Development of modeling techniques to adapt mechanical properties to the model scale

Preparation of the geomechanical model for sub-area 001

Preparation of the geomechanical model for sub-area 009

Scale dependency of stress and rock properties

In the first place stresses in rock depend on acting forces and on elastic rock properties. Measured stresses as well as elastic properties are a matter of scale. This is because natural rock is composed of spatially variable fractions of mineral types, it can comprise inclusions of variable size or variable porosity, it can exhibit fractures of different orientation and frictional properties or alternating layers of differing rock type. Typical measurement methods of both stress and elastic properties require a certain minimum scale to derive a value (e.g. ~1 m for hyrofrac tests and some ten centimetres for static compression tests) and often measurements are conducted at locations suitable for the measurement method (e.g. hydrofrac tests in stiff rock units more likely allow development of proper fractures than in pre-damaged sections or in units of low strength and intact rock facilitates rock sample preparation and permits reproducible results). Thus, measurements of stress and elastic properties generally represent an integrated value over the measurement scale or are biased due to the specific location chosen for the measuerments. Several approaches have been used to assess the impact of inhomogenous rock properties on stress.

A) Statistical distribution of rock stiffness in space

Young’s modulus is defined statistically distributed in a rock layer following a chi square distribution centering around a mean value. The resulting horizontal stress along a vertical profile fluctuates around the stress path derived for a homogneous Young’s modulus of the mean value of the distribution. The amount of deviation reaches up to 7% in the tested cases but is on average much lower. The deviation depends on element size with larger elements yielding larger deviations.

B) Statistical distribution of rock stiffness in layered stratigraphy

A succession of horizontally layered units is considered in which rock stiffness is statistically assigned to the individual layers according to a chi square distribution centering around a mean value. A maximum deviation of 21 % in the magnitude of maximum horizontal stress was derived with respect to the stress in a homogenous rock column. Grouping neighbouring layers with a stiffness corresponding to their mean value results a reduction of deviation with respect to the stress in homogneous rock column but may lead to deviations with respect to data the model was originally calibrated to taking into account the detailed stratigraphy.

C) Mixing laws for effective elastic media

The self-consistent method is used to calculate effective elastic constants for inclusions of variable shape, volume ratio and stiffness deviation between the inclusion material and the matrix material. In an example the Young’s modulus decreases by 45 % if a consolidated sandstone includes 5 % flat clay lenses of low stiffness.

D) Multiscale material model

The mean-field homogenization approach is used to numerically compute the mechanical behaviour of composite materials. Assuming a spherical porosity of 10 % the maximum horizontal stress magnitude reduces by up to 8 % compared to if the rock is without porosity. Cleavages represented by penny shaped voids of very low aspect ratios can play a greater role, particularly for the minimum horizontal stress. Depending on the orienation of the cleavages relative to the acting far-field forces the maximum horizontal stress decraeses by up to 36 % and the minimum horizontal stress by 20 % relative to the case with no cleavages. The minimum horizontal stress can also increase with other orientaions. A random orientation of the cleavgaes still reduces the maximum horizontal stress by 15 % in the tested example. These numbers refer to the mean field where strain and stress are averaged over the composite material. Stress on the micro-level in the individual constituents oft he material can differ more.

E) Jointed Material

The jointed material model is used to capture the effect of fractured rock on the stress state depending on the orientation and frictional properties of the fractures. In a test case the minimum horizontal stress is increased by 8 % and the maximum horizontal stress is decreased by 3 % relative to the case without fractures. In a trial and error approach it was inferred that this corresponds to a reduction of Young‘s modulus by 10 % relative to the intact rock without fractures.