An Alternative Approach to Predicting Reservoir Performance in a Coalbed Methane (CBM) Well Flowing Under Dominant Matrix Shrinkage Effect

Coalbed methane (CBM) reservoirs contain gas molecules in adsorbed state into the solid matrix of coal. The pressure depletion in CBM reservoir causes the matrix gas to desorb into the cleat system which leads to matrix shrinkage. The pore volume of the cleat network changes as coal matrix shrinks. Consequently, cleat porosity and permeability of reservoir change as reservoir pressure depletes. The change in cleat porosity and permeability due to shrinkage of coal matrix with depletion of reservoir pressure invalidates the underlying assumptions made in the derivation of diffusivity equation. Under the conditions of changing porosity and permeability, the utility of the standard method of inflow performance relationship (IPR), paired with \(\frac{P}{Z^{*}}\) method suggested by King (in: SPE Annual Technical Conference and Exhibition, New Orleans, 1990), for performance prediction diminishes. In this paper, an effort has been made to predict reservoir performance of such CBM reservoirs with an alternative approach. The method suggested by Upadhyay and Laik (Transp Porous Media, 2017. doi: 10.1007/s11242-016-0816-6) has been leveraged to describe pseudo-steady-state flow in the form of a new equation that relates stress-dependent pseudo-pressure function with time. The analytical equation derived in this paper is useful in predicting reservoir pressure and flowing bottom hole pressure of a CBM well under the situation when coal matrix shrinks below desorption pressure. The paper aims to predict production performance of CBM reservoirs producing under the influence of matrix shrinkage effect with an approach alternative to conventional IPR approach paired with \(\frac{P}{Z^{*}}\) method. The results of this analytical solution have been validated with the help of numerical simulator CMG–GEM as well as in-field production data. The equations and workflow suggested in this paper can be easily implemented in spreadsheet applications like Microsoft Excel tools.     

A New Cleat Volume Compressibility Determination Method and Corresponding Modification to Coal Permeability Model

Coal is known as a dual-porosity media composed of cleat and matrix pore. Methane can be stored in the cleats or adsorbed on the inner surface of matrix pore. While fluid mobility is mainly controlled by the developed cleat network, methane desorption has a significant effect on cleat deformation. In the process of coalbed methane recovery, both reservoir compaction and matrix shrinkage will occur and have opposite effects on permeability evolutions. A variety of analytical permeability models have been developed to describe the transient characteristics of permeability in coals. In this study, three common permeability models are first revisited and evaluated against the experimental data under uniaxial strain condition. Shi–Durucan (S&D) model demonstrates the best performance among these models. However, constant cleat volume compressibility was used to assume for S&D model, and the generalization of S&D model is significantly limited. For ease of generalization, the relation between cleat volume compressibility and effective horizontal stress is re-derived and introduced to the derivation of permeability model. Since coal reservoirs usually demonstrate strong anisotropy and heterogeneity, the influences of elastic and adsorption properties are further tested to reveal the overall trend of permeability. The results show that S&D model and its modification with the main variable of effective horizontal stress have the best performances in matching the experimental data under uniaxial strain. The relationship between cleat volume compressibility and effective horizontal stress can be better reflected by the inverse proportional function. In addition, the strengths of reservoir compaction effect relative to matrix shrinkage effect in different models only vary with Poisson’s ratio, while their magnitudes are also affected by Young’s modulus. For a typical coal reservoir, the C&B and P&M models will observe a stronger permeability decline at the initial, while the improved P&M model will receive an earlier and more rapid rebound than the S&D and W&Z models.     

