Rachel M. Gelet supervised by Benjamin Loret, Laboratoire 3S-R, INP Grenoble

1. Thermo-hydro-mechanical study of porous media with double porosity in local thermal non-equilibrium Rachel M. Gelet supervised by Benjamin Loret, Laboratoire 3S-R, INP Grenoble Nasser Khalili, School of Civil and Environmental Engineering, UNSW

2. Content Problem statement I. Problem statement • Thermo-hydro-mechanical (THM) coupled model • Numerical analysis • Geothermal energy recovery applications • Conclusion

3. Targeted applications Enhanced geothermal energy recovery Enhanced oil recovery Great depth igneous rock Hydraulic stimulation to increase the permeability dual porosity Full THM problem Forced convection Conventional oil recovery Hydro-Mechanical models Tertiary oil recovery • Chemical : Chemo-Hydro-Mechanical • Thermal : Thermo-Hydro-Mechanical • Miscible CO2 : Unsaturated Local thermal non-equilibrium Example : Hot water injection Modeled crustal temperature at 5 km depth (Chopra & Holgate 2005) 300 ºC 100 ºC 3 The temperature data contained in this image has been derived from proprietary information owned by Earth Energy Pty Ltd ABN 078 964 735.

4. Classification of fractured reservoirs np:porosity of the porous block [-], nf: porosity of the fracture network [-] Reservoir with all storage in the fractures Sharp decline of production rate after a short period of time nf >> np Reservoir with large storage in the matrix Production rate depends on the degree of fracturation np >> nf Reservoir with equal storage Smooth production rate nf ≈ np (Bai et al., 1993)

5. The dual porosity concept Barenblatt (1960) • Two distinct type of cavities: • Pores • Fractures • Two distinct roles: • Storage in the pores • Transport in the fractures Warren and Root (1963) kp: permeability of the porous block[m2] kf: permeability of the fracture network [m2] • Two overlapping single porous media: Fracture network Transfers of mass, momentum, heat, entropy Fractured porous medium Porous block

6. Objectives and methods • 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)

7. Content Problem statement Thermo-hydro-mechanical (THM) coupled model Numerical analysis Geothermal energy recovery applications Conclusion

8. A three phase closed mixture • Generalized diffusion • Hydraulic (Darcy) • Thermal (Fourier) Solid phase • Forced convection fracture fluid phase • Generalized transfer • Mass • Heat Pore fluid phase The significant contribution is the local thermal non-equilibrium ≠ ≠ Khalili and Loret (2001) Khalili and Selvadurai (2003)

9. Mixture displacements: Pore fluid pressure: Fracture fluid pressure: Solid temperature: Pore fluid temperature: Fracture fluid temperature: The mixture theory • Eachcontinuumpoint is occupied by all the constituents of the mixture: Truesdell (1957) • Biot’s approach of mixture: The solid phase is defined as the reference constituent, since it holds the other phases. Biot (1977)

10. The Thermodynamics of Irreversible Processes (TIP) CD inequality • The Clausius-Duhem (CD) inequality: • is used to • identify the generalized forces/stresses and fluxes/strains • restrain the constitutive equations • Three types of constitutive equations are required by the TIP • the thermo-mechanical process: • the generalized transfer: • - the generalized diffusion:

11. The CD inequality: thermo-mechanical reversible process thermo-poro-elasticity This implies the existence of an elastic potential for the mixture The generalized strains - the total strain - the pore fluid volume content - the fracture fluid volume content - the entropy of the solid The generalized stresses - the total stress - the pore fluid pressure - the fracture fluid pressure - the solid temperature

12. The thermo-mechanical constitutive laws Total strain Total stress Symmetric fully coupled matrix Pore volume Fracture volume Solid entropy Pore pressure Fracture pressure Solid temperature The entropies of fluids are defined separately: 0 The thermal response of the mixture is controlled by the solid phase alone: ΔTs = 0 ΔTs > 0

13. The effective stress concept The effective stress is - linked to the elastic deformation: • - extended to dual porous media: - based on a consistent loading decomposition Nur and Byerlee (1971) - uses three compressibilities [Pa-1] : c, cp , cs The end result is (with a continuum mechanics sign convention): Khalili and Valliappan (1996)

