Hydraulic fracturing of a saturated porous medium-II: Special cases

This investigation deals with the problem of steady state hydraulic fracture in an infinite isotropic fluid-saturated elastic porous medium induced by a uniform pressure applied to the crack surfaces. A quasi-static approach is employed in the study. A boundary value problem is formulated and then analyzed by means of the Fourier transform associated with the Wiener-Hopf technique. Stress intensity factor and potential energy release rate are found by asymptotic analysis and the superposition principle as functions of the speed of crack propagation. The material breakdown process at the crack tip is discussed based on Dugdale's model. Finally, a brief discussion of the effect of pressure drop on the hydraulic fracture process and the decrease in crack speed during crack extension is included.     

A general theory of thermoporoelastoplasticity for saturated porous materials

A general theory of thermoporoelastoplasticity for saturated porous materials is presented. The theory is derived from the thermodynamics of open systems and irreversible processes. The thermal effects, due to the saturating fluid, are taken into account through a latent heat associated with the increase of fluid mass content. The theory does not assume incompressibility nor infinitesimal displacements for the saturating fluid. To take into account the plastic compressibility of the skeleton, the notion of plastic porosity is introduced. This plastic porosity is different from the overall plastic dilatation. The usual isothermal phenomenological theories appear to be particular cases of the proposed general theory.     

Interaction between crack tip advancement and fluid flow in fracturing saturated porous media

We address stepwise crack tip advancement and pressure fluctuations, which have been observed in the field and experimentally in fracturing saturated porous media. Both fracturing due to mechanical loading and pressure driven fracture are considered. After presenting the experimental evidence and the different explanations for the phenomena put forward and mentioning briefly what has been obtained so far by published numerical and analytical methods we propose our explanation based on Biot’s theory. A short presentation of three methods able to simulate the observed phenomena namely the Central Force Model, the Standard Galerkin Finite Element Method SGFEM and extended finite element method XFEM follows. With the Central Force Model it is evidenced that already dry geomaterials break in an intermittent fashion and that the presence of a fluid affects the behavior more or less depending on the loading and boundary conditions. Examples dealing both with hydraulic fracturing and mechanical loading are shown. The conditions needed to reproduce the observed phenomena with FE models at macroscopic level are evidenced. They appear to be the adoption of a crack tip advancement/time step algorithm which interferes the least possible with the three interacting velocities, namely the crack tip advancement velocity on one side, the seepage velocity of the fluid in the domain and from the crack (leak-off), and the fluid velocity within the crack on the other side. Further the crack tip advancement algorithm must allow for reproducing jumps observed in the experiments.     

Spin-up in a saturated porous medium

It is shown that, on the Brinkman model, spin-up is confined to boundary layers whose thickness is of order k ^1/2, and the spin-up is established in a time of order k/, where k, , and denote permeability, density, porosity and dynamic viscosity, respectively.     

Linearization of dynamic equations for a saturated porous medium

Equations of the mechanics of a saturated porous medium that account for the deformation tensor of the fluid and the initial state of the medium are derived. Emphasis is on the linearization and justification of the basic nonlinear equations of the theory. The relationship between the pressure (deformation) of the fluid and the thermodynamic and kinematic parameters of the medium is established. The solutions obtained are compared with Biot’s well-known classical results and analyzed     

Thawing in saturated porous media

A mathematical model for thawing in a saturated porous medium is considered. The well-posedness of the corresponding mathematical problem is proved and similarity solutions are found.

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Sommario Si considera un modello matematico per to scongelamento in un mezzo poroso saturo. Viene dimostrata la buona posizione del corrispondente problema matematico e si trovano soluzioni di similarità.

     

Hydraulic fracturing in an inhomogeneous reservoir

A method of estimating the change in well productivity resulting from the hydraulic fracturing of a finite reservoir, piecewise-homogeneous in the horizontal and vertical directions, with an arbitrary number of cracks is proposed. A correction coefficient that enables these estimates to take into account the effect of capillary trapping of mud and fracturing fluid filtrate is derived [1, 2]. Novosibirsk. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 109–114, September–October, 1988.     

