Empirical Formula for Dynamic Biot Coefficient of Sandstone Samples from South-West of Poland

In this research, two empirical correlations have been introduced to calculate the dynamic Biot coefficients of low-porosity and high-porosity sandstone samples from two open pit mines located in South-West Poland. The experiments were conducted using an acoustic velocity measurement apparatus. Under the undrained condition, firstly, the confining pressure was increased in increments of 200 psi, and the values of pore pressure and dynamic elastic modulus were recorded. This experiment was continued until the Skempton coefficient remained in the range of 0.98–1. Secondly, an experiment on the same sample was conducted under drained conditions, and the corresponding dynamic elastic moduli were calculated. Then, using the calculated dynamic elastic moduli, the dynamic Biot coefficient was determined for each sample under different confining pressure. Finally, two empirical correlations were formulated for each sandstone category. The results demonstrate that, as the confining pressure increases, the Biot coefficient decreases from 0.79 to 0.50 and from 0.84 to 0.45 for low-porosity and high-porosity samples, respectively. Furthermore, as the porosity increases, the sandstone behavior increasingly approaches that of soil. The empirical correlations can be used for sandstone formations with the same porosity in projects where there is not a measurement method for the Biot coefficient.

A Laboratory Method for Estimation of Storage Capacity of Rock Samples under Effective Stress

Summary Fluid storage capacity measurements of core plugs in the laboratory consider pore volume as a function of effective stress. The latter is equal to applied confining pressure – n × applied pore pressure. However, the results are often reported as a function of difference in the applied pressures, because the effective stress coefficient (n) is an unknown. This creates confusion during the interpretation of laboratory data and leads to added uncertainties in the analysis of the storage capacity of the samples under in-situ conditions. In this paper, we present a new laboratory method that allows simultaneous prediction of the sample pore volume, the coefficient of isothermal pore compressibility, and the effective stress coefficient. These quantities are necessary to predict the fluid storage as a function of effective stress. The method requires two stages of gas (helium) uptake by the sample under confining pressure and pore pressure and measures pressure-volume data. Confining pressure is always kept larger than the equilibrium pore pressure, but their values at each stage are changed arbitrarily. The analysis is simple and includes simultaneous solutions of two algebraic equations including the measured pressure-volumedata. The model is validated by taking the reference pore volume near zero stress. The reference volume predicted matches with that measured independently using the standard helium porosimeter. For sandstone, shale, and carbonate samples, the estimated pore compressibility is, on average, 10−6 psi−1. The effective stress coefficient is higher than unity and is a linear function of the ratio of the applied pressure values. We present a new graphical method that predicts the Biot coefficient (α) of the rock sample, a fundamental quantity used during the strain calculations that indicates the tendency of the rock to deform volumetrically. A new fundamental rule is found between the applied pressure difference and the effective stress: σe/α = pc − pp. Interestingly, the predicted Biot coefficient values for the shale samples show values between 0.46 and 1.0. This indicates that features of the shale sample, such as mineral variability, fine-scale lamination, and fissility, come into play during the fluid storage measurements.

An Analytical Model for Fracture Initiation of Radial Drilling-Fracturing in Shale Formations

