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.     

Hydraulic Fracture Propagation with 3-D Leak-off

Hydraulic fracture models typically couple a fracture elasticity model with a geological reservoir model to forecast the rate of fluid leak-off from the propagating fracture. The most commonly used leak-off model is that originally specified by Carter, which involves the assumption that the fracture is embedded within an infinite homogenous porous medium where flow only occurs perpendicular to the fracture plane. The objectives of this paper are: (1) to show that assuming one-dimensional leak-off can lead to erroneous conclusions, (2) to present a robust numerical methodology for simulating three-dimensional leak-off from propagating hydraulic fractures, and (3) to present and compare a new analytical method based on assuming three-dimensional flow of an incompressible fluid through an incompressible porous formation from a circular planar fracture. Provided the fluid and formation compressibility can be ignored within the reservoir flow model, the three-dimensional leak-off from a circular planar fracture can be written in closed-form as a function, which depends linearly on fracture pressure and radial extent. This simple expression for leak-off can be easily coupled to a range of circular fracture elasticity models. As a comparison example, the Carter model, our new function and a three-dimensional numerical model of the full problem are coupled to the PK-radial fracture model. Comparison with the numerical model shows that our new function overestimates fracture growth during intermediate times but accurately predicts both the early and late-time asymptotic behavior. In contrast, the Carter model fails to replicate both the early and late-time asymptotic behavior. Our new function additionally improves on the Carter model by not requiring the evaluation of convolution integrals and allowing easy evaluation of both the spatial leakage flux distribution across the fracture face and the three-dimensional pressure distribution within the porous formation.     

Thermo‐hydrodynamic‐mechanical multiphase flow model for CO2 sequestration in fracturing porous media

In this paper, we introduce a fully coupled thermo‐hydrodynamic‐mechanical computational model for multiphase flow in a deformable porous solid, exhibiting crack propagation due to fluid dynamics, with focus on CO₂ geosequestration. The geometry is described by a matrix domain, a fracture domain, and a matrix‐fracture domain. The fluid flow in the matrix domain is governed by Darcy's law and that in the crack is governed by the Navier–Stokes equations. At the matrix‐fracture domain, the fluid flow is governed by a leakage term derived from Darcy's law. Upon crack propagation, the conservation of mass and energy of the crack fluid is constrained by the isentropic process. We utilize the representative elementary volume‐averaging theory to formulate the mathematical model of the porous matrix, and the drift flux model to formulate the fluid dynamics in the fracture. The numerical solution is conducted using a mixed finite element discretization scheme. The standard Galerkin finite element method is utilized to discretize the diffusive dominant field equations, and the extended finite element method is utilized to discretize the crack propagation, and the fluid leakage at the boundaries between layers of different physical properties. A numerical example is given to demonstrate the computational capability of the model. It shows that the model, despite the relatively large number of degrees of freedom of different physical nature per node, is computationally efficient, and geometry and effectively mesh independent. Copyright © 2017 John Wiley & Sons, Ltd.     

裂缝气液重力置换的流动实验及仿真

通过可视化模拟实验对真实裂缝下气液重力置换现象进行了模拟,得到了其气液界面现象与常规平板裂缝的差异;通过对压力、置换量等数据的处理证实了气液重力置换的机理;根据实验的液体漏失流量数据提出了钻井过程中漏失、置换、溢流区间;最后根据k-ε双方程模型进行了气液重力置换流动的仿真计算,得到三维裂缝流道中都是钻井液下部侵入,呈现一个近似的直角三角形形状。本文对现场钻井工程方案设计及安全钻井具有指导意义。     

Propagation of a pre-existing turbulent fluid fracture

The propagation of rough and smooth wall pre-existing turbulent fluid fractures is investigated. The laminar fluid fracture is included as a special case for comparison. Lubrication theory is assumed to apply in the fracture and turbulence is introduced through the wall shear stress. The Perkins–Kern–Nordgren approximation is made in which the fluid pressure is proportional to the half-width of the fracture. The fracture half-width satisfies a non-linear diffusion equation. By using a linear combination of the Lie point symmetries of the non-linear diffusion equation a group invariant solution for the fracture length, volume and half-width is derived. The evolution of the length, half-width and mean flow velocity is analysed for a range of working conditions at the fracture entry. It is found that the mean flow velocity increases approximately linearly along the fracture.     

