Calibrating the EGS Flow Stimulation Process for 2 Basement Rock

Abstract: We use Matlab 3D finite element fluid flow/transport modelling to simulate localized 10 wellbore temperature events of order 0.05-0.1oC logged in Fennoscandia basement rock at ~ 1.5km 11 depths. The temperature events are approximated as steady-state heat transport due to fluid 12 draining from the crust into the wellbore via naturally occurring fracture-connectivity structures. 13 Flow simulation is based on the empirics of spatially-correlated fracture-connectivity fluid flow 14 widely attested by well-log, well-core, and well-production data. Matching model wellbore15 centric radial temperature profiles to a 2D analytic expression for steady-state radial heat transport 16 with Peclet number Pe ≡ r0φv0/D (r0 = wellbore radius, v0 = Darcy velocity at r0, φ = ambient porosity, 17 D = rock-water thermal diffusivity), gives Pe ~ 10-15 for fracture-connectivity flow intersecting the 18 well, and Pe ~ 0 for ambient crust. Darcy flow for model Pe ~ 10 at radius ~ 10 meters from the 19 wellbore gives permeability estimate κ ~ 0.02Darcy for flow driven by differential fluid pressure 20 between least principal crustal stress pore pressure and hydrostatic wellbore pressure. Model 21 temperature event flow permeability κm ~ 0.02Darcy is related to well-core ambient permeability κ ~ 22 1μDarcy by empirical poroperm relation κm ~ κ exp(αmφ) for φ ~ 0.01 and αm ~ 1000. Our 23 modelling of wellbore temperature events calibrates the concept of reactivating fossilized fracture24 connectivity flow for EGS permeability stimulation of basement rock. 25


Introduction
Winning significant quantities of heat energy from the Earth's deep crystalline rock heat store requires a scientific and technical understanding of rock fluid flow properties that has yet to be established.The long-standing short-fall in accessing basement rock heat energy at drillable depths [1][2][3][4].is due principally to uncertainty how to send fluid from an injector wellbore to a producer wellbore at a sufficient rate through a sufficiently large volume of low-porosity/lowpermeability crustal rock to justify drilling costs.
We address the current wellbore-to-wellbore flow process uncertainty by focusing on local crustal flow phenomena signaled by isolated wellbore temperature log events at 1-2 km depths in Fennoscandia basement granites.The observed wellbore temperature events of order 0.05-0.1 o C can be plausibly attributed to heat advective fluid flow into a newly drilled wellbore via residual fracture-connectivity networks in otherwise effectively impermeable basement rock [5][6].The observed Fennoscandia wellbore temperature events are paralleled by instances of electrical conductivity deviations plausibly due to chemical ion transport in the same or similar flow systems.
Because of the multiplicative nature of permeability, doubling fracture-connectivity parameter α in a crustal volume can increase volume permeability by one to two orders of magnitude without having to do work against confining stresses to accommodate increased porosity.There is, therefore, an implicit energy argument that the crustal fluid-rock interaction processes that generate κ(x,y,z) ∝ exp(αφ(x,y,z)) flow heterogeneity do so as a consequence of a reduced crustal deformation energy budget.
Such fluid-rock interaction energetics may explain the persistence of rock-physical spectral scaling phenomenology S(k) ~ 1/k to the 5-9km depth crusts recorded in the KTB scientific deep well [24].In spite of decreasing porosity and permeability due to increased confining pressures at 5-10km depths, the S(k) ~ 1/k spatial fluctuation scaling attested by reservoir flow systematics at 1-5km depths do not cease at greater depths.It can further be noted that the amplitudes of well-log spatial fluctuations do not diminish with extreme depth.It is thus logical to deduce fluid-rich  and shows discrete wellbore intervals of considerable temperature fluctuation at 800-1100 meters and 1750-1850 meters.These intervals are associated with a complex, highly fractured and attenuative heterogeneous mafic gneissic rock.Our present interest lies in the more uniform granites between 1120 to 1700 meters, where the reduced temperature profile shows a series of sharp isolated positive temperature events.Our working hypothesis for the physical process leading to the Figure 2 OTN1 temperature events is that isolated fracture-connectivity structures in the crust intersect the wellbore and feed warmer crustal water into the cooler wellbore fluid column.We assume that the crustal fluid flow system is approaching a steady state in which the amplitude and spatial distribution of the temperature incursions change slowly.structures.Axial temperature distributions along wellbore axial profiles thus become a diagnostic tool for investigating wellbore-centric fluid flow processes in crustal rock.