Radial drilling-fracturing, the combination of radial drilling and hydraulic fracturing, can guide fractures toward the target area and effectively enhance the recovery of the low permeable reservoir. In this paper, based on the stress superposition principle, we establish an analytical model to predict fracture initiation pressure (FIP) and the shale failure mode for radial drilling-fracturing applied in shale formations. In contrast with the former studies, this model can additionally consider the failure from shale beddings and is more applicable in the shale reservoir. The model classifies the shale failure into three modes and, respectively, gives the criterion for each failure mode. Then, a series of sensitivity analyses is conducted by examining effects of various parameters. By analyzing the variation characteristic of the initiation pressures required for three failure modes, the main conclusions are as follows. Firstly, matrix failure and shear failure along bedding tend to take place when the azimuth of radial borehole is moderate. Small and large azimuths are favorable for the occurrence of tensile failure along bedding. Secondly, a high ratio of horizontal in situ stress predisposes shale to generate matrix failure, and bedding tensile failure and bedding shear failure are apt to occur when the ratio of horizontal in situ stress is low. Thirdly, with the increasing intersection angle of the radial borehole wall and bedding plane, the failure mode apt to occur changes from bedding tensile failure to bedding shear failure and then to matrix failure. Fourthly, shale prefers to yield bedding shear failure under a small Biot coefficient and generate the other two failure modes when Biot coefficient is large. Fifthly, permeability coefficient virtually has no influence on the failure mode of shale. The research clarifies the fracture initiation characteristics of radial drilling-fracturing in shale formations and provides a reference for the field application of radial drilling-fracturing.

Experimental Study of Stress-Seepage Coupling Properties of Sandstone under Different Loading Paths

The study on hydromechanical coupling properties of rocks is of great importance for rock engineering. It is closely related to the stability analysis of structures in rocks under seepage condition. In this study, a series of conventional triaxial tests under drained condition and hydrostatic compression tests under drained or undrained condition on sandstones were conducted. Moreover, complex cyclic loading and unloading tests were also carried out. Based on the experimental results, the following conclusions were obtained. For conventional triaxial tests, the elastic modulus, peak strength, crack initiation stress, and expansion stress increase with increased confining pressure. Pore pressure weakened the effect of the confining pressure under drained condition, which led to a decline in rock mechanical properties. It appeared that cohesion was more sensitive to pore pressure than to the internal friction angle. For complex loading and unloading cyclic tests, in deviatoric stress loading and unloading cycles, elastic modulus increased obviously in first loading stage and increased slowly in next stages. In confining pressure loading and unloading cycles, the Biot coefficient decreased first and then increased, which indicates that damage has a great impact on the Biot coefficient.

The Influence of Rate of Change in Confining and Pore Pressure on Values of the Modulus of Compressibility of the Rock Skeleton and Biot’s Coefficient

This work discusses the results of a study of the influence of rates of change of confining pressure on the result of a drained compressibility tests intended to determine the modulus of compressibility of a rock skeleton Ks. A series of cyclical compressibility tests was performed on samples of sandstone soaked in kerosene, for various rates of compression and decompression of the pressure liquid filling the cell and the pore volume of the sample. The studies showed that the deformability of the tested sample was directly proportional to the rate of change of the confining pressure. As a consequence, the value of the Ks modulus and Biot coefficient α decreased with increasing sample load rate. This phenomenon should be attributed primarily to equilibration of the liquid pressure inside the high-pressure cell with the liquid pressure in the sample pore space, caused by filtration of the pore liquid. These phenomena prove that the filtration process impacts the values of the modulus of compressibility of the rock skeleton Ks and of Biot coefficient α determined on the basis of the experiment. This is significant in the context of the use of Biot equations as constitutive equations for a porous rock medium.

On the Poroelastic Biot Coefficient for a Granitic Rock

The Biot coefficient is a parameter that is encountered in the theory of classical poroelasticity, dealing with the mechanics of a fluid-saturated porous medium with elastic grains and an elastic skeletal structure. In particular, the coefficient plays an important role in the partitioning of externally applied stresses between the pore fluid and the porous skeleton. The conventional approach for estimating the Biot coefficient relies on the mechanical testing of the poroelastic solid, in both a completely dry and a fully saturated state. The former type of tests to determine the skeletal compressibility of the rock can be performed quite conveniently. The latter tests, which determine the compressibility of the solid material constituting the porous skeleton, involve the mechanical testing of the fully saturated rock. These tests are challenging when the rock has a low permeability, since any unsaturated regions of the rock can influence the interpretation of the compressibility of the solid phase composing the porous rock. An alternative approach to the estimation of the solid grain compressibility considers the application of the multi-phasic theories for the elasticity of composite materials, to estimate the solid grain compressibility. This approach requires the accurate determination of the mineralogical composition of the rock using XRD, and the estimation of the elasticity characteristics of the minerals by appealing to published literature. This procedure is used to estimate the Biot coefficient for the Lac du Bonnet granite obtained from the western region of the Canadian Shield.

