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The prescription of a simple and robust parameterization for calving is one of the most significant open problems in ice sheet modelling. One common approach to modelling of crevasse propagation in calving in ice shelves has been to view crevasse growth as an example of linear elastic fracture mechanics. Prior work has however focused on highly idealized crack geometries, with a single fracture incised into a parallel-sided slab of ice. In this paper, we study how fractures growing from opposite sides of such an ice slab interact with each other, focusing on different simple crack arrangements: we consider either perfectly aligned cracks, or periodic arrays of laterally offset cracks. We visualize the dynamics of crack growth using simple tools from dynamical systems theory, and find that aligned cracks tend to impede each other's growth due to the torques generated by normal stresses on the crack faces, while periodically offset facilitate simultaneous growth of bottom and top cracks. For periodic cracks, the presence of multiple cracks on one side of the ice slab however also generates torques that slow crack growth, with widely spaced cracks favouring calving at lower extensional stresses than closely spaced cracks.

The goal of this paper is to develop an enhanced pseudo-3D (P3D) model for hydraulic fracturing (HF), whose predictions are more accurate compared to that of the original P3D model, but which requires significantly less computational resources than a fully planar HF simulator. We show that the lack of viscous resistance in the height growth and the local approximation in the computation of elastic interactions, which precludes the incorporation of lateral toughness, are the primary weaknesses of the original P3D model that considers symmetric stress barriers and no leak-off. To account for the viscous resistance, an apparent fracture toughness is introduced. The apparent toughness is calibrated using a one-dimensional HF model resulting in an approximate expression that captures all regimes of propagation. To incorporate non-local elastic interactions, the fracture opening in every vertical cross-section is approximated by a plane-strain solution, and then the 2D elasticity interaction integral is evaluated. To increase the computational efficiency, this 2D integral is further approximated by two one-dimensional integrals. The use of non-local elasticity allows us to include the asymptotic solution in the tip element, and, in particular, to include the effect of lateral fracture toughness. To further increase the accuracy of the P3D model, the flat fracture tip is replaced by its curved counterpart. This also permits us to capture radial behaviour at early times before the fracture has reached the stress barriers. To evaluate the accuracy of the model we have developed, the results are compared to the predictions calculated using a recently developed fully planar HF simulator, which is able to capture viscous, toughness, and intermediate propagation regimes. It is shown that the enhanced P3D model is able to approximate the propagation of hydraulic fractures accurately for various regimes of propagation, as well as for different fracture aspect ratios.

This study introduces a novel methodology for the design of the proppant pumping schedule for a hydraulic fracture, in which the fi nal proppant distribution along the crack is prescribed. The method is based on the assumption that the particles have relatively small impact on the fracture propagation, unless they reach the tip region. This makes it possible to relate the proppant velocity to the clear fluid velocity inside the fracture, which is calculated assuming no proppant. Having the history of the clear fluid velocity distribution, the prospective proppant motion can be computed. Then, volume balance is used to relate the final concentration at some point inside the fracture to the corresponding input concentration at a speci fic time instant, which helps to avoid solving an inverse problem. One exceptional feature of the approach lies in the fact that it is applicable to multiple fracture geometries and can be implemented using various hydraulic fracturing simulators. To verify the technique, two fracture geometries are considered - Khristianovich-Zheltov-Geertsma-De Klerk (KGD) and pseudo-3D (P3D). It is shown that the developed approach is capable of properly estimating the pumping schedule for both geometries. In particular, the proppant placement along the fracture at the end of the pumping period, calculated according to the adopted proppant transport model, shows close agreement with the design distribution. The comparison with Nolte's scheduling scheme shows that the latter is not always accurate, and cannot capture the essential di fferences between the schedules for the fracture geometries considered.

The aim of this study is to develop a model for proppant transport in hydraulic fractures capable of capturing both gravitational settling and tip screen-out effects, while prohibiting the particles from reaching the crack tips by imposing a width restriction based on the particle size. First, the equations that govern the propagation of hydraulic fractures and the proppant transport inside them are formulated. They are based on the solution for the steady flow of a viscous fluid, mixed with spherical particles, in a channel, which is obtained assuming an empirical constitutive model. This proppant transport model is applied to two fracture geometries – Khristianovich-Zheltov-Geertsma-De Klerk (KGD) and pseudo-3D (P3D). Numerical simulations show that the proposed method makes it possible to capture proppant plug formation and growth, as well as the gravitational settling for both geometries. A dimensionless parameter, whose magnitude reflects the intensity of the settling, is introduced for the P3D fracture. [The previous file was updated 2014-08-13 to reflect changes to publication].

One of the important hurdles in horizontal well stimulation is the generation of hydraulic fractures (HFs) from all perforation clusters within a given stage despite the challenges posed by stress shadowing and reservoir variability. In this paper we use a newly-developed, fully coupled, parallel-planar 3D HF model to investigate the potential to minimize the negative impact of stress shadowing and thereby to promote more uniform fracture growth across an array of HFs by adjusting the location of the perforation clusters. In this model the HFs are assumed to evolve in an array of parallel planes with full 3D stress coupling while the constant fluid influx into the wellbore is dynamically partitioned to each fracture so that the wellbore pressure is the same throughout the array. The model confirms the phenomenon of inner fracture suppression due to stress shadowing when the perforation clusters are uniformly distributed. Indeed, the localization of the fracture growth to the outer fractures is so dominant that the total fractured area generated by uniform arrays is largely independent of the number of perforation clusters. However, numerical experiments indicate that certain non-uniform cluster spacings promote a profound improvement in the even development of fracture growth. Identifying this effect relies on this new model’s ability to capture the full hydro-dynamical coupling between the simultaneously evolving HFs in their transition from radial to PKN-like geometries.

This MATLAB computer code implements a fourth order collocation scheme to solve the P3D nonlinear diffusion model for a hydraulic fracture propagating between two symmetric stress barriers. The model assumes equilibrium height growth that is controlled by the local fracture toughness and includes the effect of fluid leak-off into the surrounding rock.