A linear dynamic model for a saturated porous medium

A linear isothermal dynamic model for a porous medium saturated by a Newtonian fluid is developed in the paper. In contrast to the mixture theory, the assumption of phase separation is avoided by introducing a single constitutive energy function for the porous medium. An important advantage of the proposed model is it can account for the couplings between the solid skeleton and the pore fluid. The mass and momentum balance equations are obtained according to the generalized mixture theory. Constitutive relations for the stress, the pore pressure are derived from the total free energy accounting for inter-phase interaction. In order to describe the momentum interaction between the fluid and the solid, a frequency independent Biot-type drag force model is introduced. A temporal variable porosity model with relaxation accounting for additional attenuation is introduced for the first time. The details of parameter estimation are discussed in the paper. It is demonstrated that all the material parameters in our model can be estimated from directly measurable phenomenological parameters. In terms of the equations of motion in the frequency domain, the wave velocities and the attenuations for the two P waves and one S wave are calculated. The influences of the porosity relaxation coefficient on the velocities and attenuation coefficients of the three waves of the porous medium are discussed in a numerical example.     

Plane waves in a semi-infinite fluid saturated porous medium

The field equations governing the propagation of waves in an incompressible liquid-saturated porous medium are investigated and a general solution is presented. It has been revealed that coupled longitudinal and transverse waves propagate in the porous medium. The propagation of transverse waves in the fluid phase is completely due to the interaction between the solid and fluid phases. The dispersion relationship and attenuation features are discussed. Unlike other investigations, all explicit forms of the arguments are derived. The reflection of the plane harmonic waves at the plane, traction-free boundary, which shows the influence of the dissipation on the velocity, and the attenuation coefficients of the reflected waves is studied. It is of interest that pore pressure is produced in the process of reflection, even in the case of the incidence of transverse waves.     

Capillary end effect in a water saturated porous layer

The uni-directional propagation of oil injected into water flowing through a water wetted porous slab of a finite length is investigated. The inlet and outlet edges of the slab are impermeable to the oil flux. Hence, the oil accumulates within the slab, thereby leading to a saturation build-up-capillary end effect. This phenomenon is studied analytically on the basis of a nonlinear equation describing oil-water transport in porous media. A dimensionless criterion is derived, which governs the appearance and relative strength of the capillary end effect. For weak oil-water interfacial tension (large capillary number) and long porous slabs the above effect is not observed and the temporal evolution of the oil saturation is described by the Buckley-Leverett solution. Short porous slabs are found to be almost entirely subjected to the capillary end effect. Intermediate situations are identified and quantitatively described, in which the downstream part of the slab may be divided into two zones: one-characterized by the capillary end effect, and the other being a Buckley-Leverett zone.It is shown, that the oil flux injected into the slab is limited by a maximum value which depends upon the location of the injection point. The partition of the inlet flux between the upstream and downstream directions is investigated. In the upstream side of the porous slab the oil moves under the action of free imbibition only. It is found that the upstream flux is limited by the value, which is independent of the slab's length and of the location of the injection point.     

Permeability identification of a weakly permeable partially saturated porous rock

The present paper deals with the determination of permeability in partially saturated conditions for weakly permeable porous continua such as argillites or deep clayey formations. The permeability can be deduced from measurements of transient weight loss of a sample submitted to a laboratory drying test: a decrease of relative humidity is imposed by saline solution in an hermetic chamber. Assumptions of constant gas pressure equal to atmospheric pressure and of negligible Fickean diffusive transport of vapour are adopted. The only transport phenomenon taken into account inside the sample is the Darcean advective transport of the water liquid. The forward problem is solved by following two modelling approaches: a linear one and a nonlinear one. The parameter identification procedure is based upon the solution of corresponding inverse problems. In the two cases, the Levenberg–Marquardt algorithm has been used for the minimization problem. In the linear approach, the solution of the forward problem is explicit. In the non linear approach, finite volume method for the spatial discretization combined with a Newton–Raphson algorithm has been used to solve the non linear forward problem. The identification method enables variations of permeability and capillary capacity to be estimated. Comparisons between linear and non linear approaches show that the first one is useful to give mean values and order of magnitude of permeability and capacity. A more complete information is deduced from the non linear approach as variations of equivalent capacity and permeability during a test are significant in most cases. The analysis of the obtained results shows that the basic modelling assumption of constant gas pressure inside the sample would not be relevant for lower range of relative humidities and liquid permeability than those investigated.     

Pressure waves in a porous medium saturated with a gassy fluid

Experimental data on the evolution of pressure waves in a consolidated porous medium saturated with a gassy fluid are obtained. These data are generalized on the basis of a theoretical analysis.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 85–92, July–August, 1987.     

