Optimization Algorithm of Effective Stress Coefficient for Permeability

The effective stress coefficient for permeability is a significant index for characterizing the variation in permeability with effective stress. The realization of its accuracy is essential for studying the stress sensitivity of oil and gas reservoirs. The determination of the effective stress coefficient for permeability can be mainly evaluated using the cross-plotting or response surface method. Both methods preprocess experimental data and preset a specific function relation, resulting in deviation in the calculation results. To improve the calculation accuracy of the effective stress coefficient for permeability, a 3D surface fitting calculation method was proposed according to the linear effective stress law and continuity hypothesis. The statistical parameters of the aforementioned three methods were compared, and the results showed that the three-dimensional (3D) surface fitting method had the advantages of a high correlation coefficient, low root mean square error, and low residual error. The principal of using the 3D surface fitting method to calculate the effective stress coefficient of permeability was to evaluate the influence of two independent variables on a dependent variable by means of a 3D nonlinear regression. Therefore, the method could be applied to studying the relationship between other physical properties and effective stress.

Effects of Overpressure on Mechanical Properties of Unconventional Shale Reservoirs through Novel Use of a Sonic Overpressure Indicator

Summary Overpressure is a common feature among productive unconventional shale reservoirs, such as the Bone Spring (BSPG) and Wolfcamp (WFMP) Formations of the Delaware Basin (DB) of west Texas and southeastern New Mexico, and is thought to be a strong driver of well productivity. Compared with conventional reservoirs and shales in normal pressured conditions, the effects of overpressure on the mechanical properties of shales is not well understood. Here we present an analysis of overpressure in clay-bearing siliciclastic facies of the BSPG and WFMP Formations of the DB and implications for mechanical properties of the reservoir. Estimation of the effects of overpressure on mechanical properties of unconventional shale reservoirs is determined through use of the sonic overpressure indicator (SOPI). The method requires log model results that accurately characterize variations in lithology and porosity for the formations of interest. The SOPI (ΔT/ΔTN)2, where ΔT is the measured compressional sonic transit time, and ΔTN is the forward-modeled result for normally pressured conditions, can be used with elastic moduli and their interrelationships to compare estimates of mechanical properties including Poisson’s ratio ν, the Biot or effective stress coefficient α, and Young’s modulus E, in normal and overpressured conditions. Results presented here are broadly applicable to overpressured unconventional reservoirs that contain significant clay volume (>0.1 v/v) and exhibit low porosity (<0.08 v/v), comparable to that of siliciclastic-rich facies of the WFMP Formation. To account for increased VP/VS ratio, we regard overpressurization of shaly facies as an irreversible thermodynamic process that transforms a normally pressured siliciclastic system. At stress below the yield point, which is taken as the limit of normal pressure, the system responds elastically to stress; beyond this point, during overpressurization, the system responds as an elastic/plastic medium with strain hardening. We regard elastic moduli as descriptive of mechanical energy stored in this system. This perspective enables Poisson’s ratio for the overpressured system νOP to be computed from an estimate of the normally pressured system νN using (ΔT/ΔTN)2. Overpressure also results in a limited increase of the Biot or effective stress coefficient α. Moreover, recognition that overpressure results in a decrease of Young’s modulus, that is, EOP/EN < 1, provides a means of estimating the amount of strain energy stored by the formation due to overpressurization. We believe that when exposed to lower pressures by wellbore construction, this strain energy stored in overpressured unconventional reservoirs drives creep, which affects interpretations made using geomechanical models. We have developed and tested computational models based on biaxial or plane strain for vertical wells and uniaxial strain for horizontal wells that describe how creep likely affects estimation of minimum horizontal stress Shmin and pore pressure from instantaneous shut-in-pressure (ISIP) measurements. Thus, for overpressured unconventional reservoirs, ISIP determinations differ from tectonic Shmin by an amount related to ν and EOP/EN.

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.

In Situ Stress Distribution and Its Control on the Coalbed Methane Reservoir Permeability in Liulin Area, Eastern Ordos Basin, China

Permeability is one of the important factors that affect the production efficiency of coalbed methane, and it is mainly controlled by in situ stress. Therefore, it is very essential to study the in situ stress and permeability for the extraction of coalbed methane. Based on the injection/falloff well test and in situ stress measurement of 35 coalbed methane wells in the Liulin area in the east of the Ordos basin, the correlations between initial reservoir pressure, in situ stress, lateral stress coefficient, permeability, and burial depth were determined. Finally, the distribution characteristics of in situ stress and its influence on permeability were analyzed systematically. The results show that with the increase of burial depth, the initial reservoir pressure and in situ stress both increase, while the lateral stress coefficient decreases. The permeability variation is related to the type of stress field in different burial depths, and its essence is the deformation and destruction of coal pore structures caused by stress. The distribution characteristics of in situ stress at different depths and its effect on permeability are as follows: at depths < 800 m , the horizontal principal stress is dominant ( σ H ≥ σ v > σ h ) and the permeability is a simple decreasing process with the increase of the depth; at depths > 800 m , the vertical stress is dominant ( σ v ≥ σ H > σ h ). The permeability of most coal is very small due to the large in situ stresses in this depth zone. However, because of the stress release at the syncline axis, coal with high permeability is still possible at this depth zone. Due to the existence of high permeability data points at burial depth (>800 m) and the fitting relationship between permeability and vertical stress, the maximum and minimum horizontal principal stress is poor. However, the coal permeability and lateral stress coefficient show a good negative exponential relationship. This indicates that the lateral stress coefficient can be used to predict permeability better.

FEATURES OF CALCULATING THE DEFLECTION OF ROUND PLATES

A method for calculating the deflection of a round flat steel plate bearing an ax isymmetric load is considered. For practical calculations, dimensionless coefficients of deflection in the center of the plate, deflection at a certain distance from the center of the plate, the stress coefficient and the accompanying function, which depends on the deflection of the plate and the pressure force on it, are used.

CALCULATION OF THE DEFLECTION OF A PINCHED ROUND PLATE

A method for calculating the deflection of a round flat steel plate pinched along the edge, bearing an axisymmetric load, is considered. For practical calculations, dimensionless coeffi cients of deflection in the center and in the area of fixing a round flat plate pinched along the edge are applied, the stress coefficient and the accompanying function, which depends on the deflection of the plate and the pressure force on it, are taken into account