3D global mesh finite element modelling of basement rock wellbore temperature events
Empirical crustal property (I) establishes the existence of spatially-correlated poroperm structures at all scales and at all drilling-accessed depths in the crust.In parallel, empirical crustal flow properties (II)-(III) establish the poro-connectivity flow mechanics at all scales for wellboreaccessed crustal volumes.On the strength of empirics (I)-(III), we interpret the Figure 2 set of wellbore-logged basement rock thermal event structures in terms of the 'canonical' or 'type' crustto-wellbore or wellbore-to-crust fluid flow constructs sketched in Figure 4.
Given the broad evidentiary base for ambient crustal flow properties (I)-(III), it is logical to expect that when a wellbore intersects naturally-occurring flow structures, fluid at crustal confining pressures in the long-range spatially-correlated fracture-connectivity network flows into the hydrostatically under-pressured wellbore.If the intercepted flow structure is sufficiently large scale, heat will be advected into the wellbore by the inflowing fluid for a long enough period to be observed by wellbore logging.Comprehensive evidence for persistent Dm-scale crust-to-wellbore advective inflow is given in [5][6] for a Hm-scale tight gas sandstone crustal volume.
The steady-state rate at which crustal fluids flow from the crust into a wellbore, v(x,y,z), is given by Darcy's law in terms of permeability distribution κ(x,y,z) and constant dynamic viscosity of water μ, for appropriate fluid pressure boundary conditions in the crustal volume.
Conservation of mass requires that steady-state Darcy flow velocity has vanishing divergence, ∇⋅v = 0, yielding the defining flow equation for finite-element solvers, ∇⋅(κ(x,y,z)∇P(x,y,z)) = 0. ( The finite element method for solving differential equations allows for essentially arbitrary spatial variation of material properties such as κ(x,y,z) [30][31][32].As illustrated in Figure 4, our application of this finite element solver capability assumes that a single global numerical mesh spans the entire flow model.
If Darcy fluid flow carries heat through a medium of mean porosity φ, the combined steadystate conducted and advected heat energy flow is q = K∇T(x,y,z) -ρCφv(x,y,z)T(x,y,z), for K = Fourier's thermal conductivity for rock and ρC = fluid volumetric heat capacity.
Conservation of steady-state thermal energy, ∇⋅q = 0, then couples the spatially-variable temperature field T(x,y,z) to the spatially-variable Darcy fluid velocity flow field v(x,y,z) for the given crustal temperature and fluid pressure boundary conditions.This coupling leads to the defining steady-state equation for a nonlinear finite-element solver, ∇⋅∇T(x,y,z) = ρC/K ∇⋅(φv(x,y,z)T(x,y,z)) = 1/D φv(x,y,z)⋅∇T(x,y,z), where the conservation of mass condition ∇⋅v = 0 is observed, and D = K/ρC ~ 0.7⋅10 -6 m 2 /s is the essentially constant thermal diffusivity of the rock-fluid system for rock thermal conductivity Where long-range spatially-correlated fracture-connectivity percolation networks intersect a wellbore, Darcy flow can be approximated as essentially wellbore-centric radial, v(x,y,z) ~ v(r) ~ v(r 0 )r 0 /r.The 3D steady-state flow condition (4) then reduces to a 2D analytical form in wellborecentric radius r, given in terms of radial flow boundaries at r 0 < r 1 characterized by boundary temperatures T 0 and T 1 , with Peclet number Pe = r 0 φv 0 /D (Appendix A).Analytic expression (5) serves to check 3D solutions of non-linear thermal energy conservation constraint equation ( 4), while at the same time yielding an estimate of advective fluid flow rate for the crustal fracture-connectivity flow system, The degree of advection heat transport relative to conductive heat transport for a given model flow structure determines the shape to the axial temperature along the wellbore.Matching the observed axial temperature profile in turn constrains the effective steady-state flow velocity of the crustal fluid leaking into the wellbore without having to know explicitly the permeability or pressure boundary conditions.Modeling axial temperature profiles interpreted as sequences of 2D wellbore-centric radial flow Peclet numbers thus has the potential to calibrate EGS wellbore-centric flow stimulation processes.
We use Matlab 3D partial differential equation solvers to model steady-state axial temperature profiles constrained by the mass conservation (2) and heat energy conservation (4).The Matlab 3D solvers [32] have two forms for a scalar field variable u(x,y,z), and -∇⋅(c(x,y,z,u,ux,ux,uz)∇u) + a(x,y,z,u,ux,ux,uz)u = f(x,y,z,u,ux,ux,uz).(7) Eq ( 6) is equivalent to an elliptical partial differential equation, and ( 7) generalises (6) by allowing coefficients terms c(x,y,z) and a(x,y,z) and source term f(x,y,z) to depend on the field variable u(x,y,z) and its spatial derivatives, ux, ux and uz.The key feature of finite-element modelling employed here is its tractability to essentially arbitrary position-dependent coefficients, e.g., c(x,y,z) for (6), and c(x,y,z,u,ux,ux,uz) for (7), to meet the conditions imposed by stochastic poroperm media [30][31][32].6) and (7).The scalar field variables represented by finite element scale field u(x,y,z) are, for (6), pressure P(x,y,z), for solving the Darcy flow constraint equation ( 2), and, for (7), temperature T(x,y,z), for solving the thermal energy constraint equation ( 4).For pressure fields u = P(x,y,z), ( 6) is used with coefficient term c(x,y,z) representing permeability κ(x,y,z), with coefficient term a(x,y,z) and source term f(x,y,z) set to zero.For temperature fields u = T(x,y,z), (7) is used with coefficient term c(x,y,z,u,ux,ux,uz) set to unity, coefficient term a(x,y,z,u,ux,ux,uz) set to zero, and source term set to the advective flow of heat, f(x,y,z,T,Tx,Tx,Tz) = 1/D φv(x,y,z)⋅∇T(x,y,z).
The  The wellbore interior boundary surface is assigned a boundary flux distribution.As our modelling task is to compute an interior temperature distribution on the basis of an exterior temperature distribution, we assign the wellbore a heat flux boundary condition which generates an associated temperature distribution T0(r0,z) = 1/h q0(r0,z) on the basis of the computed heat delivered to the wellbore.For present purposes, the heat transfer coefficient h is taken to be a free model parameter.Model wellbore radii can range from a 5-node diameter (30cm radius) as in Figure 6 to a 50node diameter (3m radius) as in Figure 14 below.Modelling with a range of wellbore radii within the numerical volume checks for possible effects of a small wellbore defined by few nodes, and possible effects on wellbore axial temperature distribution due to the radial distance over which fluids travel from the exterior crustal boundary to the interior wellbore boundary.
To compute Darcy flow velocity v(x,y,z), the wellbore is assumed to be at hydrostatic pressure.
Fluid in the crustal volume will be subject to higher pressure, most probably given by the minimum     As we are primarily interested in interpreting wellbore axial temperature events in terms of fluid inflow from crustal domain fracture-connectivity structures, we look at the degree of spatial resolution that the Figure 2 OTN1 wellbore temperature events can give for the scale of flow structures away from the wellbore.While the steady-state heat flux temperature field (5) formally returns a value T 1 for radius r=r 1 for all values of Peclet number Pe, for radii much larger than the wellbore radius but smaller than the model radius, r 0 /r 1 << r/r 1 < 1, (5) reduces to For values of the model exterior temperature T 1 differing little from the wellbore temperature T 0 , T(r) ~ T 0 for most of the radial extent away from the wellbore, and as a result the value of the external boundary temperature T 1 has little influence on the wellbore temperature profile.
Expressing system steady-state advective heat flow (3) in terms of (5) gives, from which the wellbore temperature T 0 is directly dependent on the crustal heat inflow q 1 at the model external boundary, While ( 10) is analytically straightforward, for numerical solutions it is preferable to give a temperature rather than a heat flux boundary condition.We can, accordingly, use (10) to infer the magnitude of the heat influx at the model boundary from the given numerical value of the wellbore temperature T 0 .The physical and numerical implications of ( 8)-( 10) motivate examining the model wellbore temperature profile for a smaller model/wellbore radius ration, r1/r0 = 3 (Figure 14).Computing  Figure 11 versus Figure 15 model fits to OTN1 wellbore axial temperature profiles quantifies the effect of an order of magnitude wellbore/model radius ratio r1/r0 difference, 33 to 3.3.For 0.6m thick advective fluid inflow fracture-connectivity horizons, the 30cm wellbore temperature profiles are more sharply peaked than the 3m wellbore temperature profiles.In particular, the lower panels of Figures 11 and 15 show that model temperature profiles for adjacent 0.6m inflow horizons follow the observed temperature contours for a 30cm wellbore while model profiles for a 3m wellbore cannot follow these contours.We conclude on the basis of the latter condition that 0.6m-   The effect of thickening the fracture-connectivity structures draining into a wellbore is illustrated in Figures 18-19 for a 1m radius wellbore.Figure 18 essentially duplicates for a 1m radius wellbore what Figure 12 shows for a 30cm radius wellbore for a 0.6m-thick inflow structure.
After large Peclet numbers, Pe ~ 15, at the location of the inflow structure, at 10m offsets from the inflow structure effective advective heat transport is reduced to the level the level of thermal conduction, Pe < 1. Figure 19 shows, however, that the effect of increasing the thickness of the Stepping away from the horizon in 1.5m intervals, the Peclet numbers decline as 4.4, 2.7, and 0.7.From 2D steady-state wellbore-centric flow and temperature expressions ( 9)-( 10), r 1 q 1 = r 0 q 0 = KPeΔT 0 gives the rate at which heat energy leaves the model crustal

