupling of predominant mechanisms depending on material parameters and fracture propagation speed. In the second part, we illustrate how different near-tip regimes dictate the propagation of finite fractures of simple geometries …
High flow rate, water-driven hydraulic fractures are more common now than ever in the oil and gas industry. Although the fractures are small, the high injection rate and low viscosity of the water, lead to high Reynolds numbers and potentially turbulence in the fracture. Here we present a semi-analytical solution for a blade-shaped (PKN) geometry hydraulic fracture driven by a turbulent fluid in the limit of zero fluid leak-off to the formation. We model the turbulence in the PKN fracture using the Gaukler-Manning-Strickler parametrization, which relates the the flow rate of the water to the p…
Accurately resolving the coupled momentum transfer between the liquid and solid phases of complex fluids is a fundamental problem in multiphase transport processes, such as hydraulic fracture operations. Specifically we need to characterize the dependence of the normalized average fluid-particle force $\langle F \rangle$ on the volume fraction of the dispersed solid phase and on the rheology of the complex fluid matrix. Here we use direct numerical simulations (DNS) to study the creeping flow ($Re\ll 1$) of viscoelastic fluids through static random arrays of monodisperse spherical particles us…
4D acoustic imaging via an array of 32 sources / 32 receivers is used to monitor hydraulic fracture propagating in a 250~mm cubic specimen under a true-triaxial state of stress. We present a method based on the arrivals of diffracted waves to reconstruct the fracture geometry (and fluid front when distinct from the fracture front). Using Bayesian model selection, we rank different possible fracture geometries (radial, elliptical, tilted or not) and estimate model error. The imaging is repeated every 4 seconds and provide a quantitative measurement of the growth of these low velocity fractures.…
This work is focused on the development of a dynamic criterion for the arching and bridging of spherical particles in a 3D suspension flow through a channel with plane walls. Elasticity of the particles and the channel walls are taken into account. The carrier fluid is viscous and incompressible. Bridging occurs under the balance of the hydrodynamic force exerted from the fluid on the particles and the friction force exerted from the walls on the particles. The 3D motion of particles in fluid is analyzed by means of direct numerical simulation. The bridging criterion is formulated as a domain …
Hydraulic fractures propagating at depth are subjected to buoyant forces caused by the density contrast between fluid and solid. This paper is concerned with the analysis of the transition from an initially radial towards an elongated buoyant growth -- a critical topic for understanding the extent of vertical hydraulic fractures in the upper Earth crust. Using fully coupled numerical simulations and scaling arguments, we show that a single dimensionless number governs buoyant hydraulic fracture growth: the dimensionless viscosity of a radial hydraulic fracture at the time when buoyancy becomes…
The paper treats the propagation of a hydraulically driven crack. We explicitly write the local speed equation, which facilitates using the theory of propagating interfaces. It is shown that when neglecting the lag between the liquid front and the crack tip, the lubrication PDE yields that a solution satisfies the speed equation identically. This implies that for zero or small lag, the boundary value problem appears ill-posed when solved numerically. We suggest e - regularization, which consists in employing the speed equation together with a prescribed BC on the front to obtain a new BC formu…
The quasi-brittle nature of rocks challenges the basic assumptions of linear hydraulic fracture mechanics (LHFM): linear elastic fracture mechanics and smooth parallel plates lubrication fluid flow. We relax these hypotheses and investigate the growth of a plane-strain hydraulic fracture in an impermeable medium accounting for a rough cohesive zone and a fluid lag. In addition to a dimensionless toughness and the time-scale of coalescence of the fluid and fracture fronts as in the LHFM case, the solution now also depends on the in-situ-to-cohesive stress ratio and the intensity of the flow dev…
The paper presents analytical solution for hydraulic fracture driven by a non-Newtonian fluid and propagating under plane strain conditions in cross sections parallel to the fracture front. Conclusions are drawn on the influence of the fluid properties on the fracture propagation.
Propagation of a fluid-driven crack in an impermeable linear elastic medium under axis-symmetric conditions is investigated in the present work. The fluid exerting the pressure inside the crack is an incompressible Newtonian one and its front is allowed to lag behind the propagating fracture tip. The tip cavity is considered as filled by fluid vapors under constant pressure having a negligible value with respect to the far field confining stress. A novel algorithm is here presented, which is capable of tracking the evolution of both the fluid and the fracture fronts. Particularly, the fracture…
This research has found a novel computational efficient method of modelling flow at low Reynolds number through fracture networks. The numerical analysis was performed by connecting Hele-Shaw cells to investigate the effect of the intersection to the pressure field and hydraulic resistance for given inlet and outlet pressure values. In this analysis, the impact of intersecting length, intersecting angle and fracture aperture on the fluid flow was studied. For this purpose, two models with different topologies were established. The Hele-Shaw simulation results for hydraulic resistance, pressure…
The paper presents novel results, obtained on the basis of the modified theory of hydraulic fractures (HF). The theory underlines significance of the speed equation. When applied to numerical simulation of HF, the theory reveals three distinct issues: (i) modeling the central part of a HF; (ii) modeling the near-front zone; and (iii) tracing changes in the shape of a fracture contour. Modeling the central part leads to a stiff system of ODE in time, what strongly complicates its integration. For explicit schemes, it requires small time steps to meet the CFL condition. For implicit schemes, it …
Deep saline aquifers are promising geological reservoirs for CO2 sequestration if they do not leak. The absence of leakage is provided by the caprock integrity. However, CO2 injection operations may change the geomechanical stresses and cause fracturing of the caprock. We present a model for the propagation of a fracture in the caprock driven by the outflow of fluid from a low-permeability aquifer. We show that to describe the fracture propagation, it is necessary to solve the pressure diffusion problem in the aquifer. We solve the problem numerically for the two-dimensional domain and show th…
The paper contains an extension of the Khristianovich-Geertsma-de Klerk (KGD) model to the case when the confining rock pressure, which closes a hydraulic fracture, varies in the direction of its propagation. The extension is impelled by the need to simulate fracture hampering (acceleration) when it penetrates into a layer with increased (decreased) rock pressure. The paper presents the problem formulation, an efficient numerical method for its solving, examples of fractures propagating through layers with various stresses and general conclusions. It is established that when the fracture enter…
Magma-driven fractures are the main mechanism for magma emplacement in the crust. A fundamental question is how the released fluid controls the propagation dynamics and fracture geometry (depth and breadth) in three dimensions. Analog experiments in gelatin have shown that fracture breadth remains nearly stationary when the process in the fracture head (where breadth is controlled) is dominated by solid toughness, whereas viscous fluid dissipation is dominant in the fracture tail. We model propagation of the resulting buoyant, finger-like fracture of stationary breadth with a slowly varying op…
A universal particle velocity based algorithm for simulating hydraulic fractures, previously proposed for Newtonian fluids, is extended to the class of shear-thinning fluids. The scheme is not limited to any particular elasticity operator or crack propagation regime. The computations are based on two dependent variables: the crack opening and the reduced particle velocity. The application of the latter facilitates utilization of the local condition of Stefan type (speed equation) to trace the fracture front. The condition is given in a general explicit form which relates the crack propagation …
