Biot Coefficient Research Articles

Stress redistribution and fracture propagation during restimulation of gas shale reservoirs

Restimulation of previously hydraulically-fractured wells can restore productivity to near original levels. Understanding the stress state resulting from the original hydraulic fracturing and subsequent depletion is vital for a successful refracuring treatment. The stress obliquity in the vicinity of the wellbore, due to production from a previously introduced hydraulic fracture, promotes a new concept – that of altered-stress refracuring which allows fractures to propagate into previously unstimulated or understimulated areas and therefore enhancing recovery. In this study, a coupled poromechanical model is used to define stress redistribution and to define optimal refrac timing as defined by maximizing the size of the stress reversal region. Key factors include the time dependency of the stress reorientation, the threshold stress ratio σhmax/σhmin and the influences of permeability anisotropy/heterogeneity, pressure drawdown and rock-fluid properties. The results show that stress reorientation develops immediately as the reservoir begins to produce. This stress reversal region extends to a maximum extent before retreating as the direction of the maximum principal stress gradually returns to the initial state. The optimal refrac timing and the size of the stress reversal region are positively correlated with pressure drawdown and Biot coefficient, negatively correlated with stress ratio σhmax/σhmin ratio and Poisson’s ratio and ambiguously correlated with permeability anisotropy. Permeability magnitude and porosity have no influence on the size of the resulting zone but are negatively and positively correlated to the timing, respectively. Permeability heterogeneity has no influence on the size nor the timing. Coupled fluid flow and damage-mechanics simulations follow fracture propagation under the effect of stress redistribution during refracturing treatments. These results define the evolving path of secondary refracture as it extends perpendicular to the initial hydrofracture and ultimately turns parallel to the hydrofracture as it extends beyond the stress-reversal region. This discrete model confirms the broader findings of the continuum model.

Poroelastic behaviour of granular media with poroelastic interfaces

This paper is devoted to the modelling of stress-sensitive Biot coefficient experimentally observed in low permeable sandstones. Sandstones are described as an assemblage of rigid inclusions surrounded by poroelastic interfaces representing joints between grains and pores. A micromechanical model of Biot coefficient under loading is developed, which highlights the role of interfaces closure on the poroelastic macroscopic behaviour. Experimental results available concerning low permeability sandstones are analyzed in view of this model.

Laboratory experiments simulating poroelastic stress changes associated with depletion and injection in low‐porosity sedimentary rocks

AbstractWe characterized the poroelastic deformation of six cores from three formations associated with the Bakken play in the Williston Basin (the Lodgepole, Middle Bakken, and Three Forks formations). All are low‐porosity, low‐permeability formations, but vary considerably in clay, kerogen, and carbonate content. The experimental program simulated reservoir stress changes associated with depletion and injection via cycling both the confining pressure (Pc) and pore pressure (Pp). We measured volumetric strain, derived the corresponding bulk modulus, and calculated the Biot coefficient (α). We found α, which generally ranges between 0.3 and 0.9, to vary systematically with Pc and Pp for each of the specimens tested. The effect of pore pressure on α is much larger at low simple effective stress (σ = Pc‐Pp) during depletion than injection. The α decreases with σ for all pore pressures. For the same Pc and Pp, the Biot coefficient is consistently higher during injection than during depletion. Given the observed variations of α with Pc and Pp, the modeling of reservoir stress changes using a constant α could be problematic as poroelastic stress changes during depletion and injection are not likely to follow the same path. Scanning electron microscope examination of microstructures suggests that the variations of the bulk modulus and the Biot coefficient can be attributed to the abundance of compliant components (pores, microcracks, clays, and organic matter) and how they are distributed throughout the rock matrix.

Measurement of Biot's coefficient for COx argillite using gas pressure technique

A complete poromechanical study has been carried out on COx argillite which is currently foreseen by Andra (in France) for the storage of high-level and long-life nuclear waste. The main goal of this study was to measure Biot's coefficient for samples cured at different relative humidity. To overcome the difficulties which occur with water as the fluid to control the pore pressure of argillite, a new technique has been used with gas injection (argon) was intended to measure the coefficient in this contribution. The results unambiguously showed that Biot's tensor components were isotropic, and equal to 1 for partially saturated samples. They revealed to be independent on the level of confining pressure or pore pressure. Unexpected results were however obtained for the dry materials for which the components, larger than 1, were measured, depending on the level of pore pressure. It was therefore suspected that an additional swelling phenomenon occurred with argon due to its adsorption in the micropores of argillite. Complementary experiments were thus carried out with helium leading to the coefficients virtually equal to 1 for the dry material, as helium does not adsorb at ambient temperature. Hence, it evidenced that there was a swelling deformation due to argon adsorption leading to apparent Biot's coefficients greater than 1.