Wellbore temperature event modelling as calibration of EGS stimulation of basement rock
Fennoscandia basement rock wellbore OTN1 temperature event modelling pictured in Figures The spatially-correlated normal distributions of porosity φ(x,y,z) give the associated permeability κ(x,y,z) ∝ exp(αφ(x,y,z)) fields a lognormal distribution for sufficiently large parameter α in accord with empirical conditions (II)-(III).
Our present wellbore-centric flow modelling results can be placed in a wider basement rock permeability stimulation context: • OTN1 solute-transport galvanic well-log profiles parallel OTN1 basement rock temperature profiles.
• Deep basement well-log and well-core data supporting spatial-correlation empirics (I)-(III) imply general application to EGS basement rock stimulation.
• Crustal deformation energetics appear to favour spatially-correlated granularity over planar continuum flow structures, implying wide application of the present modelling construct.
• Wellbore-centric flow modelling provides a simple calculus for the physical scales needed for successful EGS commercial outcomes.

OTN1 basement rock solute transport events
It is plausible that the episodes of fluid influx into the OTN1 well that transport heat to the wellbore fluid also register as electrical conductivity fluctuations responding to increases solute content and/or solute transport that registers as a spontaneous potential signal.Figure 20 indicates considerable spatial correlation between the OTN1 temperature profile (black dots) and two OTN1 galvanic profiles, lateral log induction (blue trace) and spontaneous potential (red trace).As solute concentrations obey a gradient law similar to Fourier's law of heat transport, temperature modelling is similar to fluid solute concentration modelling.However, while the gradient flow principle is the same, brine concentrations are affected by local rock mineralogy with no parallel in thermal properties, hence a one-to-one correspondence between temperature and solute wellbore fluid inflow events is not expected. Zero-Mean/Unit-Variance

Crustal deformation energetics for spatially-correlated crustal fluid flow granularity
Well-log, well-core, and well-productivity empirics focus on the granular rather than the   Given the extensive evidence of well-log, well-core, and well-flow empirics (I)-(III) as the ambient condition of crustal rock, we may argue that wellbores penetrating crustal media will typically encounter localised disseminated/granularity rather than planar geometric/continuum flow surfaces associated with discrete fracture displacement structures.
We can interpret Table 1 further to argue that the ambient crustal empirics (I)-(III) result from the implied energetics of defect insertion through rock-fluid interaction.Rock stress involving fluids is more easily dissipated if fluid permeability stimulation proceeds through spatiallycorrelated fracture-connectivity granularity rather than through spatially-uncorrelated effectivemedium planar continuum displacements.For generating ambient crustal fluid-rock flow conditions, the fact that porosity increments require doing work against confining stresses means that it is energetically favourable for defect enhancement to proceed in a spatially-correlated granularity medium than in a spatially-uncorrelated continuum medium.Discrete fracture systems may thus be seen to characterize crustal tectonic settings in which solid-rock displacement rates due to far-field tectonic plate motion exceed the rate at which fluid pressures can dissipate through slower fracture-connectivity mechanisms.EGS mechanisms based on local wellborecentric fluid pressurization rather than elastic stress generated by far-field tectonics may thus couple more readily to the slower ambient-crust defect injection processes leading to spatiallycorrelated granularity than to the faster defect injection processes leading to discrete-fracture displacements.