Coal Permeability Evolution and Gas Migration Under Non-equilibrium State

Laboratory test of coal permeability is generally conducted under the condition of gas adsorption equilibrium, and the results contribute to an understanding of gas migration in the original coal seams. However, gas flow under the state of non-equilibrium, accompanied by gas adsorption and desorption, is more common in coalbed methane (CBM) recovery and \(\hbox {CO}_{2}\) geological sequestration sites. Therefore, research on gas migration under the non-equilibrium state has a greater significance with regard to CBM recovery and \(\hbox {CO}_{2}\) geological sequestration. However, most permeability models, in which only one gas pressure has been considered, cannot be used to study gas flow under the non-equilibrium state. In this study, a new mathematical model, which includes both fracture gas pressure and matrix gas pressure, and couples the gas flow with the coal deformation, has been developed and verified. With the developed model, the spatial and temporal evolution of gas flow field during gas adsorption and desorption phases has been explored. The results show that the gas pressures present nonlinear distributions in the coal core, and the matrix gas pressure is generally lower than the fracture gas pressure during adsorption, but higher than the fracture gas pressure during desorption. For gas flow during adsorption, the main factor controlling permeability varies at different points. At the initial time, the permeability is dominated by the effective stress, and at the later time, the permeability in the part close to the gas inlet is mainly controlled by the matrix swelling, whereas that in the part close to the gas outlet is still dominated by the effective stress. For gas flow during desorption, from the gas inlet to the gas outlet, the permeability deceases at the initial time, and when the time is greater than 10,000 s, it shows a decreasing and then an increasing trend. The reason is that at the initial time, the permeability is dominated by the increased effective stress caused by the sharp decrease of the fracture gas pressure. Later, desorption of the adsorbed gas results in matrix shrinkage, which further leads to an increase of the permeability.     

Drawdown Induced Changes in Permeability of Coalbeds: A New Interpretation of the Reservoir Response to Primary Recovery

A model for pore pressure-dependent cleat permeability is presented for gas-desorbing, linear elastic coalbeds under uniaxial strain conditions experienced in producing reservoirs. In the model, changes in the cleat permeability of coalbeds, which are idealised to have a bundled matchstick geometry, is controlled by the prevailing effective horizontal stresses normal to the cleats. Variations in the effective horizontal stresses under uniaxial strain conditions are expressed as a function of pore pressure reduction during drawdown, which includes a cleat compression term and a matrix shrinkage term that have competing effects on cleat permeability. A comprehensive analysis has revealed that the shape of the stress – pore pressure curve is predominantly determined by the magnitude of recovery pressure and rebound pressure relative to the initial reservoir pressure. A total of five possible scenarios have been identified with regard to response of the horizontal stress function to reservoir drawdown. When applied to four coalbed wells at two separate sites in the fairway of the San Juan basin, the model predictions at one site, where the three wells have shown increased absolute permeability during gas production, are in excellent agreement with the published pore pressure dependent permeability changes that were obtained independently from history matching the field production data. At a separate site the model correctly predicts, at least qualitatively, a strong permeability rebound at lower drawdown pressures that has been inferred through history matching the production data. An analysis of the effects of initial reservoir pressure on the response of effective horizontal stress to drawdown was carried out, with reference to the range of pressure likely to be encountered in the San Juan basin. The implications of this in terms of pore pressure dependent permeability are discussed.     

Flow of Coal-Bed Methane to a Gallery

Coal-seam methane reservoirs have a number of unique feature compared to conventional porous or fractured gas reservoirs. We propose a simplified mathematical model of methane movement in a coal seam taking into account the following features: a relatively regular cleat system, adsorptive methane storage, an extremely slow mechanism of methane release from the coal matrix into cleats and a significant change of permeability due to desorption.Parameters of the model have been combined into a few dimensionless complexes which are estimated to an order of magnitude. The simplicity of the model allows us to fully investigate the influence of each parameter on the production characteristics of the coal seam. We show that the reference time of methane release from the coal matrix into cleats – the parameter which is most poorly investigated – may have a critical influence on the overall methane production.     

Simulation of Pressure Transient Behavior for Asymmetrically Finite-Conductivity Fractured Wells in Coal Reservoirs

