# Prospects for Assessing Enhanced Geothermal System (EGS) Basement Rock Flow Stimulation by Wellbore Temperature Data

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

_{e}≡ r

_{0}φv

_{0}/D (r

_{0}= wellbore radius, v

_{0}= Darcy velocity at r

_{0}, φ = ambient porosity, D = rock-water thermal diffusivity), gives P

_{e}~ 10–15 for fracture-connectivity flow intersecting the well, and P

_{e}~ 0 for ambient crust. Darcy flow for model P

_{e}~ 10 at radius ~10 m from the wellbore gives permeability estimate κ ~ 0.02 Darcy for flow driven by differential fluid pressure between least principal crustal stress pore pressure and hydrostatic wellbore pressure. Model temperature event flow permeability κ

_{m}~ 0.02 Darcy is related to well-core ambient permeability κ ~ 1 µDarcy by empirical poroperm relation κ

_{m}~ κ exp(α

_{m}φ) for φ ~ 0.01 and α

_{m}~ 1000. Our modelling of OTN1 wellbore temperature events helps assess the prospect of reactivating fossilized fracture-connectivity flow for EGS permeability stimulation of basement rock.

## 1. Introduction

- (i)
- (ii)
- (iii)

## 2. Fennoscandia Basement Wellbore Temperature Events at 1.5 km

^{−18}–10

^{−17}m

^{2}). As discussed below, the empirical poroperm relation κ(x,y,z) ∝ exp(αφ(x,y,z)) valid for the measured value of α ~ 500 for OTN1 is similar to values found for porosity-permeability data from the KTB well and the Borrowdale volcanics at the UK Sellafield nuclear facility. This discussion places OTN1 well-log spatial fluctuation and well-core poroperm data in a wider basement rock context provided by KTB, Fennoscandia and UK Nirex metamorphic basement rock data [24,25,26,27,28].

_{e}≡ v

_{0}ρCL/K, where v

_{0}= fluid flow velocity, ρC = volumetric heat capacity of water, L = layer thickness, and K = thermal conductivity of rock, across the crustal section [29]. The relation of Peclet number to fluid flow velocity links observed temperature distributions to crustal fluid flow structures.

## 3. 3D Global Mesh Finite Element Modelling of Basement Rock Wellbore Temperature Events

**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:

**v**= κ/µ ∇P

**v**= 0, yielding the defining flow equation for finite-element solvers:

**q**= K∇T(x,y,z) − ρCφ

**v**(x,y,z)T(x,y,z)

**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:

**v**(x,y,z)T(x,y,z)) = 1/D φ

**v**(x,y,z)·∇T(x,y,z)

^{−6}m

^{2}/s is the essentially constant thermal diffusivity of the rock-fluid system for rock thermal conductivity K ~ 3 W/m∙°C and fluid volumetric heat capacity ρC ~ 4.28 MJ/m

^{3}∙°C.

**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 wellbore-centric radius r,

_{0}+ (T

_{1}− T

_{0}) ((r/r

_{0})

^{Pe}− 1)/((r

_{1}/r

_{0})

^{Pe}− 1)

_{0}< r

_{1}characterized by boundary temperatures T

_{0}and T

_{1}, with Peclet number P

_{e}= 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, φv

_{0}~ P

_{e}D/r

_{0}.

_{x},u

_{x},u

_{z})∇u) + a(x,y,z,u,u

_{x},u

_{x},u

_{z})u = f(x,y,z,u,u

_{x},u

_{x},u

_{z})

_{x}, u

_{x}and u

_{z}. 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,u

_{x},u

_{x},u

_{z}) for (7), to meet the conditions imposed by stochastic poroperm media [30,31,32].

_{x},u

_{x},u

_{z}) set to unity, coefficient term a(x,y,z,u,u

_{x},u

_{x},u

_{z}) set to zero, and source term set to the advective flow of heat, f(x,y,z,T,T

_{x},T

_{x},T

_{z}) = 1/D φ

**v**(x,y,z)·$\nabla $T(x,y,z).

_{0}(r

_{0},z) = 1/h q

_{0}(r

_{0},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.

**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 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, σ

_{h}~ 22 MPa/km·zkm, exceeds hydrostatic pressure, P

_{h}~ 10 MPa/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 σ

_{h}outside the model cube drives fluid from the crust into the wellbore.

## 4. 2D Wellbore-Centric Radial Flow Peclet Number Characterization of 3D Temperature Models

_{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 P

_{e}= r

_{0}φv

_{0}/D:

_{0}+ (T

_{1}− T

_{0}) ((r/r

_{0})

^{Pe}− 1)/((r

_{1}/r

_{0})

^{Pe}− 1)

_{0}vary with depth along the wellbore interval in crustal rock of mean porosity φ and thermal diffusivity D ~0.7 × 10

^{−6}m

^{2}/s (Appendix A).

