fully implicit mixed hybrid finite element discretization for general purpose subsurface reservoir simulation abstract we present a new fully implicit mixed hybrid finite element mhfe discretization scheme for general purpose compositional reservoir simulation the locally conservative scheme solves the coupled momentum and mass balance equations simultaneously and the fluid system is modeled using a cubic equation of state we introduce a new conservative flux approach for the mass balance equations for this fully implicit approach we discuss the nonlinear solution procedure for the proposed approach and we present extensive numerical tests to demonstrate the convergence and accuracy of the mhfe method using tetrahedral elements we also compare the method to other advanced discretization schemes for unstructured meshes and tensor permeability finally we illustrate the applicability and robustness of the method for highly heterogeneous reservoirs with unstructured grids

trust region based solver for nonlinear transport in heterogeneous porous media we describe a new nonlinear solver for immiscible two phase transport in porous media where viscous buoyancy and capillary forces are significant the flux fractional flow function f is a nonlinear function of saturation and typically has inflection points and can be non monotonic the non convexity and non monotonicity of f are major sources of difficulty for nonlinear solvers of coupled multiphase flow and transport in natural porous media we describe a modified newton algorithm that employs trust regions of the flux function to guide the newton iterations the flux function is divided into saturation trust regions delineated by the inflection unit flux and end points the updates are performed such that two successive iterations cannot cross any trust region boundary if a crossing is detected the saturation value is chopped back to the appropriate trust region boundary the proposed trust region newton solver which is demonstrated across the parameter space of viscous buoyancy and capillary effects is a significant extension of the inflection point strategy of jenny et al jcp 2009 5 for viscous dominated flows we analyze the discrete nonlinear transport equation obtained using finite volume discretization with phase based upstream weighting then we prove convergence of the trust region newton method irrespective of the timestep size for single cell problems numerical results across the full range of the parameter space of viscous gravity and capillary forces indicate that our trust region scheme is unconditionally convergent for 1d transport that is for a given choice of timestep size the unique discrete solution is found independently of the initial guess for problems dominated by buoyancy and capillarity the trust region newton solver overcomes the often severe limits on timestep size associated with existing methods to validate the effectiveness of the new nonlinear solver for large reservoir models with strong heterogeneity we compare it with state of the art nonlinear solvers for two phase flow and transport using the spe 10 model compared with existing nonlinear solvers our trust region solver results in superior convergence performance and achieves reduction in the total newton iterations by more than an order of magnitude together with a corresponding reduction in the overall computational cost

on the nonexistence of entire solutions of certain type of nonlinear differential equations abstract by utilizing nevanlinna s value distribution theory of meromorphic functions it is shown that the following type of nonlinear differential equations f n z p n 3 f p 1 e 1 z p 2 e 2 z has no nonconstant entire solutions where n is an integer 4 p 1 and p 2 are two polynomials 0 1 2 are two nonzero constants with 1 2 rational number and p n 3 f denotes a differential polynomial in f and its derivatives with polynomials in z as the coefficients of degree no greater than n 3 it is conjectured that the conclusion remains to be valid when p n 3 f is replaced by p n 1 f or p n 2 f

implicit hybrid upwinding for two phase flow in heterogeneous porous media with buoyancy and capillarity abstract we consider the numerical solution of the partial differential equations governing multiphase flow in porous media for highly nonlinear problems the temporal discretization of choice is often the unconditionally stable fully implicit method however the nonlinear systems often solved with newton s method are difficult to solve thus the computational cost is strongly dependent on the nonlinear convergence rate and enhancing this convergence property is key to speed up subsurface flow simulation we focus on the case of spatially discontinuous capillary pressure between rock regions to efficiently and accurately simulate the flow dynamics in heterogeneous porous media the flux computation combines implicit hybrid upwinding with transmission conditions between different rock regions this leads to a scheme that correctly represents the trapping mechanisms while improving the nonlinear convergence we extend our previous results hamon et al 2016 18 by generalizing the scheme to fully implicit coupled flow and transport to address realistic problems in multiple dimensions the generalized scheme is supported by an analysis of its mathematical properties our multidimensional numerical examples which range from buoyancy driven flow with capillary barriers to viscous dominated flow demonstrate that the implicit hybrid upwinding scheme improves the accuracy compared to the standard phase based upwinding scheme while leading to significant reductions in the number of nonlinear iterations in multiple dimensions