Waves in partially saturated porous media

The propagation of compressional waves in a porous medium is investigated in case the pore liquid contains a small volume fraction of gas. The effect of oscillating gas bubbles is taken into account by introducing a frequency-dependent fluid bulk modulus, which is incorporated in the Biot theory. Using a shock tube technique, new experimental data are obtained for a porous column subjected to a pressure step wave. An oscillatory behaviour is observed, consisting of two distinct frequency bands, which is predicted by the theoretical analysis.     

Fundamentals of thermomechanics of porous saturated media

Institute of Mechanics, Academy, of Sciences of the Ukrainian SSR, Kiev. Translated from Prikladnaya Mekhanika, Vol. 24, No. 4, pp. 3–13, April, 1988.     

Linear waves in saturated porous media

The peculiarity of linear and nonlinear wave propagation in porous media saturated with liquid or gas has been investigated by the methods of multiphase media mechanics. It has been shown that for the analysis and interpretation of experimental data, it is expedient to build models taking into account the nonstationary powers of interaction between the solid and liquid phases and the viscouselastic behaviour of the porous media skeleton. Inertia and inertia-viscous powers principally influence wave attentuation in porous media. Two interphase mechanisms of momentum transfer (two stress tensors — in the solid phase and liquid) lead to two types of waves. Attenuation is determined not only by interphase friction, but also by dissipation resulting from intergrain friction in the solid phase, the influence of which multiplexly exceeds the liquid viscosity influence. The real decrement of attenuation may exceed the sphere restricted by the limiting curves corresponding to the frozen and equilibrium schemes of intergrain deformation. The attenuation of momentum perturbation has been studied. The method of discrete Fourier transform has been used. The analysis of experimental data contained in the literature and their comparison with the results of calculations has been carried out.     

On localization in saturated porous continua

Localization in elastic-plastic saturated porous media is investigated here using a linear perturbation approach. The adopted localization criterion corresponds to unbounded growth of perturbations. The critical conditions are compared with those obtained by a classical band analysis. While for one phase materials these conditions coincide, in the present context the linear perturbation approach leads in the limit of unbounded growth to the singularity of the undrained acoustic tensor, while the band analysis leads to the singularity of the drained acoustic tensor. Some general results clarifying the hierarchy of these two conditions are established for a quite general set of constitutive equations.     

An equilibrium solution for a fluid saturated porous elastic solid

This work concerns the behavior of a binary mixture of a fluid and an isotropic elastic solid in static equilibrium. The displacements are assumed to be small. Thus, the governing partial differential equations are linear. The physical model is sufficiently general to allow for a nonconstant fluid pressure when the mixture is in static equilibrium. The model is applied to the problem of an arbitrary pressure distribution on an isothermal half-space. Among the results of this calculation is an explicit formula for the surface porosity. This parameter gives the fraction of the applied pressure transmitted to the fluid.

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Zusammenfassung Diese Arbeit behandelt das Verhalten einer binären Mischung eines Flüssigkeit und eines isotropen elastischen Festkorgers im statischen Gleichgewicht. Deformationen sind als klein angenommen. Die beschreibenden partiellen Differentialgleichungen sind dadurch linear. Das physikalische Modell ist allgemein genügend um einen nichtkonstanten Druck zuzulassen, wenn die Mischung im statischen Gleichgewicht ist. Das Modell wird auf das Problem einer willkürlichen druckverteilung auf dem isothermen halbraum angewandt. Unter den Ergebnissen dieser Berechnung ist eine explizite Formel für die Oberflächen porosität. Dieser Parameter gibt den Bruchteil des ausgeübten Druckes an, der auf die Flüssigkeit übertragen wird.

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Presently employed at Johnson Space Center National Aeronautics and Space Administration Houston, Texas.     

非均质饱和多孔介质弹塑性动力分析的广义耦合扩展多尺度有限元法

针对非均质饱和多孔介质弹塑性动力问题分析提出了一种广义耦合扩展多尺度有限元方法。首先,提出了基于细尺度等效刚度阵的粗尺度单元数值基函数构造方法,并给出了构造数值基函数的一般公式,所构造的耦合数值基函数有效考虑了动力相关效应与固液之间的耦合效应。其次,针对弹塑性非线性问题迭代求解,给出了基于摄动方法的位移与孔隙压强降尺度计算修正方案。最后,针对材料的强非均质特征,利用多节点粗单元技术来提高多尺度有限元方法的计算精度。通过与基于精细网格的传统有限元分析结果对比,验证了本文所提出方法的有效性与高效性。