_{0}~ 100 °C and external ambient crustal temperature T

_{1}~ 100 °C + ΔT, ΔT ~ 0.05 °C except near the advective flow horizon when ΔT = 1 °C (see Figure 7). Equation (5) is fit to the Figure 8 numerical model radial temperature distributions for the sequence of wellbore depths (a–h) in Figure 11. Beginning at the upper-left plot (a), successive axial offsets from the horizontal fracture-connectivity flow structure are described by decreasing Peclet numbers in plots (b–h) as thermal advection heat transfer decreases relative to thermal conduction heat transfer. The 2D analytic approximations (5) for optimized Peclet number closely fit the numerical model temperature profiles. The Figure 11 agreement between a sequence of 2D analytic expression (5) optimized for Peclet number, and the Figure 7, Figure 8, Figure 9 and Figure 10 3D numerical model temperature distribution, validate the finite-element numerical procedure (7). Figure 12 characterizes the effective Darcy fluid inflow parameter φv

_{0}= P

_{e}D/r

_{0}at offsets along the wellbore axis from the advective heat inflow horizon.

_{1}/r

_{0}. The heat advection flow computation for Figure 7, Figure 8, Figure 9, Figure 10, Figure 11 and Figure 12 assumes model radius r

_{1}= 10 m enclosing a wellbore radius r

_{0}= 30 cm, giving model/wellbore radius ratio r

_{1}/r

_{0}= 33. The steady-state advective heat transfer temperature field (5) indicates that for Peclet numbers of interest, e.g., P

_{e}> ~3, wellbore-centric heat transfer for large values of r

_{1}/r

_{0}is essentially decoupled from the external boundary. 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.

_{1}for radius r = r

_{1}for all values of Peclet number P

_{e}, 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:

_{0}/r

_{1}<< r/r

_{1}< 1) ~ T

_{0}(1 − (r/r

_{1})

^{Pe}) + T

_{1}(r/r

_{1})

^{Pe}~ T

_{0}+ T

_{1}(r/r

_{1})

^{Pe}

_{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.

_{e}/r (T

_{1}− T

_{0})(r

_{0}/r

_{1})

^{Pe}− T

_{0}

_{0}is directly dependent on the crustal heat inflow q

_{1}at the model external boundary:

_{0}~ T

_{1}(r

_{0}/r

_{1})

^{Pe}− r

_{1}q

_{1}/KP

_{e}

_{0}.

_{1}/r

_{0}= 3 (Figure 13). Computing the axial temperature profile for a model wellbore radius r

_{0}= 3 m in an r

_{1}= 10 m model volume tests the extent to which the r

_{1}/r

_{0}influences model axial temperature distributions. Figure 14 shows that if the external radial temperature boundary is not sufficiently far away from the wellbore heat flux boundary, the model axial temperature distribution is overly affected by the radial temperature boundary condition and the model cannot match the observed axial temperature profile. The Figure 13 wellbore radius geometry shows that the external temperature boundary must be sufficiently far from the wellbore internal heat flux boundary to allow sufficient heat to be thermally conducted away from the advective heat influx flow structure in order to match observation. Figure 15 results for the Figure 13 wellbore model are to be compared with Figure 10.

_{1}/r

_{0}difference, 33 to 3.3. For 0.6 m thick advective fluid inflow fracture-connectivity horizons, the 30 cm wellbore temperature profiles are more sharply peaked than the 3 m wellbore temperature profiles. In particular, the lower panels of Figure 10 and Figure 14 show that model temperature profiles for adjacent 0.6 m inflow horizons follow the observed temperature contours for a 30 cm wellbore while model profiles for a 3 m wellbore cannot follow these contours. We conclude on the basis of the latter condition that 0.6 m-thick fracture-connectivity inflow structures draining into an r

_{0}= 30 cm wellbore must extend deeper into the crust than r

_{1}= 1 m.

_{1}/r

_{0}= 10, does not match the OTN1 temperature profile for the adjacent inflow horizons shown in the lower-left panel of Figure 16. The 1 m model wellbore matches the lower-right panel better than does the 3 m model wellbore but does not require a narrower gauge wellbore model. From the Figure 11, Figure 15 and Figure 16 model/data matches, we can infer that, within the limits of our 161-node 3D computational spatial resolution, crustal penetration of a 0.6 m-thick fracture-connectivity inflow structures is greater than 3 m and is consistent with being a great as 10 m.