monotonicity considerations for saturated unsaturated subsurface flow it is demonstrated that monotonicity is a sufficient condition to ensure that no new nonphysical local maxima and minima can be produced in the discrete nonlinear unsaturated flow equation monotonicity conditions are derived for various types of weighting for the mobility term the basic discretization is of finite element type but the results can be extended to finite volume discretizations central weightings are only conditionally monotone while upstream weightings are unconditionally monotone sample computations are given for highly heterogeneous problems

entire functions that share a small function with its derivative abstract in this paper by studying the properties of meromorphic functions which have few zeros and poles we find all the entire functions f z which share a small and finite order meromorphic function a z with its derivative and f n z a z 0 whenever f z a z 0 n 2 this result is a generalization of several previous results

monotone difference approximations of bv solutions to degenerate convection diffusion equations we consider consistent conservative form monotone difference schemes for nonlinear convection diffusion equations in one space dimension since we allow the diffusion term to be strongly degenerate solutions can be discontinuous and in general are not uniquely determined by their data here we choose to work with weak solutions that belong to the bv in space and time class and in addition satisfy an entropy condition a recent result of wu and yin northeastern math j 5 1989 pp 395 422 states that these so called bv entropy weak solutions are unique the class of equations under consideration is very large and contains to mention only a few the heat equation the porous medium equation the two phase flow equation and hyperbolic conservation laws the difference schemes are shown to converge to the unique bv entropy weak solution of the problem in view of the classical theory for monotone difference approximations of conservation laws the main difficulty in obtaining a similar convergence theory in the present context is to show that the strongly degenerate discrete diffusion term is sufficiently smooth we provide the necessary regularity estimates by deriving and carefully analyzing a linear difference equation satisfied by the numerical flux of the difference schemes finally we make some concluding remarks about monotone difference schemes for multidimensional equations

iterative algorithms for solving the matrix equation axb cx td e in this paper we propose two iterative algorithms to solve the matrix equation axb cx td e the first algorithm is applied when the matrix equation is consistent in this case for any special initial matrix x 1 a solution the minimal frobenius norm solution can be obtained within finite iteration steps in the absence of roundoff errors the second algorithm is applied when the matrix equation is inconsistent in this case for any special initial matrix x 1 a least squares solution the minimal frobenius norm least squares solution can be obtained within finite iteration steps in the absence of roundoff errors some examples verify the efficiency of these algorithms

positive definite solutions of the nonlinear matrix equation x a h x 1 a i this paper is concerned with the positive definite solutions to the nonlinear matrix equation x a email protected 1a i where x is the unknown and a is a given complex matrix by introducing and studying a matrix operator on complex matrices it is shown that the existence of positive definite solutions of this class of nonlinear matrix equations is equivalent to the existence of positive definite solutions of the nonlinear matrix equation w b tw 1b i which has been extensively studied in the literature where b is a real matrix and is uniquely determined by a it is also shown that if the considered nonlinear matrix equation has a positive definite solution then it has the maximal and minimal solutions bounds of the positive definite solutions are also established in terms of matrix a finally some sufficient conditions and necessary conditions for the existence of positive definite solutions of the equations are also proposed

