ABSTRACT: There are a lot of uncertainty factors in rock mechanics projects. Numerical methods have been widely applied in rock mechanics.Taking into account the above considerations the paper presents: Estimation of the mean value of a random function Z(x), defined in a stochastic finite element v, (SFE), using finite basis as polynomial, exponential, trigonometric etc;
Choosing an incomplete base it is shown the function Z(x) could be presented as a linear combination of the nodes distributions; and
Applying a Monte-Carlo process in eight nodes distributions of SFE we find the distribution simulated in each point as well as its mean value.
The paper presents a simple illustration of stochastic finite element in PDE for vibrating (system) equation as well as for non stationary heat transfer (fluid flow in porous medium), and the application of stochastic finite element in geostatistics calculation (variogramme Kriking etc). Some considerations are given on uncertainty, validity land risk analysis on geology and rock mechanics. Finally it underlines the importance of SFE and geostatistics, for parameter estimation, uncertainty and risk analysis.
1 INTRODUCTION The current scientific computation paradigm in rock mechanics consists of different mathematical model soften partial differential equations (PDEs).The data required by PDE's models as resource and material parameters are in the practice subject to uncertainty, due to different errors or modeling assumptions, the lack of knowledge and information etc. The most straight forward way of doing this is the Monte Carlo method in which many realizations of the random variables are generated, each leading to a deterministic problem. The resulting sequence of solutions obtained can be processed by statistical and geostatistical procedures (Journel&Huijbreght, 1979) to obtain statistical information on the variability of the solution: mean, value, variance, covariance (covariogram), risk, (Eierman et al, 2007).