_{e}~ 15, at the location of the inflow structure, at 10 m offsets from the inflow structure effective advective heat transport is reduced to the level the level of thermal conduction, P

_{e}< 1. Figure 18 shows, however, that the effect of increasing the thickness of the inflow structure to 2–3 m reduces the Peclet number near the inflow structure, P

_{e}~ 5, and extends the axial effect of heat advective flow beyond 10 m. A thick advective heat influx structure spreads the axial temperature profile along the wellbore. The axial temperature spreading effect seen in the Peclet numbers of Figure 18 corresponds to the broader OTN1 temperature event pictured in the upper-right panels of Figure 10, Figure 15 and Figure 16.

_{1}q

_{1}= r

_{0}q

_{0}= KP

_{e}ΔT

_{0}gives the rate at which heat energy leaves the model crustal volume due to radial heat flow in Peclet number P

_{e}~ 10 fracture-connectivity structures producing OTN1 temperature excursions ΔT

_{0}~ 0.05–0.1 °C. Advective heat energy leaving the model crustal volume via wellbore-centric fluid inflow structures of thickness ℓ is then ΔQ

_{ad}= q

_{0}2πr

_{0}ℓ = 2π P

_{e}KℓΔT

_{0}. For P

_{e}= 10 and ℓ = 1 m, ΔQ

_{ad}= 2π × 10

^{3}W/m/°C 1 m 0.1 °C ~20 W for each temperature event. A similar degree of wellbore-centric heat, ΔQ

_{cd}~20 W, exits from the crustal volume due to conduction, P

_{e}~ 1, intervals of wellbore length ℓ = 10 m.

^{3}m

^{3}and heat capacity 840 J/kg/°C × 2200 kg/m

^{3}at 100 °C ambient temperature is E = 100 GJ. At a prospective loss of heat energy on order of 40 W due to OTN1 advection temperature events and associated heat conduction, the model crustal volume loses heat energy ΔE ~ 40 W·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}= KP

_{e}ΔT

_{0}/r

_{1}~ 0.3 W/m

^{2}, about 6 times the nominal crustal heat flow of 50 mW/m

^{2}. Given that the OTN1 temperature ‘events’ are an observed phenomenon that are an order of magnitude greater than can be expected from purely thermal conductivity variations, the above given numbers illustrate that the model-implied heat flows attributed by our working hypothesis for the origin of the temperature ‘events’ are not out of line with ambient crustal heat flow. I.e., 0.05–0.1 °C wellbore temperature deviations can be seen as localized advective temperature ‘events’ that are not suspiciously large deviations from geothermal gradient heat flow.

_{0}~ P

_{e}D/φr

_{0}at the wellbore and v

_{1}~ P

_{e}D/φr

_{1}at the model periphery. For a model P

_{e}~ 10 advective flow system with 10 m 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 v

_{1}~ 0.7 × 10

^{−4}m/s. These inferred model fluid flow rates can be put into context with reference to convective heat transport, which typically involves fluid flow rates in excess of 10

^{−7}m/s [33]. Fluid flow rates of order 10

^{−7}m/s can be inferred from temperature data for planar crustal fluid flow [29]. The model flow rates are thus consistent with wellbore-stimulated localized enhanced flow/transport processes giving rise to localized temperature deflections of order 0.05–0.1 °C.

_{m}~ vµ/∂

_{r}P, 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 P

_{h}~ 10 MPa/km·1.5 km and crustal pore pressure given by the minimum principal stress σ

_{h}~ P

_{p}~ 22 MPa/km·1.5 km for crustal minimum principal stress at 1.5 km depth, κ

_{m}~ 0.7 × 10

^{−4}m/s⋅0.5 ×·10

^{−3}Pa∙s × 10 m/(12 × 1.5 × 10

^{6}Pa) ~2 × 10

^{−14}m

^{2}~ 0.02 Darcy.

_{m}~ κ exp(α

_{m}φ) for φ ~ 0.01 and κ ~ 1 μDarcy is then α

_{m}~ 1000. If we associate an effective ambient value for poro-connectivity α ~ 500 (Figures 25–29 [24,25,26,27]), then an OTN1 temperature event model flow structure poro-connectivity α

_{m}~ 1000 is of order twice the ambient value is nominally characteristic of Peclet number P

_{e}~ 10 flow structures.

## 5. Wellbore Temperature Event Modelling as Assessment of EGS Stimulation of Basement Rock

_{φ}(k) that scales inversely with spatial wavenumber k, S

_{φ}(k) ∝ 1/k

^{β}, β ~ 1. 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) and (iii).

- 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.

#### 5.1. OTN1 Basement Rock Solute Transport Events

_{4}and Mg species [34], with electrical conductivities ranging between 10 mS/cm and 50 mS/cm steadily growing over a period of years since drilling [35].

#### 5.2. Basement Rock Well-Log and Well-Core Empirics for KTB, Fennoscandia and Borrowdale Metamorphic Rock

^{β}, β ~1, observed at 1–5 km depths (Figure 20 and Figure 21; [24]). It is also seen that OTN1 and Outokumpu mining district Fennoscandia basement rock subject to past high-grade metamorphism preserves S(k) ∝ 1/k

^{β}, β ~ 1, spatial correlation spectral scaling to 2.5 km depths (Figure 22 and Figure 23; [28]).