EGS couplet scale dimensions
The irreducible statement of EGS principle is to pass fluid from an injector wellbore through an ambient hot crustal volume to a producer wellbore at a sufficient rate over a sufficient length of time that the recovered heat energy covers the cost of (i) drilling sufficiently deep wells to access sufficient temperature and (ii) stimulating the wellbore-centric crustal volumes to sufficient radius to access a sufficient crustal volume.
The scale-sufficiency conditions for an EGS wellbore-centric flow doublet are simply quantified by considering the radial transfer of heat energy via wellbore fluid injected into the surrounding crust.For wellbore radius r0 and length ℓ in a crustal volume of porosity φ, fluid of for D ≡ K/ρC the thermal diffusivity of the crustal medium, ν = Pe/2 one-half the Peclet number Pe ≡ r0φv0/D, and Jν(:) and Yν(:) the order ν Bessel functions of the first and second kind respectively [37].
Peclet number Pe = r0φv0/D = r0φv0ρC/K links the crustal temperature growth to the rate at which the wellbore supplies heat, For a given injection of heat energy Q, the system Peclet number of resultant temperature field diminishes in proportion to the wellbore length ℓ and fluid temperature T0.For a maximally stimulated wellbore that supports Pe ~ 5 heat transport across every meter of wellbore length for crustal rock temperature 100 o C, the total wellbore-centric heat extraction rate is Q ~ 10MW per km of wellbore.Peer-reviewed version available at Energies 2017, 10,1979; doi:10.3390/en10121979

Summary/Conclusions
Despite abundant well-log, well-core, and well-flow counter-evidence, the working assumption that fluid flow in geological formations is effectively uniform due to spatial averaging over uncorrelated poroperm fluctuations has remained in place since the mid-19 th century observations and innovations of Darcy and Dupuit [39][40][41][42][43][44][45][46][47][48].This stasis is essentially a practical matter: for both groundwater and hydrocarbon crustal fluids, it has been found cost effective to ignore manifest fluid flow heterogeneity by the simple practice of drilling more wells.
Groundwater wells are shallow and therefore inexpensive.The considerable value of hydrocarbon fluids means that, first, a relatively few high-flow/high-profit oil/gas wells tend to cover the cost of many low-flow oil/gas wells, and, second, the population of many low-flow wells that accompany the few high-flow wells are generally profitable for considerable lengths of time [19].
Geothermal energy provision, in contrast, lacks the well-cost offsets afforded groundwater and hydrocarbon reservoir fluids.In natural convective geothermal systems with high fluid temperatures and permeabilities, the excessively large flow-rate demand to power turbines means that crustal flow heterogeneity cannot be cost-effectively solved by the rubric of spatial averaging [21].Subsurface imaging is required to achieve 'smart-drilling' practice to keep well costs down in tapping naturally convective geothermal flow systems [18].In low-porosity/low-permeability basement rock geothermal systems, the high cost of drilling and the limited technical means of EGS permeability enhancement for wellbore-to-wellbore flow mean that net fluid through-put remains generally non-commercial for power-production [1][2].
Balancing the negatives of basement rock EGS geothermal heat energy provision are the positives of great abundance of carbon-free heat energy for direct use purposes.The accessible heat energy in a 50m thick crustal section 5km below any of Finland's district heating plants with a 5-km municipal service radius is of the order E = πR 2 ℓ ρC T ~ 0.75⋅10 18   It is seen in Figure A1 that for Pe > ~ 3, the radial thermal gradient vanishes near the wellbore.
For Pe > ~ 3 all heat transport near the wellbore is by advective flow.We can thus expect observed

Figures 1 -
Figures 1-2 display the OTN1 pilot well temperature and reduced temperature profiles.The upper panel of Figure 1 shows a uniform gradient perturbed at the near-surface due to groundwater circulation, and at depth due to a section of unusually heterogeneous rock (with possible depth-related instrument effects).Removing the uniform temperature gradient, the lower Figure 1 panel highlights the effects of near-surface groundwater circulation to depth ~ 600 meters,

Figure 2 Figure 2 .
Figure 2 expands the Figure 1 wellbore depth scale over the OTN1 1120-1700m granite interval, detailing the discrete 0.05-0.1 o C temperature events of interest.A dozen or so 0.05-0.10o C positive temperature excursions of ~5-meter axial depth extent occur over a 600-meter span.The lower panel of Figure 2 displays the discrete temperature deviations relative to the upper-panel polynomial curve fit to the reduced temperature trace.

Figure 3 .
Figure 3. Representative temperature fluctuations for characteristic spatial variation in Fennoscandia basement rock thermal conductivity.Computation based on Hm scale thermal conductivity sequences from 2.5 km wellbore [28].

Figure 4
Figure 4 illustrates steady-state fluid mechanical heat transport incursion using a 2D vertical planar pressure front moving fluid across a horizontal rock-fluid interface.Warm colors representthe hotter low-permeability crustal rock at higher fluid pressure, and cool colors represent the colder high-permeability wellbore fluid at lower fluid pressure.In the center of the Figure4crustal block, a narrow channel of higher crustal permeability intersects the wellbore to permit warmer crustal fluid at higher pressure to flow into the cooler wellbore fluid at lower pressure.The wellbore fluid temperature is elevated above its background level at the site where crustal waters enter the wellbore fluid; model side boundaries are set to zero-flow conditions.The resultant steady-state interface wellbore temperature profile is given in the lower panel.The lower panel also shows that away from the localized fracture-connectivity channel, fluid can leak from the crustal interior into the wellbore to generate small temperature fluctuations flanking the central temperature event.Such disseminated temperature fluctuations along the wellbore may be present in the Figure2wellbore temperature data.

Figure 4 C
Figure 4 also illustrates in 2D the global mesh nature of our 3D finite element computations discussed below.As access to deep crustal fluids is generally via a wellbore, slow or rapid percolation passage of wellbore fluid into or out of a surrounding crustal rock volume can be defined entirely by a stochastic permeability distribution within a global volumetric mesh.For

Figure 4 .
Figure 4. (Upper) Simulation of Figure 2 temperature events as heat transport from a hotter lowpermeability medium (yellow-orange layer representing crustal rock) to a colder high-permeability medium (green-blue layer representing wellbore fluid).(Middle) Velocity vectors representing fluid flow in Figure 2 fracture-connectivity structure entering the wellbore; additional flow from the crust into the wellbore can be seen along the entire interface.(Lower) Temperature profile measured along the wellbore-crust interface in simulation of Figure 2 wellbore axial temperature profiles.

