Hydraulic fracture crack propagation driven by a non-Newtonian fluid

The problem of hydraulic fracture crack propagation in a porous medium is considered. The fracture is driven by an incompressible viscous fluid with a power-law rheology of the pseudoplastic type. The fluid seepage is described by an equation generalizing the Darcy law in the hydraulic approximation. It is shown that the system of governing equations has a power-law self-similar solution, whereas, in the limiting cases of low and high fluid saturation in the porous medium, there are some families of power-law or exponential self-similar solutions. The complete self-similar solution is constructed. The effect of the nonlinear rheology of the fracturing fluid on the behavior of the solution is studied. The problem is solved analytically for an arbitrary boundary condition at the crack inlet when the viscous stresses in the non-Newtonian fluid are close to a constant.     

Analysis of power-law self-similar solutions to the problem of hydraulic fracture crack formation

The problem of hydraulic fracture crack propagation in a porous medium is studied in the approximation of small crack opening and the inertialess flow of an incompressible Newtonian hydraulic fracturing fluid inside the crack. A one-parameter family of power-law self-similar solutions is considered in order to determine the crack width evolution, the fluid velocity in the crack, and the seepage depth in the case of high and low seepage rates through the soil when a fluid flow rate is given at the crack inlet.     

Problem of propagation of a gas fracture in a porous medium

The problem of gas fracture formation in a porous medium is investigated. An inertialess viscous polytropic gas flow along the fracture is considered. The assumption of small fracture width with respect to the height and length makes it possible to adopt the vertical plane cross-section hypothesis on the basis of which the dependence of the gas pressure inside the fracture on its width can be reduced to a linear law. Initially, the soil surrounding the fracture is soaked with oil-bearing fluid. During fracturing the reservoir gas penetrates into the soil mass and displaces the fluid. A closed system of equations, which describes the evolution of the fracture opening, the depth of gas penetration into the reservoir, and the gas velocities inside the fracture, is constructed. The limiting regimes of gas seepage into the surrounding reservoir are considered and a one-parameter family of self-similar solutions of the system is given for each. The asymptotics of the solution in the neighborhood of the fracture nose is investigated and analytic expressions for the fracture length are obtained. The solution of the problem of gas fracture is compared with the hydraulic fracturing problem in an analogous formulation within the framework of the plane cross-section hypothesis.     

Self-similar solutions of non-Newtonian fluid boundary layer in MHD

The self-similar solutions of the boundary layer for a non-Newtonian fluid in MHD were considered in [1, 2] for a power-law velocity distribution along the outer edge of the layer and constant electrical conductivity through the entire flow. However, the MHD flows of many conducting media, which are solutions or molten metals, cannot be described by the MHD equations for non-Newtonian fluids.The self-similar solutions of the boundary layer for a non-Newtonian fluid without account for interaction with the electromagnetic field were studied in [3].In the following we present the self-similar solutions for the boundary layer of pseudoplastic and dilatant fluids with account for the interaction with an electromagnetic field for the case of a power-law velocity distribution along the outer edge of the layer, when the conductivity of the fluid is constant throughout the flow and the magnetic Reynolds number is small.Izv. AN SSSR. Mekhanika Zhidkosti i Gaza, Vol. 2, No. 6, pp. 77–82, 1967The author wishes to thank S. V. Fal'kovich for his interest in this study.     

A Self-Similar Solution to the Problem of Lava Dome Growth on an Arbitrary Conical Surface

Within the framework of the asymptotic thin-film equations for a highly viscous heavy Newtonian fluid, a hydrodynamic model of non-axisymmetric lava dome growth on a conical surface is constructed. A new class of self-similar solutions describing the flow on a conical surface with finite inclination to the horizontal and a point mass supply at the apex is found analytically for power-law or exponential growth of the liquid volume with time. For a conical surface with a small inclination to the horizontal, the free-surface shape is found numerically. The asymptotics of this solution are compared with the solutions describing the flow on a horizontal plane and a conical surface with finite inclination to the horizontal.     