the transcendental meromorphic solutions of a certain type of nonlinear differential equations abstract in this paper we study the differential equations of the following form w 2 r z w k 2 q z where r z q z are nonzero rational functions we proved the following three conclusions 1 if either p z or q z is a nonconstant polynomial or k is an even integer then the differential equation w 2 p z 2 w k 2 q z has no transcendental meromorphic solution if p z q z are constants and k is an odd integer then the differential equation has only transcendental meromorphic solutions of the form f z a cos b z c 2 if either p z or q z is a nonconstant polynomial or k 1 then the differential equation w 2 z z 0 p z 2 w k 2 q z has no transcendental meromorphic solution furthermore the differential equation w 2 a z z 0 w 2 b where a b are nonzero constants has only transcendental meromorphic solutions of the form f z a cos b z z 0 where a b are constants such that a b 2 1 a 2 b 3 if the differential equation w 2 1 p z 2 w k 2 q z where p is a nonconstant polynomial and q is a nonzero rational function has a transcendental meromorphic solution then k is an odd integer and q is a polynomial furthermore if k 1 then q z c constant and the solution is of the form f z b cos q z where b is a constant such that b 2 c and q z p z

on the sylvester equation ax x b c abstract we consider the solution of the sylvester equations ax x b c for t h and a b c n n and the related linear matrix equations axb x c axb cx d e and ax x a c solvability conditions and numerical methods are considered in terms of the generalized and periodic schur and qr decompositions we emphasize the square cases where m n but the rectangular cases will be considered

convergence of implicit monotone schemes with applications in multiphase flow in porous media phase based upstreaming which is a commonly used spatial discretization for multiphase flow in reservoir simulation naturally gives rise to implicit monotone schemes when implicit time stepping is used the nonlinear gauss seidel and jacobi algorithms are shown to converge to a unique bounded solution when applied to the resulting system of equations thus for one dimensional problems we obtain an alternate constructive proof that such schemes are well defined and converge to the entropy solution of the conservation law for arbitrary cfl numbers the accuracy of phase based upstream solutions is studied for various spatial and temporal grids revealing the importance of unconditional stability when nonuniform grids and or variable porosity is involved

potential based reduced newton algorithm for nonlinear multiphase flow in porous media we present a phase based potential ordering that is an extension of the cascade ordering introduced by appleyard and cheshire john r appleyard ian m cheshire the cascade method for accelerated convergence in implicit simulators in european petroleum conference 1982 pp 113 122 the proposed ordering is valid for both two phase and three phase flow and it can handle countercurrent flow due to gravity and or capillarity we show how this ordering can be used to reduce the nonlinear algebraic system that arises from the fully implicit method fim into one with only pressure dependence the potential based reduced newton algorithm is then obtained by applying newton s method to this reduced order system numerical evidence shows that our potential based reduced newton solver is able to converge for time steps that are much larger than what the standard newton s method can handle in addition whenever standard newton converges so does the reduced newton algorithm and the number of global nonlinear iterations required for convergence is significantly reduced compared with the standard newton s method

simulation of incompressible two phase flow in porous media with large timesteps abstract multiphase flow in porous media occurs in several disciplines including petroleum reservoir engineering petroleum systems analysis and co 2 sequestration while simulations often use a fully implicit discretization to increase the time step size restrictions on the time step often exist due to non convergence of the nonlinear solver e g newton s method here this problem is addressed for the buckley leverett equations which model incompressible immiscible two phase flow with no capillary potential the equations are recast as a gradient flow using the phase field method and a convex energy splitting scheme is applied to enable large timesteps even for high degrees of heterogeneity in permeability and viscosity by using the phase field formulation as a homotopy map the underlying hyperbolic flow equations can be solved with large timesteps for a heterogeneous test problem the new homotopy method allows the timestep to be increased by more than six orders of magnitude relative to the unmodified equations while maintaining convergence

vibration of fast trains palindromic eigenvalue problems and structure preserving doubling algorithms the vibration of fast trains is governed by a quadratic palindromic eigenvalue problem l 2a 1 email protected 0 a 1 x 0 where a 0 a email protected c n x n and a 0 t a 0 accurate and efficient solution can only be obtained using algorithms which preserve the structure of the eigenvalue problem this paper reports on the successful application of the structure preserving doubling algorithms

