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GEM  International Journal on Geomathematics
Journal Prestige (SJR): 0.321 Citation Impact (citeScore): 1 Number of Followers: 1 Hybrid journal (It can contain Open Access articles) ISSN (Print) 18692672  ISSN (Online) 18692680 Published by SpringerVerlag [2626 journals] 
 Numerical aspects of hydromechanical coupling of fluidfilled fractures
using hybriddimensional element formulations and nonconformal meshes Abstract: Abstract In the field of porous and fractured media, subsurface flow provides insight into the characteristics of fluid storage and properties connected to underground matter and heat transport. Subsurface flow is precisely described by many diffusion based models in the literature. However, diffusionbased models lack to reproduce important hydromechanical coupling phenomena like inverse waterlevel fluctuations (Noordbergum effect). In theory, contemporary modeling approaches, such as direct numerical simulations (DNS) of surfacecoupled fluidsolid (fracture) interactions or coarsegrained continuum approaches like Biot’s theory, are capable of capturing such phenomena. Nevertheless, during modeling processes of fractures with high aspect ratios, DNS methods with the explicit discretization of the fluid domain fail, and coarsegrained continuum approaches require a nonlinear formulation for the fracture deformation since large deformation can be reached easily within fractures. Hence a hybriddimensional approach uses a parabolic velocity profile to avoid an explicit discretization of the fluid domain within the fracture. For fracture flow, the primary variable is the pressure field only, and the fracture domain is reduced by one dimension. The interaction between the fracture and the surrounding matrix domain, respectively, is realized by modified balance equations. The coupled system is numerically stiff when fluids are described with a low compressibility modulus. Two algorithms are proposed within this work, namely the weak coupling scheme, which uses an implicit staggerediterative algorithm to find the residual state and the strong coupling scheme which directly couples both domains by implementing interface elements. In the course of this work, a consistent implementation scheme for the coupling of hybriddimensional elements with a surrounding bulk matrix is proposed and validated and tested throughout different numerical experiments.
PubDate: 20190210
 Abstract: Abstract In the field of porous and fractured media, subsurface flow provides insight into the characteristics of fluid storage and properties connected to underground matter and heat transport. Subsurface flow is precisely described by many diffusion based models in the literature. However, diffusionbased models lack to reproduce important hydromechanical coupling phenomena like inverse waterlevel fluctuations (Noordbergum effect). In theory, contemporary modeling approaches, such as direct numerical simulations (DNS) of surfacecoupled fluidsolid (fracture) interactions or coarsegrained continuum approaches like Biot’s theory, are capable of capturing such phenomena. Nevertheless, during modeling processes of fractures with high aspect ratios, DNS methods with the explicit discretization of the fluid domain fail, and coarsegrained continuum approaches require a nonlinear formulation for the fracture deformation since large deformation can be reached easily within fractures. Hence a hybriddimensional approach uses a parabolic velocity profile to avoid an explicit discretization of the fluid domain within the fracture. For fracture flow, the primary variable is the pressure field only, and the fracture domain is reduced by one dimension. The interaction between the fracture and the surrounding matrix domain, respectively, is realized by modified balance equations. The coupled system is numerically stiff when fluids are described with a low compressibility modulus. Two algorithms are proposed within this work, namely the weak coupling scheme, which uses an implicit staggerediterative algorithm to find the residual state and the strong coupling scheme which directly couples both domains by implementing interface elements. In the course of this work, a consistent implementation scheme for the coupling of hybriddimensional elements with a surrounding bulk matrix is proposed and validated and tested throughout different numerical experiments.
 Comparative verification of discrete and smeared numerical approaches for
the simulation of hydraulic fracturing Abstract: Abstract The numerical treatment of propagating fractures as embedded discontinuities is a challenging task for which an analyst has to select a suitable numerical method from a range of options. Since their inception in the mid80s, smeared approaches for fracture simulation such as nonlocal damage, gradient damage or more lately phasefield modelling have steadily gained popularity. One of the appeals of a smeared implicit fracture representation, the ability to handle complex topologies with unknown crack paths in relatively coarse meshes as well as multiplecrack interaction and multiphysics, is a fundamental requirement for the numerical simulation of hydraulic fracturing in complex situations which is technically more difficult to achieve with many other methods. However, in hydraulic fracturing simulations, not only the prediction of the fracture path but also the computation of fracture width and propagation pressure (frac pressure) is crucial for reliable and meaningful applications of the simulation tool; how to determine some of these quantities in smeared representations is not immediately obvious. In this study, two of the most popular smeared approaches of recent, namely nonlocal damage and phasefield models, and an approach in which the solution space is locally enriched to capture a strong discontinuity combined with a cohesivezone model are verified against fundamental hydraulic fracture propagation problems in the toughnessdominated regime. The individual theoretical foundations of each approach are discussed and differences in the treatment of physical and numerical properties of the methods when applied to the same physical problems are highlighted through examples.
