Summary Poroelastic constitutive relations play an important role in modelling the static and dynamic behaviour of fluid-saturated rocks. However, the various poroelastic theories use different sets of kinematic variables to describe deformation and their interrelations require further study. We use the volume averaging approach to elaborate on the interrelation between volumetric strain and porosity changes and to highlight the importance of the interfacial strain. In particular, we show that the deviatoric interfacial strain describes the fabric changes of the porous medium in response to an applied stress field and gives rise to an anisotropic stress-strain relation at the macroscale. This notion of poroelastic anisotropy via interfacial motion is entirely different from Biot’s original way of describing poroelastic anisotropy.

Summary In Biot's theory of poroelasticity the potential energy is a function of two kinematic variables which does not include the porosity. Under this assumption the number of independent poroelastic parameters can be reduced by one. This is sometimes referred to as reciprocity property. A thermodynamic consideration shows that the porosity is a kinematic variable in poroelasticity. Since the potential energy function has to be constructed from all kinematic variables, this implies for the specific case of a poroelastic medium that the porosity must be included in the deformational potential energy expression. It also implies that the parameter reduction assumed by Biot does not hold and that one more poroelastic parameter is required to fully describe the volumetric deformation of porous rocks. In the framework of the generalized poroelasticity framework this additional parameter appears in the form of the microinhomogeneity parameter. It can be expressed in terms of measurable poroelastic parameters.

Summary The Biot bulk coefficient and the effective pressure coefficient for the bulk volume are important deformation parameters of porous rocks. There are different ways to measure these coefficients using quasi-static deformation experiments in the laboratory. However, only if the rock deforms in a self-similar manner then these two coefficients should coincide. In the presence of micro-inhomogeneities, one can expect that these two coefficients differ. We interpret laboratory measurements where such a difference has been reported and link the difference to the micro-inhomogeneity parameter. The latter is a measure for the deviation from the Biot-Gassmann rock behavior wherein self-similar deformation is assumed.

The spatio-temporal changes of the stress state in a geothermal reservoir are of key importance for the understanding of induced seismicity and planning of injection and depletion strategies. In particular the poro-elastic effects on the stress state due to re-injection or depletion of water are of interest for both geothermal projects and hydrocarbon exploitation. In addition to the conventionally used effective stress concept, poro-elasticity affects the stress tensor components differently as a function of changes in pore pressure. Here, we provide an analytical base for the long-term changes of the 3D stress tensor components as a function of pore pressure changes. Results indicate that for a constant rate of injection or depletion the coupling between pore pressure and all stress tensor components depends on the location in the reservoir with respect to the re-injection/depletion point as well as the time since the beginning of pore pressure changes. Our systematic analysis suggests that poro-elastic stress changes can even locally modify the given tectonic stress regime. Furthermore, the results predict that localized changes of maximum shear stress can lead to different fracture orientations than those expected when poro-elastic effects are not considered. These results indicate a need for 3D geomechanical-numerical studies of more realistic reservoir settings in order to study the 3D effects of pore pressure/stress coupling. Our generic 3D geomechanical-numerical study shows that less than two years of production of a single well changes shear stresses by 0.2 MPa. Thus, in reservoirs with decades of production shear stress change can reach sufficiently high values to re-activate pre-existing faults or even generate new fractures with unexpected orientations. (C) 2014 Elsevier Ltd. All rights reserved.

We provide an overview on recent developments on the acoustics of partially saturated porous rocks. The focus is on the mesoscopic flow induced by seismic waves and leading to wave attenuation and dispersion. At the laboratory scale recent core plug imbibition experiments with simultaneous acquisition of X-ray CT and ultrasonic waveforms allow us to retrieve the saturation dependence of velocities and attenation. Fluid patches and their evolution at the milimetre scale are observed. To model these relations in a consistent manner we invoke the concept of fluid patch membrane stiffness. The latter accounts for the net effect of capillary forces at the macroscale. We further extract the velocity saturation relation from time-lapse sonic logs aquired during CO2 injection into a sandstone formation. It is shown that this velocity-saturation relation can be also modelled using the mesoscopic wave-induced flow effect. Simulation results give further support that mescopic fluid patches on the centimetre scale have a first-order effect on seismic amplitudes provided that the fluid bulk modulus contrast across the patch boundaries is suffiently large. This is typically the case in the presence of a gas phase. We conclude that fluid patches on the milimetre-to-centimetre scale have important implications for attenuation estimates.