unconditionally convergent nonlinear solver for hyperbolic conservation laws with s shaped flux functions this paper addresses the convergence properties of implicit numerical solution algorithms for nonlinear hyperbolic transport problems it is shown that the newton raphson nr method converges for any time step size if the flux function is convex concave or linear which is in general the case for cfd problems in some problems e g multiphase flow in porous media the nonlinear flux function is s shaped not uniformly convex or concave as a result a standard nr iteration can diverge for large time steps even if an implicit discretization scheme is used to solve the nonlinear system of equations in practice when such convergence difficulties are encountered the current time step is cut previous iterations are discarded a smaller time step size is tried and the nr process is repeated the criteria for time step cutting and selection are usually based on heuristics that limit the allowable change in the solution over a time step and or nr iteration here we propose a simple modification to the nr iteration scheme for conservation laws with s shaped flux functions that converges for any time step size the new scheme allows one to choose the time step size based on accuracy consideration only without worrying about the convergence behavior of the nonlinear solver the proposed method can be implemented in an existing simulator e g for co 2 sequestration or reservoir flow modeling quite easily the numerical analysis is confirmed with simulation studies using various test cases of nonlinear multiphase transport in porous media the analysis and numerical experiments demonstrate that the modified scheme allows for the use of arbitrarily large time steps for this class of problems

an extended hamiltonian algorithm for the general linear matrix equation abstract a second order learning algorithm based on differential geometry is used to numerically solve the linear matrix equation q x i 1 m a i t x a i i 1 n b i t x b i an extended hamiltonian algorithm is proposed based on the manifold of symmetric positive definite matrices the algorithm is compared with traditional coupled fixed point algorithm numerical experiments illustrate that the convergence speed of the provided algorithm is faster than that of the coupled fixed point algorithm

nonlinear analysis of multiphase transport in porous media in the presence of viscous buoyancy and capillary forces nonlinear convergence problems in numerical reservoir simulation can lead to unacceptably large computational time and are often the main impediment to performing simulation studies of large scale problems we analyze the nonlinearity of the discrete transport mass conservation equation for immiscible incompressible two phase flow in porous media in the presence of viscous buoyancy and capillary forces although simulation problems are multi dimensional with large numbers of cells and variables we find that the essence of the nonlinear behavior can be understood by studying the discretized numerical flux function for the interface between two cells the numerical flux is expressed in terms of the saturations of the two cells discontinuities in the first order derivative of the flux function referred to as kinks and inflection lines are identified as the cause of convergence difficulty these critical features kinks and inflections change the curvature of the numerical flux function abruptly and can lead to overshoots oscillations or divergence in newton iterations based on our understanding of the nonlinearity a nonlinear solver is developed referred to as the numerical trust region ntr solver the solver is able to guide the newton iterations safely and efficiently through the different saturation trust regions delineated by the kinks and inflections specifically overshoots and oscillations that often lead to convergence failure are avoided numerical examples demonstrate that our ntr solver has superior convergence performance compared with existing methods in particular convergence is achieved for a wide range of timestep sizes and courant friedrichs lewy cfl numbers spanning several orders of magnitude in addition a discretization scheme is proposed for handling heterogeneities in capillary pressure saturation relationship the scheme has less degree of nonlinearity compared with the standard single point phase based upstream weighting scheme leading to an improved nonlinear convergence performance especially when used together with our ntr solver our proposed numerical solution strategy that is based on the numerical flux and handles capillarity extends the previous work by jenny et al 2009 6 and wang and tchelepi 2013 7 significantly the nonlinearity of multiphase transport in porous media is analyzed the analysis considers viscous buoyancy and capillary forces the critical features that cause nonlinear convergence failures are identified a nonlinear solver that converges without oscillations and divergence is developed the solver converges faster than existing methods and allows large timesteps