PubDate: 20190201
 Abstract: Abstract The numerical treatment of propagating fractures as embedded discontinuities is a challenging task for which an analyst has to select a suitable numerical method from a range of options. Since their inception in the mid80s, smeared approaches for fracture simulation such as nonlocal damage, gradient damage or more lately phasefield modelling have steadily gained popularity. One of the appeals of a smeared implicit fracture representation, the ability to handle complex topologies with unknown crack paths in relatively coarse meshes as well as multiplecrack interaction and multiphysics, is a fundamental requirement for the numerical simulation of hydraulic fracturing in complex situations which is technically more difficult to achieve with many other methods. However, in hydraulic fracturing simulations, not only the prediction of the fracture path but also the computation of fracture width and propagation pressure (frac pressure) is crucial for reliable and meaningful applications of the simulation tool; how to determine some of these quantities in smeared representations is not immediately obvious. In this study, two of the most popular smeared approaches of recent, namely nonlocal damage and phasefield models, and an approach in which the solution space is locally enriched to capture a strong discontinuity combined with a cohesivezone model are verified against fundamental hydraulic fracture propagation problems in the toughnessdominated regime. The individual theoretical foundations of each approach are discussed and differences in the treatment of physical and numerical properties of the methods when applied to the same physical problems are highlighted through examples.
 A Hybrid HighOrder method for passive transport in fractured porous media
 Abstract: Abstract In this work, we propose a model for the passive transport of a solute in a fractured porous medium, for which we develop a Hybrid HighOrder (HHO) space discretization. We consider, for the sake of simplicity, the case where the flow problem is fully decoupled from the transport problem. The novel transmission conditions in our model mimic at the discrete level the property that the advection terms do not contribute to the energy balance. This choice enables us to handle the case where the concentration of the solute jumps across the fracture. The HHO discretization hinges on a mixed formulation in the bulk region and on a primal formulation inside the fracture for the flow problem, and on a primal formulation both in the bulk region and inside the fracture for the transport problem. Relevant features of the method include the treatment of nonconforming discretizations of the fracture, as well as the support of arbitrary approximation orders on fairly general meshes.
PubDate: 20190201
 Abstract: Abstract In this work, we propose a model for the passive transport of a solute in a fractured porous medium, for which we develop a Hybrid HighOrder (HHO) space discretization. We consider, for the sake of simplicity, the case where the flow problem is fully decoupled from the transport problem. The novel transmission conditions in our model mimic at the discrete level the property that the advection terms do not contribute to the energy balance. This choice enables us to handle the case where the concentration of the solute jumps across the fracture. The HHO discretization hinges on a mixed formulation in the bulk region and on a primal formulation inside the fracture for the flow problem, and on a primal formulation both in the bulk region and inside the fracture for the transport problem. Relevant features of the method include the treatment of nonconforming discretizations of the fracture, as well as the support of arbitrary approximation orders on fairly general meshes.
 Flow and transport in fractured poroelastic media
 Abstract: Abstract We study flow and transport in fractured poroelastic media using Stokes flow in the fractures and the Biot model in the porous media. The Stokes–Biot model is coupled with an advection–diffusion equation for modeling transport of chemical species within the fluid. The continuity of flux on the fracturematrix interfaces is imposed via a Lagrange multiplier. The coupled system is discretized by a finite element method using Stokes elements, mixed Darcy elements, conforming displacement elements, and discontinuous Galerkin for transport. The stability and convergence of the coupled scheme are analyzed. Computational results verifying the theory as well as simulations of flow and transport in fractured poroelastic media are presented.
PubDate: 20190129
 Abstract: Abstract We study flow and transport in fractured poroelastic media using Stokes flow in the fractures and the Biot model in the porous media. The Stokes–Biot model is coupled with an advection–diffusion equation for modeling transport of chemical species within the fluid. The continuity of flux on the fracturematrix interfaces is imposed via a Lagrange multiplier. The coupled system is discretized by a finite element method using Stokes elements, mixed Darcy elements, conforming displacement elements, and discontinuous Galerkin for transport. The stability and convergence of the coupled scheme are analyzed. Computational results verifying the theory as well as simulations of flow and transport in fractured poroelastic media are presented.
 A cut finite element method for elliptic bulk problems with embedded
surfaces Abstract: Abstract We propose an unfitted finite element method for flow in fractured porous media. The coupling across the fracture uses a Nitsche type mortaring, allowing for an accurate representation of the jump in the normal component of the gradient of the discrete solution across the fracture. The flow field in the fracture is modelled simultaneously, using the average of traces of the bulk variables on the fractures. In particular the Laplace–Beltrami operator for the transport in the fracture is included using the average of the projection on the tangential plane of the fracture of the trace of the bulk gradient. Optimal order error estimates are proven under suitable regularity assumptions on the domain geometry. The extension to the case of bifurcating fractures is discussed. Finally the theory is illustrated by a series of numerical examples.
PubDate: 20190129
 Abstract: Abstract We propose an unfitted finite element method for flow in fractured porous media. The coupling across the fracture uses a Nitsche type mortaring, allowing for an accurate representation of the jump in the normal component of the gradient of the discrete solution across the fracture. The flow field in the fracture is modelled simultaneously, using the average of traces of the bulk variables on the fractures. In particular the Laplace–Beltrami operator for the transport in the fracture is included using the average of the projection on the tangential plane of the fracture of the trace of the bulk gradient. Optimal order error estimates are proven under suitable regularity assumptions on the domain geometry. The extension to the case of bifurcating fractures is discussed. Finally the theory is illustrated by a series of numerical examples.