Investigation of the influence of geomechanical and hydrogeological properties on surface uplift at In Salah

Coupled reservoir and geomechanical simulations are significantly important to understand the long-term behavior of geologic carbon storage (GCS) systems. In this study, we performed coupled fluid flow and geomechanical modeling of CO2 storage using available field data to (1) validate our existing numerical model and (2) perform parameter estimation via inverse modeling to identify the impact of key geomechanical (Young's modulus and Biot's coefficient) and hydrogeological (permeability and anisotropy ratio) properties on surface uplift and the pore pressure buildup at In Salah in Algeria. Two sets of surface uplift data featuring low and high uplifts above two injection wells and the maximum change in the pore pressure due to CO2 injection were used to constrain the inverse model. Forward simulation results with representative parameter values from the literature match both low and high surface uplifts reasonably well and predicted the maximum change in the pore pressure. In particular, forward modeling results with estimated Biot's coefficients for reservoir and caprock layers, match the observed uplift well, highlighting the significance of Biot's coefficient in coupled reservoir and geomechanical models. Parameter estimation with 12 parameter sets for both low and high uplift data demonstrates that multiple sets of parameters can match the observed data equally well and the inclusion of the pore pressure data is critically important to constrain the parameter solution during inverse modeling. For a majority of cases, estimation results for both low and high uplift data show the vertical intrinsic permeability and Young's modulus of the reservoir remained close to 13mD (1.3×10−14m2) and 10GPa, respectively, suggesting that these parameters may represent the actual effective properties. Additionally, higher correlations between reservoir permeability and caprock's Biot's coefficient with high surface uplift data were observed consistently under the pore pressure constraint, suggesting the inclusion of the pore pressure constraint is required to estimate the proper values of coupled flow and geomechanical properties associated with different surface uplift data. Overall, this study suggests that given limited data, including Biot's coefficient, in addition to permeability and Young's modulus can enhance parameter estimation of the geomechanical response during GCS.

Study on the ratio of pore-pressure/stress changes during fluid injection and its implications for CO2 geologic storage

The success of fluid injection into geological formations, which is the main operation during both carbon dioxide (CO2) geologic storage and wastewater injection, is contingent on the geomechanical integrity of the site. A key task that allows us to evaluate the risk of geomechanical failure is the precise prediction of pore-pressure buildup and subsequent change in the state of stresses during and after the fluid injection. Contrary to traditional approaches, where total stresses are assumed to remain constant, recent studies have ascertained that total stresses in fact change in every direction as fluid extraction/injection disturbs the pore-pressure field and causes deformations. In this study, we conduct an in-depth investigation of the ratio of change in total stress to that in pore-pressure, ∆σ/∆P, which has been denoted in the literature as the pore-pressure/stress coupling. We employ a numerical simulation method that couples single-phase fluid flow in porous media with poroelasticity to explore the spatiotemporal evolution of the ∆σ/∆P ratio for various conditions. These numerical experiments allow us to examine how different material properties and structural geometries would influence the evolution of ∆σ/∆P in both vertical and horizontal directions. These ratios of pore-pressure/stress changes exhibit different spatiotemporal evolutions depending on key factors that include the hydraulic boundary condition, Biot's coefficient, Poisson's ratio, and the hydraulic diffusivity of both the injection zone and caprock. On the basis of observations, we suggest firsthand guidelines for analytically determining the ratio of pore-pressure/stress changes, ∆σ/∆P. Finally, we use examples and case studies to illustrate how the ∆σ/∆P ratio can be incorporated into an analytic calculation for determining a maximum sustainable pressure limit.

Elastic and inelastic deformation of fluid-saturated rock.