a natural gradient descent algorithm for the solution of discrete algebraic lyapunov equations based on the geodesic distance a new framework to calculate the numerical solution of the discrete algebraic lyapunov equation is proposed by using the geometric structures on the riemannian manifold specifically two algorithms based on the manifold of positive definite symmetric matrices are provided one is a gradient descent algorithm with an objective function of the classical euclidean distance the other is a natural gradient descent algorithm with an objective function of the geodesic distance on the curved riemannian manifold furthermore these two algorithms are compared with a traditional iteration method simulation examples show that the convergence speed of the natural gradient descent algorithm is the fastest one among three algorithms

iterative algorithms for the minimum norm solution and the least squares solution of the linear matrix equations abstract in this paper two iterative algorithms are proposed to solve the linear matrix equations a 1 xb 1 c 1 x t d 1 m 1 a 2 xb 2 c 2 x t d 2 m 2 when the matrix equations are consistent by the first algorithm a solution x can be obtained within finite iterative steps in the absence of roundoff error for any initial value furthermore the minimum norm solution can be got by choosing a special kind of initial matrix additionally the unique optimal approximation solution to a given matrix x 0 can be derived by finding the minimum norm solution of a new matrix equations a 1 x b 1 c 1 x t d 1 m 1 a 2 x b 2 c 2 x t d 2 m 2 when the matrix equations are inconsistent we present the second algorithm to find the least squares solution with the minimum norm finally two numerical examples are tested by matlab the results show that these iterative algorithms are efficient

operator based linearization approach for modeling of multiphase multi component flow in porous media a new approach for the linearization of governing equations that describe flow and transport in porous media is proposed in this work it is based on an approximate representation of the exact physics of the problem which is similar to an approximate representation of space and time discretization performed in conventional simulation the governing equations are introduced as a combination of operators dependent on spatially altered properties and operators fully controlled by nonlinear properties of fluid and rock next a parametrization in the physics space of the problem is introduced the property based operators are approximated using direct interpolation in the space of nonlinear unknowns the discrete version of the governing equations is constructed as a combination of operators that approximate both nonlinear physics and discretization in time and space this approach is applied to the reservoir simulation of miscible and immiscible displacement processes the performance of the method demonstrates a convergence of simulation results by resolution in physical space with the improved performance

global fom and gmres algorithms for a class of complex matrix equations abstract in this paper a generalized global arnoldi process is given to produce an f orthonormal basis of complex matrix space then we propose the global full orthogonalization method gl fom and global generalized minimum residual gl gmres method for solving a class of complex matrix equations the new methods can be implemented by the original coefficient matrices by using schur complement we give the expressions of the approximate solutions and the corresponding residuals some convergence results of the proposed methods are also derived finally numerical examples are given to illustrate the effectiveness with comparison to other existing methods

entire solutions of certain type of differential equations ii abstract we analyze the transcendental entire solutions of the following type of nonlinear differential equations f n z p f p 1 e 1 z p 2 e 2 z in the complex plane where p 1 p 2 and 1 2 are nonzero constants and p f denotes a differential polynomial in f of degree at most n 1 with small functions of f as the coefficients

extended hamiltonian learning on riemannian manifolds theoretical aspects this paper introduces a general theory of extended hamiltonian second order learning on riemannian manifolds as an instance of learning by constrained criterion optimization the dynamical learning equations are derived within the general framework of extended hamiltonian stationary action principle and are expressed in a coordinate free fashion a theoretical analysis is carried out in order to compare the features of the dynamical learning theory with the features exhibited by the gradient based ones in particular gradient based learning is shown to be an instance of dynamical learning and the classical gradient based learning modified by a momentum term is shown to resemble discrete time dynamical learning moreover the convergence features of gradient based and dynamical learning are compared on a theoretical basis this paper discusses cases of learning by dynamical systems on manifolds of interest in the scientific literature namely the stiefel manifold the special orthogonal group the grassmann manifold the group of symmetric positive definite matrices the generalized flag manifold and the real symplectic group of matrices