 Convolutions on the sphere: commutation with differential operators
 Abstract: Abstract We generalize the definition of convolution of vectors and tensors on the 2sphere, and prove that it commutes with differential operators. Moreover, vectors and tensors that are normal/tangent to the spherical surface remain so after the convolution. These properties make the new filtering operation particularly useful to analyzing and modeling nonlinear dynamics in spherical systems, such as in geophysics, astrophysics, and in inertial confinement fusion applications. An essential tool we use is the theory of scalar, vector, and tensor spherical harmonics. We then show that our generalized filtering operation is equivalent to the (traditional) convolution of scalar fields of the Helmholtz decomposition of vectors and tensors.
PubDate: 20190122
 Abstract: Abstract We generalize the definition of convolution of vectors and tensors on the 2sphere, and prove that it commutes with differential operators. Moreover, vectors and tensors that are normal/tangent to the spherical surface remain so after the convolution. These properties make the new filtering operation particularly useful to analyzing and modeling nonlinear dynamics in spherical systems, such as in geophysics, astrophysics, and in inertial confinement fusion applications. An essential tool we use is the theory of scalar, vector, and tensor spherical harmonics. We then show that our generalized filtering operation is equivalent to the (traditional) convolution of scalar fields of the Helmholtz decomposition of vectors and tensors.
 Fast and robust flow simulations in discrete fracture networks with GPGPUs
 Abstract: Abstract A new approach for flow simulation in very complex discrete fracture networks based on PDEconstrained optimization has been recently proposed in Berrone et al. (SIAM J Sci Comput 35(2):B487–B510, 2013b; J Comput Phys 256:838–853, 2014) with the aim of improving robustness with respect to geometrical complexities. This is an essential issue, in particular for applications requiring simulations on geometries automatically generated like the ones used for uncertainty quantification analyses and hydromechanical simulations. In this paper, implementation of this approach in order to exploit Nvidia Compute Unified Device Architecture is discussed with the main focus to speed up the linear algebra operations required by the approach, being this task the most computational demanding part of the approach. Furthermore, two different approaches for linear algebra operations and two storage formats for sparse matrices are compared in terms of computational efficiency and memory constraints.
PubDate: 20190120
 Abstract: Abstract A new approach for flow simulation in very complex discrete fracture networks based on PDEconstrained optimization has been recently proposed in Berrone et al. (SIAM J Sci Comput 35(2):B487–B510, 2013b; J Comput Phys 256:838–853, 2014) with the aim of improving robustness with respect to geometrical complexities. This is an essential issue, in particular for applications requiring simulations on geometries automatically generated like the ones used for uncertainty quantification analyses and hydromechanical simulations. In this paper, implementation of this approach in order to exploit Nvidia Compute Unified Device Architecture is discussed with the main focus to speed up the linear algebra operations required by the approach, being this task the most computational demanding part of the approach. Furthermore, two different approaches for linear algebra operations and two storage formats for sparse matrices are compared in terms of computational efficiency and memory constraints.
 Twophase Discrete Fracture Matrix models with linear and nonlinear
transmission conditions Abstract: Abstract This work deals with twophase Discrete Fracture Matrix models coupling the twophase Darcy flow in the matrix domain to the twophase Darcy flow in the network of fractures represented as codimension one surfaces. Two classes of such hybriddimensional models are investigated either based on nonlinear or linear transmission conditions at the matrix–fracture interfaces. The linear transmission conditions include the cellcentred upwind approximation of the phase mobilities classically used in the porous media flow community as well as a basic extension of the continuous phase pressure model accounting for fractures acting as drains. The nonlinear transmission conditions at the matrix–fracture interfaces are based on the normal flux continuity equation for each phase using additional interface phase pressure unknowns. They are compared both in terms of accuracy and numerical efficiency to a reference equidimensional model for which the fractures are represented as fulldimensional subdomains. The discretization focuses on Finite Volume cellcentred TwoPoint Flux Approximation which is combined with a local nonlinear solver allowing to eliminate efficiently the additional matrix–fracture interfacial unknowns together with the nonlinear transmission conditions. 2D numerical experiments illustrate the better accuracy provided by the nonlinear transmission conditions compared to their linear approximations with a moderate computational overhead obtained thanks to the local nonlinear elimination at the matrix–fracture interfaces. The numerical section is complemented by a comparison of the reduced models on a 3D test case using the Vertex Approximate Gradient scheme.
PubDate: 20190120
 Abstract: Abstract This work deals with twophase Discrete Fracture Matrix models coupling the twophase Darcy flow in the matrix domain to the twophase Darcy flow in the network of fractures represented as codimension one surfaces. Two classes of such hybriddimensional models are investigated either based on nonlinear or linear transmission conditions at the matrix–fracture interfaces. The linear transmission conditions include the cellcentred upwind approximation of the phase mobilities classically used in the porous media flow community as well as a basic extension of the continuous phase pressure model accounting for fractures acting as drains. The nonlinear transmission conditions at the matrix–fracture interfaces are based on the normal flux continuity equation for each phase using additional interface phase pressure unknowns. They are compared both in terms of accuracy and numerical efficiency to a reference equidimensional model for which the fractures are represented as fulldimensional subdomains. The discretization focuses on Finite Volume cellcentred TwoPoint Flux Approximation which is combined with a local nonlinear solver allowing to eliminate efficiently the additional matrix–fracture interfacial unknowns together with the nonlinear transmission conditions. 2D numerical experiments illustrate the better accuracy provided by the nonlinear transmission conditions compared to their linear approximations with a moderate computational overhead obtained thanks to the local nonlinear elimination at the matrix–fracture interfaces. The numerical section is complemented by a comparison of the reduced models on a 3D test case using the Vertex Approximate Gradient scheme.