In situ rock is often saturated with fluid, the presence of which affects both elastic parameters and inelastic deformation processes. Techniques were developed for testing fluid-saturated porous rock under the limiting conditions of drained (long-term), undrained (short-term) and unjacketed (solid matrix) response in hydrostatic, axisymmetric and plane-strain compression. Drained and undrained poroelastic parameters, including bulk modulus, Biot and Skempton coefficients, of Berea sandstone were found to be stress dependent up to 35 MPa mean stress, and approximately constant at higher levels of loading. The unjacketed bulk modulus was measured to be constant for pressure up to 60 MPa, and it appears to be larger than the unjacketed pore bulk modulus. An elasto-plastic constitutive model calibrated with parameters from drained tests provided a first-order approximation of undrained inelastic deformation: dilatant hardening was observed due to pore pressure decrease during inelastic deformation of rock specimens with constant fluid content.This article is part of the themed issue 'Energy and the subsurface'.

Study on CO2 huff-n-puff of horizontal wells in continental tight oil reservoirs

A poro-elasticity reservoir simulation model was developed to analyze the huff-n-puff operations in the Fuyu continental tight oil layer. The model was based on the “two-way” coupling of a geomechanical model and a compositional model. The pressures induced from the compositional model were set as input for the geomechanical model, and the effective stresses calculated from the geomechanical model were set as input for the compositional model. The results show that under specific assumptions, effective stress increases during the depletion and production period and decreases during the injection and soaking period, resulting in corresponding changes to the displacement and effective stress. Biot coefficient also has a significant and systematic influence on the resulting oil recovery factor of CO2 huff-n-puff. In addition, a sensitivity analysis of the injection parameters, including injection timing, production time, and soaking time, was performed in detail. This work provides a better understanding of the mechanisms of the CO2 huff-n-puff operations of horizontal wells in a continental tight oil sandstone formation.

Effective stresses and shear failure pressure from in situ Biot's coefficient, Hejre Field, North Sea

ABSTRACTWe propose a combination of Biot's equations for effective stress and the expression for shear failure in a rock to obtain an expression for minimum pore pressure in a stable vertical well bore. We show that a Biot's coefficient calculated from logging data in the Hejre Field, North Sea, is significantly different from 1. The log‐derived Biot's coefficient is above 0.8 in the Shetland Chalk Group and in the Tyne Group, and 0.6–0.8 in the Heno Sandstone Formation. We show that the effective vertical and horizontal stresses obtained using the log‐derived Biot's coefficient result in a drilling window for a vertical well larger than if approximating Biot's coefficient by 1. The estimation of the Biot's coefficient is straightforward in formations with a stiff frame, whereas in formations such as shales, caution has to be taken. We discuss the consequence of assumptions made on the mineral composition of shales as unphysical results could be obtained when choosing inappropriate mineral moduli.

A new method to determine Biot's coefficients of Bakken samples

In recent years, the development of oil and gas from shale has proceeded quickly in the world through the use of multistage fracturing technology in horizontal well. However, without the knowledge of rock poroelastic characteristics, the successful rate of hydraulic fracturing will be low. Among those poroelastic characteristics, effective stress is required for creating artificial fractures in the shale formation. In this study a new method is proposed to measure the Biot's coefficient, which is one of the key poroelastic parameters for calculating the effective stress.The Biot's coefficient is obtained after the variations of both the confining and the pore pressures are recorded with a simplified measuring procedures. The Bakken shale samples from Williston Basin are tested. The experiment results show that the Biot's coefficient of Bakken samples obtained from horizontal drilling and vertical drilling are significantly different from each other. This provides a solid base to scientists and engineers for Bakken in-situ stress analysis during multistage hydraulic fracturing and reservoir depletion due to production.

Biot coefficient is distinct from effective pressure coefficient

Within the Biot poroelasticity theory, the effective pressure coefficient for the bulk volume of a fluid-saturated rock and the Biot coefficient are one and the same quantity. The effective pressure coefficient for the bulk volume is the change of confining pressure with respect to fluid-pressure changes when the bulk volume is held constant. The Biot coefficient is the fluid volume change induced by bulk volume changes in the drained condition. However, there is experimental evidence showing a difference between these two coefficients, arguably caused by microinhomogeneities, such as microcracks and other compliant pore-scale features. In these circumstances, we advocate using the generalized constitutive pressure equations recently developed by Sahay wherein the effective pressure coefficient and the Biot coefficient enter as distinct quantities. Therein, the difference is attributed to the porosity effective pressure coefficient that serves as a measure for the deviation from the Biot prediction and accounts for microinhomogeneities. We have concluded that these generalized constitutive pressure equations offer a meaningful alternative to model observed rock behavior.