upstream differencing for multiphase flow in reservoir simulation upstream weighting for multiphase flow in reservoir simulation is analyzed the associated numerical flux is shown to be well defined monotone lipschitz continuous and consistent in the case of a two phase flow the corresponding numerical flux is compared to that of godunov and engquist and osher finally simple way to obtain a higher order scheme is outlined

difference analogue of the lemma on the logarithmic derivative with applications to difference equations the lemma on the logarithmic derivative of a meromorphic function has many applications in the study of meromorphic functions and ordinary dierential equa tions in this paper a dierence analogue of the logarithmic derivative lemma is presented and then applied to prove a number of results on meromorphic solutions of complex dierence equations these results include a dierence analogue of the clunie lemma as well as other results on the value distribution of solutions

gradient based and least squares based iterative algorithms for matrix equations axb cxtd f abstract this paper develops a gradient based and a least squares based iterative algorithms for solving matrix equation axb cx t d f the basic idea is to decompose the matrix equation system under consideration into two subsystems by applying the hierarchical identification principle and to derive the iterative algorithms by extending the iterative methods for solving ax b and axb f the analysis shows that when the matrix equation has a unique solution under the sense of least squares the iterative solution converges to the exact solution for any initial values a numerical example verifies the proposed theorems

the solution of the matrix equations axb cxd e and ya dz yc bz e f the conditions for the existence of a unique solution of the matrix equation axb cxd e are proved to be that i the pencils a c and d b are regular and ii the spectra of the pencils have an empty intersection a numerical algorithm for solving the equation is proposed the possibility of a least squares type solution is briefly discussed the set of equations ya dz yc bz e f is proved to be equivalent to the aforementioned equation and its solution is also investigated a numerical algorithm is proposed

implicit hybrid upwind scheme for coupled multiphase flow and transport with buoyancy abstract numerical simulation of coupled multiphase flow and transport in porous media is used in many scientific and industrial applications such as groundwater management and oil recovery to solve the governing partial differential equations we focus on the fully implicit backward euler finite volume method fim in this numerical scheme large algebraic systems must be solved at each time step using newton s method this can be quite expensive especially for highly nonlinear problems with buoyancy in this work we present a new first order approximation of the flux in the fim leading to improved nonlinear convergence rate and radius this approximation is based on implicit hybrid upwinding and relies on separate treatments of the viscous and buoyancy parts of the numerical flux moreover in the viscous part the total velocity discretization is key as it controls the coupling between the elliptic flow problem and the hyperbolic multiphase transport problem to achieve robustness and efficiency we propose a differentiable total velocity discretization that adapts to the balance of forces at each interface we analyze the theoretical properties of the resulting numerical scheme and prove that saturations remain between physical bounds challenging two and three phase test cases illustrate that our numerical scheme brings a significant reduction in the number of nonlinear iterations compared to a finite volume discretization relying on the widely used phase per phase upstream weighting our numerical scheme therefore reduces the simulation cost while providing similar accuracy

analysis of hybrid upwinding for fully implicit simulation of three phase flow with gravity the systems of algebraic equations arising from implicit backward euler finite volume discretization of the conservation laws governing multiphase flow in porous media are quite challenging for nonlinear solvers in the presence of countercurrent flow due to buoyancy the numerical flux obtained with single point phase potential upwinding ppu is not differentiable which causes convergence difficulties for nonlinear solvers recently lee efendiev and tchelepi adv water resour 82 2015 pp 27 38 proposed a hybrid upwinding strategy for two phase flow that gives rise to a differentiable numerical flux across the entire viscous gravity parameter space here we first present an implicit hybrid upwinding ihu scheme for hyperbolic conservation laws extending the work of lee efendiev and tchelepi to an arbitrary number of fluid phases we show that the numerical flux obtained with the ihu is consistent and a monotone function of its own saturation it is also a differentiable function of t

