Effective-Stress Coefficients of Porous Rocks Involving Shocks and Loading/Unloading Hysteresis

SPE Journal ◽  
2020 ◽  
pp. 1-24
Author(s):  
Faruk Civan
Keyword(s):  
Effective Stress ◽  
Power Law ◽  
Rock Properties ◽  
Porous Rocks ◽  
Limit Values ◽  
Biot Coefficient ◽  
Rock Formations ◽  
Porosity Ratio ◽  
Basic Power ◽  
Power Law Equation

Summary A critical review, examination, and clarification of the various issues and problems concerning the definition and dependence of the effective-stress coefficients of porous-rock formations is presented. The effective-stress coefficients have different values for different rock properties because the physical mechanisms of rock deformation can affect the various rock properties differently. The alteration of petrophysical properties occurs by the onset of various rock-deformation/damaging processes, including pore collapsing and grain crushing, and affects the values of the effective-stress coefficients controlling the different petrophysical properties of rock formations. The slope discontinuity observed in the effective-stress coefficients of naturally or induced fractured-rock formations during loading/unloading, referred to as a shock effect, is essentially related to deformation of fractures at less than the critical effective stress and deformation of matrix at greater than the critical effective stress. The hysteresis observed in the effective-stress coefficients of heterogeneous porous rocks during loading/unloading is attributed to elastic deformation under the fully elastic predamage conditions, and/or irreversible pore-structure-alteration/deformation processes. A proper correlation of the Biot-Willis coefficient controlling the bulk volumetric strain is developed using the data available from various sources in a manner to meet the required endpoint-limit conditions of the Biot-Willis coefficient, ranging from zero to unity. The modified power-law equation presented in this paper yields a physically meaningful correlation because it successfully satisfies the low-end- and high-end-limit values of the Biot-Willis coefficient and also provides a better quality match of the available experimental data than the semilogarithmic equation and the popular basic power-law equation. It is shown that the semilogarithmic correlation cannot predict the values of the Biot coefficient beyond the range of the data because it generates unrealistic values approaching the negative infinity for the Biot coefficient for the low-permeability/porosity ratio and unrealistically high values approaching the positive infinity for the high-permeability/porosity ratio. The basic power-law equation is not adequate either because it can only satisfy the low-end value but cannot satisfy the high-end value of the Biot coefficient. The correlation developed in this paper from the modified power-law equation is effectively applicable over the full range of the Biot-Willis coefficient, extending from zero to unity. To the best of the author’s knowledge, this paper is the first to present an effective theory and formulation of the convenient correlation of the Biot-Willis poroelastic coefficient that not only satisfies both of the two endpoint-limit values of the Biot-Willis coefficient but also produces the best match of the available experimental data.

Effective Correlation of Stress and Thermal Effects on Porosity and Permeability of Naturally Fractured Formations by a Modified Power Law

SPE Journal ◽  
2019 ◽  
Vol 24 (05) ◽  
pp. 2378-2397
Author(s):  
Faruk Civan
Keyword(s):  
Experimental Data ◽  
Effective Stress ◽  
Power Law ◽  
Fractured Rock ◽  
Petrophysical Properties ◽  
Fractured Reservoir ◽  
Rock Formations ◽  
Naturally Fractured ◽  
Porosity And Permeability ◽  
Modified Power Law

Summary Effective theory and methodology are proposed and validated for accurate correlation of stress–dependent petrophysical properties of naturally fractured or induced–fractured reservoir formations by means of a matrix/fracture dual–compressibility treatment. Inspection of various experimental data indicates a sudden change in trends at a certain critical net effective stress in the stress dependence of petrophysical properties of porous rocks as a result of a stress shock caused by the opening or closing of fractures. The variation of petrophysical properties in fractured–rock formations subjected to stress loading/unloading and thermally induced stress occurs mainly by deformation of the fractures below the critical effective stress and the deformation of the matrix above the critical effective stress. The alteration of petrophysical properties and a slope discontinuity might also be experienced when the stress exceeds the onset of other rock–alteration/damaging processes, such as pore collapsing and grain crushing. Proper formulations of the relevant processes and special correlation methods are presented in a manner to capture this nature of the petrophysical experimental data obtained by testing of cores extracted from naturally fractured or induced–fractured reservoir–rock formations. The dependency of porosity and permeability of fractured–rock samples under stress because of thermal, hydraulic, and mechanical effects is represented accurately by a modified–power–law equation derived from a kinetics model as confirmed by effective correlations of various experimental data. It is shown that this new model represents the thermal effect better than the frequently used Arrhenius (1889) equation and Vogel–Tammann–Fulcher (VTF) equation (Vogel 1921; Fulcher 1925; Tammann and Hesse 1926).