Based on Fick’s law in matrix and Darcy flow in cleats and hydraulic fractures, a new semi-analytical model considering the effects of boundary conditions was presented to investigate pressure transient behavior for asymmetrically fractured wells in coal reservoirs. The new model is more accurate than previous model proposed by Anbarci and Ertekin, SPE annual technical conference and exhibition, New Orleans, 27–30 Sept 1998 because new model is expressed in the form of integral expressions and is validated well through numerical simulation. (1) In this paper, the effects of parameters including fracture conductivity, coal reservoir porosity and permeability, fracture asymmetry factor, sorption time constant, fracture half-length, and coalbed methane (CBM) viscosity on bottomhole pressure behavior were discussed in detail. (2) Type curves were established to analyze both transient pressure behavior and flow characteristics in CBM reservoir. According to the characteristics of dimensionless pseudo pressure derivative curves, the process of the flow for fractured CBM wells was divided into six sub-stages. (3) This paper showed the comparison of transient steady state and pseudo steady state models. (4) The effects of parameters including transfer coefficient, wellbore storage coefficient, storage coefficient of cleat, fracture conductivity, fracture asymmetry factor, and rate coefficient on the shape of type curves were also discussed in detail, indicating that it is necessary to keep a bigger fracture conductivity and fracture symmetry for enhancing well production and reducing pressure depletion during the hydraulic fracturing design.     

Laboratory Investigation of LNAPL Migration in Double-Porosity Soil Under Fractured Condition Using Digital Image Analysis

Gas production from shale gas reservoirs plays a significant role in satisfying increasing energy demands. Compared with conventional sandstone and carbonate reservoirs, shale gas reservoirs are characterized by extremely low porosity, ultra-low permeability and high clay content. Slip flow, diffusion, adsorption and desorption are the primary gas transport processes in shale matrix, while Darcy flow is restricted to fractures. Understanding methane diffusion and adsorption, and gas flow and equilibrium in the low-permeability matrix of shale is crucial for shale formation evaluation and for predicting gas production. Modeling of diffusion in low-permeability shale rocks requires use of the Dusty gas model (DGM) rather than Fick’s law. The DGM is incorporated in the TOUGH2 module EOS7C-ECBM, a modified version of EOS7C that simulates multicomponent gas mixture transport in porous media. Also included in EOS7C-ECBM is the extended Langmuir model for adsorption and desorption of gases. In this study, a column shale model was constructed to simulate methane diffusion and adsorption through shale rocks. The process of binary \(\hbox {CH}_{4}{-}\hbox {N}_{2}\) diffusion and adsorption was analyzed. A sensitivity study was performed to investigate the effects of pressure, temperature and permeability on diffusion and adsorption in shale rocks. The results show that methane gas diffusion and adsorption in shale is a slow process of dynamic equilibrium, which can be illustrated by the slope of a curve in \(\hbox {CH}_{4}\) mass variation. The amount of adsorption increases with the pressure increase at the low pressure, and the mass change by gas diffusion will decrease due to the decrease in the compressibility factor of the gas. With the elevated temperature, the gas molecules move faster and then the greater gas diffusion rates make the process duration shorter. The gas diffusion rate decreases with the permeability decrease, and there is a limit of gas diffusion if the permeability is less than \(1.0\,\times \,10^{-15}\, \hbox { m}^{2}\). The results can provide insights for a better understanding of methane diffusion and adsorption in the shale rocks so as to optimize gas production performance of shale gas reservoirs.     

Pore-Scale Numerical Investigation on Chemical Stimulation in Coal and Permeability Enhancement for Coal Seam Gas Production

Permeability is a controlling factor for gas migration in coal seam reservoirs and has invariably been the barrier to economically viable gas production in certain deposits. Cleats are the main conduits for gas flow in coal seams though cleat mineralisation is known to significantly reduce permeability. Cleat demineralisation by the use of acids may enhance the effective cleat aperture and therefore permeability. This modelling study examines how acids transport through coal subject to reactive cleat mineralisation, and develops a fundamental understanding of the mechanisms controlling permeability change from pore scale to sample scale. A novel Lattice Boltzmann Method (LBM)-based numerical model for the simulation, prediction, and visualisation of the reaction transport is proposed to numerically investigate relationships between physio-chemical changes and permeability during coal stimulation. In particular, the work studies the interaction of acidic fluids (HCl) with reactive mineral (e.g. calcite) and assumed non-reactive mineral (e.g. coal) surfaces, mineral dissolution and mass transfer, and resultant porosity change. The reaction of a calcite cemented core sub-plug from the Bandanna Formation of Bowen Basin (Australia), is used as a study case. LBM simulations revealed a permeability enhancement (27.15 times of the pre-flooding permeability) along the x-axis after 20 min HCl flooding of a \(5.3~\hbox {cm} \times 5.3~\hbox {cm} \times 1.3~\hbox {cm}\) sub-section. The analysis and evaluation of the 4D permeability evolution is conducted as a contribution work for the fluid flow modelling in the subsurface petrophysical conditions, at the micron to centimetre scales. The simulation results demonstrate the proposed algorithm is capable for studies of multiple mineral reactions with disparate reaction rates.     