On the existence of self-similar solutions of the equations governing unsteady flow through a porous medium

In this paper the conditions for the existence of self-similar solutions of the equations governing unsteady flows through a porous medium are presented and discussed. The first two sections deal with the case of non-Newtonian fluids of power-law behavior; the third section analyzes the case of non-Darcy gas flows. The boundary and initial conditions occuring currently in a large class of fluid mechanics problems, of practical interest in engineering, are considered.     

Propagation of a plane-strain hydraulic fracture with a fluid lag: Early-time solution

This paper studies the propagation of a plane-strain fluid-driven fracture with a fluid lag in an elastic solid. The fracture is driven by a constant rate of injection of an incompressible viscous fluid at the fracture inlet. The leak-off of the fracturing fluid into the host solid is considered negligible. The viscous fluid flow is lagging behind an advancing fracture tip, and the resulting tip cavity is assumed to be filled at some specified low pressure with either fluid vapor (impermeable host solid) or pore-fluids infiltrating from the permeable host solid. The scaling analysis allows to reduce problem parametric space to two lumped dimensionless parameters with the meaning of the solid toughness and of the tip underpressure (difference between the specified pressure in the tip cavity and the far field confining stress). A constant lumped toughness parameter uniquely defines solution trajectory in the parametric space, while time-varying lumped tip underpressure parameter describes evolution along the trajectory. Further analysis identifies the early and large time asymptotic states of the fracture evolution as corresponding to the small and large tip underpressure solutions, respectively. The former solution is obtained numerically herein and is characterized by a maximum fluid lag (as a fraction of the crack length), while the latter corresponds to the zero-lag solution of Spence and Sharp [Spence, D.A., Sharp, P.W., 1985. Self-similar solution for elastohydrodynamic cavity flow. Proc. Roy. Soc. London, Ser. A (400), 289–313]. The self-similarity at small/large tip underpressure implies that the solution for crack length, crack opening and net fluid pressure in the fluid-filled part of the crack is a given power-law of time, while the fluid lag is a constant fraction of the increasing fracture length. Evolution of a fluid-driven fracture between the two limit states corresponds to gradual expansion of the fluid-filled region and disappearance of the fluid lag. For small solid toughness and small tip underpressure, the fracture is practically devoid of fluid, which is localized into a narrow region near the fracture inlet. Corresponding asymptotic solution on the fracture lengthscale corresponds to that of a crack loaded by a pair of point forces which magnitude is determined from the coupled hydromechanical solution in the fluid-filled region near the crack inlet. For large solid toughness, the fluid lag is vanishingly small at any underpressure and the solution is adequately approximated by the zero-lag self-similar large toughness solution at any stage of fracture evolution. The small underpressure asymptotic solutions obtained in this work are sought to provide initial condition for the propagation of fractures which are initially under plane-strain conditions.     

Flow and heat transfer of an electrically conducting fluid of second grade in a porous medium over a stretching sheet subject to a transverse magnetic field

An analysis is performed for flow and heat transfer of a steady laminar boundary layer flow of an electrically conducting fluid of second grade in a porous medium subject to a transverse uniform magnetic field past a semi-infinite stretching sheet with power-law surface temperature or power-law surface heat flux. The effects of viscous dissipation, internal heat generation of absorption and work done due to deformation are considered in the energy equation. The variations of surface temperature gradient for the prescribed surface temperature case (PST) and surface temperature for the prescribed heat flux case (PHF) with various parameters are tabulated. The asymptotic expansions of the solutions for large Prandtl number are also given for the two heating conditions. It is shown that, when the Eckert number is large enough, the heat flow may transfer from the fluid to the wall rather than from the wall to the fluid when Eckert number is small. A physical explanation is given for this phenomenon.     