Conductivity, Dielectric and Modulus study of NH4PF6 based Zwitterionic polymer electrolyte

2020 ◽  
Vol 13 ◽  
Author(s):  
Manindra Kumar ◽  
Neelabh Srivastava
Keyword(s):  
Polymer Electrolyte ◽  
Power Law ◽  
Loss Tangent ◽  
Electric Modulus ◽  
Relaxation Behavior ◽  
Conductivity Data ◽  
Frequency Range ◽  
Zwitterionic Polymer ◽  
Power Law Equation ◽  
Jonscher’S Power Law

Background and Objective: Zwitterionic polymer electrolyte has been successfully synthesized using NH4PF6 salt. The conductivity of the synthesized polymer membrane is found to be of the order of 10-3Scm-1. Dielectric and Modulus properties of the polymer electrolyte have also been studied which showed well relaxation peaks with both temperature and salt concentrations. Result: This is well depicted with the loss tangent curve. Debye type relaxation behavior has observed from the electric modulus. Conclusion: Frequency dependent conductivity data (fitted with Jonscher's power law equation) confirmed the presence of NCL/SLPL type behavior in the studied frequency range.

scholarly journals Derivation and evaluation of landslide-triggering thresholds by a Monte Carlo approach

2014 ◽  
Vol 18 (12) ◽  
pp. 4913-4931 ◽  
Author(s):  
D. J. Peres ◽  
A. Cancelliere
Keyword(s):  
Monte Carlo ◽  
Slope Stability ◽  
Power Law ◽  
Rainfall Intensity ◽  
Antecedent Rainfall ◽  
Rainfall Events ◽  
Physically Based ◽  
Power Law Equation ◽  
Triggering Rainfall ◽  
Landslide Triggering

Abstract. Assessment of landslide-triggering rainfall thresholds is useful for early warning in prone areas. In this paper, it is shown how stochastic rainfall models and hydrological and slope stability physically based models can be advantageously combined in a Monte Carlo simulation framework to generate virtually unlimited-length synthetic rainfall and related slope stability factor of safety data, exploiting the information contained in observed rainfall records and field-measurements of soil hydraulic and geotechnical parameters. The synthetic data set, dichotomized in triggering and non-triggering rainfall events, is analyzed by receiver operating characteristics (ROC) analysis to derive stochastic-input physically based thresholds that optimize the trade-off between correct and wrong predictions. Moreover, the specific modeling framework implemented in this work, based on hourly analysis, enables one to analyze the uncertainty related to variability of rainfall intensity within events and to past rainfall (antecedent rainfall). A specific focus is dedicated to the widely used power-law rainfall intensity–duration (I–D) thresholds. Results indicate that variability of intensity during rainfall events influences significantly rainfall intensity and duration associated with landslide triggering. Remarkably, when a time-variable rainfall-rate event is considered, the simulated triggering points may be separated with a very good approximation from the non-triggering ones by a I–D power-law equation, while a representation of rainfall as constant–intensity hyetographs globally leads to non-conservative results. This indicates that the I–D power-law equation is adequate to represent the triggering part due to transient infiltration produced by rainfall events of variable intensity and thus gives a physically based justification for this widely used threshold form, which provides results that are valid when landslide occurrence is mostly due to that part. These conditions are more likely to occur in hillslopes of low specific upslope contributing area, relatively high hydraulic conductivity and high critical wetness ratio. Otherwise, rainfall time history occurring before single rainfall events influences landslide triggering, determining whether a threshold based only on rainfall intensity and duration may be sufficient or it needs to be improved by the introduction of antecedent rainfall variables. Further analyses show that predictability of landslides decreases with soil depth, critical wetness ratio and the increase of vertical basal drainage (leakage) that occurs in the presence of a fractured bedrock.