Static and Dynamic Moduli of Malm Carbonate: A Poroelastic Correlation

The static and poroelastic moduli of a porous rock, e.g., the drained bulk modulus, can be derived from stress–strain curves in rock mechanical tests, and the dynamic moduli, e.g., dynamic Poisson’s ratio, can be determined by acoustic velocity and bulk density measurements. As static and dynamic elastic moduli are different, a correlation is often required to populate geomechanical models. A novel poroelastic approach is introduced to correlate static and dynamic bulk moduli of outcrop analogues samples, representative of Upper-Malm reservoir rock in the Molasse basin, southwestern Germany. Drained and unjacketed poroelastic experiments were performed at two different temperature levels (30 and 60 $$^\circ$$ C). For correlating the static and dynamic elastic moduli, a drained acoustic velocity ratio is introduced, corresponding to the drained Poisson’s ratio in poroelasticity. The strength of poroelastic coupling, i.e., the product of Biot and Skempton coefficients here, was the key parameter. The value of this parameter decreased with increasing effective pressure by about 56 $$~\%$$ from 0.51 at 3 MPa to 0.22 at 73 MPa. In contrast, the maximum change in P- and S-wave velocities was only 3 % in this pressure range. This correlation approach can be used in characterizing underground reservoirs, and can be employed to relate seismicity and geomechanics (seismo-mechanics).

Homogenization of coupled flow and deformation in a porous material

In this paper we present a framework for computational homogenization of the fluid–solid interaction that pertains to the coupled deformation and flow of pore fluid in a fluid-saturated porous material. Large deformations are considered and the resulting problem is established in the material setting. In order to ensure a proper FE-mesh in the fluid domain of the RVE, we introduce a fictitious elastic solid in the pores; however, the adopted variational setting ensures that the fictitious material does not obscure the motion of the (physical) solid skeleton. For the subsequent numerical evaluation of the RVE-response, hyperelastic properties are assigned to the solid material, whereas the fluid motion is modeled as incompressible Stokes’ flow. Variationally consistent homogenization of the standard first order is adopted. The homogenization is selective in the sense that the resulting macroscale (upscaled) porous media model reminds about the classical one for a quasi-static problem with displacements and pore pressure as the unknown macroscale fields. Hence, the (relative) fluid velocity, i.e. seepage, “lives” only on the subscale and is part of the set of unknown fields in the RVE-problem. As to boundary conditions on the RVE, a mixture of Dirichlet and weakly periodic conditions is adopted. In the numerical examples, special attention is given to an evaluation of the Biot coefficient that occurs in classical phenomenological models for porous media.

Meso-hydro-mechanically informed effective stresses and effective pressures for saturated and unsaturated porous media

Abstract Based on the meso-structured Voronoi cell model and the meso-macro homogenization procedure between the discrete particle assembly and the porous continuum for wet granular materials, meso-hydro-mechanically informed effective pressure and effective stress measures for saturated and unsaturated porous media are defined. The meso-hydro-mechanically informed generalized effective stress for saturated porous continua taking into account the volumetric deformation of solid grains due to pore liquid pressure is derived. The Biot coefficient associated to the meso-hydro- mechanically informed generalized effective stress for saturated porous media is formulated. The differences of the definitions for proposed generalized effective stress and Biot coefficient compared with those defined in the generalized Biot theory of saturated porous continua and in averaging theories are discussed. The wet meso-structured Voronoi cell model, consisting of three immiscible and interrelated (i.e. solid grains, interstitial liquid and gas) phases, at low bulk saturation (below about 30%) is proposed. A meso-structural pattern with the binary bond mode of pendular liquid bridges is assumed in particular to derive the meso-hydro- mechanically informed macroscopic anisotropic effective pressure and effective stress tensors for unsaturated porous media. As the isotropic case of the wet meso-structured Voronoi cell model is considered, the meso-hydro-mechanically informed effective pressure tensor degrades to the scalar variable in the same form as in the theory of macroscopic unsaturated porous continua. The proposed meso-hydro-mechanically informed Bishop's parameter is derived and obtained as a function of saturation, porosity, and meso-structural parameters, without need to introduce any macroscopic phenomenological assumptions for the description of hydro-mechanical constitutive behavior.