 A hybriddimensional discrete fracture model for nonisothermal twophase
flow in fractured porous media Abstract: Abstract We present a hybriddimensional numerical model for nonisothermal twophase flow in fractured porous media, in which the fractures are modeled as entities of codimension one embedded in a bulk domain. Potential fields of applications of the model could be radioactive waste disposal or geothermal energy production scenarios in which a twophase flow regime develops or where \(\hbox {CO}_2\) is used as working fluid. We test the method on synthetic test cases involving compressible fluids and strongly heterogeneous, full tensor permeability fields by comparison with a reference solution obtained from an equidimensional discretization of the domain. The results reveal that especially for the case of a highly conductive fracture, the results are in good agreement with the reference. While the model qualitatively captures the involved phenomena also for the case of a fracture acting as both hydraulic and capillary barrier, it introduces larger errors than in the highlyconductive fracture case, which can be attributed to the lowerdimensional treatment of the fracture. Finally, we apply the method to a threedimensional showcase that resembles setups for the determination of upscaled parameters of fractured blocks.
PubDate: 20190120
 Abstract: Abstract We present a hybriddimensional numerical model for nonisothermal twophase flow in fractured porous media, in which the fractures are modeled as entities of codimension one embedded in a bulk domain. Potential fields of applications of the model could be radioactive waste disposal or geothermal energy production scenarios in which a twophase flow regime develops or where \(\hbox {CO}_2\) is used as working fluid. We test the method on synthetic test cases involving compressible fluids and strongly heterogeneous, full tensor permeability fields by comparison with a reference solution obtained from an equidimensional discretization of the domain. The results reveal that especially for the case of a highly conductive fracture, the results are in good agreement with the reference. While the model qualitatively captures the involved phenomena also for the case of a fracture acting as both hydraulic and capillary barrier, it introduces larger errors than in the highlyconductive fracture case, which can be attributed to the lowerdimensional treatment of the fracture. Finally, we apply the method to a threedimensional showcase that resembles setups for the determination of upscaled parameters of fractured blocks.
 Artificial intelligence for determining systematic effects of laser
scanners Abstract: Abstract Artificial intelligence is interpreted by a machine learning algorithm. Its realization is applied for a twodimensional grid of points and depends on six parameters which determine the limits of loops. The outer loop defines the width of the grid, the most inner loop the number of scans, which result from the threedimensional coordinate system for the \(x_i\) , \(y_i\) , \(z_i\) coordinates of a laser scanner. The \(y_i\) coordinates approximate the distances measured by the laser scanner. The minimal standard deviations of the measurements distorted by systematic effects for the \(y_i\) coordinates are computed by the Monte Carlo estimate of Sect. 6. The minimum of these minimal standard deviations is found in the grid of points by the machine learning algorithm and used to judge the outcome. Two results are given in Sect. 7. They differ by the widths of the grid and show that only for precise applications the systematic effects of the laser scanner have to be taken care of. Instead of assuming a standard deviation for the systematic effects from prior information as mentioned in Sect. 1, the \(x_i\) , \(y_i\) , \(z_i\) coordinates are repeatedly measured by the laser scanner. However, there are too few repetitions to fulfill the conditions of the multivariate model of Sect. 2 for all measured coordinates. The variances of the measurements plus systematic effects computed by the Monte Carlo estimate of Sect. 6 can therefore be obtained for a restricted number of points only. This number is computed by random variates. For two numbers, the variations of the standard deviations of the \(y_i\) coordinates, the variations of the standard deviations of the \(x_i\) , \(y_i\) , \(z_i\) coordinates from the multivariate model, the variations of the standard deviations of the systematic effects and the variations of the confidence intervals are presented. The repeated measurements define time series whose auto and crosscorrelation functions are applied as correlations for the systematic effects of the measurements. The ergodicity of the time series is shown.
PubDate: 20190120
 Abstract: Abstract Artificial intelligence is interpreted by a machine learning algorithm. Its realization is applied for a twodimensional grid of points and depends on six parameters which determine the limits of loops. The outer loop defines the width of the grid, the most inner loop the number of scans, which result from the threedimensional coordinate system for the \(x_i\) , \(y_i\) , \(z_i\) coordinates of a laser scanner. The \(y_i\) coordinates approximate the distances measured by the laser scanner. The minimal standard deviations of the measurements distorted by systematic effects for the \(y_i\) coordinates are computed by the Monte Carlo estimate of Sect. 6. The minimum of these minimal standard deviations is found in the grid of points by the machine learning algorithm and used to judge the outcome. Two results are given in Sect. 7. They differ by the widths of the grid and show that only for precise applications the systematic effects of the laser scanner have to be taken care of. Instead of assuming a standard deviation for the systematic effects from prior information as mentioned in Sect. 1, the \(x_i\) , \(y_i\) , \(z_i\) coordinates are repeatedly measured by the laser scanner. However, there are too few repetitions to fulfill the conditions of the multivariate model of Sect. 2 for all measured coordinates. The variances of the measurements plus systematic effects computed by the Monte Carlo estimate of Sect. 6 can therefore be obtained for a restricted number of points only. This number is computed by random variates. For two numbers, the variations of the standard deviations of the \(y_i\) coordinates, the variations of the standard deviations of the \(x_i\) , \(y_i\) , \(z_i\) coordinates from the multivariate model, the variations of the standard deviations of the systematic effects and the variations of the confidence intervals are presented. The repeated measurements define time series whose auto and crosscorrelation functions are applied as correlations for the systematic effects of the measurements. The ergodicity of the time series is shown.
