Magma transport in brittle rock occurs by diking. Understanding the dynamics of diking and its observable consequences is essential to deciphering magma propagation in volcanic areas. Furthermore, diking plays a key role in tectonic phenomena such as continental rifting and plate divergence at mid-ocean ridges. Physics-based models of propagating dikes usually involve coupled transport of a viscous fluid with rock deformation and fracture. But the behaviour of dikes is also affected by the exchange of heat with the surroundings and by interaction with rock layering, pre-existing cracks, and the external stress field, among other factors. This complexity explains why existing models of propagating dikes are still relatively rudimentary: they are mainly 2D, and generally include only a subset of the factors described above. Here, we review numerical models on dike propagation focusing on the most recent studies (from the last 15-20 years). We track the influence of two main philosophies, one in which fluid dynamics are taken to control the behavior and the other which focuses on rock fracturing. It appear that uncertainties in the way that rock properties such as fracture toughness vary from laboratory to field scale remains one of the critical issues to be resolved. Finally, we present promising directions of research that include emerging approaches to numerical modeling and insights from hydraulic fracturing as an industrial analogue.

This paper considers the problem of a hydraulic fracture in which an incompressible Newtonian fluid is injected at a constant rate to drive a fracture in a permeable, infinite, brittle elastic solid. The two cases of a plane strain and a penny-shaped fracture are considered. The fluid pressure is assumed to be uniform and thus the lag between the fracture front and the fluid is taken to be zero. The validity of these assumptions is shown to depend on a parameter, which has the physical interpretation of a dimensionless fluid viscosity. It is shown that when the dimensionless viscosity is negligibly small, the problem depends only on a single parameter, a dimensionless time. Small and large time asymptotic solutions are derived which correspond to regimes dominated by storage of fluid in the fracture and infiltration of fluid into the rock, respectively. Evolution from the small to the large time asymptotic solution is obtained using a fourth order Runge-Kutta method.

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[1] The use of elastic plate theory to model the emplacement of laccoliths and large mafic sills has been debated for nearly 40 years. These intrusions typically attain a horizontal width that is large relative to the emplacement depth. Provided that large-scale plasticity and/or heterogeneity is not observed in the overlying host rock, it should then be valid to approximate its deformation based on analysis of a thin elastic plate with effective properties that are the consequence of interaction among heterogeneities that are small relative to the size of the intrusion. But the predictions that are usually cited from elastic plate theory are characterized by bell-shaped geometry, in contrast to the flat-topped, steep-sided geometry typical of many laccoliths or the nearly uniform thickness typical of large mafic sills. This fact has motivated several alternate explanations of laccolith and large mafic sill emplacement. Nonetheless, elastic plate theory should be revisited in light of the fact that previous elastic plate-based predictions have, in general, not taken into account an appropriate fracture propagation condition, fluid flow in the growing intrusion, and, importantly, the influence of the weight of the magma on intrusion growth. We present a model for the growth of circular intrusions that accounts for all of these factors. The model predicts the appropriate geometry for both laccoliths and large mafic sills. The predicted thickness to length relationships are also consistent with field data. Hence, while it may sometimes be appropriate, there is, in general, no fundamental need to appeal to large-scale rock plasticity in order to explain observed intrusion geometries, and it may, in fact, be appropriate to understand the growth of laccoliths and large sills in light of a single underlying mechanical model. Citation: Bunger, A. P., and A. R. Cruden (2011), Modeling the growth of laccoliths and large mafic sills: Role of magma body forces,

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Summary Placing multiple hydraulic fractures at intervals along horizontal wells has proved to be a highly effective method for stimulation. However, the mechanical interaction between a growing hydraulic fracture and one or more previous hydraulic fractures can affect the fracture geometry such that the final fracture array is suboptimal for stimulation. If the fracture-array geometry is idealized as a set of regular and planar fractures, history matching and production forecasting may be inaccurate. During the treatments, the fractures can curve toward or away from one another, potentially intersecting one another. A detailed parametric study of this phenomenon using a coupled 2D numerical fracturing simulator shows that the curving is associated with a combination of opening and sliding along the previously placed hydraulic fracture, as well as the previous fracture's disturbance of the local stress field because of its propped width. Dimensional analysis and scaling techniques are used to identify the key parameters that are associated with suppression of each mechanism that can lead to hydraulic-fracture curving. The analysis, which is in agreement with available data, results in a clarification of the conditions under which attractive and repulsive curving are expected, as well as the conditions under which curving is expected to be negligible or even completely suppressed. This last case of planar hydraulic-fracture growth is of practical importance and will usually be considered desirable. We present a straightforward method for determining whether planar fracture growth is expected that additionally gives insight into how design parameters can be modified to promote planar hydraulic-fracture growth.

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A field experiment was carried out to measure hydraulic fracture growth in naturally fractured rock. Hydraulic fracture interactions with pre-existing natural fractures, shear zones, veins, and adjacent hydraulic fractures were measured and mapped during the project. Tiltmeter and microseismic arrays were installed to test the performance of these monitoring methods in determining the fracture geometry, which was eventually revealed by the mine-through mapping. The physically mapped fractures were oriented approximately horizontally, perpendicular to the minimum stress direction. They crossed natural fractures and shear zones, but were offset by some shear zones, most often oriented with an approximate 45° dip. The analysis of the tiltmeter data correctly predicted fractures to be horizontal. Microseismic monitoring, although a proven method for imaging hydraulic fractures, did not resolve the fracture orientation or size for conditions at the E48 Northparkes site because of a lack of recorded micro-seismic events. The hydraulic fractures grew through solid rock, along natural fractures and stepped along inclined shear zones. Proppant was distributed throughout the fractures, including in the offset portions. Initial modeling indicates higher treatment pressure and slower extension rate for a stepped 2D hydraulic fracture compared to a straight fracture. Introduction Hydraulic fracture growth through naturally fractured reservoirs presents theoretical, design, and application challenges. High treating pressure, unplanned screenout, and shorter-than-designed propped fractures are some problems that result (Thiercelin and Makkhyu 2007; Zhang and Jeffrey 2008; Beugelsdijk et al. 2000; Wu et al. 2004). An experiment to measure hydraulic fracture growth in a naturally fractured rock at a mine site was, therefore, carried out to obtain details of fracture geometry from physical mapping during and after mining. Northparkes Mines, located 300 km west of Sydney, Australia, provided the site for this experiment. The mine is developing a new copper-gold porphyry orebody called E48, which will be mined by block caving methods. The rock contains numerous veins, natural fractures, and shear zones. In 2006, prior to preconditioning, to verify fracture growth and interaction with shear zones in the rock mass, a mine-through of several hydraulic fractures placed ahead of a development tunnel was undertaken. The monitored and mined fracture project described in this paper was developed to enable additional fracture monitoring and analysis to be included in the project. The orebody was preconditioned by hydraulic fracturing in 2008 under a separate project. Results from this work are pertinent to the application of hydraulic fracturing to stimulation of naturally fractured oil and gas reservoirs (Warpinski 1991; Settari 1988; Warpinski and Teufel 1987), to stimulation of geothermal and hot dry rock reservoirs (Sanyal et al. 2000), and to preconditioning of ore bodies prior to mining (Brown 2003; van As and Jeffrey 2002). Of special interest are the interaction of the hydraulic fractures with shear zones that exist in the rock mass, and the direct comparison of fracture geometry obtained by remote monitoring and by direct physical mapping.