Approximate Semi-analytical Solution for Injection–Falloff–Production Well Test: An Analytical Tool for the In Situ Estimation of Relative Permeability Curves

An injection–falloff–production test (IFPT) was originally proposed in Chen et al. (in: SPE conference paper, 2006. doi: 10.2118/103271-MS, SPE Reserv Eval Eng 11(1):95–107, 2008) as a well test for the in situ estimation of two-phase relative permeability curves to be used for simulating multiphase flows in porous media. Hence, we develop an approximate semi-analytical solution for the two-phase saturation distribution in an oil–water system during the flowback period of an IFPT according to the mathematical theory of waves. In fact, we show that the weak solution we construct for the saturation equation for the flowback period satisfies the Oleinik entropy condition and hence is unique. In addition, we allow the governing relative permeabilities during the flowback period to be different from the relative permeabilities during injection. Using the saturation solution with the steady-state pressure theory of Thompson and Reynolds, we obtain a solution for the wellbore pressure during the flowback period. By comparing results from our solution with those from a commercial numerical simulator, we show that our approximate semi-analytical solution yields accurate saturation profiles and bottom hole pressures history. The use of very small time steps and a highly refined radial grid is necessary to generate a good solution from a reservoir simulator. The approximate analytical pressure solution developed is used as a forward model to match pressure and water flow rate data from an IFPT in order to estimate reservoir rock absolute permeability and skin factor in conjunction with in situ imbibition and drainage water–oil relative permeabilities.     

分形理论在煤层气非平衡吸附模型中的应用

传统的煤层气动力学模型均是建立在欧几里得几何基础上的,难以描述煤层孔隙结构的复杂性及形状的不规则性。本文以分形理论为基础,通过引入分形维数来刻画煤层孔隙结构的复杂性并考虑煤层的吸附特性、双重介质特征及介质的变形,建立基于Fick第二定律的分形介质煤层气非稳态渗流数学模型。由于流动方程的强非线性,结合各类边界条件用正则摄动法和Laplace变换得到模型在拉氏空间上的近似解析解,再利用Laplace数值反演求得实空间上的数值解。对参数进行敏感性分析并绘制了典型压力曲线,这些结果为煤层气开采提供了理论依据和试井方法。     

An Analytical Solution for Capillary Gravity Drainage with Dominant Viscous Forces

The development of the capillary fringe during gravity drainage has a significant influence on saturation and pressure distributions in porous formations (Sarkarfarshi et al. in Int J Greenh Gas Control 23:61–71, 2014). This paper introduces an analytical solution for gravity drainage in an axisymmetric geometry with significant capillary pressure. The drainage process results from the injection of a lighter and less viscous injectant into a porous medium saturated with a heavier and more viscous pore fluid. If the viscous force dominates the capillary and the buoyancy forces, then the flow regime is approximated by differential equations and the admissible solution comprises a front shock wave and a trailing simple wave. In contrast to existing analytical solutions for capillary gravity drainage problems (e.g., Nordbotten and Dahle in 47(2) 2011; Golding et al. in J Fluid Mech 678:248–270 2011), this solution targets the saturation distribution during injection at an earlier point in time. Another contribution of this analytical solution is the incorporation of a completely drained flow regime close to the injection well. The analytical solution demonstrates the strong dependency of the saturation distribution upon relative permeability functions, gas entry capillary pressure, and residual saturation. The analytical results are compared to results from a commercial reservoir engineering software package (\(\hbox {CMG } \hbox {STARS}^{\mathrm{TM}}\)).     