Oscillatory fluid flow in deformable porous media induced by seismic waves is associated with dissipation. Though this dissipation is typically described through macroscopic equations, it has its origin in the fluid phase enclosed in the pore space of the rock. Therefore simulating fluid flow at the pore-scale provides a means to get further insight into the mechanism of wave-induced flow. In this paper we use the finite volume method to simulate oscillatory fluid flow in a small sub-volume of rock sample digitized at micro-meter resolution. The aim is to quantify the amount of dissipation at pore-scale and to relate it to the frequency of the periodic input signal (a proxy for the propagating wave).

Poroelasticity theory provides a general framework for seismic wave propagation in fluid-saturated porous media. This framework entails four wave modes. The fast P- and S-waves describe the in-phase compressional and shear motions. The slow P- and S-waves describe the corresponding out-of-phase motions. In heterogeneous poroelastic media wave mode conversions can occur. The fast to fast wave conversion scattering is important for the quantification of scattering attenuation. The fast to slow wave conversion describes dissipation processes. Analysis of the slow to slow wave conversion process allows us to understand the relation between permeability and seismic attenuation.

Biot’s theory of poroelasticity implicitely contains a porosity equation that predicts changes in the porosity due to stress loading to be governed by the difference between the macroscopic solid and fluid pressures. This means that the effective stress coefficient for porosity equals 1. However, there is experimental and theoretical evidence that this porosity effective stress coefficient can be smaller and also greater than unity. In order to model such poroelastic rock behavior we make use of a generalized poroelasticity framework. We analyze a generalized porosity equation that depends on an effective pressure and, in particular, we obtain bounds for the porosity effective stress coefficient. These results are then used to obtain new bounds on various other effective stress coefficients.

A theory for attenuation and dispersion of elastic waves due to wave-induced generation of vorticity at pore-scale heterogeneities in a macroscopically homogeneous porous medium is developed. The diffusive part of the vorticity field associated with a viscous wave in the pore space—the so-called slow shear wave—is linked to the porous medium acoustics through incorporation of the fluid strain rate tensor of a Newtonian fluid in the poroelastic constitutive relations. The method of statistical smoothing is then used to derive dynamic-equivalent elastic wave velocities accounting for the conversion scattering process into the diffusive slow shear wave in the presence of randomly distributed pore-scale heterogeneities. The result is a simple model for wave attenuation and dispersion associated with the transition from viscosity- to inertia-dominated flow regime.

Ultrasonic attenuation measurements were performed on a brine saturated sandstone and the results were analysed using a new spectral ratio method. The results indicate that a generalised standard linear solid model was a good fit to the data and that the model parameters were sensitive to pore pressure.

Based on time-lapse sonic and neutron porosity logs from the Nagaoka CO2 sequestration experiment, a P-wave velocity-saturation relation at reservoir depth is retrieved. It does not coincide with either of the end-member models of uniform and patchy saturation but falls in between even if realistic error estimates for the host rock properties are considered. Assuming a random distribution of CO2 patches it is shown that the mechanism of wave-induced flow can be evoked to explain this velocity-saturation relation. Characteristic CO2 patch size estimates range from 1 to 5 mm. Such mesoscopic heterogeneity can be responsible for attenuation and dispersion in the well logging frequency band. Citation: Caspari, E., T. M. Muller, and B. Gurevich (2011), Time-lapse sonic logs reveal patchy CO2 saturation in-situ, Geophys. Res. Lett., 38, L13301, doi:10.1029/2011GL046959.

Understanding of physical rock properties is currently of great importance both for industry and fundamental science, for it allows improving interpretation and reducing risks. Digital rock physics provides us with a promising opportunity for rock analysis and quantification. Moreover it allows running simulations on rock samples unsuitable for laboratory experiments. A detailed computational rock physics workflow including 3D rock imaging, processing and simulations of physical experiments has been created and tested on different rock samples. Here we describe this workflow and demonstrate the results of elastic simulation in comparison with experimental data obtained in physical laboratory for a sandstone sample.

The state of stress within and around a reservoir is a key parameter for fluid flow, fracture stimulation, design of wellbore arrays and wellbore stability. Therefore pore pressure induced stress changes (pore pressure/stress coupling) have immediate implications for the reservoir management. We analyze the effects of pore pressure changes on the individual components of the principal stress tensor and not only the minimum horizontal stress component. The results show that pore pressure stress coupling has a tensor character and can cause significant changes in the stress field within the reservoir as a function of distance to the injection point and tectonic regime. The tensor character of pore pressure/stress coupling leads to changes in the differential stress of the system which is essential for fault reactivation. It is demonstrated that injection as well as depletion can lead to fault reactivation, and that the rock stability depends on the tectonic regime and on the location with respect to the injection (depletion) point. Furthermore, the tectonic regime can be locally modified within the reservoir. Thus an improved understanding of pore pressure/stress coupling effects contributes to fault seal prediction, optimized placement of in-fill wells, stimulation operations and wellbore stability assessment.

