1. Oral defense, the 22 nd September 2011 Krafla geothermal power plant (Island) Rachel M. Gelet supervised by Benjamin Loret, Laboratoire 3S-R, INP Grenoble Nasser Khalili, School of Civil and Environmental Engineering, UNSW Thermo-hydro-mechanical study of porous media with double porosity in local thermal non-equilibrium

2. /50 2 Content II.Thermo-hydro-mechanical (THM) coupled model III.Numerical analysis IV. Geothermal energy recovery applications V.Conclusion I.Problem statement

3. /50 I. Problem statement II. THM coupled model III. Numerical analysis IV. Geothermal energy V. Conclusion Great depth igneous rock Hydraulic stimulation to increase the permeability dual porosity Full THM problem Forced convection Targeted applications Example : Hot water injection Conventional oil recovery Tertiary oil recovery Hydro-Mechanical models Chemical : Chemo-Hydro-Mechanical Thermal : Thermo-Hydro-Mechanical Miscible CO 2 : Unsaturated Enhanced oil recovery Enhanced geothermal energy recovery Modeled crustal temperature at 5 km depth (Chopra & Holgate 2005) The temperature data contained in this image has been derived from proprietary information owned by Earth Energy Pty Ltd ABN 078 964 735. 100 ºC 300 ºC Local thermal non-equilibrium 3

4. /50 I. Problem statement II. THM coupled model III. Numerical analysis IV. Geothermal energy V. Conclusion 4 Classification of fractured reservoirs Reservoir with all storage in the fractures Sharp decline of production rate after a short period of time Reservoir with large storage in the matrix Production rate depends on the degree of fracturation Reservoir with equal storage Smooth production rate n p : porosity of the porous block [-], n f : porosity of the fracture network [-] n f >> n p n p >> n f n f ≈ n p (Bai et al., 1993)

5. /50 I. Problem statement II. THM coupled model III. Numerical analysis IV. Geothermal energy V. Conclusion 5 The dual porosity concept Barenblatt (1960) Two distinct type of cavities: Two distinct roles: 1.Storage in the pores 2.Transport in the fractures Warren and Root (1963) Two overlapping single porous media: Fractured porous medium Fracture network Transfers of mass, momentum, heat, entropy Porous block k p : permeability of the porous block [m 2 ] k f : permeability of the fracture network [m 2 ] 1.Pores 2.Fractures

6. /50 I. Problem statement II. THM coupled model III. Numerical analysis IV. Geothermal energy V. Conclusion 6 Problematics: Fractured reservoirs under coupled THM loading Specifications: - The model must be general and fully coupled - Saturation with one fluid (ex: water) Framework: Thermodynamics of Irreversible Processes Computational aspect: Finite Element Method Background: Extension of a HM model to account for thermal properties Khalili and Valliappan (1996) Objectives and methods

7. /50 7 Content I.Problem statement II.Thermo-hydro-mechanical (THM) coupled model III.Numerical analysis IV. Geothermal energy recovery applications V.Conclusion

8. /50 I. Problem statement II. THM coupled model III. Numerical analysis IV. Geothermal energy V. Conclusion Generalized transfer - Mass - Heat ≠≠ 8 A three phase closed mixture Solid phase fracture fluid phase Pore fluid phase The significant contribution is the local thermal non-equilibrium Generalized diffusion - Hydraulic (Darcy) - Thermal (Fourier) Forced convection Khalili and Loret (2001) Khalili and Selvadurai (2003)

9. /50 I. Problem statement II. THM coupled model III. Numerical analysis IV. Geothermal energy V. Conclusion 9 The mixture theory Each continuum point is occupied by all the constituents of the mixture: Biot’s approach of mixture: The solid phase is defined as the reference constituent, since it holds the other phases. Mixture displacements: Pore fluid pressure: Fracture fluid pressure: Solid temperature: Pore fluid temperature: Fracture fluid temperature: Truesdell (1957) Biot (1977)

10. /50 I. Problem statement II. THM coupled model III. Numerical analysis IV. Geothermal energy V. Conclusion 10 The Thermodynamics of Irreversible Processes (TIP) The Clausius-Duhem (CD) inequality: is used to i.identify the generalized forces/stresses and fluxes/strains ii.restrain the constitutive equations Three types of constitutive equations are required by the TIP - the thermo-mechanical process: - the generalized transfer: - the generalized diffusion: CD inequality