A gas-mechanical coupled constitutive equation for fractured coal containing gas

A gas-mechanical coupled constitutive equation is developed for coal containing gas or CO2 to predict the failure behavior based on fracture mechanics theory. A novel parameter of |∂cosα/∂Σ1| has been chosen as the failure characteristic parameter for coal containing gas. In addition, a closed-form gas-mechanical coupled constitutive equation is derived as σ1=σ3+fκ+fχpσtσcσtσcσ3+σc2. The effects of the coal friction coefficient f, the gas pore pressure p, Biot's coefficient χ and the fracture parameter κ have been coupled in the proposed model. Without considering the effect of pore pressure, the equation wonderfully coincides with the original Hoek-Brown Empirical Strength Criterion. Sensitivity analysis indicates that under a determining confining pressure, the failure stress is an increasing function of f, in contrast, a decreasing function of p, κ and χ. Due to the low tensile strength of coal, the effects of different fracture parameter κ from different fracture criterions can nearly be ignored when the pore pressure p is high. At last, in order to further verify the gas-mechanical coupled model proposed in this study, the comparison between some laboratory data and theoretical values for coal containing gas have been carried out. It seems that the new model is in good agreement with the existing experimental data.

Effect of kerogen on rock physics of immature organic-rich chalks

We study the rock physics of organic-rich chalk from the Shefela basin, central Israel, based on laboratory measurements of core material from the Zoharim well. This deposit is an immature source rock that includes the Late Cretaceous Ghareb and Mishash formations. The studied rock is composed of three dominating phases: minerals, kerogen, and pores. We investigate how porosity and kerogen influence the physical behavior of the rock, based on measurements of porosity, density, acoustic velocities, elastic moduli, tensile strength and compressive strength. We find that the dynamic bulk modulus and the static Young’s modulus decrease with increasing kerogen content as well as with increasing porosity. To describe the stiffness of the solid skeleton we use two different poroelasticity models for fluid substitution assuming material isotropy: Biot-Gassmann model and Marion’s BAM model. The obtained poroelasticity coefficients indicate a soft matrix composed of kerogen-micrite mixture, characterized by a low BAM normalized stiffness factor (w ∼ 0.2) and a high Biot coefficient (β ∼ 0.9). These values appear to vary with changes in kerogen content, grain size and degree of cementation. Porosity, which typically decreases with depth in the studied basin, does not show a clear correlation with kerogen content. Moreover, both compressive and tensile strengths decrease linearly with increasing porosity, regardless of kerogen content. We use a density–kerogen relationship to create a porosity depth profile, by combining data from continuous cores and density well logs. By comparing sonic logs with laboratory measurements of P-wave velocities on fully-saturated cores, we are able to detect a gas zone in the shallow interval of the sequence in the studied well.

Investigation of possible wellbore cement failures during hydraulic fracturing operations

Abstract We model and assess the possibility of shear failure along the vertical well by using the Mohr–Coulomb failure model and employing a rigorous coupled flow-geomechanic analysis. To this end, we take various values of cohesion between the well casing and the surrounding cement to represent different quality levels of cementing operation (low cohesion corresponds to low-quality cement and/or incomplete cementing). The simulation results show that there is very little fracturing when the cement is of high quality. Conversely, incomplete cementing and/or weak cement can cause significant shear failure and evolution of long fractures/cracks along the vertical well. Specifically, low cohesion between the well and cemented areas can cause significant shear failure along the well, while high cohesion does not cause shear failure. The Biot and thermal dilation coefficients strongly affect shear failure along the well casing, and low Young's modulus causes fast failure propagation. Still, for the high quality of the cementing job, failure propagates very little. When the hydraulic fracturing pressure is high or when permeability increases significantly, low cohesion of the cement can cause fast propagation of shear failure and of the resulting fracture/crack, but a high-quality cement with no weak zones exhibits limited shear failure that is only concentrated near the bottom of the vertical part of the well. Thus, high-quality cement and complete cementing along the vertical well appears to be the strongest protection against shear failure of the wellbore cement and, consequently, against contamination hazards to drinking water aquifers during hydraulic fracturing operations.