#### 5.3. Crustal Deformation Energetics for Spatially-Correlated Crustal Fluid Flow Granularity

^{2}/s) = P’ Δ

^{3}/12 μ (Pa/m m

^{3}/Pa·s). The corresponding fluid velocity is v (m/s) = P’ Δ

^{2}/12 μ. 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.

^{m}are modelled to increase from 3 to ~5–10 [38,39]. The increased exponent heightens the effect of increments in defect population on fluid flow in tight rock.

#### 5.4. EGS Couplet Scale Dimensions

_{0}and length ℓ in a crustal volume of porosity φ, fluid of temperature T

_{0}and volumetric heat capacity ρC injected or produced at radial velocity v

_{0}steadily transfers Q = 2πr

_{0}φv

_{0}ℓ ρCT

_{0}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 the initial crustal temperature as:

_{0}(r/r

_{0})

^{ν}∫dk/k exp(−Dk

^{2}t) [J

_{ν}(kr)Y

_{ν}(kr

_{0}) − Y

_{ν}(kr)J

_{ν}(kr

_{0})]/[J

_{ν}

^{2}(kr

_{0}) + Y

_{ν}

^{2}(kr

_{0})]

_{e}/2 one-half the Peclet number P

_{e}≡ r

_{0}φv

_{0}/D, and J

_{ν}(:) and Y

_{ν}(:) the order ν Bessel functions of the first and second kind respectively [40].

_{e}= r

_{0}φv

_{0}/D = r

_{0}φv

_{0}ρC/K links the crustal temperature growth to the rate at which the wellbore supplies heat:

_{e}= Q/2πKℓT

_{0}

_{0}. For a maximally stimulated wellbore that supports P

_{e}~ 5 heat transport across every meter of wellbore length for crustal rock temperature 100 °C, the total wellbore-centric heat extraction rate is Q ~ 10 MW per km of wellbore.

## 6. Summary/Conclusions

^{2}ℓ ρC T ~0.75 × 10

^{18}J ~ 150 million BOE, the energy content of a sizeable conventional oil field [49]. The heat energy content accessible to a single fully-stimulated P

_{e}~ 5 EGS doublet of 50 m offset and 3 km horizontal extent to produce 30 MW heat energy extraction is ~1000 times the annual energy consumption of currently operating district heating plants. From Figure 29, conductive recharge of an EGS 30 MW doublet heat extraction crustal volume can supply heat to the injector-producer wellbore pair for ~30 years.

_{e}~ 5–10 fracture-connectivity flow path between the doublet pair. Model estimates of natural flow stimulation through naturally enhanced poro-connectivity parameter α indicates that doubling the poro-connectivity from ambient crust value α ~500 derived from well-core data (e.g., Figure 24, Figure 25, Figure 26, Figure 27 and Figure 28, [24,25,26,27]) to α ~1000 achieves the necessary stimulation goal for wellbore-centric flow systems of order 10-meter radius.

_{e}~ 10–15 flow structure or broader P

_{e}~ 5 flow structures, but radial cross-well fluid flow must occur along the entire length ℓ of the injection-production wellbore pair to achieve full EGS heat production. In this definition of fully-stimulated EGS wellbore production Q, we use a conservative estimate Q = 2πP

_{e}Kℓ’T

_{0}where the effective length unit of wellbore heat production is 3 m, so that ℓ’ = ℓ/3. Additionally, to produce ~10 MW from fully-stimulated wellbore-pairs for 30 years, the radial cross-well flow-connectivity flow structures must be of order 50 m path length, equating to ~25 m stimulation radius for each wellbore. Development of the full axial-stimulation and the 25 m radial-stimulation conditions indicated by OTN1 temperature event assessment for deep crustal basement EGS are currently very much works in progress.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A. 2D Steady-State Wellbore-Centric Heat Transport Analytics

**v**(r) ~ r

_{0}v

_{0}/r in a wellbore-centric radial domain r

_{0}≤ r ≤ r

_{1}, the conservation of heat energy condition:

^{2}T(r) = 1/D φ

**v**(r) ∇T(r)

**A**(r) = 1/r ∂

_{r}(rA

_{r}), (A1) becomes:

_{r}

^{2}T(r) + (1 − P

_{e})/r ∂

_{r}T(r) = 0

_{e}= r

_{0}φv

_{0}/D, from which it follows that:

_{0}+ (T

_{1}− T

_{0}) ((r/r

_{0})

^{Pe}− 1)/((r

_{1}/r

_{0})

^{Pe}− 1)

_{0}+ (T

_{L}− T

_{0})∙(e

^{β}

^{z/L}− 1)/(e

^{β}− 1)

_{0}ρCL/K for a layer of thickness L with thermal constants ρC/K = 1/D traversed by a fluid front moving at velocity v

_{0}.