11. /50 I. Problem statement II. THM coupled model III. Numerical analysis IV. Geothermal energy V. Conclusion 11 The CD inequality: thermo-mechanical This implies the existence of an elastic potential for the mixture reversible process thermo-poro-elasticity The generalized stresses - the total stress - the pore fluid pressure - the fracture fluid pressure - the solid temperature The generalized strains - the total strain - the pore fluid volume content - the fracture fluid volume content - the entropy of the solid

12. /50 I. Problem statement II. THM coupled model III. Numerical analysis IV. Geothermal energy V. Conclusion 12 The thermo-mechanical constitutive laws Pore volume Fracture volume Solid entropy Pore pressure Fracture pressure Solid temperature Total stress Total strain Symmetric fully coupled matrix The entropies of fluids are defined separately: The thermal response of the mixture is controlled by the solid phase alone: ΔT s = 0ΔT s > 0 0

13. /50 I. Problem statement II. THM coupled model III. Numerical analysis IV. Geothermal energy V. Conclusion 13 The effective stress concept - extended to dual porous media: - based on a consistent loading decomposition Nur and Byerlee (1971) - uses three compressibilities [Pa -1 ] : c, c p, c s - linked to the elastic deformation: The end result is (with a continuum mechanics sign convention): Khalili and Valliappan (1996) The effective stress is

14. /50 I. Problem statement II. THM coupled model III. Numerical analysis IV. Geothermal energy V. Conclusion 14 The CD inequality: generalized transfer irreversible processes Mass transferEntropy transferHeat transfer Rate of mass transfer Scaled chemical potential Rate of momentum and energy transfers Coldness Rate of entropy transfer 00

15. /50 I. Problem statement II. THM coupled model III. Numerical analysis IV. Geothermal energy V. Conclusion 15 The generalized transfer constitutive law Closure relations The transfer forces are identified as differences Mass Heat Scaled chemical potential Coldness Symmetric uncoupled matrix Barenblatt (1960) Warren and Root (1963) Bowen and Chen (1975) The heat transfer parameters between the phases [W/m 3.K] The leakage parameter [Pa.s]

16. /50 I. Problem statement II. THM coupled model III. Numerical analysis IV. Geothermal energy V. Conclusion 16 The CD inequality: generalized diffusion irreversible processes Gradient / Force Flow ‘Temperature’ ‘Pressure’ Heat Thermal diffusion Fourier’s law Isothermal heat flow FluidThermo-osmosis Hydraulic diffusion Darcy’s law Mitchell and Soga (1993)

17. /50 I. Problem statement II. THM coupled model III. Numerical analysis IV. Geothermal energy V. Conclusion 17 The generalized diffusion constitutive law Direct diffusion of heat Direct diffusion of mass Isothermal heat transfer Thermo-osmosis Generalized fluxes Generalized forces Special features: The partial coupling is due to the space separation of the phases The symmetry of the matrix is enforced (Onsager’s reciprocity principal). Symmetric partially coupled matrix

18. /50 I. Problem statement II. THM coupled model III. Numerical analysis IV. Geothermal energy V. Conclusion  18 Summary of the constitutive equations Identification of the constitutive laws Coupled thermo-mechanics - Thermo-poro-elasticity - Hydraulic coupling - Entropy exchange Un-coupled generalized transfer - Mass transfer - Entropy transfer - Heat transfer Coupled generalized diffusion - Seepage, Thermo-osmosis, - Conduction, Isothermal flow

19. /50 I. Problem statement II. THM coupled model III. Numerical analysis IV. Geothermal energy V. Conclusion 19 The coupled model: 6 field equations - Mass for the fluid phases (k = p, f) - Momentum for the mixture - Energy for the solid phase - Energy for the fluid phases (k = p, f) Mass transfer Heat transfer Convection

20. /50 20 Content I.Problem statement II.Thermo-hydro-mechanical (THM) coupled model III.Numerical analysis IV. Geothermal energy recovery applications V.Conclusion

21. /50 I. Problem statement II. THM coupled model III. Numerical analysis IV. Geothermal energy V. Conclusion 21 Finite element analysis: hints Primary unknowns: - u the mixture displacement - p p and p f the fluid pressures - temperatures T s, T p and T f Spatial discretization: - Galerkin method - Streamline Upwind / Petrov-Galerkin method for the field equations holding convective terms Time discretization: generalized trapezoidal method with α = 2/3 Full Newton-Raphson algorithm