14. The CD inequality: generalized transfer irreversible processes Mass transfer Entropy transfer Heat transfer 00 Rate of mass transfer Scaled chemical potential Rate of momentum and energy transfers Coldness Rate of entropy transfer

15. The generalized transfer constitutive law Closure relations The transfer forces are identified as differences Scaled chemical potential Symmetric uncoupled matrix Mass Heat Coldness Bowen and Chen (1975) Barenblatt (1960) Warren and Root (1963) The heat transfer parameters between the phases [W/m3.K] The leakage parameter [Pa.s]

16. The CD inequality: generalized diffusion irreversible processes Mitchell and Soga (1993)

17. The generalized diffusion constitutive law Generalized fluxes Generalized forces Isothermal heat transfer Direct diffusion of heat Symmetric partially coupled matrix Thermo-osmosis Direct diffusion of mass • 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).

18. Summary of the constitutive equations 

19. The coupled model: 6 field equations - Momentum for the mixture - Mass for the fluid phases (k = p, f) Mass transfer - Energy for the solid phase Heat transfer - Energy for the fluid phases (k = p, f) Heat transfer Convection

20. Content Problem statement Thermo-hydro-mechanical (THM) coupled model Numerical analysis Geothermal energy recovery applications Conclusion

21. Finite element analysis: hints • Primary unknowns: • - u the mixture displacement • - pp and pf the fluid pressures • - temperatures Ts, Tp and Tf • 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. A domestic code • 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 22

23. Axi-symmetric mesh: r - zplane,θ= 0° top z = 1 m well right z = 0 m bottom r1= 0.1 m r2 = 800 m >> r1 0 T Insights in double porosity pp 0 0 pf • Investigation of a borehole stability problem (1T + no convection) 0 σθ 0 σr σr 0 0 σz z θ 1 km r Reservoir εz εz 0 0 • In-situ conditions(prior to drilling) • Boundary conditions: pw = 12MPa, Tw = 100°C σr = σθ= -23.5 MPa σz = -29MPa pp= pf = 9.8MPa T = 50°C 0 0 -pw 0 0 pw Jp = 0 pw Jf= 0 0 q = 0 Tw

24. Insights in double porosity cont’d • Comparison of the diffusivity ratios 2P with hydraulic equilibrium ≡ McTigue (1986) 1. The heat diffuses faster than the flow in the porous blocks 2. The flow diffuses faster than the heat 24

25. Insights in double porosity cont’d Influence of the mass transfer: Zero leakage parameter Average leakage parameter High leakage parameter Associated single porosity (A1P) x • The A1P model underestimates the failure potential compared to the 2P model • The 2P model is capable of finer predictions than the A1P model

26. Implementation of convection • Balance of energy of the fluid phase: Numerical wiggles with FEM = requires stabilization Diffusion contribution Transient contribution Convective contribution • The Streamline Upwind / Petrov-Galerkin methodBrooks and Hughes (1982) 1. Galerkin method 2. SUPG method 26

27. Implementation of convection cont’d • 1D transient problem: Hot side vf Cold side φ = 1 φ = 0 Grid Péclet number = 42.5 x 10 m • 2D transient problem: SUPG FEM response Galerkin FEM response Grid Péclet number = 106 φ = 1 vf φ = 0 27

28. Summary of testing steps 28

29. Content Problem statement Thermo-hydro-mechanical (THM) coupled model Numerical analysis Geothermal energy recovery applications Conclusion

30. Context Electricity • 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 Cold water Injection (ex: 70ºC) Hot water production (ex: 178ºC)

31. Objectives • Research 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 • 1st Objective: • Calibrate the parameters of the model from field experiments data • Understand favourable conditions for LTNE to arise • 2nd Objective: Investigate the influence of - The dual porosity on the effective stress • - The fracture spacing (controlling the magnitude of transfers) 1P-2T Single porosity Two temperatures 2P-2T Double porosity Two temperatures

32. Experimental data 1P-2T • The Fenton Hill Hot Dry Rock (HDR) reservoir: • 3 parameters need to be fitted: • The fracture network permeability • kf [m2] • The fracture network porosity • nf [-] • The solid fluid heat transfer • which is related to the solid fluid specific surface Unknown fracture path Zyvoloski et al. (1981) 2b: average fracture aperture [m]