_{e}< 10. The red curve denotes pure conduction. The central straight-line blue curve denotes P

_{e}= 1, for which thermal conduction transport equal fluid advective transport. Blue curves to the left of the P

_{e}= 1 curve have P

_{e}< 1 while blue curves to right of the P

_{e}= 1 curve have 1 < P

_{e}< 10.

**Figure A1.**Steady-state wellbore-centric radial temperature profiles T(r) = T

_{0}+ (T

_{1}− T

_{0})((r/r

_{0})

^{Pe}− 1)/((r

_{1}/r

_{0})

^{Pe}− 1) for range of fluid flow velocities associated with Peclet numbers P

_{e}= r

_{0}φv

_{0}/D. Fluid fow is in a cylindrical section with radius r

_{1}at fixed temperature T

_{1}and a central wellbore of radius r

_{0}at temperature T

_{0}. Fluid advection velocity fields are determined by fluid velocity v

_{0}at the central wellbore, v(r) ~ r

_{0}v

_{0}/r. The amount of heat carried by the fluid is proportional to the porosity φ of the flow medium.

_{e}> ~3, the radial thermal gradient vanishes near the wellbore. For P

_{e}> ~3 all heat transport near the wellbore is by advective flow. We can thus expect observed wellbore temperature fluctuations to correspond to P

_{e}> ~3 fluid advection along the radial fracture-connectivity pathway. The numerical model result shown in Figure 8 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.

_{0}≤ T(r) ≤ T

_{1}for radial shells centered on a central source/sink cavity is:

_{0}+ (T

_{1}− T

_{0}) (exp(P

_{e}(r

_{0}/r − 1)) − 1)/(exp(P

_{e}(r

_{0}/r

_{1}− 1)) − 1) (sphere)

_{0}r

_{0}/r, (cylinder)

_{0}r

_{0}

^{2}/r

^{2}, (sphere)

_{r}T(r) − ρCv(r)T(r) has the form of thermal conduction with heat flux magnitude Q

_{0}= v

_{0}ρC(T

_{1}− T

_{0}):

_{0}r

_{0}/r, (cylinder)

_{0}r

_{0}

^{2}/r

^{2}, (sphere)

^{2}T(r) = 0 in the radial domain r

_{0}≤ r ≤ r

_{1}:

_{0}+ (T

_{1}− T

_{0})/ln(r

_{0}/r

_{1})∙ln(r

_{0}/r), Q(r) = Q

_{0}r

_{0}/r, (cylinder)

_{0}+ (T

_{1}− T

_{0})/(r

_{0}/r

_{1}− 1)∙(r

_{0}/r − 1), Q(r) = Q

_{0}r

_{0}

^{2}/r

^{2}(sphere)

_{0}is the flux at the inner radius r

_{0}, Q

_{0}= K(T

_{1}− T

_{0})/r

_{0}/ln(r

_{0}/r

_{1}) < 0 for cylindrical flow and Q

_{0}= K(T

_{1}− T

_{0})/r

_{0}

^{2}/(1/r

_{1}− 1/r

_{0}) < 0; the negative signs for the heat flow magnitudes indicate inward heat flow.

^{x}as the limit of (1 + x/n)

^{n}as n → ∞ gives the limiting case α → 0 for the ratio (x

^{α}− 1)/(x

_{0}

^{α}− 1) as:

^{α}− 1)/(x

_{0}

^{α}− 1) = (e

^{αlnx}− 1)/(e

^{α}

^{lnx0}− 1) ~ ((1 + αlnx/n)

^{n}− 1)/((1 + αlnx

_{0}/n)

^{n}− 1)

_{0}− 1) = ln(1/x)/ln(1/x

_{0})

_{e}→ 0, (exp(P

_{e}(r

_{0}/r − 1)) − 1)/(exp(P

_{e}(r

_{0}/r

_{1}− 1)) − 1) ~ (1 + P

_{e}(r

_{0}/r − 1) − 1)/(1 + P

_{e}(r

_{0}/r

_{1}− 1) − 1) = (r

_{0}/r − 1)/(r

_{0}/r

_{1}− 1).