Strength and deformation of frozen saturated sand at −30 °C

1982 ◽  
Vol 19 (1) ◽  
pp. 104-107 ◽  
Author(s):  
V. R. Parameswaran ◽  
M. Roy
Keyword(s):  
Strain Rate ◽  
Power Law ◽  
Quartz Sand ◽  
Peak Strength ◽  
Strain Rates ◽  
Saturated Sand ◽  
Ideal Plastic ◽  
Strength And Deformation ◽  
Power Law Equation

Frozen saturated quartz sand containing 20% moisture by weight, when deformed at −30 °C at various strain rates, showed various modes of behaviour such as visco-plastic, almost ideal plastic, and brittle with little plasticity with increasing order of strain rate. The values of peak strength observed for strain rates between 5 × 10−7 and 6 × 10−3 s−1 were in the range 15–42 MPa, stress being related to strain rate by a power-law equation.

scholarly journals RANDOM WALKS ON DIRECTED NETWORKS: THE CASE OF PAGERANK

2007 ◽  
Vol 17 (07) ◽  
pp. 2343-2353 ◽  
Author(s):  
SANTO FORTUNATO ◽  
ALESSANDRO FLAMMINI
Keyword(s):  
Random Walk ◽  
Power Law ◽  
Degree Distribution ◽  
Directed Graphs ◽  
Damping Factor ◽  
Web Pages ◽  
Stationary Probability ◽  
Exact Results ◽  
Limit Values ◽  
Underlying Graph

PageRank, the prestige measure for Web pages used by Google, is the stationary probability of a peculiar random walk on directed graphs, which interpolates between a pure random walk and a process where all nodes have the same probability of being visited. We give some exact results on the distribution of PageRank in the cases in which the damping factor q approaches the two limit values 0 and 1. When q → 0 and for several classes of graphs the distribution is a power law with exponent 2, regardless of the in-degree distribution. When q → 1 it can always be derived from the in-degree distribution of the underlying graph, if the out-degree is the same for all nodes.

Measuring the onset of mine tailings erosion

2007 ◽  
Vol 44 (4) ◽  
pp. 473-489 ◽  
Author(s):  
M Haneef-Mian ◽  
Ernest K Yanful ◽  
Robert Martinuzzi
Keyword(s):  
Shear Stress ◽  
Power Law ◽  
Mine Tailings ◽  
Laser Doppler Anemometry ◽  
Critical Shear Stress ◽  
Laser Doppler ◽  
Shear Stresses ◽  
Cohesive Sediments ◽  
Fine Grained ◽  
Power Law Equation

The present study gives details of a methodology for estimating the critical shear stress for erosion of mine tailings and other naturally occurring cohesive sediments. Erosion of a cohesive sediments bed occurs when the critical shear stress is exceeded to break the interparticle bond. Experiments were conducted in a 30 cm diameter laboratory column and calibrated using laser Doppler anemometry. The results showed that the erosion pattern of mine tailings particles was similar to those of fine-grained cohesive sediments. A power-law relation of the form E = α[(τ – τcr)/τcr]n is suggested for mine tailings, where E is the erosion rate, α is a coefficient, τ is the shear stress, τcr is the critical shear stress, and n is an exponent. The computed values of α, n, and τcr in the power-law equation were found to be comparable to values derived from experiments in a rotating circular flume. The derived expression for rate of erosion may be incorporated in resuspension and transport models for fine mine tailings of a similar nature.Key words: mine tailings, laser Doppler velocimetry, wall shear stresses, critical shear stress for erosion, erosion – shear stress relationship.