22. /50 I. Problem statement II. THM coupled model III. Numerical analysis IV. Geothermal energy V. Conclusion A domestic code 22 Language: Fortran Background: double porosity HM code New implemented features - Thermo-poro-elasticity 1T + 1P / 2P - Thermo-poro-elasticity 2T + 1P / 2P Stabilization of convection - Streamline Upwind / Petrov-Galerkin + DCM

23. /50 I. Problem statement II. THM coupled model III. Numerical analysis IV. Geothermal energy V. Conclusion 23 Insights in double porosity Investigation of a borehole stability problem (1T + no convection) σ r = σ θ = -23.5 MPa σ z = -29 MPa p p = p f = 9.8 MPa T = 50°C welltop + bottomright Mechanical Hydro pore Hydro fracture Thermo Boundary conditions: p w = 12 MPa, T w = 100°C In-situ conditions (prior to drilling) r z θ 1 km Reservoir σrσr 0 σθσθ 0 σzσz 0 0 0 0 00 εzεz 0 εzεz 0 σrσr 0 pwpw - pw- pw TwTw pwpwp 0 pfpf 0 T 0 J p = 0 J f = 0 q = 0 bottom right top well Axi-symmetric mesh: r - z plane, θ = 0° z = 1 m z = 0 m r 1 = 0.1 m r 2 = 800 m >> r 1

24. /50 I. Problem statement II. THM coupled model III. Numerical analysis IV. Geothermal energy V. Conclusion Insights in double porosity cont’d 2PA1P2P Parameter Pore matrixMixturefracture network Porosity [-]0.140.1540.014 Permeability [m 2 /10 -19 ]5.05.55.0 Hydraulic diffusivity [m 2 /s.10 -6 ]69.07.49.0 Thermal diffusivity [m 2 /s.10 -6 ]1.6 0.66 <1 < 2.16 < 2.38 1. The heat diffuses faster than the flow in the porous blocks 2. The flow diffuses faster than the heat McTigue (1986) Comparison of the diffusivity ratios 24 2P with hydraulic equilibrium ≡

25. /50 I. Problem statement II. THM coupled model III. Numerical analysis IV. Geothermal energy V. Conclusion 25 Insights in double porosity cont’d The A1P model underestimates the failure potential compared to the 2P model The 2P model is capable of finer predictions than the A1P model Zero leakage parameter Average leakage parameter Influence of the mass transfer: x High leakage parameter Associated single porosity (A1P)

26. /50 I. Problem statement II. THM coupled model III. Numerical analysis IV. Geothermal energy V. Conclusion 26 Implementation of convection The Streamline Upwind / Petrov-Galerkin method Brooks and Hughes (1982) Balance of energy of the fluid phase: Diffusion contributionTransient contribution Convective contribution 1. Galerkin method 2. SUPG method Numerical wiggles with FEM = requires stabilization

27. /50 I. Problem statement II. THM coupled model III. Numerical analysis IV. Geothermal energy V. Conclusion Implementation of convection cont’d 2D transient problem: 1D transient problem: Galerkin FEM response SUPG FEM response x 10 m Hot side Cold side φ = 1φ = 0 Grid Péclet number = 42.5 φ = 1 φ = 0 vfvf Grid Péclet number = 10 6 vfvf 27

28. /50 I. Problem statement II. THM coupled model III. Numerical analysis IV. Geothermal energy V. Conclusion 28 Summary of testing steps Checking of the code Coupled THM - Single porosity - Local Thermal Equilibrium - No convection Analytical McTigue (1986) Coupled THM - Double porosity - Local Thermal Equilibrium - No convection Satisfying FEM results Forced convection with SUPG - Un-coupled 1D - Un-coupled 2D Analytical Brooks and Hughes (1982)

29. /50 29 Content I.Problem statement II.Thermo-hydro-mechanical (THM) coupled model III.Numerical analysis IV. Geothermal energy recovery applications V.Conclusion

30. /50 I. Problem statement II. THM coupled model III. Numerical analysis IV. Geothermal energy V. Conclusion 30 Context Procedure to recover geothermal energy  Identify a site with a high geothermal gradient  Drill the injection well  Hydraulic fracturation  Drill the production well next  Perform a circulation test Electricity  Thermal depletion of the reservoir  Fluid loss Electricity Cold water Injection (ex: 70ºC) Hot water production (ex: 178ºC)