 Phasefield modeling through iterative splitting of hydraulic fractures in
a poroelastic medium Abstract: Abstract We study the propagation of hydraulic fractures using the fixed stress splitting method. The phase field approach is applied and we study the mechanics step involving displacement and phase field unknowns, with a given pressure. We present a detailed derivation of an incremental formulation of the phase field model for a hydraulic fracture in a poroelastic medium. The mathematical model represents a linear elasticity system with fading elastic moduli as the crack grows that is coupled with an elliptic variational inequality for the phase field variable. The convex constraint of the variational inequality assures the irreversibility and entropy compatibility of the crack formation. We establish existence of a minimizer of an energy functional of an incremental problem and convergence of a finite dimensional approximation. Moreover, we prove that the fracture remains small in the third direction in comparison to the first two principal directions. Computational results of benchmark problems are provided that demonstrate the effectiveness of this approach in treating fracture propagation. Another novelty is the treatment of the mechanics equation with mixed boundary conditions of Dirichlet and Neumann types. We finally notice that the corresponding pressure step was studied by the authors in Mikelić et al. (SIAM Multiscale Model Simul 13(1):367–398, 2015a).
PubDate: 20190120
 Abstract: Abstract We study the propagation of hydraulic fractures using the fixed stress splitting method. The phase field approach is applied and we study the mechanics step involving displacement and phase field unknowns, with a given pressure. We present a detailed derivation of an incremental formulation of the phase field model for a hydraulic fracture in a poroelastic medium. The mathematical model represents a linear elasticity system with fading elastic moduli as the crack grows that is coupled with an elliptic variational inequality for the phase field variable. The convex constraint of the variational inequality assures the irreversibility and entropy compatibility of the crack formation. We establish existence of a minimizer of an energy functional of an incremental problem and convergence of a finite dimensional approximation. Moreover, we prove that the fracture remains small in the third direction in comparison to the first two principal directions. Computational results of benchmark problems are provided that demonstrate the effectiveness of this approach in treating fracture propagation. Another novelty is the treatment of the mechanics equation with mixed boundary conditions of Dirichlet and Neumann types. We finally notice that the corresponding pressure step was studied by the authors in Mikelić et al. (SIAM Multiscale Model Simul 13(1):367–398, 2015a).
 A stabilized Lagrange multiplier finiteelement method for flow in porous
media with fractures Abstract: Abstract In this work we introduce a stabilized, numerical method for a multidimensional, discretefracture model (DFM) for singlephase Darcy flow in fractured porous media. In the model, introduced in an earlier work, flow in the \((n1)\) dimensional fracture domain is coupled with that in the ndimensional bulk or matrix domain by the use of Lagrange multipliers. Thus the model permits a finite element discretization in which the meshes in the fracture and matrix domains are independent so that irregular meshing and in particular the generation of small elements can be avoided. In this paper we introduce in the numerical formulation, which is a saddlepoint problem based on a primal, variational formulation for flow in the matrix domain and in the fracture system, a weakly consistent stabilizing term which penalizes discontinuities in the Lagrange multipliers. For this penalized scheme we show stability and prove convergence. With numerical experiments we analyze the performance of the method for various choices of the penalization parameter and compare with other numerical DFM’s.
PubDate: 20190120
 Abstract: Abstract In this work we introduce a stabilized, numerical method for a multidimensional, discretefracture model (DFM) for singlephase Darcy flow in fractured porous media. In the model, introduced in an earlier work, flow in the \((n1)\) dimensional fracture domain is coupled with that in the ndimensional bulk or matrix domain by the use of Lagrange multipliers. Thus the model permits a finite element discretization in which the meshes in the fracture and matrix domains are independent so that irregular meshing and in particular the generation of small elements can be avoided. In this paper we introduce in the numerical formulation, which is a saddlepoint problem based on a primal, variational formulation for flow in the matrix domain and in the fracture system, a weakly consistent stabilizing term which penalizes discontinuities in the Lagrange multipliers. For this penalized scheme we show stability and prove convergence. With numerical experiments we analyze the performance of the method for various choices of the penalization parameter and compare with other numerical DFM’s.