Figure 6
Figure6pictures the model geometry for wellbore-centric flow/transport simulation of Figure2OTN1 temperature data.Internally, the model volume is characterized by a spatially-correlated stochastic distribution of porosity φ(x,y,z) and its associated permeability field κ(x,y,z) ~ exp(αφ(x,y,z)).The crustal volume has essentially uniform pressure and temperature conditions on each side; zero-flux boundary conditions are set on the top and bottom faces.As illustrated in Figure4for 2D, we can inset at will in the Figure6crustal volume geometric flow structures to give enhanced percolation via greater fracture connectivity along the flow structure.Where inserted flow structures intersect the model volume external boundaries, we assume the structures connect to the surrounding crust to deliver heat energy at the external boundary.As displayed in Figure8below, the external crustal heat energy is represented by a fixed temperature increment at the intersection of the flow structure with the model external faces.

Figure 6 .
Figure 6.Wellbore-centric computational volume data cube 161 nodes on a side for node spacing ~ 12cm.A 30cm radius wellbore is denoted by shaded vertical shaft with interior surface face F3.External boundary faces are F1-F2 and F4-F7.Pressure, temperature, and/or flux boundary conditions at faces F1-F7 define the steady-state flow/advection simulation performed by Matlab finite-element solvers (6) and(7).

Figure 6
computation volume is discretized by 161 nodes on a side.With a notional physical dimension of 20 meters per side, the nodal spatial resolution is Δx = Δy = Δz = 12.5 cm.Ambient poroperm properties within the crustal domain are numerical realisations of a 3D stochastic spatial connectivity distribution representing porosity φ(x,y,z) and associated permeability κ(x,y,z) ~ exp(αφ(x,y,z)) given by crustal empirics (I)-(III).Figure 7 shows the degree of spatial heterogeneity typical of the power-law scaling spatial fluctuation amplitudes: ~ 6 octaves for the normally distributed porosity about mean porosity φ ~ 1.2% and ~ 9 octaves for the lognormally distributed permeability generated by fracture-connectivity parameter α ~ 300.

Figure 7 .
Figure 7. Frequency distributions typical of ambient porosity and permeability fluctuations embedded in Figure 6 numerical crustal volume characterized by spatially correlated porosity and permeability empirics (I)-(III).
principal stress.Given the fluid-flow empirics (I)-(III), almost all fluid will ultimately be connected through a global fracture-connectivity pathway, and hence the fluid pressure will be approximately in equilibrium with the minimum principal stress.As the minimum horizontal principal stress, σh ~ 22MPa/km • zkm, exceeds hydrostatic pressure, Ph ~ 10MPa/km • zkm, wherever in the crustal volume there is a geometric feature of elevated fracture-connectivity parameter α that connects the surrounding crust to the interior wellbore, we can assume that fluid pressure σh outside the model cube drives fluid from the crust into the wellbore.In consideration of wellbore temperature log data in Figures1-2and illustrative 2D heat transport modelling of Figure4, we can suppose that over time episodes of fluid flow in the wellbore have removed heat from the wellbore and its immediately proximate crust[5][6].It follows that fracture-connectivity pathways leaking fluid into the wellbore bring crustal heat into the wellbore to generate positive temperature events at sites where fracture-connectivity structures intersect the wellbore (e.g., Figure4).A model external boundary temperature distribution is illustrated in Figure8.The incremented boundary temperature field corresponds to a 0.6m-thick horizontal flow structure of higher poro-connectivity parameter α which conveys fluids from the surrounding crustal volume to the central wellbore.Figures 9-10 display quadrant section contour plots for the temperature and heat transport solutions to(7)  generated by the 0.6m-thick flow structure pictured in Figure8.

Figure 11 Figure 8 .Figure 9 .Figure 10 .Figure 11 .
Figure 11 shows model wellbore axial temperature profile matches for the Figure 2 significant OTN1 temperature events in the 1120m-1700m interval of granite basement.Except for the upperright temperature event of Figure 11, the Figure 8-10 model flow structures have 0.6m thicknesses.A 2-3m thickness flow structure is required to match the 1405m-depth temperature event shown in the Figure 11 upper-right profile.

Figure 12 shows
Figure12shows the axial changes in model radial temperature profiles (blue dots) for the 1218m OTN1 temperature event given in the upper left plot of Figure11.Model radial temperature profiles T 0 < T(r) < T 1 for r 0 < r < r 1 at successive depths along the wellbore axis are approximated by best-fits (red traces) to steady-state radial temperature distribution (5) for free parameter Peclet number Pe = r 0 φv 0 /D,

Figure 12 . 5 Preprints
Figure 12. 2D analytic wellbore-centric radial temperature profiles (red traces) for given Peclet number superposed on 3D numerical model radial temperature distributions (blue dots) for 30cm radius wellbore embedded in 10-meter radius crustal volume as seen in Figure 8. Horizontal axes are wellbore-centric radius in meters; vertical axes are temperatures from nominal wellbore ambient T 0 ~ 100 to ambient crust T 1 ~ 100 o C + ΔT for ΔT ~ 0.05 o C except near the advective flow horizon when ΔT = 1 o C. From upper left of left quartet to lower right of right quartet, the 3D model radial temperature distributions are fit to 2D steady-state radial temperature distributions (5) governed by numbers Pe = r 0 φv 0 /D.The Peclet number for the fracture-connectivity horizon is 16.Stepping away from the horizon in 1.5m intervals, the Peclet numbers decline as 6.2, 4.2, and 2.1.At wellbore axial offsets greater than 6m from the fracture-connectivity horizon, thermal conductivity values of Peclet numbers Pe < 1 dominate.

Figure 13 .
Figure 13.Effective inflow velocity φv 0 = PeD/r 0 at successive wellbore offsets from the inflow horizon given by the Figures 8-12 model of OTN1 temperature event at depth 1218 meters.For crustal rock of mean porosity φ = 0.01, the computed inflow Darcy fluid velocity at wellbore radius 30cm is v 0 ~ 4 mm/s across a 1m wellbore depth interval.