Impact of Effective Stress and Matrix Deformation on the Coal Fracture Permeability

The permeability of coal is an important parameter in mine methane control and coal bed methane exploitation because it determines the practicability of methane extraction. We developed a new coal permeability model under tri-axial stress conditions. In our model, the coal matrix is compressible and Biot’s coefficient, which is considered to be 1 in existing models, varies between 0 and 1. Only a portion of the matrix deformation, which is represented by the effective coal matrix deformation factor $f_\mathrm{m}$ , contributes to fracture deformation. The factor $f_\mathrm{m}$ is a parameter of the coal structure and is a constant between 0 and 1 for a specific coal. Laboratory tests indicate that the Sulcis coal sample has an $f_\mathrm{m}$ value of 0.1794 for $\hbox {N}_{2}$ and $\hbox {CO}_{2}$ . The proposed permeability model was evaluated using published data for the Sulcis coal sample and is compared to three popular permeability models. The proposed model agrees well with the observed permeability changes and can predict the permeability of coal better than the other models. The sensitivity of the new model to changes in the physical, mechanical and adsorption deformation parameters of the coal was investigated. Biot’s coefficient and the bulk modulus mainly affect the effective stress term in the proposed model. The sorption deformation parameters and the factor $f_\mathrm{m}$ affect the coal matrix deformation term.     

On the Validation of a Compositional Model for the Simulation of $$\text {CO}_2$$ Injection into Saline Aquifers

In this work we apply a recently proposed Bayesian Markov chain Monte Carlo framework (Akbarabadi et al. in Comput Geosci 19(6):1231–1250, 2015) to quantify uncertainty in the three-dimensional permeability field of a rock core. This process establishes the credibility of a compositional two-phase flow model to describe the displacement of brine by \(\text {CO}_2\) and \(\text {CO}_2\) storage in saline aquifers. We investigate the predictive capabilities of the compositional model in the context of an unsteady-state \(\text {CO}_2\)-brine drainage experiment at the laboratory scale, performed at field-scale aquifer conditions. We employ forward models consisting of a system of discretized partial differential equations along with relative permeability curves obtained by a curve fitting of experimental measurements. We consider a forward model to be validated when: (1) numerical simulations reveal that the Bayesian framework has accurately characterized the core’s permeability and (2) Monte Carlo predictions show excellent agreement between measured and simulated data. A large set of numerical studies with an accurate compositional simulator shows that forward models have been successfully validated. For such models, our numerical results show that we are able to capture all the dominant features and general trends of the \(\text {CO}_2\) saturation fields observed in the core. Our study is consistent with the design and findings of real experiments. Fluid properties, relative permeability data, measured porosity field, physical dimensions, and thermodynamic conditions are the same as those reported in Akbarabadi and Piri (Adv Water Resour 52:190–206, 2013). However, the measured saturation data are from flow experiments different from those reported in Akbarabadi and Piri (2013), and will be presented here.     

Laboratory and Simulation Investigation of Enhanced Coalbed Methane Recovery by Gas Injection

Methane/carbon dioxide/nitrogen flow and adsorption behavior within coal is investigated simultaneously from a laboratory and simulation perspective. The samples are from a coalbed in the Powder River Basin, WY. They are characterized by methane, carbon dioxide, and nitrogen sorption isotherms, as well as porosity and permeability measurements. This coal adsorbs almost three times as much carbon dioxide as methane and exhibits significant hysteresis among pure-component adsorption and desorption isotherms that are characterized as Langmuir-like. Displacement experiments were conducted with pure nitrogen, pure carbon dioxide, and various mixtures. Recovery factors are greater than 94% of the OGIP. Most interestingly, the coal exhibited ability to separate nitrogen from carbon dioxide due to the preferential strong adsorption of carbon dioxide. Injection of a mixture rich in carbon dioxide gives slower initial recovery, increases breakthrough time, and decreases the volume of gas needed to sweep out the coalbed. Injection gas rich in nitrogen leads to relatively fast recovery of methane, earlier breakthrough, and a significant fraction of nitrogen in the produced gas at short times. A one-dimensional, two-phase (gas and solid) model was employed to rationalize and explain the experimental data and trends. Reproduction of binary behavior is characterized as excellent, whereas the dynamics of ternary systems are predicted with less accuracy. For these coals, the most sensitive simulation input were the multicomponent adsorption–desorption isotherms, including scanning loops. Additionally, the coal exhibited a two-porosity matrix that was incorporated numerically.     