 Translation matrix elements for spherical Gauss–Laguerre basis
functions Abstract: Abstract Spherical Gauss–Laguerre (SGL) basis functions, i.e., normalized functions of the type \(L_{nl1}^{(l + 1/2)}(r^2) r^{l} Y_{lm}(\vartheta ,\varphi ), m \le l < n \in {\mathbb {N}}\) , constitute an orthonormal polynomial basis of the space \(L^{2}\) on \({\mathbb {R}}^{3}\) with radial Gaussian weight \(\exp (r^{2})\) . We have recently described reliable fast Fourier transforms for the SGL basis functions. The main application of the SGL basis functions and our fast algorithms is in solving certain threedimensional rigid matching problems, where the center is prioritized over the periphery. For this purpose, socalled SGL translation matrix elements are required, which describe the spectral behavior of the SGL basis functions under translations. In this paper, we derive a closedform expression of these translation matrix elements, allowing for a direct computation of these quantities in practice.
PubDate: 20190120
 Abstract: Abstract Spherical Gauss–Laguerre (SGL) basis functions, i.e., normalized functions of the type \(L_{nl1}^{(l + 1/2)}(r^2) r^{l} Y_{lm}(\vartheta ,\varphi ), m \le l < n \in {\mathbb {N}}\) , constitute an orthonormal polynomial basis of the space \(L^{2}\) on \({\mathbb {R}}^{3}\) with radial Gaussian weight \(\exp (r^{2})\) . We have recently described reliable fast Fourier transforms for the SGL basis functions. The main application of the SGL basis functions and our fast algorithms is in solving certain threedimensional rigid matching problems, where the center is prioritized over the periphery. For this purpose, socalled SGL translation matrix elements are required, which describe the spectral behavior of the SGL basis functions under translations. In this paper, we derive a closedform expression of these translation matrix elements, allowing for a direct computation of these quantities in practice.
 Mathematical analysis, finite element approximation and numerical solvers
for the interaction of 3D reservoirs with 1D wells Abstract: Abstract We develop a mathematical model for the interaction of a threedimensional reservoir with the flow through wells, namely narrow cylindrical channels cutting across the reservoir. Leak off or sink effects are taken into account. To enable the simulation of complex configurations featuring multiple wells, we apply a model reduction technique that represents the wells as onedimensional channels. The challenge in this case is to account for the interaction of the reservoir with the embedded onedimensional wells. The resulting problem consists of coupled partial differential equations defined on manifolds with heterogeneous dimensionality. The existence and regularity of weak solutions of such problem is thoroughly addressed. Afterwards, we focus on the numerical discretization of the problem in the framework of the finite element method. We notice that the numerical scheme does not require conformity between the computational mesh of the reservoir and the one of the wells. From the standpoint of the solvers, we discuss the application of multilevel algorithms, such as the algebraic multigrid method. Finally, the reduced mathematical model and the discretization method is applied to a few configurations of reservoir with wells, with the purpose of verifying the theoretical properties and to assess the generality of the approach.
PubDate: 20190120
 Abstract: Abstract We develop a mathematical model for the interaction of a threedimensional reservoir with the flow through wells, namely narrow cylindrical channels cutting across the reservoir. Leak off or sink effects are taken into account. To enable the simulation of complex configurations featuring multiple wells, we apply a model reduction technique that represents the wells as onedimensional channels. The challenge in this case is to account for the interaction of the reservoir with the embedded onedimensional wells. The resulting problem consists of coupled partial differential equations defined on manifolds with heterogeneous dimensionality. The existence and regularity of weak solutions of such problem is thoroughly addressed. Afterwards, we focus on the numerical discretization of the problem in the framework of the finite element method. We notice that the numerical scheme does not require conformity between the computational mesh of the reservoir and the one of the wells. From the standpoint of the solvers, we discuss the application of multilevel algorithms, such as the algebraic multigrid method. Finally, the reduced mathematical model and the discretization method is applied to a few configurations of reservoir with wells, with the purpose of verifying the theoretical properties and to assess the generality of the approach.
 Simulation of settlement and bearing capacity of shallow foundations with
soft particle code (SPARC) and FE Abstract: Abstract In this study we investigate the development of shear zones due to the settlement of shallow foundations and their loadsettlement behavior. Firstly, a welldocumented experiment of shallow penetration into sand is used for the validation of the soft particle code (SPARC). For these simulations a hypoplastic material model for sand with calibration for the model sand is implemented in SPARC. In order to deliver a more comprehensive investigation, the shape of the shear zones predicted by SPARC is also compared with the analytical solution. Secondly, the penetration of shallow foundation into clay is investigated by means of SPARC and the finite element method. For this purpose, barodesy for clay with the calibration for Dresden clay is implemented in both numerical methods. The simulations are carried out for six different surcharges, corresponding to a range of overconsolidated clay to normalconsolidated clay. Furthermore, the loadsettlement behavior and the shape of shear zones for both methods are compared and the weaknesses and strengths of each numerical approach are discussed. Finally, the peaks of the loadsettlement curves for all surcharges are compared with the analytical solution. Results show that SPARC performs better at predicting the trajectories of particles under the foundation, which consequently leads to better estimation of the loadsettlement behavior.
