Author
Lihua Zuo
Other affiliations: Texas A&M University–Texarkana, Texas A&M University–Kingsville
Bio: Lihua Zuo is an academic researcher from Texas A&M University. The author has contributed to research in topics: Fracture (geology) & Oil shale. The author has an hindex of 17, co-authored 37 publications receiving 803 citations. Previous affiliations of Lihua Zuo include Texas A&M University–Texarkana & Texas A&M University–Kingsville.
Papers
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TL;DR: In this article, a theoretical evaluation model of coal brittleness has been established based on the statistical damage theory and the energy evolution law of rock failure process, the calculated results were in good agreement with the experimental results, which proved that the theoretical analysis method of coal BRittleness proposed in this paper can be used to analyze the applicability of coal seam fracturing.
102 citations
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TL;DR: In this paper, the authors considered a reaction diffusion problem with an unknown nonlinear source function that has to be determined from overposed data and derived a uniqueness result and a numerical algorithm with some theoretical qualification.
Abstract: In this paper, we consider a reaction‐diffusion problem with an unknown nonlinear source function that has to be determined from overposed data. The underlyingmodelisintheformofatime-fractionalreaction‐diffusionequation and the work generalizes some known results for the inverse problems posed for PDEs of parabolic type. For the inverse problem under consideration, a uniqueness result is proved and a numerical algorithm with some theoretical qualification is presented in the one-dimensional case. The key both to the uniqueness result and to the numerical algorithm relies on the maximum principle which has recently been shown to hold for the fractional diffusion equation. In order to show the effectiveness of the proposed method, results of numerical simulations are presented. (Some figures may appear in colour only in the online journal)
83 citations
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TL;DR: In this article, a 3D displacement discontinuity method (3D DDM) is introduced to model multiple fractures in a stage in 3D. The fracture geometry is prescribed, which combined with vertical and horizontal fractures.
72 citations
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TL;DR: In this article, a coupled 3D hydraulic-fracture-propagation model considering the effects of weak bedding planes (BPs) is introduced. But the model is limited to estimating the fracture-height containment or predicting the fracture geometry under the influence of multiple BPs.
Abstract:
Weak bedding planes (BPs) that exist in many tight oil formations and shale–gas formations might strongly affect fracture–height growth during hydraulic–fracturing treatment. Few of the hydraulic–fracture–propagation models developed for unconventional reservoirs are capable of quantitatively estimating the fracture–height containment or predicting the fracture geometry under the influence of multiple BPs. In this paper, we introduce a coupled 3D hydraulic–fracture–propagation model considering the effects of BPs. In this model, a fully 3D displacement–discontinuity method (3D DDM) is used to model the rock deformation. The advantage of this approach is that it addresses both the mechanical interaction between hydraulic fractures and weak BPs in 3D space and the physical mechanism of slippage along weak BPs. Fluid flow governed by a finite–difference methodology considers the flow in both vertical fractures and opening BPs. An iterative algorithm is used to couple fluid flow and rock deformation. Comparison between the developed model and the Perkins–Kern–Nordgren (PKN) model showed good agreement.
I–shaped fracture geometry and crossing–shaped fracture geometry were analyzed in this paper. From numerical investigations, we found that BPs cannot be opened if the difference between overburden stress and minimum horizontal stress is large and only shear displacements exist along the BPs, which damage the planes and thus greatly amplify their hydraulic conductivity. Moreover, sensitivity studies investigate the impact on fracture propagation of parameters such as pumping rate (PR), fluid viscosity, and Young's modulus (YM). We investigated the fracture width near the junction between a vertical fracture and the BPs, the latter including the tensile opening of BPs and shear–displacement discontinuities (SDDs) along them. SDDs along BPs increase at the beginning and then decrease at a distance from the junction. The width near the junctions, the opening of BPs, and SDDs along the planes are directly proportional to PR. Because viscosity increases, the width at a junction increases as do the SDDs. YM greatly influences the opening of BPs at a junction and the SDDs along the BPs. This model estimates the fracture–width distribution and the SDDs along the BPs near junctions between the fracture tip and BPs and enables the assessment of the PR required to ensure that the fracture width at junctions and along intersected BPs is sufficient for proppant transport.
71 citations
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01 Aug 2016TL;DR: The URTeC Technical Program Committee accepted this presentation on the basis of information contained in an abstract submitted by the author(s), despite the fact that the contents of this paper have not been reviewed byURTeC.
Abstract: The URTeC Technical Program Committee accepted this presentation on the basis of information contained in an abstract submitted by the author(s). The contents of this paper have not been reviewed by URTeC and URTeC does not warrant the accuracy, reliability, or timeliness of any information herein. All information is the responsibility of, and, is subject to corrections by the author(s). Any person or entity that relies on any information obtained from this paper does so at their own risk. The information herein does not necessarily reflect any position of URTeC. Any reproduction, distribution, or storage of any part of this paper without the written consent of URTeC is prohibited.