Gas Sorption and the Consequent Volumetric and Permeability Change of Coal II: Numerical Modeling

The permeability of coalbed methane reservoirs may evolve during the recovery of methane and injection of gas, due to the change of effective stress and gas adsorption and desorption. Experimental and numerical studies were conducted to investigate the sorption-induced permeability change of coal. This paper presents the numerical modeling part of the work. It was found that adsorption of pure gases on coal was well represented by parametric adsorption isotherm models in the literature. Based on the experimental data of this study, adsorption of pure \(\hbox {N}_2\) was modeled using the Langmuir equation, and adsorption of pure \(\hbox {CO}_2\) was well represented by the N-Layer BET equation. For the modeling of CO \(_2\) & N \(_2\) binary mixture adsorption, the ideal adsorbed solution (IAS) model and the real adsorbed solution (RAS) model were used. The IAS model estimated the total amount of mixture adsorption and the composition of the adsorbed phase based on the pure adsorption isotherms. The estimated total adsorption and adsorbed-phase composition were very different from the experimental results, indicating nonideality of the CO \(_2\) –N \(_2\) –Coal-adsorption system. The measured sorption-induced strain was linearly proportional to the total amount of adsorption despite the species of the adsorbed gas. Permeability reduction followed a linear correlation with the volumetric strain with the adsorption of pure \(\hbox {N}_2\) and the tested CO \(_2\) & N \(_2\) binary mixtures, and an exponential correlation with the adsorption of pure \(\hbox {CO}_2\) .     

Unstable Immiscible Displacements in Oil-Wet Rocks

Displacement of a viscous fluid by a lower viscosity immiscible fluid (such as waterflood of a viscous oil) in a porous medium is unstable. The displacement front generates viscous fingers which lead to low oil recovery efficiency. These fingers are much smaller in width than typical reservoir simulation grid blocks, and capturing their effect in reservoir simulation is important. A dimensionless scaling group (viscous finger number) had been suggested in the past, which has a power-law relationship with the breakthrough recovery and cumulative recovery in unstable core floods. The relative permeability used in large grid block simulations had been modified to so-called pseudo-relative permeability on the basis of the dimensionless group, thus incorporating the effect of fingers in waterflood predictions. However, the previous proposed models were constructed from experiments in only water-wet rocks. This paper extends the recent viscous fingering models to oil-wet systems. Sandstone cores were treated to alter the wettability to oil-wet. Adverse viscosity water floods were performed in oil-wet cores. Viscosity ratio, velocity and diameter were varied. It is shown that the previously developed viscous finger number does not work for the oil-wet experiments. The correlating dimensionless number is modified for oil-wet systems; it is also different from the dimensionless group identified by Peters and Flock (Soc Petroleum Eng, 1981. doi: 10.2118/8371-PA) for oil-wet cores. A pseudo-relative permeability model has been developed for oil-wet cores. Corefloods have been matched by the new pseudo-relative permeability model to determine the model parameters. This pseudo-relative permeability model can be used in reservoir simulations of water and polymer floods in viscous oil-wet reservoirs.     

Comment on “New method to obtain periodic solutions of period two and three of a rational difference equation” [Nonlinear Dyn 79:241–250]

In this paper, we discuss the results of Elsayed (Nonlinear Dyn 79:241–250, 2015) for periodic solution of period two and three of rational difference equation. Also, we study the existence of periodic solutions of some difference equation by using old method and new method and comparison of results. The results obtained here correct and improve some known results in Elsayed (Nonlinear Dyn 79:241–250, 2015). Some numerical examples will be given to illustrate our results.     

