One‐dimensional analytical solution for mesoscopic flow induced damping in a double‐porosity dual‐permeability material

Heterogeneities, such as fractures and cracks, are ubiquitous in porous rocks. Mesoscopic heterogeneities, that is, heterogeneities on length scales much larger than typical pore size but much smaller than the wavelength, are increasingly believed to be responsible for significant wave energy loss in the seismic frequency band. When a compressional wave stresses a material containing mesoscopic heterogeneities, the more compliant parts of the material (e.g., fractures and cracks) respond with a greater fluid pressure than the stiffer portions (e.g., matrix pores). The induced fluid flow, resulting from the pressure gradients developed on such scale, is called mesoscopic flow. In the present study, the double‐porosity dual‐permeability model is adopted to incorporate mesoscopic heterogeneities into rock models to account for the attenuation of wave energy. Based on the model, the damping effect due to mesoscopic flow in a one‐dimensional porous structure is investigated. Analytical solutions for several boundary‐value problems are obtained in the frequency domain. The dynamic responses of infinite and finite porous layer are examined. Numerical calculations show that the damping effect of mesoscopic flow is significant on the pore pressure response and the resulting effective stress. For the displacement, the effect is seen only at the very low frequency range or near the resonance frequencies. Copyright © 2017 John Wiley & Sons, Ltd.     

Three-dimensional dynamic response of multilayered poroelastic media

ABSTRACT

The dynamic response of a multilayered poroelastic medium under 3D time-harmonic loading is studied using an exact stiffness method. The poroelastic medium under consideration consists of N layers of different thicknesses and properties and an underlying poroelastic half-space. The exact stiffness matrices for each layer and the underlying half-space are derived explicitly using Biot’s poroelastodynamic theory and double-dimensional Fourier transforms. Selected numerical results are presented to demonstrate the influence of various parameters on dynamic response of multilayered poroelastic media under traction and fluid loading. The application of proposed solution scheme to soil–structure interaction problems is also presented.     

Hydraulic crack propagation in a porous medium

We develop a model for the propagation of a fluid-filled crack in a porous medium. The problem is motivated by the mechanism whereby drainage networks may form in partially molten rock below the Earth's lithosphere. Other applications include the propagation of hydraulic fractures in jointed rocks and in oil drilling operations, and the formation of dessication cracks in soils. Motivated by the lithospheric problem, we study a situation in which gravity acts in the direction of crack propagation. The model couples the elastoliydrodynamic equations of crack propagation with a pore pressure field in the porous rock, which drives the fluid flow which supplies the crack. The effect of the pore flow is to include a diffusional term in the evolution equation for the crack width, thus allowing a crack initiated at the base of the lithosphere to propagate down into the asthenosphere. Asymptotic and numerical solutions are presented for this crack evolution. However, the predicted drainage of melt into this crack is tiny compared with the upward percolative melt migration, and the predicted width of cracks (millimetres) is much too small to allow propagation of melt into the lithosphere without freezing. As a mechanism to explain magma fracturing in the lithosphere, the process described here therefore requires further refinement.     

Induced stresses due to fluid extraction from axisymmetric reservoirs

Earthquakes can be induced by fluid extraction, as well as by fluid injection.Segall (1989) proposed that poroelastic stresses are responsible for inducing earthquakes associated with fluid extraction. Here, I present methods for computing poroelastic stress changes due to fluid extraction for general axisymmetric reservoir geometries. The results ofGeertsma (1973) for a thin disk reservoir with uniform pressure drop are recovered as a special case. Predicted surface subsidence agrees very well with measured leveling changes over the deep Lacq gas field in southwestern France. The induced stresses are finite if the reservoir pressure changes are continuous. Computed stress changes are on the order of several bars, suggesting that the preexisting stress states in regions of extraction induced seismicity are very close to frictional instability prior to production.     

Dynamic poroelastic soil column and borehole problem analysis

The one-dimensional dynamic column and borehole problems of soil mechanics formulated on the basis of the poroelastic theory of Vardoulakis and Beskos are solved analytically-numerically. The quasi-static counterparts of these problems are analysed as special cases of the dynamic ones. Use of Laplace transform with respect to time reduces the column and borehole problems to ordinary differential equations with constant and variable coefficients, respectively. The transformed solution of these problems is obtained analytically for the column and by finite differences for the borehole problem, and after, a numerical Laplace transform inversion produces the time domain response. Both a suddenly applied and a harmonically varying with time load are considered. It is concluded that the significance of inertial effects depends on the kind of loading and that the degree of saturation for the nearly saturated case greatly affects the response.     

2D dynamic response of unlined and lined tunnels in poroelastic soil to harmonic body waves

The problem of harmonic wave diffraction by tunnels in an infinite poroelastic saturated soil obeying Biot's theory is studied numerically under conditions of plane strain and the effect of poroelasticity on the response is assessed through some parametric studies. The method is based on the theory of Mei and Foda, which considers the total field to be approximated by the superposition of an elastodynamic problem with modified elastic constants and mass density for the whole domain and a diffusion problem for the pore fluid pressure confined to a boundary layer at the free boundaries. Both problems are solved numerically by the boundary element method in the frequency domain. Results dealing with the response of a circular tunnel with and without an elastic concrete liner in an infinite poroelastic medium to incident harmonic P and SV plane waves are provided and compared against analytical ones as well as to those corresponding to linear elastic soil behaviour. Copyright © 2002 John Wiley & Sons, Ltd.     

Determination of porosity and saturation using seismic waveform inversion

Characterization of a reservoir model requires determination of its petrophysical parameters, such as porosity and saturation. We propose a new method to determine these parameters directly from seismic data. The method consists of the computation and inversion of seismic waveforms. A high frequency method is presented to model wave propagation through an attenuative and dispersive poroelastic medium. The high frequency approximation makes it possible to efficiently compute sensitivity functions. This enables the inversion of seismic waveforms for porosity and saturation. The waveform inversion algorithm is applied to two laboratory crosswell datasets of a water saturated sand. The starting models were obtained using travel time tomography. The first dataset is inverted for porosity. The misfit reduction for this dataset is approximately 50%. The second dataset was obtained after injection of a nonaqueous-phase liquid (NAPL), possibly with some air, which made the medium more heterogeneous. This dataset was inverted for NAPL and air saturation using the porosity model obtained from the first inversion. The misfit reduction of the second experiment was 70%. Regions of high NAPL and high air saturation were found at the same location. These areas correlate well with the position of one of the injection points as well as regions of higher NAPL concentrations found after excavation of the sand. It is therefore possible to directly invert waveforms for pore fluid saturation by taking into account the attenuation and dispersion caused by the poroelasticity.     

Poroelastic behaviour of saturated brittle rock with anisotropic damage

A new anisotropic poroelastic damage model is proposed for saturated brittle porous materials. The model is formulated in the framework of the continuum damage mechanics. A second‐rank symmetric tensor is used to characterize material damage due to oriented microcracks. The classic Biot poroelastic theory is then extended to include poroelastic damage coupling. Both the deterioration of elastic properties and poroelastic coefficients is taken into account. A suitable procedure for determination of model parameters from standard laboratory tests is presented. The validity of the model is tested through comparison between numerical predictions and experimental data in various loading conditions. The overall performance of the model is evaluated. The choice of relevant effective stress for the microcrack propagation criterion in saturated cohesive geomaterials is discussed. Copyright © 2000 John Wiley & Sons, Ltd.