69 citations
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11 Jun 2010
Abstract: The validity of the cubic law for laminar flow of fluids through open fractures consisting of parallel planar plates has been established by others over a wide range of conditions with apertures ranging down to a minimum of 0.2 µm. The law may be given in simplified form by Q/Δh = C(2b)3, where Q is the flow rate, Δh is the difference in hydraulic head, C is a constant that depends on the flow geometry and fluid properties, and 2b is the fracture aperture. The validity of this law for flow in a closed fracture where the surfaces are in contact and the aperture is being decreased under stress has been investigated at room temperature by using homogeneous samples of granite, basalt, and marble. Tension fractures were artificially induced, and the laboratory setup used radial as well as straight flow geometries. Apertures ranged from 250 down to 4µm, which was the minimum size that could be attained under a normal stress of 20 MPa. The cubic law was found to be valid whether the fracture surfaces were held open or were being closed under stress, and the results are not dependent on rock type. Permeability was uniquely defined by fracture aperture and was independent of the stress history used in these investigations. The effects of deviations from the ideal parallel plate concept only cause an apparent reduction in flow and may be incorporated into the cubic law by replacing C by C/ƒ. The factor ƒ varied from 1.04 to 1.65 in these investigations. The model of a fracture that is being closed under normal stress is visualized as being controlled by the strength of the asperities that are in contact. These contact areas are able to withstand significant stresses while maintaining space for fluids to continue to flow as the fracture aperture decreases. The controlling factor is the magnitude of the aperture, and since flow depends on (2b)3, a slight change in aperture evidently can easily dominate any other change in the geometry of the flow field. Thus one does not see any noticeable shift in the correlations of our experimental results in passing from a condition where the fracture surfaces were held open to one where the surfaces were being closed under stress.
1,557 citations
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TL;DR: In this article, the degree of ill-posedness of fractional diffusion inverse problems was examined using the two-parameter Mittag-Leffler function and singular value decomposition.
Abstract: Over the last two decades, anomalous diffusion processes in which the mean squares variance grows slower or faster than that in a Gaussian process have found many applications. At a macroscopic level, these processes are adequately described by fractional differential equations, which involves fractional derivatives in time or/and space. The fractional derivatives describe either history mechanism or long range interactions of particle motions at a microscopic level. The new physics can change dramatically the behavior of the forward problems. For example, the solution operator of the time fractional diffusion diffusion equation has only limited smoothing property, whereas the solution for the space fractional diffusion equation may contain weak singularity. Naturally one expects that the new physics will impact related inverse problems in terms of uniqueness, stability, and degree of ill-posedness. The last aspect is especially important from a practical point of view, i.e., stably reconstructing the quantities of interest. In this paper, we employ a formal analytic and numerical way, especially the two-parameter Mittag-Leffler function and singular value decomposition, to examine the degree of ill-posedness of several 'classical' inverse problems for fractional differential equations involving a Djrbashian–Caputo fractional derivative in either time or space, which represent the fractional analogues of that for classical integral order differential equations. We discuss four inverse problems, i.e., backward fractional diffusion, sideways problem, inverse source problem and inverse potential problem for time fractional diffusion, and inverse Sturm–Liouville problem, Cauchy problem, backward fractional diffusion and sideways problem for space fractional diffusion. It is found that contrary to the wide belief, the influence of anomalous diffusion on the degree of ill-posedness is not definitive: it can either significantly improve or worsen the conditioning of related inverse problems, depending crucially on the specific type of given data and quantity of interest. Further, the study exhibits distinct new features of 'fractional' inverse problems, and a partial list of surprising observations is given below. (a) Classical backward diffusion is exponentially ill-posed, whereas time fractional backward diffusion is only mildly ill-posed in the sense of norms on the domain and range spaces. However, this does not imply that the latter always allows a more effective reconstruction. (b) Theoretically, the time fractional sideways problem is severely ill-posed like its classical counterpart, but numerically can be nearly well-posed. (c) The classical Sturm–Liouville problem requires two pieces of spectral data to uniquely determine a general potential, but in the fractional case, one single Dirichlet spectrum may suffice. (d) The space fractional sideways problem can be far more or far less ill-posed than the classical counterpart, depending on the location of the lateral Cauchy data. In many cases, the precise mechanism of these surprising observations is unclear, and awaits further analytical and numerical exploration, which requires new mathematical tools and ingenuities. Further, our findings indicate fractional diffusion inverse problems also provide an excellent case study in the differences between theoretical ill-conditioning involving domain and range norms and the numerical analysis of a finite-dimensional reconstruction procedure. Throughout we will also describe known analytical and numerical results in the literature.
226 citations
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01 Jan 2009TL;DR: The heat equation also shows what a PDE is in general: we are given some kind of relationship between a function u(x,t) and/or its partial derivatives as discussed by the authors.
Abstract: The heat equation also shows what a PDE is in general: We are given some kind of relationship between a function u(x,t) and/or its partial derivatives. In the one dimensional case this can be partial derivatives in time t or in space x. The problem is to find functions u(x,t) which satisfy the given relationship and sometimes satisfy additional conditions (initital value conditions or boundary conditions).
190 citations
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TL;DR: In this paper, a general framework for the rigorous numerical analysis of time-fractional nonlinear parabolic partial differential equations, with a fractional derivative of order α ∆ in(0, 1)$ in time, is presented.
Abstract: We present a general framework for the rigorous numerical analysis of time-fractional nonlinear parabolic partial differential equations, with a fractional derivative of order $\alpha\in(0,1)$ in time It relies on three technical tools: a fractional version of the discrete Gronwall type inequality, discrete maximal regularity, and regularity theory of nonlinear equations We establish a general criterion for showing the fractional discrete Gronwall inequality and verify it for the L1 scheme and convolution quadrature generated by backward difference formulas Further, we provide a complete solution theory, eg, existence, uniqueness, and regularity, for a time-fractional diffusion equation with a Lipschitz nonlinear source term Together with the known results of discrete maximal regularity, we derive pointwise $L^2(\Omega)$ norm error estimates for semidiscrete Galerkin finite element solutions and fully discrete solutions, which are of order $O(h^2)$ (up to a logarithmic factor) and $O(\tau^\alpha)$,
161 citations