PubDate: 20181101
 Abstract: Abstract In this study we investigate the development of shear zones due to the settlement of shallow foundations and their loadsettlement behavior. Firstly, a welldocumented experiment of shallow penetration into sand is used for the validation of the soft particle code (SPARC). For these simulations a hypoplastic material model for sand with calibration for the model sand is implemented in SPARC. In order to deliver a more comprehensive investigation, the shape of the shear zones predicted by SPARC is also compared with the analytical solution. Secondly, the penetration of shallow foundation into clay is investigated by means of SPARC and the finite element method. For this purpose, barodesy for clay with the calibration for Dresden clay is implemented in both numerical methods. The simulations are carried out for six different surcharges, corresponding to a range of overconsolidated clay to normalconsolidated clay. Furthermore, the loadsettlement behavior and the shape of shear zones for both methods are compared and the weaknesses and strengths of each numerical approach are discussed. Finally, the peaks of the loadsettlement curves for all surcharges are compared with the analytical solution. Results show that SPARC performs better at predicting the trajectories of particles under the foundation, which consequently leads to better estimation of the loadsettlement behavior.
 Advanced computation of steadystate fluid flow in Discrete
FractureMatrix models: FEM–BEM and VEM–VEM fractureblock coupling Abstract: Abstract In this note the issue of fluid flow computation in a Discrete FractureMatrix (DFM) model is addressed. In such a model, a network of percolative fractures delimits porous matrix blocks. Two frameworks are proposed for the coupling between the two media. First, a FEM–BEM technique is considered, in which finite elements on nonconforming grids are used on the fractures, whereas a boundary element method is used on the blocks; the coupling is pursued by a PDEconstrained optimization formulation of the problem. Second, a VEM–VEM technique is considered, in which a 2D and a 3D virtual element method are used on the fractures and on the blocks, respectively, taking advantage of the flexibility of VEM in using arbitrary meshes in order to ease the meshing process and the consequent enforcement of the matching conditions on fractures and blocks.
PubDate: 20181101
 Abstract: Abstract In this note the issue of fluid flow computation in a Discrete FractureMatrix (DFM) model is addressed. In such a model, a network of percolative fractures delimits porous matrix blocks. Two frameworks are proposed for the coupling between the two media. First, a FEM–BEM technique is considered, in which finite elements on nonconforming grids are used on the fractures, whereas a boundary element method is used on the blocks; the coupling is pursued by a PDEconstrained optimization formulation of the problem. Second, a VEM–VEM technique is considered, in which a 2D and a 3D virtual element method are used on the fractures and on the blocks, respectively, taking advantage of the flexibility of VEM in using arbitrary meshes in order to ease the meshing process and the consequent enforcement of the matching conditions on fractures and blocks.
 Algorithm of microseismic source localization based on asymptotic
probability distribution of phase difference between two random stationary
Gaussian processes Abstract: Abstract The article is devoted to the problem of estimating the coordinates of microseismic sources (localizing the sources) using multichannel data recorded by a surface seismic array. A new statistical algorithm is proposed for the source localization, which is mainly based on the phases of the Discrete Finite Fourier Transforms of the array sensor seismograms. This algorithm was constructed using the Maximum Likelihood concept under the following constraints: (a) noise components of the array seismograms are statistically independent stationary Gaussian processes with different power spectral densities; (b) the signaltonoise ratios in the array seismograms are small, but the duration of signals generated by a microseismic source in the sensors is quite large; (c) the time function of a microseismic source can be approximated by a stationary Gaussian random process. The asymptotic probability density function was obtained in the paper for the phase differences of two Gaussian stationary random time series. This function provided a theoretical basis for constructing the new statistical phase algorithm. The algorithm requires evaluation of the coherence functions for all pairs of the sensor seismograms. For this reason, it inquires more calculations for the source localization than the known phase algorithms. But Monte Carlo simulation has shown that the new phase algorithm provides a more accurate estimation of microseismic source coordinates compared to the most popular phase algorithm.
PubDate: 20181101
 Abstract: Abstract The article is devoted to the problem of estimating the coordinates of microseismic sources (localizing the sources) using multichannel data recorded by a surface seismic array. A new statistical algorithm is proposed for the source localization, which is mainly based on the phases of the Discrete Finite Fourier Transforms of the array sensor seismograms. This algorithm was constructed using the Maximum Likelihood concept under the following constraints: (a) noise components of the array seismograms are statistically independent stationary Gaussian processes with different power spectral densities; (b) the signaltonoise ratios in the array seismograms are small, but the duration of signals generated by a microseismic source in the sensors is quite large; (c) the time function of a microseismic source can be approximated by a stationary Gaussian random process. The asymptotic probability density function was obtained in the paper for the phase differences of two Gaussian stationary random time series. This function provided a theoretical basis for constructing the new statistical phase algorithm. The algorithm requires evaluation of the coherence functions for all pairs of the sensor seismograms. For this reason, it inquires more calculations for the source localization than the known phase algorithms. But Monte Carlo simulation has shown that the new phase algorithm provides a more accurate estimation of microseismic source coordinates compared to the most popular phase algorithm.