Experimental Investigation of Depletion- and Injection-Induced Changes in Poromechanical, Transport, and Strength Properties of High-Porosity Sandstone

SPE Journal ◽  
2021 ◽  
pp. 1-21
Author(s):  
Saeed Rafieepour ◽  
Stefan Z. Miska ◽  
Evren M. Ozbayoglu ◽  
Nicholas E. Takach ◽  
Mengjiao Yu ◽  
...  
Keyword(s):  
Pore Pressure ◽  
Effective Stress ◽  
Microstructural Analysis ◽  
Optical Microscope ◽  
Strength Properties ◽  
Rock Properties ◽  
Absolute Permeability ◽  
High Porosity ◽  
Confining Stress ◽  
Stress Changes

Summary In this paper, an extensive series of experiments was performed to investigate the evolution of poromechanical (dry, drained, undrained, and unjacketed moduli), transport (permeability), and strength properties during reservoir depletion and injection in a high-porosity sandstone (Castlegate). An overdetermined set of eight poroelastic moduli was measured as a function of confining pressure (Pc) and pore pressure (Pp). The results showed larger effect on pore pressure at low Terzaghi’s effective stress (nonlinear trend) during depletion and injection. Moreover, the rock sample is stiffer during injection than depletion. At the same Pc and Pp, Biot’s coefficient and Skempton’s coefficient are larger in depletion than injection. Under deviatoric loading, absolute permeability decreased by 35% with increasing effective confining stress up to 20.68 MPa. Given these variations in rock properties, modeling of in-situ-stress changes using constant properties could attain erroneous predictions. Moreover, constant deviatoric stress-depletion/injection failure tests showed no changes or infinitesimal variations of strength properties with depletion and injection. It was found that failure of Castlegate sandstone is controlled by simple effective stress, as postulated by Terzaghi. Effective-stress coefficients at failure (effective-stress coefficient for strength) were found to be close to unity (actual numbers, however, were 1.03 for Samples CS-5 and CS-9 and 1.04 for Sample CS-10). Microstructural analysis of Castlegate sandstone using both scanning electron microscope (SEM) and optical microscope revealed that the changes in poroelastic and transport properties as well as the significant hysteresis between depletion and injection are attributed to the existence and distribution of compliant components such as pores, microcracks, and clay minerals.

Determination of an effect of 3-chlorophenol on hydroxyl radical generation during ozonation through kinetic study in the power law type

2004 ◽  
Vol 4 (5-6) ◽  
pp. 305-311
Author(s):  
Y.-H. Han ◽  
I. Koshiishi ◽  
H. Utsumi
Keyword(s):  
Power Law ◽  
Ozone Decomposition ◽  
Oh Radical ◽  
Catalytic Intermediates ◽  
Stable Adduct ◽  
Solution Proceeds ◽  
Radical Generation ◽  
Power Law Equation ◽  
The Relationship

Ozone decomposition in aqueous solution proceeds through a radical type chain mechanism. These reactions involve the very reactive and catalytic intermediates hydroxyl (OH) radical, O2− radical, HO2 radical, OH−, H2O2, etc. OH radical is proposed as an important factor in the ozonation of water. In the previous study, generation of OH radical in the ozonation of water containing 3-chlorophenol was mathematically evaluated. In this study, we estimated the kinetic equation for the effect of 3-chlorophenol on OH radical generation during ozonation using the power law equation, in order to analyze it more correctly. The OH radical was trapped with a 5,5-dimethyl-1-pyrroline-N-oxide (DMPO) as a stable adduct, DMPO-OH. The relationship between the ozone concentration, 3-chlorophenol content, and the initial velocity (ν0) of DMPO-OH generation was analyzed mathematically, and the following equation was obtained: ν0 (10−6 M/s)=(1.58×10−5)×[3-chlorophenol (10−6 M)]×[ozone (10−6 M)]2.40+(3.09×10−5)×[ozone (10−6 M)]1.72. The equation fitted very well with the experimental results, and square of the correlation coefficient was larger than 0.9.

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