An Analytical Model for Fracture Initiation from a Particular Radial Borehole in Hydraulic Fracturing Guided by Multiradial Boreholes

Radial drilling-fracturing, the combination of the hydraulic fracturing and radial borehole, is a technology that can guide the hydraulic fractures to directionally propagate and efficiently develop low permeability reservoir. In this paper, an analytical model of two radial boreholes (a basic research unit) is established to predict fracture initiation pressure (FIP) from one particular radial borehole and the interference between radial boreholes when the hydraulic fracturing is guided by multi-radial boreholes. The model is based on the stress superposition principle and the maximum tensile stress criterion. The effects of in situ stress, wellbore pressure, and fracturing fluid percolation are considered. Then, sensitivity analysis is performed by examining the impact of the intersection angle between radial boreholes, the depth difference between radial boreholes, the radius of radial boreholes, Biot coefficient, and the number of radial boreholes. The results show that FIP declines with the increase of radial boreholes number and the decrease of intersection angle and depth difference between radial boreholes. Meanwhile, the increase of radial borehole number and the reduction of intersection angle and depth difference strengthen the interference between radial boreholes, which conduce to the formation of the fracture network connecting radial boreholes. Besides, FIP declines with the increase of radial borehole radius and the decrease of Biot coefficient. Large radius and low Biot coefficient can enlarge the influence range of additional stress field produced by radial boreholes, enhance the mutual interference between radial boreholes, and guide fracture growth between radial boreholes. In hydraulic fracturing design, in order to reduce FIP and strengthen the interference between radial boreholes, the optimization design can be carried out by lowering intersection angle, increasing radius and number of boreholes, and reducing the depth difference between boreholes when the conditions permit. The research clarifies the interference between radial boreholes and provides the theoretical basis for optimizing radial boreholes layout in hydraulic fracturing guided by multi-radial boreholes.

Fault slip in hydraulic stimulation of geothermal reservoirs: Governing mechanisms and process-structure interaction

Hydraulic stimulation of geothermal reservoirs in low-permeability basement and crystalline igneous rock can enhance permeability by reactivation and shear dilation of existing fractures. The process is characterized by interaction between fluid flow, deformation, and the fractured structure of the formation. The flow is highly affected by the fracture network, which in turn is deformed because of hydromechanical stress changes caused by the fluid injection. This process-structure interaction is decisive for the outcome of hydraulic stimulation, and, in analysis of governing mechanisms, physics-based modeling has potential to complement field and experimental data. Here, we show how recently developed simulation technology is a valuable tool to understand governing mechanisms of hydromechanical coupled processes and the reactivation and deformation of faults. The methodology fully couples flow in faults and matrix with poroelastic matrix deformation and a contact mechanics model for the faults, including dilation because of slip. Key elements are high aspect ratios of faults and strong nonlinearities in highly coupled governing equations. Example simulations using our open-source software illustrate direct and indirect hydraulic fault reactivation and corresponding permeability enhancement. We investigate the effect of the fault and matrix permeability and the Biot coefficient. A higher matrix permeability leads to more leakage from a permeable fault and thus suppresses reactivation and slip of the fault compared to the case with a lower matrix permeability. If a fault is a barrier to flow, increase of pressure because of the fluid injection results in stabilization of the fault; the situation is opposite if the fault is highly permeable compared to the matrix. For the given setup, lowering the Biot coefficient results in more slip than the base case. While conceptually simple, the examples illustrate the strong hydromechanical couplings and the prospects of physics-based numerical models in investigating the dynamics.