 Inverse gravimetry: background material and multiscale mollifier
approaches Abstract: Abstract This paper represents an extended version of the publications Freeden (in: Freeden, Nashed, Sonar (eds) Handbook of Geomathematics, 2nd edn, vol 1, Springer, New York, pp 3–78, 2015) and Freeden and Nashed (in: Freeden, Nashed (eds) Handbook of Mathematical Geodesy, Geosystems Mathematics, Birkhäuser, Basel, pp 641–685, 2018c). It deals with the illposed problem of transferring input gravitational potential information in the form of Newtonian volume integral values to geological output characteristics of the density contrast function. Some essential properties of the Newton volume integral are recapitulated. Different methodologies of the resolution of the inverse gravimetry problem and their numerical implementations are examined including their dependence on the data source. Three types of input information may be distinguished, namely internal (borehole), terrestrial (surface), and/or external (spaceborne) gravitational data sets. Singular integral theory based inversion of the Newtonian integral equation such as a Haartype solution is handled in a multiscale framework to decorrelate specific geological signal signatures with respect to inherently given features. Reproducing kernel Hilbert space regularization techniques are studied (together with their transition to certain mollified variants) to provide geological contrast density distributions by “downward continuation” from terrestrial and/or spaceborne data. Numerically, reproducing kernel Hilbert space spline solutions are formulated in terms of Gaussian approximating sums for use of gravimeter data systems.
PubDate: 20181101
 Abstract: Abstract This paper represents an extended version of the publications Freeden (in: Freeden, Nashed, Sonar (eds) Handbook of Geomathematics, 2nd edn, vol 1, Springer, New York, pp 3–78, 2015) and Freeden and Nashed (in: Freeden, Nashed (eds) Handbook of Mathematical Geodesy, Geosystems Mathematics, Birkhäuser, Basel, pp 641–685, 2018c). It deals with the illposed problem of transferring input gravitational potential information in the form of Newtonian volume integral values to geological output characteristics of the density contrast function. Some essential properties of the Newton volume integral are recapitulated. Different methodologies of the resolution of the inverse gravimetry problem and their numerical implementations are examined including their dependence on the data source. Three types of input information may be distinguished, namely internal (borehole), terrestrial (surface), and/or external (spaceborne) gravitational data sets. Singular integral theory based inversion of the Newtonian integral equation such as a Haartype solution is handled in a multiscale framework to decorrelate specific geological signal signatures with respect to inherently given features. Reproducing kernel Hilbert space regularization techniques are studied (together with their transition to certain mollified variants) to provide geological contrast density distributions by “downward continuation” from terrestrial and/or spaceborne data. Numerically, reproducing kernel Hilbert space spline solutions are formulated in terms of Gaussian approximating sums for use of gravimeter data systems.
 A greedy algorithm for nonlinear inverse problems with an application to
nonlinear inverse gravimetry Abstract: Abstract Based on the Regularized Functional Matching Pursuit (RFMP) algorithm for linear inverse problems, we present an analogous iterative greedy algorithm for nonlinear inverse problems, called RFMP_NL. In comparison to established methods for nonlinear inverse problems, the algorithm is able to combine very diverse types of basis functions, for example, localized and global functions. This is important, in particular, in geoscientific applications, where global structures have to be distinguished from local anomalies. Furthermore, in contrast to other methods, the algorithm does not require the solution of large linear systems. We apply the RFMP_NL to the nonlinear inverse problem of gravimetry, where gravitational data are inverted for the shape of the surface or inner layer boundaries of planetary bodies. This inverse problem is described by a nonlinear integral operator, for which we additionally provide the Fréchet derivative. Finally, we present two synthetic numerical examples to show that it is beneficial to apply the presented method to inverse gravimetric problems.
PubDate: 20181101
 Abstract: Abstract Based on the Regularized Functional Matching Pursuit (RFMP) algorithm for linear inverse problems, we present an analogous iterative greedy algorithm for nonlinear inverse problems, called RFMP_NL. In comparison to established methods for nonlinear inverse problems, the algorithm is able to combine very diverse types of basis functions, for example, localized and global functions. This is important, in particular, in geoscientific applications, where global structures have to be distinguished from local anomalies. Furthermore, in contrast to other methods, the algorithm does not require the solution of large linear systems. We apply the RFMP_NL to the nonlinear inverse problem of gravimetry, where gravitational data are inverted for the shape of the surface or inner layer boundaries of planetary bodies. This inverse problem is described by a nonlinear integral operator, for which we additionally provide the Fréchet derivative. Finally, we present two synthetic numerical examples to show that it is beneficial to apply the presented method to inverse gravimetric problems.
 Describing the singular behaviour of parabolic equations on cones in
fractional Sobolev spaces Abstract: Abstract In this paper, the Dirichlet problem for parabolic equations in a wedge is considered. In particular, we study the smoothness of the solutions in the fractional Sobolev scale \(H^s\) , \(s\in \mathbb {R}\) . The regularity in these spaces is related with the approximation order that can be achieved by numerical schemes based on uniform grid refinements. Our results provide a first attempt to generalize the wellknown \(H^{3/2}\) Theorem of Jerison and Kenig (J Funct Anal 130:161–219, 1995) to parabolic PDEs. As a special case the heat equation on radialsymmetric cones is investigated.
PubDate: 20181101
 Abstract: Abstract In this paper, the Dirichlet problem for parabolic equations in a wedge is considered. In particular, we study the smoothness of the solutions in the fractional Sobolev scale \(H^s\) , \(s\in \mathbb {R}\) . The regularity in these spaces is related with the approximation order that can be achieved by numerical schemes based on uniform grid refinements. Our results provide a first attempt to generalize the wellknown \(H^{3/2}\) Theorem of Jerison and Kenig (J Funct Anal 130:161–219, 1995) to parabolic PDEs. As a special case the heat equation on radialsymmetric cones is investigated.