Model of stepwise propagation of the tip of a hydraulic fracture in the absence of filtration

Quasi-static stepwise propagation of a hydraulic fracture in rock with a regular structure in the absence of filtration is considered. It is proposed to use a brittle fracture diagram taking into account the hydraulic fracturing fluid pressure and the confining pressure. Fracture curves describing the brittle rock fracture where the hydraulic fracturing fluid partially fills the fracture are constructed and used to predicted the possibility of stepwise propagation of hydraulic fracturing in the case where the fluid gradually flows into the fracturing crack. The regularity of the structure of the brittle rocks fracture is estimated from the results of two full-scale experiments: the critical stress intensity factor and the tensile strength limit of the rock. Experiments on pulsed loading of polymethylmethacrylate samples with stepwise crack propagation along concentric circular arcs were performed. The results of the experiments are consistent with theoretical predictions.     

New Correlative Models to Improve Prediction of Fracture Permeability and Inertial Resistance Coefficient

Presence of fracture roughness and occurrence of nonlinear flow complicate fluid flow through rock fractures. This paper presents a qualitative and quantitative study on the effects of fracture wall surface roughness on flow behavior using direct flow simulation on artificial fractures. Previous studies have highlighted the importance of roughness on linear and nonlinear flow through rock fractures. Therefore, considering fracture roughness to propose models for the linear and nonlinear flow parameters seems to be necessary. In the current report, lattice Boltzmann method is used to numerically simulate fluid flow through different fracture realizations. Flow simulations are conducted over a wide range of pressure gradients through each fracture. It is observed that creeping flow at lower pressure gradients can be described using Darcy’s law, while transition to inertial flow occurs at higher pressure gradients. By detecting the onset of inertial flow and regression analysis on the simulation results with Forchheimer equation, inertial resistance coefficients are determined for each fracture. Fracture permeability values are also determined from Darcy flow as well. According to simulation results through different fractures, two parametric expressions are proposed for permeability and inertial resistance coefficient. The proposed models are validated using 3D numerical simulations and experimental results. The results obtained from these two proposed models are further compared with those obtained from the conventional models. The calculated average absolute relative errors and correlation coefficients indicate that the proposed models, despite their simplicity, present acceptable outcomes; the models are also more accurate compared to the available methods in the literature.     

Numerical investigation of dual-porosity model with transient transfer function based on discrete-fracture model

Based on the characteristics of fractures in naturally fractured reservoir and a discrete-fracture model, a fracture network numerical well test model is developed.Bottom hole pressure response curves and the pressure field are obtained by solving the model equations with the finite-element method. By analyzing bottom hole pressure curves and the fluid flow in the pressure field, seven flow stages can be recognized on the curves. An upscaling method is developed to compare with the dual-porosity model(DPM). The comparisons results show that the DPM overestimates the inter-porosity coefficient λ and the storage factor ω. The analysis results show that fracture conductivity plays a leading role in the fluid flow. Matrix permeability influences the beginning time of flow from the matrix to fractures. Fractures density is another important parameter controlling the flow. The fracture linear flow is hidden under the large fracture density.The pressure propagation is slower in the direction of larger fracture density.     

Similarity solution of a penny-shaped fluid-driven fracture in a zero-toughness linear elastic solid

This note deals with the problem of a penny-shaped hydraulic fracture propagating in an impermeable elastic solid. Growth of the fracture is driven by injection of an incompressible Newtonian fluid at the center of the fracture. The solution is restricted to the so-called viscosity-dominated regime where it can be assumed that the solid has zero toughness. The paper describes the construction of a semi-analytical similarity solution, which incorporates the known singularity of the fluid pressure at the center of the fracture and at the tip and which is based on series expansions of the fracture opening and fluid pressure in terms of Jacobi polynomials.     

