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Modeling hydraulic fracturing in the presence of a natural fracture network is a challenging task, owing to the complex interactions between fluid, rock matrix, and rock interfaces, as well as the interactions between propagating fractures and existing natural interfaces. Understanding these complex interactions through numerical modeling is critical to the design of optimum stimulation strategies. In this paper, we present an explicitly integrated, fully coupled discrete-finite element approach for the simulation of hydraulic fracturing in arbitrary fracture networks. The individual physical processes involved in hydraulic fracturing are identified and addressed as separate modules: a finite element approach for geomechanics in the rock matrix, a finite volume approach for resolving hydrodynamics, a geomechanical joint model for interfacial resolution, and an adaptive remeshing module. The model is verified against the Khristianovich-Geertsma-DeKlerk closed-form solution for the propagation of a single hydraulic fracture and validated against laboratory testing results on the interaction between a propagating hydraulic fracture and an existing fracture. Preliminary results of simulating hydraulic fracturing in a natural fracture system consisting of multiple fractures are also presented. Figure 3. Typical mesh arrangement around a fracture tip. A polar coordinate system is established with its origin at the tip. The reference points used in Equations (13) and (14) are denoted as small circles, whereas alternative reference points shown as diamonds can also be used with modified formulations as elaborated in [24].

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The onset of convection is observed in a cylindrical annulus which is heated from the outside, cooled from the inside and rotating about its vertical axis of symmetry. The dynamical constraint exerted by the dominating Coriolis force inhibits the instability when the top and bottom boundaries of the annulus are conical so as to make the vertical height vary with distance from the axis. The experimental observations are in good agreement with the theoretical predictions by Busse (1970a). This confirmation indicates the absence of subcritical finite amplitude instabilities and suggests that the annulus experiment provides a close dynamical model for convection in the liquid core of the earth.

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Because of dynamical constraints in a rotating system, the component of gravity perpendicular to the axis of rotation is the dominant driving force of convection in liquid planetary cores and in stars. Except for the sign, the centrifugal force closely resembles the perpendicular component of gravity. Convection processes in stars and planets can therefore be modeled in laboratory experiments by using the centrifugal force with a reversed temperature gradient.

Finite element calculations of magma flow in dikelike channels with length‐to‐width ratios of 1000:1 or more have been used to investigate the coupling between thermal and dynamical regimes due to temperature‐dependent viscosity and dissipation. Steady state solutions with realistic thermal and dynamical parameter values have been obtained. The models show that the onset of solidification on the boundaries of a basaltic or andesitic dike, as predicted by idealized laminar flow models, can be prevented or significantly delayed by a small amount of transverse flow induced by rising bubbles, boundary roughness, or turbulence. This effect will reduce the critical initial widths of dikes estimated by Bruce and Huppert (1989). In the absence of transverse flow, the bulk temperature of the magma may actually increase slightly with distance along the dike as a result of viscous dissipation even while solidification is occurring on the walls of the dike. With converging or necking dike walls it is found that boundary temperatures fall to a minimum and then increase with distance along a dike even if viscous heating is neglected. For viscous heating to offset significant rates of heat loss (2 kW/m2) in a plane‐parallel 1‐m‐wide basaltic dike, an average flow velocity of 2.7 m/s driven by a pressure gradient of 1.7 MPa/km is required. A 7% void fraction caused by exsolution of volatiles or chamber gas in the magma will produce this pressure gradient. The ease of producing such a gradient by reducing the density of the magmatic column with addition of a gas phase makes it likely that flows of basaltic magma could be maintained in dikes tens of kilometers long. Furthermore, a gas phase may be important for the propagation of the fracture prior to the initial injection of magma during dike emplacement. Rapid transport of more silicic magmas through dikes is inhibited by the requirement of large driving pressure gradients exceeding several hundred megapascals per kilometer. However, the pressure gradient can be substantially reduced if the magma is a heterogenous mixture of a predominantly silicic component with a more mafic component. Pipe flow experiments involving molten polymers that exhibit dynamical similarity to magmas strongly suggest that unmixing occurs when a two‐component magma mixture, in which the components have different viscosities, rises within a dike. Within a few dike widths of the inlet the less viscous mafic component encapsulates the more viscous silicic part, effectively lubricating the passage of the more viscous component. Compositional variations in the Obsidian Dome volcano support the occurrence of this pressure reduction or self‐lubrication mechanism. Such a self‐lubrication process often may be necessary to permit very viscous magmas to reach Earth's surface. If so, chemical and lithologic zoning would be anticipated as a common feature of near‐surface intrusions or lava flows that are characterized by a high silica and/or high crystal content such as at Deadman Dome in Long Valley, California.