Pressure transient response of stochastically heterogeneous fractured reservoirs

A stochastic model for flow through inhomogeneous fractured reservoirs of double porosity, based on Barenblattet al.'s continuum approach, is presented. The fractured formation is conceptualized as an interconnected fracture network surrounding porous blocks, and amenable to the continuum approach. The block permeability is negligible in comparison to that of the fractures, and therefore the reservoir permeability is represented by the permeability of the fracture network. The fractured reservoir inhomogeneity is attributed to the fracture network, while the blocks are considered homogeneous. The mathematical model is represented by a coupled system of partial differential random equations, and a general solution for the average and for the correlation moments of the fracture pressure are obtained by the Neumann expansion (or Adomian decomposition). The solution for pressure is represented by an infinite series and an approximate solution for radial flow, is obtained by retaining the first two terms of the series. The purpose of this investigation is to get an insight on the pressure behavior in inhomogeneous fractured reservoirs and not to obtain type curves for determination of reservoir properties, which owing to the nonuniqueness of the solution, is impossible. For the present analysis we assumed an ideal reservoir with cylindrical symmetric inhomogeneity around the well. Fractured rock reservoirs being practically inhomogeneous, it is of interest to compare the pressure behavior of such reservoirs, with Warren and Roots's solution for (ideal) homogeneous reservoirs, used as a routine for determining the fractured reservoir characteristic parameters and, using the results of well tests. The comparison of the results show that inhomogeneous and homogeneous reservoirs exhibit a similar pressure behavior. While the behavior is identical, the same drawdown or a build-up pressure curve may be fitted by different characteristic dimensionless parameters and, when attributed to an inhomogeneous or a homogeneous reservoir. It is concluded that the ambiguity in determining the fractured reservoir and, makes questionable the usefulness of determination of these parameters. Computations were also carried out to determine the correlation between the fracture pressure at the well and the fracture pressure at different points.     

Two-subsystem thermodynamics for the mechanics of aging amorphous solids

The effect of physical aging on the mechanics of amorphous solids as well as mechanical rejuvenation is modeled with nonequilibrium thermodynamics, using the concept of two thermal subsystems, namely a kinetic one and a configurational one. Earlier work (Semkiv and Hütter in J Non-Equilib Thermodyn 41(2):79–88, 2016) is extended to account for a fully general coupling of the two thermal subsystems. This coupling gives rise to hypoelastic-type contributions in the expression for the Cauchy stress tensor, that reduces to the more common hyperelastic case for sufficiently long aging. The general model, particularly the reversible and irreversible couplings between the thermal subsystems, is compared in detail with models in the literature (Boyce et al. in Mech Mater 7:15–33, 1988; Buckley et al. in J Mech Phys Solids 52:2355–2377, 2004; Klompen et al. in Macromolecules 38:6997–7008, 2005; Kamrin and Bouchbinder in J Mech Phys Solids 73:269–288 2014; Xiao and Nguyen in J Mech Phys Solids 82:62–81, 2015). It is found that only for the case of Kamrin and Bouchbinder (J Mech Phys Solids 73:269–288, 2014) there is a nontrivial coupling between the thermal subsystems in the reversible dynamics, for which the Jacobi identity is automatically satisfied. Moreover, in their work as well as in Boyce et al. (Mech Mater 7:15–33, 1988), viscoplastic deformation is driven by the deviatoric part of the Cauchy stress tensor, while for Buckley et al. (J Mech Phys Solids 52:2355–2377, 2004) and Xiao and Nguyen (J Mech Phys Solids 82:62–81, 2015) this is not the case.     

Second Order Corrections to Mean Field Evolution for Weakly Interacting Bosons in The Case of Three-body Interactions

In this paper, we consider the Hamiltonian evolution of N weakly interacting bosons. Assuming triple collisions, its mean field approximation is given by a quintic Hartree equation. We construct a second order correction to the mean field approximation using a kernel k(t, x, y) and derive an evolution equation for k. We show global existence for the resulting evolution equation for the correction and establish an a priori estimate comparing the approximation to the exact Hamiltonian evolution. Our error estimate is global and uniform in time. Comparing with the work of Rodnianski and Schlein (Commun Math Phys 291:31–61, 2009), and Grillakis, Machedon and Margetis (Commun Math Phys 294:273–301, 2010; Adv Math 288:1788–1815, 2011), where the error estimate grows in time, our approximation tracks the exact dynamics for all time with an error of the order \({O(1/\sqrt{N}).}\)