Water Injection into a Low-Permeability Rock - 1 Hydrofracture Growth

In this paper, we model water injection through a growing vertical hydrofracture penetrating a low-permeability reservoir. The results are useful in oilfield waterflood applications and in liquid waste disposal through reinjection. Using Duhamel's principle, we extend the Gordeyev and Entov (1997) self-similar 2D solution of pressure diffusion from a growing fracture to the case of variable injection pressure. The flow of water injected into a low-permeability rock is almost perpendicular to the fracture for a time sufficiently long to be of practical interest. We revisit Carter's model of 1D fluid injection (Howard and Fast, 1957) and extend it to the case of variable injection pressure. We express the cumulative injection through the injection pressure and effective fracture area. Maintaining fluid injection above a reasonable minimal value leads inevitably to fracture growth regardless of the injector design and the injection policy. The average rate of fracture growth can be predicted from early injection. A smart injection controller that can prevent rapid fracture growth is needed.     

On the theory of seepage waves of pressure in a fracture in a porous permeable medium

Seepage pressure waves in fractures in a porous permeable medium are studied. The effects of the reservoir and fracture porosity and permeability, the fracture width, and the rheological properties of the saturating fluid on the perturbation dynamics in the fracture are analyzed. It is shown that in porous permeable reservoirs, fractures are wave channels through which low-frequency fluctuations of borehole pressure propagate. Accurate solutions are obtained which describe the evolution of pressure fields in a fracture with an instantaneous change in the borehole pressure by a constant value. Based on these solutions, dependences of the fluid flow rate on time and interface pressure are determined.     

On the problem of fluid leakoff during hydraulic fracturing

While a hydraulic fracture is propagating, fluid flow and associated pressure drops must be accounted for both along the fracture path and perpendicularly, into the formation that is fractured, because of fluid leakoff. The accounting for the leakoff shows that it is the main factor that determines the crack length. The solved problem is useful for the technology of hydraulic fracturing and a good example of mass transport in a porous medium. To find an effective approach for the solution, the thin crack is represented here as the boundary condition for pore pressure spreading in the formation. Earlier such model was used for heat conduction into a rock massif from a seam under injection of hot water. Of course, the equations have other physical sense and mathematically they are somewhat different. The new plane solution is developed for a linearized form that permits the application of the integral transform. The linearization itself is analogous to the linearization of the natural gas equation using the real gas pseudo-pressure function and where the flux rates are held constant and approximations are introduced only into the time derivatives. The resulting analytical solution includes some integrals that can be calculated numerically. This provides rigorous tracking of the created fracture volume, leakoff volume and increasing fracture width. The solutions are an advance over existing discreet formulations and allow ready calculations of the resulting fracture dimensions during the injection of the fracturing fluid.     

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.     

Propagation of a penny-shaped fluid-driven fracture in an impermeable rock: asymptotic solutions

This paper presents an analysis of the propagation of a penny-shaped hydraulic fracture in an impermeable elastic rock. The fracture is driven by an incompressible Newtonian fluid injected from a source at the center of the fracture. The fluid flow is modeled according to lubrication theory, while the elastic response is governed by a singular integral equation relating the crack opening and the fluid pressure. It is shown that the scaled equations contain only one parameter, a dimensionless toughness, which controls the regimes of fracture propagation. Asymptotic solutions for zero and large dimensionless toughness are constructed. Finally, the regimes of fracture propagation are analyzed by matching the asymptotic solutions with results of a numerical algorithm applicable to arbitrary toughness.     

Self-similar solutions of the problem of formation of a hydraulic fracture in a porous medium

The problem of hydraulic fracture formation in a porous medium is investigated in the approximation of small fracture opening and inertialess incompressible Newtonian fluid fracture flow when the seepage through the fracture walls into the surrounding reservoir is asymptotically small or large. It is shown that the system of equations describing the propagation of the fracture has self-similar solutions of power-law or exponential form only. A family of self-similar solutions is constructed in order to determine the evolution of the fracture width and length, the fluid velocity in the fracture, and the length of fluid penetration into the porous medium when either the fluid flow rate or the pressure as a power-law or exponential function of time is specified at the fracture entrance. In the case of finite fluid penetration into the soil the system of equations has only a power-law self-similar solution, for example, when the fluid flow rate is specified at the fracture entrance as a quadratic function of time. The solutions of the self-similar equations are found numerically for one of the seepage regimes.     