31. /50 I. Problem statement II. THM coupled model III. Numerical analysis IV. Geothermal energy V. Conclusion 31 Rese arch niche: Use our continuum model with local thermal non-equilibrium Research challenges: Predict accurately the thermal time course of a given site Understand the impact of a circulation test on the stresses and pressures Identify the mechanism of fluid loss  1 st Objective: - Calibrate the parameters of the model from field experiments data - Understand favourable conditions for LTNE to arise  2 nd Objective: Investigate the influence of - The dual porosity on the effective stress - The fracture spacing (controlling the magnitude of transfers) Objectives 1P-2T Single porosity Two temperatures 2P-2T Double porosity Two temperatures

32. /50 I. Problem statement II. THM coupled model III. Numerical analysis IV. Geothermal energy V. Conclusion 32 Experimental data 1P-2T The Fenton Hill Hot Dry Rock (HDR) reservoir: Zyvoloski et al. (1981) 3 parameters need to be fitted: 1.The fracture network permeability k f [m 2 ] 2.The fracture network porosity n f [-] 3.The solid fluid heat transfer which is related to the solid fluid specific surface Unknown fracture path 2b: average fracture aperture [m]

33. /50 I. Problem statement II. THM coupled model III. Numerical analysis IV. Geothermal energy V. Conclusion 33 A generic HDR reservoir 1P-2T Model assumptions - Single porosity model (n p = 0) - The material properties remain constant - No natural convection nor gravity - No initial stiffness - The flow regime is laminar Insight from characteristic times Inflow Outflow k f : fracture permeability [m 2 ] 2b: average fracture aperture [m] x E : average fracture spacing [m] 230 m 200 m

34. /50 I. Problem statement II. THM coupled model III. Numerical analysis IV. Geothermal energy V. Conclusion 34 Boundary conditions 1P-2T Initial conditions - Local thermal equilibrium: - In-situ pressure: - Over burden vertical stress: - Earth lateral stress: Loading conditions Thermal Hydraulic Mechanical

35. /50 I. Problem statement II. THM coupled model III. Numerical analysis IV. Geothermal energy V. Conclusion 35 The double step pattern 1P-2T The thermal drawdown is successfully calibrated with field data: 1 st stage2 nd stage 3 rd stage Convection dominance of the fluid Heat transfer Final depletion of the mixture Local thermal equilibrium ≡ κ sf > 100 mW/m 3.K n f = 0.005 k f = 3.06 10 -14 m 2 = 2626 m ≈ 2660 m = 2673 m = 2703 m Fenton Hill experimental data κ sf : The solid-to-fracture fluid heat transfer parameter

36. /50 I. Problem statement II. THM coupled model III. Numerical analysis IV. Geothermal energy V. Conclusion 36 A THM-1P-2T coupled response Inflow Outflow Cooling of the solid phase arises during the late period only Fluid temperature Solid temperature

37. /50 I. Problem statement II. THM coupled model III. Numerical analysis IV. Geothermal energy V. Conclusion 37 A THM-1P-2T coupled response cont’d Inflow Outflow The fracture fluid pressure is not affected by the circulation test

38. /50 I. Problem statement II. THM coupled model III. Numerical analysis IV. Geothermal energy V. Conclusion 38 A THM-1P-2T coupled response cont’d Inflow Outflow Thermally induced tensile stress Thermal contraction Vertical strain Vertical effective stress

39. /50 I. Problem statement II. THM coupled model III. Numerical analysis IV. Geothermal energy V. Conclusion 39 A generic HDR reservoir 2P-2T Model assumptions Same as before. Boundary conditions are extended to double porosity Thermal Hydraulic Inflow Outflow + T s = T p

40. /50 I. Problem statement II. THM coupled model III. Numerical analysis IV. Geothermal energy V. Conclusion 40 The double step pattern holds for 2P-2T The calibration is successful on two different sites 30 < κ sf < 120 mW/m 3.K n f = 0.005 k f = 1.0 10 -14 m 2 = 2626 m ≈ 2660m = 2673 m = 2703 m n f = 0.005 k f = 1.0 10 -13 m 2 ≈ 2152 m (Zyvoloski et al., 1981) (Kolditz and Clauser, 1998)

41. /50 I. Problem statement II. THM coupled model III. Numerical analysis IV. Geothermal energy V. Conclusion 41 Bounds for the THM-2P-2T response The double porosity model (2P) Average response - solid, pore fluid, fracture fluid - T s = T p ≠ T f - Average diffusive flow in the pores: k p = 10 -21 m 2 Average mass transfer A single porosity model (1P) Upper bond - solid, fracture fluid - A motion less pore fluid (no mass transfer nor hydraulic diffusion) - T s ≠ T f An ‘unconnected’ double porosity model (2P) Lower bond - solid, pore fluid, fracture fluid - T s = T p ≠ T f - Slow diffusive flow in the pores: k p = 10 -23 m 2 Small mass transfer Theoretical response