Figure 14 .
Figure 14.Following Figure 6, a wellbore-centric data cube of 161 nodes with node spacing ~ 12cm and a 3m radius wellbore denoted by shaded vertical shaft with interior surface face F5.External boundary faces are F1-F4 and F6-F7.Pressure, temperature, and/or flux boundary conditions at faces F1-F7 define the steady-state flow/advection simulation performed by Matlab finite-element solvers (6) and (7).

Preprints
(www.preprints.org)| NOT PEER-REVIEWED | Posted: 18 September 2017 doi:10.20944/preprints201709.0079.v1Peer-reviewed version available at Energies 2017, 10, 1979; doi:10.3390/en10121979 the axial temperature profile for a model wellbore radius r0 = 3m in an r1 = 10m model volume tests the extent to which the r1/r0 influences model axial temperature distributions.The Figure 15 results of the Figure 14 wellbore model are to be compared with Figure 11.

Figure 15 .
Figure 15.Following Figure 11, overlays of Figure 14 model wellbore axial temperature profiles (red traces) for 3m radius wellbore embedded in a 10-meter radius crustal volume superposed on OTN1 wellbore temperature inflexions (black traces).Horizontal axes give wellbore depth in meters; vertical axes are wellbore temperature profiles normalised to zero-mean/unit-variance.The favoured model flow channel thickness is 0.6m, as illustrated in Figure 8. (Upper left) Model temperature profile computed for single 0.6m thick flow channel.(Upper right) Model temperature profile for 2-3m thick flow channel.(Lower left) Model temperature profile for 4-m spaced pair of 0.6m thick flow channels.(Lower right) Model temperature profile for 3-m spaced pair of 0.6m thick flow channels.

Figures 16 -
Figures[16][17] indicate that a 1m radius wellbore model, r1/r0 = 10, does not match the OTN1 temperature profile for the adjacent inflow horizons shown in the lower-left panel of Figure17.The 1m model wellbore matches the lower-right panel better than does the 3m model wellbore but does not require a narrower gauge wellbore model.From the Figures11, 15, and 17 model/data matches, we can infer that, within the limits of our 161-node 3D computational spatial resolution, crustal penetration of a 0.6m-thick fracture-connectivity inflow structures is greater than 3 meters and is consistent with being a great as 10 meters.

inflow structure to 2 -Figure 18 .
Figure 18.Following Figure 12, 2D analytic wellbore-centric radial temperature profiles (red traces) for given Peclet number superposed on 3D numerical model radial temperature distributions (blue dots) for 1m radius wellbore embedded in 10-meter radius crustal volume (Figure 16) with a 0.6m fractureconnectivity structure.From upper left of left quartet to lower right of right quartet, the 3D model radial temperature distributions are fit to 2D steady-state radial temperature distributions (5) governed by numbers Pe = r 0 φv 0 /D.The Peclet number for the fracture-connectivity horizon is of order 14-19.

Figure 19 . 5 Preprints
Figure 19.Following Figure 18, 2D analytic wellbore-centric radial temperature profiles (red traces) for given Peclet number superposed on 3D numerical model radial temperature distributions (blue dots) for 1m radius wellbore embedded in 10-meter radius crustal volume with a 2-3m fracture-connectivity structure.From upper left of left quartet to lower right of right quartet, the 3D model radial temperature distributions are fit to 2D steady-state radial temperature distributions (5) governed by numbers Pe = r 0 φv 0 /D.The Peclet number for the fracture-connectivity horizon is 7. Stepping away from the horizon in 1.5m intervals, the Peclet numbers decline as 6.2, 4.2, and 2.1.At wellbore axial offsets greater than 6m from the fracture-connectivity horizon, thermal conductivity values of Peclet numbers Pe < 1 dominate.

3 W/m 2 ,
volume due to radial heat flow in Peclet number Pe ~ 10 fracture-connectivity structures producing OTN1 temperature excursions ΔT 0 ~ 0.05-0.1 o C. Advective heat energy leaving the model crustal volume via wellbore-centric fluid inflow structures of thickness ℓ is then ΔQad = q 0 2πr0 ℓ = 2π Pe KℓΔT0.For Pe = 10 and ℓ =1m, ΔQad = 2π 10 3W/m/ o C 1m 0.1 o C ~ 20W for each temperature event.A similar degree of wellborecentric heat, ΔQcd ~ 20W, exits from the crustal volume due to conduction, Pe ~ 1, intervals of wellbore length ℓ =10m.The heat energy of a crustal volume of size 10 3 m 3 and heat capacity 840J/kg/ o C ⋅ 2200kg/m 3 at 100 o C ambient temperature is E = 100GJ.At an estimated 40W loss of heat energy due to OTN1 advection temperature events and associated heat conduction, the model crustal volume loses heat energy ΔE ~ 40W ⋅ 3⋅10 7 s ~ 1.2 GJ at the rate of 1% per year.As the advectively lost heat of the OTN1 temperature events is easily replaced by thermal conduction at the model boundaries, the observed heat rate loss is consistent with a steady-state thermal condition.Advective heat flow into the model volume via a wellbore-centric flow structure occurs at a rate, q 1 = KPeΔT 0 /r 1 ~ 0.about 6 times the nominal crustal heat flow rate of 50mW/m 2 .Spatially averaged fluid flow velocity estimates proceed from model Peclet numbers, v0 ~ PeD/φr0 at the wellbore and v1 ~ PeD/φr1 at the model periphery.For a model Pe ~ 10 advective flow system with 10m external radius in a crustal medium of mean ambient porosity φ ~ 0.01 and thermal diffusivity D ~ 0.7 ⋅10 -6 m 2 /s, the external crustal inflow fluid flow velocity v1 ~ .7⋅10 - m/s.From Darcy's law (1), the associated model permeability of the fracture-conductivity flow structure generating OTN1 temperature events, κm ~ vμ/∂rP, is given by wellbore-crust pressure differential ΔP/Δr and fluid viscosity μ ~ 0.5⋅ 10 -3 Pa•s.Estimating the wellbore-crust pressure differential ΔP as the difference between wellbore hydrostatic pressure Ph ~ 10MPa/km • 1.5km and crustal pore pressure given by the minimum principal stress σh ~ Pp ~ 22MPa/km • 1.5km for crustal minimum principal stress at 1.5km depth, κm ~ .7 10 -4 m/s ⋅ .5⋅ 10 -3 Pa•s ⋅ 10m/(12 ⋅ 1.5 ⋅ 10 6 Pa) ~ 2⋅ 10 -14 m 2 ~ 0.02Darcy.The fracture connectivity parameter associated with the fracture-connectivity flow structures generating observed temperature events is then κm ~ κ exp(αmφ) for φ ~ 0.01 and κ ~ 1µDarcy is then αm ~ 1000.If we associate an effective ambient value for poro-connectivity α ~ 500 (Figures 25-29 [24-27]), then an OTN1 temperature event model flow structure poro-connectivity αm ~ 1000 is of order twice the ambient value is nominally characteristic of Peclet number Pe ~ 10 flow structures.