_{0}becomes effectively very large, r

_{0}→ ∞, the above expressions revert to the plane flow geometry case for P

_{e}/r

_{0}= v

_{0}/D = v

_{0}ρC/K and β ≡ v

_{0}ρCL/K:

_{0}+ δr) ∝ (1 + δr/r

_{0})

^{α}= (1 + δr/r

_{0})

^{v0r0/D}~ exp(v

_{0}δr/D) = exp(βδr/L) (cylinder)

_{0}+ δr) = v

_{0}/(1 + δr/r

_{0}) ~ v

_{0}

_{0}+ δr) = Q

_{0}/(1 + δr/r

_{0}) ~ Q

_{0}≡ v

_{0}ρC(T

_{1}− T

_{0})

_{0}+ δr) ∝ exp(αδr/r

_{0}) = exp(δrv

_{0}ρC/K) = exp(βδr/L)(sphere)

_{0}+ δr) = v

_{0}/(1 + 2δr/r

_{0}) ~ v

_{0}

_{0}+ δr) = Q

_{0}/(1 + 2δr/r

_{0}) ~ Q

_{0}≡ v

_{0}ρC(T

_{1}− T

_{0})

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**Figure 1.**OTN1 temperatures (

**Upper**) and reduced temperatures (

**Lower**) between 100 and 1900 m; in upper plot, black line denotes constant-gradient fit to temperature data in red.

**Figure 2.**Reduced temperature log positive excursions (

**Upper**) and deviations from local temperature trending (

**Lower**) logged for the granitic rocks between 1120 and 1700 m in the OTN1 pilot well; in upper plot the black line denotes a polynomial fit to longer-term temperature trends. Short term temperature deviations in lower plot can be plausibly associated with independent local fracture-connectivity flow structures that intersect the wellbore at the given depths.

**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.**(

**Upper**) Velocity field representing fluid flow and heat transport in Figure 2 fracture-connectivity structure entering a wellbore fluid; background flow from the crust into the fluid is seen along the interface. (

**Middle**) 2D representation of Figure 2 temperature events as heat transport from a hotter low-permeability medium (red-magenta layers representing hotter crustal rock) to a colder high-permeability medium (yellow-oranges layer representing wellbore fluid). (

**Lower**) Temperature profile measured along the fluid-crust interface representing Figure 2 wellbore axial temperature profiles.

**Figure 5.**Wellbore-centric computational volume data cube 161 nodes on a side for node spacing ~12 cm. A 30 cm 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.**Frequency distributions typical of ambient porosity and permeability fluctuations embedded in Figure 5 numerical crustal volume characterized by spatially correlated porosity and permeability empirics (I)–(III).

**Figure 7.**Model crustal volume external boundary temperature distribution over wellbore-centric computational volume for 0.6 m-thick isolated fracture-connectivity percolation fluid flow horizon that intersects an internal vertical wellbore. The existence of such isolated fracture-connectivity percolation fluid flow structures in the ambient crust is a consequence of the crustal rock-fluid interaction empirics (I)–(III) given in the introduction. Following Figure 5, the model volume is 20 m on a side and with an embedded central vertical wellbore. By model hypothesis, the crustal volume is at a constant ambient fluid pressure and temperature (nominally 100 °C), with the interior wellbore at lower fluid pressure and temperature. A horizonal crustal section of elevated poro-connectivity permeability κ(x,y,z) ~ exp(αφ(x,y,z)) feeds fluid of incrementally higher temperature through the model volume into the wellbore. The elevated model boundary temperature is fixed at nominal 1 °C above the ambient temperature. Steady-state fluid inflow from the crust at incremented boundary temperature transports heat to the wellbore, creating a localized temperature deviation from the ambient wellbore temperature. With these boundary conditions, the steady-state finite-element solver (7) generates a 3D temperature field that obeys the conservation of energy constraint Equation (4). The model axial temperature distribution at the central wellbore is compared to OTN1 temperature profiles in Figure 2. The resulting 3D temperature field is nominally a series of radially symmetric wellbore-centric temperature distributions that vary as a function of depth along the wellbore. The finite-element model 2D approximations to wellbore-centric temperature distributions can be compared with the analytic solution (5) for strictly 2D wellbore-centric flow characterized by Peclet number P

_{e}= r

_{0}φv

_{0}/D.

**Figure 8.**Quarter section of temperature distribution over wellbore-centric computational volume for thin fracture-connectivity plane illustrated in Figure 7. Wellbore axis is located at (x,y) = (0,0). Model dimensions in meters. Model temperatures are nominal 100 °C with 1 °C of elevated temperature at z = 4 horizon of advective flow entry from enclosing crustal volume.

**Figure 9.**Quarter section of normalised advective heat flow distribution over wellbore-centric computational volume for thin fracture-connectivity plane illustrated in Figure 7. The temperature field associated with the displayed advective flow distribution is shown in Figure 8. Wellbore axis is located at (x,y) = (0,0). Model dimensions in meters.