42. /50 I. Problem statement II. THM coupled model III. Numerical analysis IV. Geothermal energy V. Conclusion 42 A THM-2P-2T coupled response Cooling during the late period, t = 3.17 years, and for k f = 1.10 -14 m 2 Tensile stresses are damped by the pore pressure contribution Inflow Outflow Porous block temperature Fracture fluid temperature Pore fluid pressure Vertical effective stress

43. /50 I. Problem statement II. THM coupled model III. Numerical analysis IV. Geothermal energy V. Conclusion 43 A THM-2P-2T coupled response cont’d Small fracture spacing x E leads to local thermal equilibrium  The solid-fracture-fluid heat transfer κ sf parameter increase if x E decreases Materials Site Inflow Outflow Porous block temperature Fracture fluid temperature

44. /50 I. Problem statement II. THM coupled model III. Numerical analysis IV. Geothermal energy V. Conclusion 44 A THM-2P-2T coupled response cont’d The induced tensile stress is reduced for large fracture spacing x E Small fracture spacing x E leads to hydraulic equilibrium  The leakage parameter η increase if x E decreases with Inflow Outflow Pore fluid pressure Vertical effective stress

45. /50 I. Problem statement II. THM coupled model III. Numerical analysis IV. Geothermal energy V. Conclusion 45 A THM-2P-2T coupled response cont’d Fluid loss is large during the early period and reduces with time  The mass transfer force is function of the difference between the fluid pressures and between the fluid temperatures Early period t = 34.72 days Mass transfer force

46. /50 I. Problem statement II. THM coupled model III. Numerical analysis IV. Geothermal energy V. Conclusion 46 A THM-2P-2T coupled response cont’d Fluid loss is large during the early period and reduces with time  The mass transfer force is function of the difference between the fluid pressures and between the fluid temperatures This model response matches typical field observations (Murphy et al., 1981) Mass transfer force Late period t = 3.2 years

47. /50 I. Problem statement II. THM coupled model III. Numerical analysis IV. Geothermal energy V. Conclusion 47 Summary of the geothermal results Calibration with field tests 1P response - Double step pattern - No influence on the fracture pressure - Thermally induced tensile effective stress Fenton Hill 2P response - Double step pattern - Pore pressure reduction - Damping effect on the tensile effective stress - Water loss reduces with time Fenton Hill Rosemanowes Recommendation The average fracture spacing x E should be large to minimize the thermally induced tensile effective stress and uncontrolled water loss

48. /50 48 Content I.Problem statement II.Thermo-hydro-mechanical (THM) coupled model III.Numerical analysis IV. Geothermal energy recovery applications V.Conclusion

49. /50 I. Problem statement II. THM coupled model III. Numerical analysis IV. Geothermal energy V. Conclusion 1.An original constitutive model Dual porosity Full THM coupling with generalized diffusion, transfer and convection Local thermal non-equilibrium between the phases 2.A domestic THM non-linear finite element code Forced convection stabilized with the SUPG method 3.A borehole stability analysis of fractured media: 1 st paper (Revised) 4.Applications to enhanced geothermal systems: 2 nd and 3 rd paper (In progress) Identification of a double step pattern for LTNE Prediction of THM response and of water loss over time Recommendations on the fracture network geometry 49 The main contributions

50. /50 I. Problem statement II. THM coupled model III. Numerical analysis IV. Geothermal energy V. Conclusion Conclusions 50 Assets The model is thermodynamically admissible The model parameters were all interpreted The convective terms were successfully stabilized for the late period The FEM allows rapid simulations The domestic code can be enhanced for further research Drawbacks  The initial stress state of the mixture does not influence the results (ex: initial stiffness)  Some parameters of the model could not be identified (ex: thermo-osmosis, etc.)  The time integration scheme should be enhanced to capture stiff thermal loadings

51. /50 I. Problem statement II. THM coupled model III. Numerical analysis IV. Geothermal energy V. Conclusion 51 Perspectives Investigate the effects of forced convection for a borehole stability problem The thermally induced pore pressure due to LTNE should extend instability issues over a larger time scale The impact of the coupled mass transfer law should affect the effective stress Characterization of the specific surface area between the solid and the fracture network Comparison between the empirical formula used throughout and experimental data would help characterize LTNE Experimental tests so as to validate the new mass transfer law

52. /50 52 Thank you ! Cham Jorge from www.phdcomics.com