Figure 20 . 5 . 2 .Figure 21 .
Figure 20.OTN1 galvanic property well-log data from granitic interval 1400m-1600m.Data are scaled to zero-mean/unit-variance format.Lateral log induction shown in blue; spontaneous potential shown in red; reference temperatures shown as black dots.The temperature data are shifted 10 meters due to depth discrepancy between separate well-log operations.

Figure 29 .
Figure 29.Porosity & permeability distributions for Figure 25 synthetic normal probability plot BVG wellbore recharge data.BVG field-scale permeability distribution minimum permeability κ ~ 3 10 -20 m 2 for crustal volumes with no evidence of passing fluid through extent fracture structures versus minimum BVG permeability κ ~ 3 10 -18 m 2 for crustal volumes with evidence of passing fluid through extent fracture structures.
spatially-averaged effective continuum nature of crustal rock.Key features of rock granularity emerge from considering the empirical spatial fluctuation relation between well-core porosity and well-core permeability, δφ ∝ δlog(κ)[5][6][14][15][16][17].Consider characterizing the porosity of a sample rock volume by a number n of grain-scale defects associated with the pore population.The grain-scale defects are cement bond failures that allow pores to communicate their fluid content with adjacent pores.Such defects are logically associated with pores as pores have the lowest elastic modulus and hence are subject to the greatest local strains within a crustal volume undergoing deformation during tectonic stress loading.In this granularity construct, grain-scale defect connectivity offers a simple quantitative account of crustal rock permeability associated with fluid percolation between pores.For a sample rock volume with n defects, spatial connectivity between defects within the sample scales as the combinatorial factor n! = n(n-1)(n-2)(n-3)……1.Permeability associated with defect connectivity is thus quantified as κ ∝ n!.A well-known mathematical relation, log(n!) ~ n(log(n)-1), quantifies the effect of incrementing the defect population of a sample volume by a small number δn << n.For porosity increment n → n + δn, the incremented permeability is δlog(n!) = log((n+ δn)!) -δlog(n!) ~ (n+ δn)log(n+ δn) -1) -n(log(n)-1), giving δlog(n!) ~ δn log(n).This relation exactly expresses the empirical property of well-core poroperm sequences, δφ ∝ δlog(κ).For porosity proportional to defect number, φ ∝ n, and permeability proportional to defect version available at Energies 2017,10, 1979; doi:10.3390/en10121979connectivity factor n!, κ ∝ n!, cement bond defect connectivity in a granular medium gives a direct mechanism for the essentially universally observed properties of well-core poroperm spatial correlation (II) and its associated lognormality of crustal permeability distributions at core scale and well-production at field scale (III).Placing EGS-type stimulation in wellbore-centric fluid pressurization context, the above granularity construct quantifies the effect on inter-granular fluid flow of introducing a single grainscale defect into a poroperm structure, n → n +1.The incremental effect differs for Poiseuille flow versus granular percolation: adding a single defect to a conduit between continuum dislocation surfaces has a smaller effect than adding a single defect to disseminated/granular-connectivity structures of empirics (I)-(III).For a fluid of dynamic viscosity µ driven by pressure gradient P', Poiseuille volumetric flow per unit breadth of the flow front is Q [m 2 /s] = P' Δ 3 /12µ [Pa/m m 3 /Pa•s].The corresponding fluid velocity is v [m/s] = P' Δ 2 /12µ.For a gap Δ comprising a number n defects in the continuum flow structure, the mean gap increment is δΔ ~ Δ/n.It follows from (v + δv)/v = 1 + δv/v = (Δ+ δΔ) 2 /Δ 2 ~ 1 + 2δΔ/Δ, that adding a single defect to the medium increases the gap by Δδ and increases fluid velocity by δv/v ~ 2/n.For the disseminated empirical granular medium with fluid velocity v ∝ exp(αφ), the equivalent increment gives (κ + δκ)/κ = 1 + δκ/κ ∝ exp(αδφ) ~ 1 + αδφ, whence for φ = nδφ, δκ/κ = δv/v ~ αδφ = αφ/n.For standard reservoir formations with porosity in the range .1 < φ < .3, the empirical values of α, 20 < α < 40, give αφ ~ 6 ± 2 fluid velocity increment factor for aquifer formations.For basement rock with porosity an order of magnitude smaller, φ ~ .01, the value of α increases by an order of magnitude, 300 < α < 700, giving an empirical estimate of fluid velocity increment factor αφ ~ 5 for basement formations.Allowing for rough surfaces at large confining stresses in the deep crust, the effective exponent characteristic of Poiseuille flow mechanics significantly increases [34].If fracture surface roughness is included within the flow conduit, exponents for the effective Poiseuille flow factor Δ m are modelled to increase from 3 to ~ 5-10 [35-36].The increased exponent heightens the effect of increments in defect population on fluid flow in tight rock.

