Computations in Real Rock - Heat Transport in 1/K-Noise Fractures

Abstract

Heretofore the concept of geothermal heat transport has followed a rigid script dating from the early 70s. It features a mechanically quasi-uniform low-porosity/low-permeability clastic rock matrix that is either naturally or artificially riven by sundry quasi-planar megafractures supporting Poiseuille flow of in situ fluids. The origin and persistence of this fracture/flow concept is likely traceable to the visual identification of large-scale finite-width joints in basement rock outcrops. Three serious problems, however, exist for joint control in situ fluid flow. First, joint expansion seen in mechanical free-surface outcrops does not obviously apply at 3-5km-depth confining pressures. Second, attempts to identify wellbore evidence of active interwell fracture-borne fluid pathways have not been conspicuously successful. Third, there is little evidence in well-log and/or well-core data for large-scale discrete mechanical fractures in an otherwise mechanically quasi-uniform rock matrix. Rather well-log and well-core evidence strongly favours a quite different physical relationship between the rock matrix, rock fractures, and in situ fluid flow. By joint prescription, wellbore microresistivity logs in an reservoir volume should record abrupt high-conductivity spikes as the resistivity sensor passes over conductive-fluid-containing joints. The power-spectrum of any spike sequence is essentially flat, S(k) ~ 1/k0 ~ const. In conspicuous contrast, microresistivity well-log data have, in common with the vast majority of geophysical well logs for most rock types and environments, Fourier power-spectra that scale inversely with wave number, S(k) ~ 1/k1, over ~ 3 decades of scale range (generally ~1/Km less than k less than ~1/m, ~1/Dm less than k less than ~1/cm for microresistivity). The systematic well-log power-law spectral scaling S(k) ~ 1/k phenomenon that negates the matrix + megafracture reservoir concept of in situ flow opens the door on a scale-independent physical concept of in situ fractures, fluid flow and heat transport. Power-law scaling phenomena occur notably in thermodynamic binary-population order-disorder continuous phase transitions when an order parameter reaches a critical value. In rock, the plausible critical-valued order parameter is grain-scale fracture density determined by the number of cement-disrupted/percolating grain-grain contact defects with in a host population of intact/non-percolating cement-bonded grain-grain contacts. As tectonic finite-strain deformation continually induces grain-scale failure of cement bonds in (clastic) rock, a critical-state percolation threshold value is maintained, rock volumes become percolation-permeable on scale lengths from mm to km, and well-log geophysical properties such as sonic wave speeds, electrical resistivity, soluble chemical species density, neutron porosity, and mass density fluctuations attain the power-law scaling fluctuation power-spectra S(k) ~ 1/k observed over ~1/Km less than k less than ~1/cm. At the same time and by a closely related percolating grain-scale-fracture mechanism, in situ clastic rock attains a well-attested ~85% correlation ƒÂƒÓ ~ ƒÂlog(ƒÈ) recorded for spatial fluctuations in clastic reservoir well-core porosity ƒÓ and the logarithm of well-core permeability ƒÈ. The observed physical conditions on in situ fractures and fluid flow S(k) ~ 1/k and ƒÂƒÓ ~ ƒÂlog(ƒÈ) are straightforwardly incorporated into finite-element flow simulators. Flow simulations for 2D dense-grid realisations of permeable media obeying these spatial correlation systematics suggest that flow can proceed along epipe-likef structures. Laminar flow heat transport in pipes has a fixed Nusselt number Nu ~ 4. For a pipe of radius r flowing water at temperature ƒ¢T from ambient, Newtonfs law of convective heat transport is Qconv = hƒ¢T, [h] = W/m2/oC. Fourierfs law of conductive heat transport for the same system is Qcond = K/rƒ¢T, [K] = W/m/oC. As Nu ß Qconv/Qcond, the coefficient of heat convection h = 4K/r scales inversely with pipe radius r. Fracture-borne percolation could thus carry substantially more heat per flow mass than the large-pipe equivalents of reservoir planar-fracture flow models. Efficient heat transfer via pipe-like percolation could lead to much more compact reservoir volumes. Finite-element 3D numerical flow/transport codes like Sutra can model a range of well-to-well percolation flow heat transport schema. Based on the forgoing considerations, we compact well-to-well heat transport models to the scale of a pair or quartet of horizontal parallel wellbores of length. and offset 2a obey .a2 ~ 106m3, 100-1000 times smaller than generally conceived reservoir volumes. A pair of 1km-long parallel wellbores at ~50m offset can circulate fluid at wellbore flux ~25L/s and mean heat exchange volume percolation flow velocity ~10-8m/s to notionally sustain ~1MWe power production per wellbore pair.