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Uncertainty analysis of thermo-hydro-mechanical coupled processes in heterogeneous porous media

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Abstract

In this paper we present an uncertainty analysis of thermo-hydro-mechanical (THM) coupled processes in a typical geothermal reservoir in crystalline rock. Fracture and matrix are treated conceptually as an equivalent porous medium, and the model is applied to available data from the Urach Spa and Falkenberg sites (Germany). The finite element method (FEM) is used for the numerical analysis of fully coupled THM processes, including thermal water flow, advective–diffusive heat transport, and thermoelasticity. Non-linearity in system behavior is introduced via temperature and pressure dependent fluid properties. Reservoir parameters are considered as spatially random variables and their realizations are generated using conditional Gaussian simulation. The related Monte-Carlo analysis of the coupled THM problem is computationally very expensive. To enhance computational efficiency, the parallel FEM based on domain decomposition technology using message passing interface (MPI) is utilized to conduct the numerous simulations. In the numerical analysis we considered two reservoir modes: undisturbed and stimulated. The uncertainty analysis we apply captures both the effects of heterogeneity and hydraulic stimulation near the injection borehole. The results show the influence of parameter ranges on reservoir evolution during long-term heat extraction, taking into account fully coupled thermo-hydro-mechanical processes. We found that the most significant factors in the analysis are permeability and heat capacity. The study demonstrates the importance of taking parameter uncertainties into account for geothermal reservoir evaluation in order to assess the viability of numerical modeling.

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Abbreviations

a :

Range of a variogram model (m)

B :

Strain displacement matrix

C :

Sill in a variogram model (–)

\({{\mathbb C}}\) :

Fourth-order material tensor

c p :

Specific heat capacity of porous medium (J/kg K)

\({c_p^l}\) :

Specific heat capacity of liquid (J/kg K)

\({c_p^s}\) :

Specific heat capacity of solid (J/kg K)

D :

Matrix form of constitutive tensor \({{\mathbb C}}\)

E :

Young’s modulus (Pa)

f :

Right-hand-side (RHS) term

G :

Shear modulus (Pa)

g :

Acceleration of gravity (m/s2)

g :

Vector form of g

h :

Lag distance in variogram model (m)

I :

Identity tensor

k :

Intrinsic permeability (m2)

K :

Laplace matrix

L :

Differential operator

m :

Specific unit tensor

M :

Mass matrix

n :

Porosity (–)

n :

Normal vector

N :

Shape function

p :

Pressure (Pa)

q T :

Heat flux vector (J/m2 s)

Q T :

Heat sources

q f :

Flow flux vector (m3/m2 s)

Q f :

Flow source/sink terms

S s :

Specific storage (Pa−1)

t :

Time (s)

T :

Temperature (K)

u :

Solid displacement vector (m)

v :

Fluid phase velocity (m/s)

α T :

Linear thermal expansion coefficient for the solid (K−1)

δ :

Kronecker delta

ΔT :

Temperature increment (K)

\({{\bf \epsilon}}\) :

Strain tensor

γ :

Semivariogram (–)

Γ:

Domain boundary

λ :

First Lamé constant (Pa)

λ e :

Heat conductivity of porous media (W/m K)

λ l :

Heat conductivity of liquid (W/m K)

λ s :

Heat conductivity of solid (W/m K)

μ :

Dynamic viscosity (Pa s)

ν :

Poisson ratio (–)

ω :

Linear test function

ω :

Quadratic test function

ω T :

Thermal gradient (K/m)

Ω:

The model domain

ρ :

Density of porous medium (kg/m3)

ρ l :

Density of liquid (kg/m3)

ρ s :

Density of solid (kg/m3)

σ :

Total stress tensor (Pa)

σ′:

Effective stress tensor (Pa)

θ :

Relaxation parameter

in:

Value at the injection borehole

out:

Value at the production borehole

T :

Transpose of matrix

0:

Value at the initial condition (t = 0)

H:

Hydraulic process

M:

Mechanical process

n :

Time step index

r :

Reference value

T:

Heat transport process

\({\dot{A}}\) :

dA/dt

 :

Vector of nodal values of the unknown A

∇A :

Gradient of a scalar

∇ · A :

Divergence of a vector

∇A :

Gradient of a vector

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Authors and Affiliations

  1. Department of Environmental Informatics, UFZ-Helmholtz Centre for Environmental Research, Permoserstrasse 15, 04318, Leipzig, Germany

    Norihiro Watanabe, Wenqing Wang & Olaf Kolditz

  2. Applied Environmental System Analysis, Dresden University of Technology, Helmholtzstrasse 10, 01069, Dresden, Germany

    Norihiro Watanabe & Olaf Kolditz

  3. Edinburgh Collaborative of Subsurface Science and Engineering (ECOSSE), School of Geoscience, University of Edinburgh, West Mains Road, Edinburgh, EH9 3JW, Scotland, UK

    Christopher I. McDermott

  4. Graduate School of Environmental Science, Okayama University, 1-1, Tsushima-Naka, 3-chome, Okayama, 700-8530, Japan

    Takeo Taniguchi

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  1. Norihiro Watanabe
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  2. Wenqing Wang
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Correspondence to Norihiro Watanabe.

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Watanabe, N., Wang, W., McDermott, C.I. et al. Uncertainty analysis of thermo-hydro-mechanical coupled processes in heterogeneous porous media. Comput Mech 45, 263–280 (2010). https://doi.org/10.1007/s00466-009-0445-9

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