Figure 30 shows
Figure30shows the temperature growth curve T(r,t) as a function of Peclet number.A kmlong EGS wellbore doublet capable of producing Q ~ 10MW heat energy for 30 years from crust at 100 o C requires a wellbore doublet radially offset by a distance ~ 50 meters.An EGS wellbore doublet separated by 50 meters thus requires that each wellbore be effectively stimulated to a 25m radius.

Figure 30 .
Figure30.For Peclet number Pe = r0φv0/D applied to a wellbore-centric heat exchange flow system, an EGS wellbore couplet requires a crustal volume of order given by the length of the wellbores and by the radial offset of the wellbore couplet (vertical axis) for heat supply over a given period of time (horizontal axis).
GJ ~ 150 million BOE, the energy content of a sizeable conventional oil field[48].The heat energy content accessible to a single fully-stimulated Pe ~ 5 EGS doublet of 50m offset and 3 km horizontal extent to produce 30MW heat energy extraction is ~ 1000 times the annual energy consumption of currently operating district heating plants.From Figure30, conductive recharge of an EGS 30MW doublet heat extraction crustal volume can supply heat to the injector-producer wellbore pair for ~ 30 years.Our OTN1 wellbore temperature event modeling calibration of naturally occurring basement rock fracture-connectivity flow systems suggests that the necessary condition of 'full stimulation' for EGS wellbore doublets is that every 3-5m interval of wellbore supports a Pe ~ 5-10 fractureconnectivity flow path between the doublet pair.Model estimates of natural flow stimulation through naturally enhanced poro-connectivity parameter α indicates that doubling the poroconnectivity from ambient crust value α ~ 500 derived from well-core data (e.g.,[24][25][26][27]) to α ~ 1000 achieves the necessary stimulation goal for wellbore-centric flow systems of order 10-meter radius.Preprints (www.preprints.org)| NOT PEER-REVIEWED | Posted: 18 September 2017 doi:10.20944/preprints201709.0079.v1Peer-reviewed version available at Energies 2017, 10, 1979; doi:10.3390/en10121979withPeclet number given as β ≡ v0ρCL/K for a layer of thickness L with thermal constants ρC/K = 1/D traversed by a fluid front moving at velocity v0.The role of crustal porosity φ in weighting the fluid heat transport is omitted from the derivation of (A3), doubtless in consequence of there being little observational control over fluid flow through large-scale sequences of crustal layering and the likelihood that many such layers have moderate to high porosity.As neither of these factors come into play for wellbore-centric flow in basement rock, we carry porosity as an explicit factor for characterizing heat transport.Wellbore-centric radial temperature distributions (A3) are shown in Fig A1 for a sequence of Peclet numbers 0 < Pe < 10.The red curve denotes pure conduction.The central straight-line blue curve denotes Pe = 1, for which thermal conduction transport equal fluid advective transport.Blue curves to the left of the Pe = 1 curve have Pe < 1 while blue curves to right of the Pe = 1 curve have 1 < Pe < 10.

Figure A1 .
Figure A1.Steady-state wellbore-centric radial temperature profiles T(r) = T0 + (T1 -T0)((r/r0) Pe -1)/((r1/r0) Pe -1) for range of fluid flow velocities associated with Peclet numbers Pe = r0φv0/D.Fluid fow is in a cylindrical section with radius r1 at fixed temperature T1 and a central wellbore of radius r0 at temperature T0.Fluid advection velocity fields are determined by fluid velocity v0 at the central wellbore, v(r) ~ r0v0/r.The amount of heat carried by the fluid is proportional to the porosity φ of the flow medium.
wellbore temperature fluctuations to correspond to Pe > ~ 3 fluid advection along the radial fractureconnectivity pathway.The numerical model result shown in Figure 9 indicates that the finite element solution to conservation of thermal energy constraint (2) conforms to the 2D analytic solution (A3) for spatially-averaged radial flow in a wellbore-centric geometry.The spherical analogue temperature field T0 ≤ T(r) ≤ T1 for radial shells centered on a central source/sink cavity is Preprints (www.preprints.org)| NOT PEER-REVIEWED | Posted: 18 September 2017 doi:10.20944/preprints201709.0079.v1Peer-reviewed version available at Energies 2017, 10, 1979; doi:10.3390/en10121979

preprints.org) | NOT PEER-REVIEWED | Posted: 18 September 2017 doi:10.20944/preprints201709.0079.v1 Peer
6-19 suggests a canonical or type concept of basement rock fracture-connectivity permeability that is amenable to detailed numerical investigation of wellbore-centric fluid processes in Dm-scale crustal volumes characterised by spatially-correlated poroperm distributions.Porosity spatial correlation is defined by empirical condition (I) that a wellbore porosity sequence φ(ξ) within the crustal volume has a Fourier power-spectrum Sφ(k) that scales inversely with spatial wavenumber k, Preprints (www.-reviewedversion available at Energies 2017, 10, 1979; doi:10.3390/en10121979

preprints.org) | NOT PEER-REVIEWED | Posted: 18 September 2017 doi:10.20944/preprints201709.0079.v1
10,r-reviewed version available at Energies 2017,10, 1979; doi:10.3390/en10121979scale defect in a crustal volume produces greater flow effect if the defect is embedded in a granular percolation flow structure than if the defect contributes to a conduit gap in the smooth continuum flow structures of the discrete fracture concept of EGS stimulation.It follows that energy expended by wellbore pressurization is more effective in dissipating wellbore fluid pressure fronts if defects generated by fluid pressures contribute to granularity flow structures than if they contribute to continuum flow structures.If the observed effect of large confining stresses is taken into account, the empirical granularity picture is consistent with rough fracture surfaces expected at the limit of thin flow conduit gaps.

Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 18 September 2017 doi:10.20944/preprints201709.0079.v1
10,r-reviewed version available at Energies 2017,10, 1979; doi:10.3390/en10121979temperature T0 and volumetric heat capacity ρC injected or produced at radial velocity v0 steadily transfers Q = 2πr0φv0 ℓ ρCT0 watts of heat energy from/to the wellbore to/from the crust.The crustal temperature surrounding an injector wellbore grows as a function of radius and time from