**Figure 10.**Overlays of Figure 5 model wellbore axial temperature profiles (red traces) for 30 cm radius wellbore embedded in a 10-m radius crustal volume superposed on OTN1 wellbore temperature events A, B, C, and D (black traces). OTN1 event depths are given on horizontal axes; event/model temperatures are normalized to zero-mean/unit-variance format. The favoured model flow channel thickness is 0.6 m, as illustrated in Figure 7. The observed flow channel events occur either singly or in pairs. (

**Upper Left**) Model temperature profile computed for single 0.6 m thick flow channel. (

**Upper Right**) Model temperature profile for 2–3 m thick flow channel. (

**Lower Left**) Model temperature profile for 4-m spaced pair of 0.6 m thick flow channels. (

**Lower Right**) Model temperature profile for 3-m spaced pair of 0.6 m thick flow channels.

**Figure 11.**2D analytic wellbore-centric radial temperature profiles (Equation (5), red traces) for given Peclet number matched to 3D numerical model radial temperature distributions (blue dots) for 30 cm radius wellbore embedded in 10-m radius crustal volume (Figure 6 and Figure 7). 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 °C + ΔT for ΔT ~ 0.05 °C except near the advective flow structure when ΔT = 1 °C. From upper left to lower right, plots (

**a**–

**h**) show the 3D model radial temperature distributions fit to 2D steady-state radial temperature distributions governed by Peclet numbers P

_{e}= r

_{0}φv

_{0}/D at successive axial distances from the fluid influx fracture-connectivity structure. The Peclet number (

**a**) at the fracture-connectivity structure is 16. Stepping away from the influx structure in 1.5 m intervals, Peclet numbers (

**b**–

**d**) decline as 6.2, 4.2, and 2.1; at wellbore axial offsets greater than 6 m from the fracture-connectivity structure, thermal conductivity values of Peclet numbers (

**e**–

**h**) edge to P

_{e}= 0.

**Figure 12.**Effective inflow velocity φv

_{0}= P

_{e}D/r

_{0}at successive wellbore offsets from the inflow horizon given by the Figure 7, Figure 8, Figure 9, Figure 10 and Figure 11 model of OTN1 temperature event at depth 1218 m. For crustal rock of mean porosity φ = 0.01, the computed inflow Darcy fluid velocity at wellbore radius 30 cm is v

_{0}~ 4 mm/s across a 1 m wellbore depth interval.

**Figure 13.**Following Figure 5, a wellbore-centric data cube of 161 nodes with node spacing ~12 cm and a 3 m 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). The effect of the large wellbore radius is to reduce the model area over which to axially conduct heat away from the model radial inflow structure.

**Figure 15.**Following Figure 10, overlays of Figure 13 model wellbore axial temperature profiles (red traces) for 300 cm radius wellbore embedded in a 10-m radius crustal volume superposed on OTN1 wellbore temperature events A, B, C, and D (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 0.6 m, as illustrated in Figure 7. (

**Upper Left**) Model temperature profile computed for single 0.6 m thick flow channel. (

**Upper Right**) Model temperature profile for 2–3 m thick flow channel. (

**Lower Left**) Model temperature profile for 4-m spaced pair of 0.6 m thick flow channels. (

**Lower Right**) Model temperature profile for 3-m spaced pair of 0.6 m thick flow channels.

**Figure 16.**Following Figure 11 and Figure 14, overlays of Figure 14 model wellbore axial temperature profiles (red traces) for 100 cm radius wellbore embedded in a 10-m radius crustal volume superposed on OTN1 wellbore temperature events A, B, C, and D (black traces). Horizontal axes give wellbore depth in meters; vertical axes are wellbore temperature profiles normalised to zero-mean/unit-variance. (

**Upper Left**) Model temperature profile computed for single 0.6 m thick flow channel. (

**Upper Right**) Model temperature profile for 2–3 m thick flow channel. (

**Lower Left**) Model temperature profile for 4-m spaced pair of 0.6 m thick flow channels. (

**Lower Right**) Model temperature profile for 3-m spaced pair of 0.6 m thick flow channels.

**Figure 17.**Following Figure 11, 2D analytic wellbore-centric radial temperature profiles (red traces) for given Peclet number superposed on 3D numerical model radial temperature distributions (blue dots) for 1 m radius wellbore embedded in 10-m radius crustal volume (Figure 14) with a 0.6 m fracture-connectivity structure. From upper left, plots (

**a**–

**h**) show 3D model radial temperature distributions fit to 2D steady-state radial temperature distributions (5) governed by numbers P

_{e}= r

_{0}φv

_{0}/D. The Peclet number for the fracture-connectivity horizon in plot (

**a**) is 19. Stepping away from the horizon in 1.5 m intervals, Peclet numbers decline for plots (

**b**–

**d**) indicating fluid advection P

_{e}= 14, 4.4 and 2.7 and plots (

**e**–

**h**) with P

_{e}= 0.7 to 0 indicating thermal conduction.