Self-similar problems of propagation of an extended hydrofracture in a permeable medium

The propagation of an extended hydrofracture in a permeable elastic medium under the influence of an injected viscous fluid is considered within the framework of the model proposed in [1, 2]. It is assumed that the motion of the fluid in the fracture is turbulent. The flow of the fluid in the porous medium is described by the filtration equation. In the quasisteady approximation and for locally one-dimensional leakage [3] new self-similarity solutions of the problem of the hydraulic fracture of a permeable reservoir with an exponential self-similar variable are obtained for plane and axial symmetry. The solution of this two-dimensional evolution problem is reduced to the integration of a one-dimensional integral equation. The asymptotic behavior of the solution near the well and the tip of the fracture is analyzed. The difficulties of using the quasisteady approximation for solving problems of the hydraulic fracture of permeable reservoirs are discussed. Other similarity solutions of the problem of the propagation of plane hydrofractures in the locally one-dimensional leakage approximation were considered in [3, 4] and for leakage constant along the surface of the fracture in [5–7].Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.2, pp. 91–101, March–April, 1992.     

Choice of a self-similar solution in boundary-layer theory

If the speed of the outer flow at the edge of the boundary layer does not depend on the time and is specified in the form of a power-law function of the longitudinal coordinate, then a self-similar solution of the boundary-layer equations can be found by integrating a third-order ordinary differential equation (see [1–3]). When the exponent of the power in the outerflow velocity distribution is negative, a self-similar solution satisfying the equations and the usually posed boundary conditions is not uniquely determinable [4], A similar result was obtained in [5] for flows of a conducting fluid in a magnetic field. In the present paper we study the behavior of non-self-similar perturbations of a self-similar solution, enabling us to provide a basis for the choice of a self-similar solution.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 42–46, July–August, 1974.     

A solution to the hydraulic fracture problem in the traveling wave form

Solutions to the system of equations describing the propagation of hydraulic fracture cracks in a porous medium are obtained in the traveling wave form. The only sought solution is the separatrix of integral curves on the “penetration depth-crack width” plane. Some necessary dependencies that should be given at the crack inlet are found for the fluid flow rate and the fluid pressure. The crack width and the fluid penetration depth are related by power laws in the limiting cases when the crack propagation processes or the fluid penetration processes are dominant.     

The Stokes boundary layer for a power-law fluid

We develop semi-analytical, self-similar solutions for the oscillatory boundary layer (‘Stokes layer’) in a semi-infinite power-law fluid bounded by an oscillating wall (the so-called Stokes problem). These solutions differ significantly from the classical solution for a Newtonian fluid, both in the non-sinusoidal form of the velocity oscillations and in the manner at which their amplitude decays with distance from the wall. In particular, for shear-thickening fluids the velocity reaches zero at a finite distance from the wall, and for shear-thinning fluids it decays algebraically with distance, in contrast to the exponential decay for a Newtonian fluid. We demonstrate numerically that these semi-analytical, self-similar solutions provide a good approximation to the flow driven by a sinusoidally oscillating wall.     

Computational Modeling of Fluid Flow through a Fracture in Permeable Rock

Laminar, single-phase, finite-volume solutions to the Navier–Stokes equations of fluid flow through a fracture within permeable media have been obtained. The fracture geometry was acquired from computed tomography scans of a fracture in Berea sandstone, capturing the small-scale roughness of these natural fluid conduits. First, the roughness of the two-dimensional fracture profiles was analyzed and shown to be similar to Brownian fractal structures. The permeability and tortuosity of each fracture profile was determined from simulations of fluid flow through these geometries with impermeable fracture walls. A surrounding permeable medium, assumed to obey Darcy’s Law with permeabilities from 0.2 to 2,000 millidarcies, was then included in the analysis. A series of simulations for flows in fractured permeable rocks was performed, and the results were used to develop a relationship between the flow rate and pressure loss for fractures in porous rocks. The resulting friction-factor, which accounts for the fracture geometric properties, is similar to the cubic law; it has the potential to be of use in discrete fracture reservoir-scale simulations of fluid flow through highly fractured geologic formations with appreciable matrix permeability. The observed fluid flow from the surrounding permeable medium to the fracture was significant when the resistance within the fracture and the medium were of the same order. An increase in the volumetric flow rate within the fracture profile increased by more than 5% was observed for flows within high permeability-fractured porous media.     