Water Injection into a Low-Permeability Rock - 2: Control Model

In Part 1, we have demonstrated the inevitable growth of the fluid injection hydrofractures in low-permeability rocks. Thus, a smart controller that manages fluid injection in the presence of hydrofracture extension is highly desirable. Such a controller will be an essential part of automated waterflood project surveillance and control. Here we design an optimal injection controller using methods of optimal control theory. The controller inputs are the history of the injection pressure and the cumulative injection, along with the fracture size. The output parameter is the injection pressure and the control objective is the injection rate. We demonstrate that the optimal injection pressure depends not only on the instantaneous measurements, but it is determined by the whole history of the injection and of the fracture area growth. We show the controller robustness when the inputs are delayed and noisy and when the fracture undergoes abrupt extensions. Finally, we propose a procedure that allows estimation of the hydrofracture size at no additional cost.     

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.     

Darcy Flow in a Wavy Channel Filled with a Porous Medium

Flow in channels bounded by wavy or corrugated walls is of interest in both technological and geological contexts. This paper presents an analytical solution for the steady Darcy flow of an incompressible fluid through a homogeneous, isotropic porous medium filling a channel bounded by symmetric wavy walls. This packed channel may represent an idealized packed fracture, a situation which is of interest as a potential pathway for the leakage of carbon dioxide from a geological sequestration site. The channel walls change from parallel planes, to small amplitude sine waves, to large amplitude nonsinusoidal waves as certain parameters are increased. The direction of gravity is arbitrary. A plot of piezometric head against distance in the direction of mean flow changes from a straight line for parallel planes to a series of steeply sloping sections in the reaches of small aperture alternating with nearly constant sections in the large aperture bulges. Expressions are given for the stream function, specific discharge, piezometric head, and pressure.     

零重力条件下低温射流抑制大尺寸液氢储罐热分层的数值研究

液氢是一种常用的沸点低、易蒸发的空间低温推进剂. 空间微重力环境中浮力对流被极大减弱,当推进剂储罐壁面存在局部漏热时,储罐内部气液两相流体系会出现环绕漏热源的热分层现象,引起局部过热沸腾,导致储罐内部压力急剧增大,危害系统结构安全. 利用低温射流抑制储罐热分层现象是一种有效手段. 低温流体通过设置在储罐内部的射流喷嘴与储罐内部的流体混合,消减局部高温,实现温度的均匀化. 采用全充满的二维大尺寸储罐模型,对零重力条件下液氢储罐内局部漏热引起的热分层现象开展了数值模拟,主要分析了位于靠近储罐底部的漏热带以及出口衔接段漏热带漏热形成的局部热分层现象的抑制和消除,并研究了不同低温射流条件对于消除零重力条件下液氢储罐内部热分层效果的影响. 研究结果表明对于大尺寸储罐,当采用圆形射流喷嘴且低温射流条件相同时,射流喷嘴的位置对罐体内部热分层消除效果影响不是很明显. 当射流喷嘴位于储罐内部同一相对位置且入射流量相同时,圆形射流喷嘴因出流方向更集中,罐内流场演变更快,消除热分层比半球形射流喷嘴更有效.      

Fractured water injection wells: Pressure transient analysis

In this paper we study the pressure drop in a hydraulic fracture after shut-in of a water injection well. The pressure transient behavior depends on fracture closure, lateral stress, rock elasticity and fracture fluid leak-off. Under the assumption that horizontal cross-sections of a vertical fracture do not depend on the vertical variable, we formulate a mathematical model which allows for determination of both pore pressure and elastic rock displacements jointly with the fracture aperture and fracture fluid pressure. An analytical consideration is performed for the case of an ideal very long fracture with the same aperture along its full length. In the general case, fracture closure is analyzed numerically.