The importance of the viscous boundary layer flow for porous medium acoustics has been recognized by Biot who analyzed this effect for cylindrical tubes and subsequently extended his low-frequency theory to the full frequency range. In this paper we develop an alternative approach to model attenuation and dispersion associated with the transition from the viscosity- to inertia-dominated regimes. Instead of analyzing the oscillatory Stokes flow in elastically-rigid tubes, we base our analysis on the visocosity-exdended Biot framework. We show that the conversion scattering process from a P-wave into the diffusive slow shear wave is a descriptor of the vorticity diffusion process occurring within the viscous boundary layer. We make use of previous results for conversion scattering from P- to slow S-wave and derive a dynamic-equivalent P-wavenumber from which we deduce attenuation and velocity dispersion. Comparison with the predictions of other models shows that the conversion scattering approach can model attenuation and dispersion associated with the transition from the viscosity- to inertia-dominated regimes.

We model microseismicity triggered by fluid injection on the basis of the theory of poroelasticity accounting for the external stress field. Consideration of the fully coupled poroelastic field equations enables us to apply a Coulomb failure criterion using pore fluid pressure and stress tensor as well as the coefficient of friction. The poroelastic fields are calculated with the finite-element method simulating fluid injection with constant injection rate into a 2-D domain. The influence of diffusivity, injection rate and stress field on the occurrence of microseismicity is analysed and compared to simulations based on pore fluid pressure diffusion only. We show that an anisotropic initial stress field causes elongated microseismic clouds. These clouds are indistinguishable from those generated in poroelastic solids under isotropic stress but exhibiting anisotropic hydraulic diffusivity. This similarity shows that microseismicity distributions dependent on both, the hydraulic properties and the coupling of pore fluid pressure to the stress field. In particular, neglecting the influence of the external stress field may lead to overestimation of the anisotropy of diffusivity tensor components. Furthermore, the results of our numerical simulations are strongly sensitive to changes of fluid injection rate.

We present a methodology to assess rock properties by using X-ray computed tomography (CT) images along with the ultrasonic P-wave velocity measurements in laboratory under different confining pressures, and borehole logs from a stratigraphic interval in Campos Basin, offshore Brazil. The higher resolution of sampling in the core data overcomes the aliasing and smoothing effect of log-based measurements and enables the delineation of stratigraphic cycles. In addition, we observe a good correlation between X-ray CT value and ultrasonic P-wave velocity. Large variation in measured ultrasonic P-wave velocity between different samples is related to variations in calcium content. This improved understanding of the cyclic nature of physical property variations may help to correlate stratigraphy between wells and provide valuable information for paleo-climate studies. This study also helps to understand the facies heterogeneity in fine scale which is needed for choosing the correct rock physics model for further undergoing studies such as time lapse in this field. Given the periodic nature of sedimentary facies, the unique combination of X-ray CT images, ultrasonic P-wave velocities, and high resolution well logs in this field could be used as an example for seismic wave propagation studies in periodic media.

Within the viscosity-extended Biot framework the existence of a slow shear wave mode is predicted. At high frequencies this wave mode degenerates to a diffusion wave. Using the method of statistical smoothing in randomly heterogeneous poroelastic solids we analyze the process of conversion scattering from fast compressional waves into slow shear diffusion waves. In particular, we compute a dynamic-equivalent wave number from which attenuation and dispersion characteristics are inferred. We observe that this conversion scattering process results into fast compressional wave attenuation having same nature as the Biot global flow attenuation mechanism.

Accurate numerical modeling tools are of practical importance for understanding of seismic wave propagation in reservoir rocks. A 2D finite-difference (FD) poroelastic code is used for simulation of acoustic wave propagation through water-saturated medium with regular distributed circular gas inclusions. The results of the numerical simulations are validated with a newly derived theoretical solution for the same geometry of patches. Finally, the numerical results for patchy-saturated media are used for qualitative interpretation of the results of a laboratory experiment in which saturation and compressional velocity are simultaneously measured in a sandstone sample.

The fluid transport properties of fractured porous rocks usually show a strong dependence on the effective pressure and thus on the pore pressure. Consequently the process of pore pressure diffusion initiated by fluid injection experiments is governed by the non-linear diffusion equation. The spatio-temporal evolution of microseismicity associated with the process of nonlinear pore pressure diffusion is analyzed. In particular, it is shown that the process of nonlinear pore pressure diffusion allows to define two triggering fronts. One corresponding to the triggering of microseismic events without changing the fluid transport properties significantly, another one that triggers the majority of microseismic events and that alters the fluid transport properties significantly. The applicability of this model for the interpretation of microseismicity observed during hydraulic fracturing treatments is discussed.