Convective-Region Top Front Propagation in a Uniform Medium under the Action of Point, Linear, and Plane Heat and Momentum Sources

A relation between the height of a convective front rising in an unstratified medium and the momentum and heat fluxes released on the substrate surface is proposed for point, linear, and uniform plane sources arbitrarily dependent on time. This relation makes it possible to determine the integral power of a plume on the basis of optical observations of the height of the propagating convective front. As particular solutions, three classes of self-similar regimes related with the heat and momentum sources, whose rate is a step-shaped, power-law, or exponential function of time, are obtained. A one-dimensional integral model of a rising convective jet is constructed. The classes of self-similar jets corresponding to power or exponential heat and momentum sources are described. It is shown that all self-similar jets corresponding to heat and momentum sources governed by a power law with a fairly large exponent are characterized by the same temperature and velocity profiles.     

Radiation effect on mixed convection from a horizontal surface in a porous medium

The effect of thermal radiation on the non-Darcy mixed convection flow over a non-isothermal horizontal surface immersed in a saturated porous medium has been studied. The wall temperature is assumed to have a power-law variation with the distance measured from the leading edge of the plate. The non-linear coupled parabolic partial differential equations governing the flow have been solved numerically using a finite-difference scheme. For some particular cases, the self-similar solution has also been obtained. The heat transfer is found to be strongly influenced by the radiative flux number, buoyancy parameter, variation of wall temperature, non-Darcy parameter and the nature of the free stream velocity.     

Description of weakly compressible fluid flows in porous media for a nonlinear seepage law

The equations that describe weakly compressible fluid flows through a weakly deformable porous skeleton are analyzed for the nonlinear seepage law with a limiting (initial) pressure gradient. With reference to approximate and numerical solutions of the well and well-gallery start problems, it is shown that taking into account in the continuity equation the quadratic term usually discarded when obtaining the elastic regime equations may qualitatively change the behavior of large spatial scale solutions.     

Non-steady MHD flow for a non-Newtonian fluid with variable conductivity

This paper deals with a similarity solution for the unsteady flow of a conducting non-Newtonian power-law in-compressible fluid, when a porous plate is moving uniformly in the presence of a transverse magnetic field, assuming that the electrical conductivity is a function of the velocity. The aim of this analysis is to determine the velocity and the effect of variation of the electrical conductivity on the solution. The basic equations have been solved by applying the perturbation method for small and large values of the magnetic interaction parameterN. The main features of the exact solution is that it represents shear flow.     

Effects of viscous dissipation on a power-law fluid over plate embedded in a porous medium

The present study is devoted to investigate the influences of viscous dissipation on buoyancy induced flow over a horizontal or a vertical flat plate embedded in a non-Newtonian fluid saturated porous medium. The Ostwald-de Waele power-law model is used to characterize the non-Newtonian fluid behavior. Similarity solutions for the transformed governing equations are obtained with prescribed variable surface temperature (PT) or with prescribed variable surface heat flux (PHF) for the horizontal plate case. While, the similarity solutions are obtained with prescribed variable surface heat flux for the vertical plate case. Different similar transformations, for each case, are used. Numerical results for the details of the velocity and temperature profiles are shown on graphs. Nusselt number associated with temperature distributions and excess surface temperature associated with heat flux distributions which are entered in tables have been presented for different values of the power-law index n and the exponent as well as Eckert number.     

Effects of variable fluid viscosity on flow past a heated stretching sheet embedded in a porous medium in presence of heat source/sink

The boundary layer flow and heat transfer of a fluid through a porous medium towards a stretching sheet in presence of heat generation or absorption is considered in this analysis. Fluid viscosity is assumed to vary as a linear function of temperature. The symmetry groups admitted by the corresponding boundary value problem are obtained by using a special form of Lie group transformations viz. scaling group of transformations. These transformations are used to convert the partial differential equations corresponding to the momentum and the energy equations into highly non-linear ordinary differential equations. Numerical solutions of these equations are obtained by shooting method. It is found that the horizontal velocity decreases with increasing temperature-dependent fluid viscosity parameter up to the crossing-over point but increases after that point and the temperature decreases in this case. With the increase of permeability parameter of the porous medium the fluid velocity decreases but the temperature increases at a particular point of the sheet. Effects of Prandtl number on the velocity boundary layer and on the thermal boundary layer are studied and plotted.