**Figure 18.**Following Figure 12 and Figure 17, 2D analytic wellbore-centric radial temperature profiles (red traces) for given Peclet number superposed on 3D numerical model radial temperature distributions (blue dots) for 1 m radius wellbore embedded in 10-m radius crustal volume with a 2–3 m fracture-connectivity structure. From upper left, plots (

**a**–

**h**) show 3D model radial temperature distributions fit to 2D steady-state radial temperature distributions (5) governed by numbers P

_{e}= r

_{0}φv

_{0}/D. The Peclet number for the fracture-connectivity horizon in plot (

**c**) is 7. Stepping away from the horizon in 1.5 m intervals, the Peclet numbers decline below the influx layer, plots (

**a**,

**b**) with P

_{e}= 5.7 and 6.1, and above the influx layer, plots (

**d**–

**h**) with P

_{e}= 5.2 to 0.0001.

**Figure 19.**OTN1 galvanic property well-log data from granitic interval 1400 m–1600 m. For purposes of comparison, the three data sequences are scaled to zero-mean/unit-variance format. As seen in Figure 2, the reference temperature deflections (black dots) range from −0.05 °C to +0.15 °C relative to the local temperature trend. Lateral log induction variations are shown in blue; spontaneous potential variations are shown in red. The temperature data are shifted 10 m due to depth discrepancy between separate well-log operations.

**Figure 20.**KTB well-log fluctuation power-spectral scaling inversely with spatial frequency, S(k) 1/k

^{β}, β ~ 1; depth 6900–8130 m; LLD = resistivity; DTCO = P-wave sonic; RHOB = mass density; PEF = photo-electric absorption; NPHI = neutron porosity; VPEF = volumetric photo-electric absorption [24].

**Figure 21.**KTB well-log fluctuation amplitude preservation to 6.5 km depth for P-wave sonic velocity (

**Upper**panel) and neutron porosity (

**Lower**panel) [24].

**Figure 23.**Sample Outokumpu well-log sonic-velocity spectral scaling S(k) ~ k

^{β}, β ~ 1.2 ± 0; depths 950–1500 m and 1900–2500 m [28].

**Figure 24.**KTB well-core spatial correlation δφ ∝ δlog(κ); zero-mean/univariance format; φ = blue, δlog(κ) = red; well-core from depth interval 4000–5500 m [24].

**Figure 25.**KTB well-core relation κ ∝ exp(αφ); data blue circles; exponent fit for poro-connectivity parameter α (red); well-core from depth interval 4000–5500 m [24].

**Figure 26.**Well-core porosity, UK Nirex Borrowdale volcanics, Sellafield nuclear facility [27].

**Figure 27.**Synthetic data reproduction of field-scale wellbore hydrologic recharge distribution, for the Borrowdale Volcanics Group (BVG) metamorphic rock suite at the Sellafield nuclear facility [25].

**Figure 28.**Porosity & permeability distributions for Figure 24 synthetic normal probability plot Borrowdale Volcanics Group 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 active fracture structures versus minimum BVG permeability κ ~ 3 × 10

^{−18}m

^{2}for crustal volumes with evidence of passing fluid through active fracture structures.

**Figure 29.**For Peclet number P

_{e}= r

_{0}φv

_{0}/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).

Flow Geometry | Flow Law | Flow Velocity | Velocity Increment | Increment Factor |
---|---|---|---|---|

Smooth Continuum | Q = P’ Δ^{3}/12 μ | v ∝ Δ^{2} | δv/v ~ 2/n | 2 |

Granular | v = κ/μ P’ | v ∝ exp(αφ) | δv/v ~ αφ/n | ~5 |

Rough Continuum | Q = P’ Δ^{m}/12 μ | v ∝ Δ^{m−1}, m ~ 5–10 | δv/v ~ (m − 1)/n | ~4–9 |

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Leary, P.; Malin, P.; Saarno, T.; Kukkonen, I. Prospects for Assessing Enhanced Geothermal System (EGS) Basement Rock Flow Stimulation by Wellbore Temperature Data. *Energies* **2017**, *10*, 1979.
https://doi.org/10.3390/en10121979

**AMA Style**

Leary P, Malin P, Saarno T, Kukkonen I. Prospects for Assessing Enhanced Geothermal System (EGS) Basement Rock Flow Stimulation by Wellbore Temperature Data. *Energies*. 2017; 10(12):1979.
https://doi.org/10.3390/en10121979

**Chicago/Turabian Style**

Leary, Peter, Peter Malin, Tero Saarno, and Ilmo Kukkonen. 2017. "Prospects for Assessing Enhanced Geothermal System (EGS) Basement Rock Flow Stimulation by Wellbore Temperature Data" *Energies* 10, no. 12: 1979.
https://doi.org/10.3390/en10121979