Page 271 - Geothermal Energy Systems Exploration, Development, and Utilization

5.1 Introduction 247
From the mathematical point of view, THM processes lead to a nonlinear coupled
initial boundary value problem (IBVP), which needs to be solved numerically.
Among the available numerical methods, finite differences, volumes and elements
(FEM) are mainly used (Noorishad et al., 1984; Kohl, 1992; Lewis and Schrefler,
1998; de Boer, 2005; Wang and Kolditz, 2007; Rutqvist et al., 2008). Irrespective of
the specific numerical method, the calculation of coupled THM problems is very
expensive. This is mainly due to two reasons: degree of freedom (i.e., number of
field variables) and strong coupling among nonlinear processes.
There are several ways to improve the computational efficiency, for example,
more efficient numerical algorithms, optimization of memory management in the
code, and parallelization techniques. Among them, parallel computing provides the
most powerful speed-up. Thanks to the decreasing hardware cost in the past years,
parallel computation is becoming very attractive for applied research (Topping and
Khan, 1996; Schrefler et al., 2000; Shioya and Yagawa, 2005; Wang et al., 2009).

5.1.2
Uncertainty Analysis

Data uncertainty is one of the major problems in subsurface reservoir analysis.
Direct borehole measurements are very limited due to technical issues and costs.
Normally, data are available from core samples and wellbore logging for the local
scale, and geophysical measurements (e.g., microseismic monitoring) for a larger
scale (e.g., Tenzer et al. (2000) for Urach Spa site and (Weidler et al., 2002) for Soultz
site). Thus, subsurface models are derived from limited information and include
uncertainties. Dealing with uncertainty analysis is a common tool, for example, for
safety assessment of nuclear waste repositories (Rautman and Treadway, 1991).
For HDR geothermal systems, aspects of uncertainty have been investigated in
the framework of sensitivity analysis and parameters identification so far. Fractal
and statistical DFN models have been developed, for example, by Watanabe and
Takahashi (1995); Willis-Richards, (1995); Tezuka and Watanabe (2000). DFN
models can represent the structural reservoir geology, but they are still restricted to
simplified processes. Inversion methods have been used to identify physical rock
parameters in order to reproduce the observed reservoir behavior (Finsterle and
Pruess, 1997; Lehmann et al., 1998).
Monte Carlo analysis is one common method to quantify parameter uncer-
tainty and the corresponding system evolution. Using geostatistical techniques,
for example, sequential Gaussian simulation and indicator simulation, enables
the generation of multiple stochastically equivalent realizations, which take into
account the status of the incomplete knowledge (Goovaerts, 1997; Pebesma and
Wesseling, 1998; Chil´ es and Delfiner, 1999; Deutsch, 2002). Statistical properties of
sequential Gaussian processes are described, for example, by Rouhani et al. (1996).
The conditional distribution of an unknown statistical variable at a particular
location, given a set of known values at nearby locations, is the Gaussian (normal).
Therefore, the distribution of sample data has to be transformed to the Gaussian
distribution beforehand. The mean and variance of the conditional distribution can