33. A generic HDR reservoir 1P-2T • Model assumptions • Single porosity model (np = 0) • The material properties remain constant • No natural convection nor gravity • No initial stiffness • The flow regime is laminar • Insight from characteristic times Outflow 230 m Inflow 200 m kf: fracture permeability [m2] 2b: average fracture aperture [m] xE: average fracture spacing [m]

34. Boundary conditions 1P-2T • Initial conditions • Local thermal equilibrium: • In-situ pressure: • Over burden vertical stress: • Earth lateral stress: • Loading conditions • ThermalHydraulicMechanical

35. = 2626 m ≈ 2660 m = 2673 m = 2703 m The double step pattern 1P-2T • The thermal drawdown is successfully calibrated with field data: κsf: The solid-to-fracture fluid heat transfer parameter nf = 0.005 kf = 3.06 10-14 m2 Fenton Hill experimental data Local thermal equilibrium ≡κsf > 100 mW/m3.K 1st stage 2nd stage 3rd stage Convection dominance of the fluid Heat transfer Final depletion of the mixture

36. A THM-1P-2T coupled response Outflow Inflow Fluid temperature Cooling of the solid phase arises during the late period only Solid temperature

37. A THM-1P-2T coupled response cont’d Outflow Inflow • The fracture fluid pressure is not affected by the circulation test

38. A THM-1P-2T coupled response cont’d Outflow Thermal contraction Inflow Vertical strain Thermally induced tensile stress Vertical effective stress

39. A generic HDR reservoir 2P-2T • Model assumptions • Same as before • . • Boundary conditions • are extended to double porosity • ThermalHydraulic + Ts = Tp Outflow Inflow

40. The double step pattern holds for 2P-2T • The calibration is successful on two different sites nf = 0.005 kf = 1.0 10-14 m2 = 2626 m ≈ 2660m = 2673 m = 2703 m nf = 0.005 kf = 1.0 10-13 m2 ≈ 2152 m (Zyvoloski et al., 1981) (Kolditz and Clauser , 1998) 30 < κsf < 120 mW/m3.K

41. Bounds for the THM-2P-2T response The double porosity model (2P) Average response - solid, pore fluid, fracture fluid - Ts = Tp ≠ Tf - Average diffusive flow in the pores: kp = 10-21 m2 Average mass transfer A single porosity model (1P) Upper bond - solid, fracture fluid - A motion less pore fluid (no mass transfer nor hydraulic diffusion) - Ts ≠ Tf An ‘unconnected’ double porosity model (2P) Lower bond - solid, pore fluid, fracture fluid - Ts = Tp ≠ Tf - Slow diffusive flow in the pores: kp = 10-23 m2 Small mass transfer Theoretical response

42. A THM-2P-2T coupled response • Coolingduring the late period, t = 3.17 years, andfor kf = 1.10-14 m2 Outflow Inflow Fracture fluid temperature Porous block temperature Tensile stresses are damped by the pore pressure contribution Pore fluid pressure Vertical effective stress

43. A THM-2P-2T coupled response cont’d • Small fracture spacing xE leads to local thermal equilibrium • The solid-fracture-fluid heat transfer κsfparameter increase if xE decreases Site Materials Outflow Inflow Fracture fluid temperature Porous block temperature

44. A THM-2P-2T coupled response cont’d • Small fracture spacing xE leads to hydraulic equilibrium • The leakage parameter η increase if xE decreases with Outflow Inflow Vertical effective stress Pore fluid pressure • The induced tensile stress is reduced for large fracture spacing xE

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. 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 Late period t = 3.2 years Mass transfer force • This model response matches typical field observations (Murphy et al., 1981)

47. Summary of the geothermal results Recommendation The average fracture spacing xE should be large to minimize the thermally induced tensile effective stress and uncontrolled water loss

48. Content Problem statement Thermo-hydro-mechanical (THM) coupled model Numerical analysis Geothermal energy recovery applications Conclusion

49. The main contributions • An original constitutive model • Dual porosity • Full THM coupling with generalized diffusion, transfer and convection • Local thermal non-equilibrium between the phases • A domestic THM non-linear finite element code • Forced convection stabilized with the SUPG method • A borehole stability analysis of fractured media:1st paper (Revised) • Applications to enhanced geothermal systems:2nd and3rd 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

50. Conclusions • 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 50