Experimental investigation and hybrid numerical analytical hydraulic mechanical simulation of supercritical CO2 flowing through a natural fracture in caprock

Recent work on analogue CO2 storage sites has shown that the single most defining natural factor as to whether the store can be successfully utilised to retain 99% of the injected CO2 for 1000 years is the behaviour of fractures within the low permeability strata acting as caprock. Here we present experimental and numerical investigation of the hydro-mechanical behaviour of a natural fracture in a caprock during the flow of supercritical CO2 through it. The caprock is a naturally fractured dolomitic limestone sample recovered from a depth of ∼1500m, and is the primary seal to the natural CO2 storage analogue, the Fizzy field, in the Southern North Sea. For the first time the hydro mechanical behaviour of the fracture is examined using unique experimental equipment applying multiple high pressure single phase supercritical CO2 fluid flow experiments at representative in situ reservoir pressures (10–30MPa, with confining pressures from 35MPa to 55MPa) and a temperature of 40°C. The fracture surfaces are scanned to provide high resolution images both prior to and after the experimental investigation. The results are modelled through the further development of a hybrid numerical analytical approach to fluid flow through a discrete fracture, implemented in the open source code OpenGeoSys. The work indicates that through the statistical approximation of the fracture surface and combination of the application of standard nonlinear flow models and analytical mechanical solutions, the key features of the hydro-mechanical behaviour of the supercritical fluid flow through the sample can be replicated. The results provide insight into erroneous effective stress assumptions at higher fluid pressures and the importance of understanding the coupled process multi-physics behaviour of fractures in a CO2 storage setting. Over-simplistic approximations using the effective stress law lead to a Biot's coefficient greater than 1 being predicted under varying fluid and confining pressures.

Effect of Biot's coefficient and fluid properties on isothermal H-M coupled consolidation analysis of porous media

In this study, a hydraulic-mechanical (H-M) coupling model was implemented, which considers porous media as a deforming porous continuum where fluid flow is taken into account. The primary variables are the time-dependent pore pressure and displacement of the solid skeleton. Physical laws, including the linear momentum balance and mass conservation, are applied to describe the corresponding processes as mechanics and fluid flow occurring within the porous media during the consolidation progress. The general field equations were derived with the aid of a finite element formulation and generalized trapezoidal method to deal with the governing equations in spatial and time domains. In the numerical analysis, the massive diversity of time domain of coupled problems was discussed in detail. In addition, a convergence condition for the partitioned approach applied to deal with a typical matter was proposed. A numerical example was executed to simulate a one-dimensional isothermal consolidation in which a hydraulic-mechanical coupling is mathematically adopted. Finally, a series of parametric studies was carried out to indicate the effect of Biot's coefficient and fluid properties on the isothermal consolidation of porous media. The results of parametric studies show that the consolidation of porous media can be expedited with compressible particles and less compressible fluids than those with incompressible particles and more compressible fluids, respectively.

Integrating damage zone heterogeneities based on stochastic realizations of fracture networks for fault stability analysis

In the domain of reservoir engineering, fault stability assessment is traditionally performed by assimilating fault systems to surfaces. However, faults are complex and heterogeneous geological systems, whose compartmentalized architecture generally corresponds to an inner core surrounded by outer, often fractured, damage zones (DZ). To date, few studies have assessed the potential role of the DZ characteristics (anisotropy, complex spatial distribution, etc.) on the shear behavior of the whole system, the main challenge being the proper (numerical) integration of fracture networks in reservoir simulators. The present study was motivated by the stochastically generated fracture networks representative of DZ investigated on outcrops (Cirques de Navacelles, South of France). We propose here a methodology relying on homogenization techniques for deriving the spatial distribution of effective anisotropic hydro-poro-elastic DZ properties from these stochastic fracture networks. Using them as inputs of 2D finite-element coupled hydro-mechanical models of a reservoir-scale fault zone, the key role of the spatially-varying DZ stiffness and Biot's coefficients are highlighted: neglecting them might lead to a high over-estimation of the maximum sustainable injection pressure (by nearly 50% in the considered cases).