The validity of the cubic law for laminar flow of fluids through open fractures consisting of parallel planar plates has been established by others over a wide range of conditions with apertures ranging down to a minimum of 0.2 µm. The law may be given in simplified form by Q/Δh = C(2b)3, where Q is the flow rate, Δh is the difference in hydraulic head, C is a constant that depends on the flow geometry and fluid properties, and 2b is the fracture aperture. The validity of this law for flow in a closed fracture where the surfaces are in contact and the aperture is being decreased under stress has been investigated at room temperature by using homogeneous samples of granite, basalt, and marble. Tension fractures were artificially induced, and the laboratory setup used radial as well as straight flow geometries. Apertures ranged from 250 down to 4µm, which was the minimum size that could be attained under a normal stress of 20 MPa. The cubic law was found to be valid whether the fracture surfaces were held open or were being closed under stress, and the results are not dependent on rock type. Permeability was uniquely defined by fracture aperture and was independent of the stress history used in these investigations. The effects of deviations from the ideal parallel plate concept only cause an apparent reduction in flow and may be incorporated into the cubic law by replacing C by C/ƒ. The factor ƒ varied from 1.04 to 1.65 in these investigations. The model of a fracture that is being closed under normal stress is visualized as being controlled by the strength of the asperities that are in contact. These contact areas are able to withstand significant stresses while maintaining space for fluids to continue to flow as the fracture aperture decreases. The controlling factor is the magnitude of the aperture, and since flow depends on (2b)3, a slight change in aperture evidently can easily dominate any other change in the geometry of the flow field. Thus one does not see any noticeable shift in the correlations of our experimental results in passing from a condition where the fracture surfaces were held open to one where the surfaces were being closed under stress.

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VALIDITY OF CUBIC LAW FOR FLUID FLOW IN A DEFORMABLE ROCK FRACTURE - eScholarship

Hot dry rock geothermal energy extraction experiments in artificial reservoirs created by hydraulic stimulation in naturally fractured crystalline rocks have been undertaken in several countries over the last 20 years. The experiments have had mixed results in terms of fluid recovery, system impedance, and heat extraction. Numerical models have not yet delivered a generally agreed understanding of the processes and sensitivities involved in reservoir creation and circulation. In this paper a two-dimensional fracture network model is described, which attempts to address the problems of both reservoir creation and circulation using rock mass characterization and in situ stress data as the primary inputs with a view to encapsulating our present understanding of how such systems work. The model is a prototype for a three-dimensional version currently under development and is intended mainly for engineering sensitivity studies. The basis of the model lies in approximations of fracture mechanical behavior drawn from the rock mechanics literature, a very simplified analysis of the operative physical processes, and mapping of the connectivity of fracture networks to a fine resolution regular grid. Taken together, these permit the approximate resolution of what is normally a supercomputer problem on a personal computer. The model is applied to field data gathered at Fenton Hill, New Mexico; Hijiori, Yamagata Prefecture, Japan; and Rosemanowes, Cornwall, England.

Progress toward a stochastic rock mechanics model of engineered geothermal systems

Thermal drawdown-induced flow channeling in a single fracture in EGS

A coupled thermo-hydro-mechanical modeling of fracture aperture alteration and reservoir deformation during heat extraction from a geothermal reservoir

[1] The constitutive thermo-hydro-mechanical equations of fractured media are embodied in the theory of mixtures applied to three-phase poroelastic media. The solid skeleton contains two distinct cavities filled with the same fluid. Each of the three phases is endowed with its own temperature. The constitutive relations governing the thermomechanical behavior, generalized diffusion and transfer are structured by, and satisfy, the dissipation inequality. The cavities exchange both mass and energy. Mass exchanges are driven by the jump in scaled chemical potential, and energy exchanges by the jump in coldness. The finite element approximation uses the displacement vector, the two fluid pressures and the three temperatures as primary variables. It is used to analyze a generic hot dry rock geothermal reservoir. Three parameters of the model are calibrated from the thermal outputs of Fenton Hill and Rosemanowes HDR reservoirs. The calibrated model is next applied to simulate circulation tests at the Fenton Hill HDR reservoir. The finer thermo-hydro-mechanical response provided by the dual porosity model with respect to a single porosity model is highlighted in a parameter analysis. Emphasis is put on the influence of the fracture spacing, on the effective stress response and on the permeation of the fluid into the porous blocks. The dual porosity model yields a thermally induced effective stress that is less tensile compared with the single porosity response. This effect becomes significant for large fracture spacings. In agreement with field data, fluid loss is observed to be high initially and to decrease with time.

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2012JB009161

S. Xu

Papers

A hydro-thermo-mechanical numerical model for HDR geothermal reservoir evaluation

Modelling of a european prototype hdr reservoir