Abstract
Development of high-pressure, high-temperature (HPHT) petroleum reservoirs situated at depths exceeding 5 km and in situ temperature of 170 °C increases the demand for theories and supporting experimental data capable of describing temperature effects on rock stiffness. With the intention of experimentally investigating temperature effects on stiffness properties, we investigated three sandstones from the deep North Sea Basin. As the North Sea Basin is presently undergoing substantial subsidence, we assumed that studied reservoir sandstones have never experienced higher temperature than in situ. We measured ultrasonic velocities in a low- and high-stress regime, and used mass density and stress–strain curves to derive, respectively, dynamic and static elastic moduli. We found that in both regimes, the dry sandstones stiffens with increasing testing temperature and assign expansion of minerals as a controlling mechanism. In the low-stress regime with only partial microcrack closure, we propose closure of microcracks as the stiffening mechanism. In the high-stress regime, we propose that thermal expansion of constituting minerals increases stress in grain contacts when the applied stress is high enough for conversion of thermal strain to thermal stress, thus leading to higher stiffness at in situ temperature. We then applied an extension of Biot’s effective stress equation including a non-isothermal term from thermoelastic theory and explain test results by adding boundary conditions to the equations.
Highlights
List of Symbols Biot’s coefficient Volumetric thermal expansion coefficient Linear thermal expansion coefficient d Dry density m Grain density Porosity k Permeability Biot Biot’s effective stress dra Total stress P Pore pressure reduced by Biot’s coefficient eff Non-isothermal effective stress H Hydrostatic stress
The following load cycles do not show a significant effect on the stiffness properties (Fig. 9), confirmed by Analysis Of Variance (ANOVA) analysis
Shear modulus for O-samples increases with temperature but is 4% lower for the triaxial stress symmetry as compared to hydrostatic stress (Figs. 10, 11)
Summary
List of Symbols Biot’s coefficient Volumetric thermal expansion coefficient Linear thermal expansion coefficient d Dry density m Grain density Porosity k Permeability Biot Biot’s effective stress dra Total stress P Pore pressure reduced by Biot’s coefficient eff Non-isothermal effective stress H Hydrostatic stress. A Axial stress R Radial stress P Pore pressure T Temperature Biot Strain resulting from Biot’s effective stress dra Strain resulting from total stress P Strain resulting from pore pressure T Strain resulting from temperature Strain resulting from the non-isothermal effective stress Strain resulting in volume expansion Potential strain Δ Change Γ Δ∕ΔT tP First arrival time of P-wave tS First arrival time of S-wave VP P-wave velocity VS Shear wave velocity Esta Static Young’s modulus Edyn Dynamic Young’s modulus Ks Solid bulk modulus KQ Quartz bulk modulus. The effective stress is primarily a function of the overburden load and the pore pressure, where the contribution from the latter is limited by the elastic stiffness properties of the rock as expressed in Biot’s coefficient. The effective stress field and elastic moduli of the rock are linked, and in order to provide reliable estimates, we need to understand how rock stiffness of given lithology depends on temperature and stress
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