meromorphic solutions of certain type of differential difference equations abstract we consider meromorphic solutions to general differential difference equation of type l z f j 1 q c j z f n j z j j 1 r j e j z where l z f 0 is a linear differential difference polynomial of f with small function coefficients and 1 n 1 n q j 1 j r are nonvanishing constants c j z 1 j q are nonvanishing small functions if the equation admits a transcendental meromorphic solution f of finite order satisfying that f f 1 f f then representation of f and relations between coefficients in z f and i j can be obtained

a numerical method for a generalized algebraic riccati equation in this paper we develop a numerical method for computing the semistabilizing solution of a generalized algebraic riccati equation gare the semistabilizing solution of such a gare has been used to characterize the solvability of the j j prime spectral factorization problem for general rational matrices which have poles and zeros on the extended imaginary axis the main difficulty for solving such a gare is that its associated skew hamiltonian hamiltonian pencil has eigenvalues on the extended imaginary axis consequently it is not clear which eigenspace of the associated skew hamiltonian hamiltonian pencil can characterize the desired semistabilizing solution i e it is not clear which eigenvectors and principal vectors corresponding to the eigenvalues on the extended imaginary axis should be contained in the eigenspace that we wish to compute and hence the well known generalized eigenvalue approach for the classical algebraic riccati equations cannot be directly employed for it our proposed method consists of computations of the eigendecomposition of the system pencil corresponding to the eigenvalues on the extended imaginary axis and the stable eigenspace of an augmented matrix pencil hence it is a generalization of the generalized eigenvalue approach for the classical algebraic riccati equations

on h2 model reduction of bilinear systems the h 2 model reduction problem for continuous time bilinear systems is studied in this paper by defining the h 2 norm of bilinear systems in terms of the state space matrices the h 2 model reduction error is computed via the reachability or observability gramian necessary conditions for the reduced order bilinear models to be h 2 optimal are given the gradient flow approach is used to obtain the solution of the h 2 model reduction problem the formulation allows certain properties of the original models to be preserved in the reduced order models the model reduction procedure developed can also be applied to finite dimensional linear time invariant systems a numerical example is employed to illustrate the effectiveness of the proposed method

uniqueness and value distribution of differences of entire functions abstract we consider the existence of transcendental entire solutions of certain type of non linear difference equations as an application we investigate the value distribution of difference polynomials of entire functions in particular we are interested in the existence of zeros of f n z f m z c f m z a where f is an entire function n m are two integers such that n m 0 and are non zero complex numbers we also obtain a uniqueness result in the case where shifts of two entire functions share a small function

mathematical structure of the black oil model for petroleum reservoir simulation this model incorporates compressibility and general mass transfer between phases it consists of the conditions of thermodynamic equilibrium an equation of state for the volume balance between the fluidand the rock void darcy s law for the volumetric flow rates and component conservation equations these relations are manipulated to form a pressure equation and a modified system of component conservation equations it is shown that the pressure equation is parabolic and that in the absence of diffusive forces such as capillary pressure and mixing the component conservation equations are hyperbolic subject to technical conditions on the relative permeabilities this sequential formulation of the flow equations forms types of wave structures that occur for the system

vertex approximate gradient discretization preserving positivity for two phase darcy flows in heterogeneous porous media abstract in this article a new nodal discretization is proposed for two phase darcy flows in heterogeneous porous media the scheme combines the vertex approximate gradient vag scheme for the approximation of the gradient fluxes with an hybrid upwind hu approximation of the mobility terms in the saturation equation the discretization in space incorporates naturally nodal interface degrees of freedom d o f allowing to capture the transmission conditions at the interface between different rock types for general capillary pressure curves it is shown to guarantee the physical bounds for the saturation unknowns as well as a nonnegative lower bound on the capillary energy flux term numerical experiments on several test cases exhibit that the scheme is more robust compared with previous approaches allowing the simulation of 3d large discrete fracture matrix dfm models

numerical approximation of a two phase flow problem in a porous medium with discontinuous capillary forces we consider a simplified model of a two phase flow through a heterogeneous porous medium focusing on the capillary forces motion a nonlinear degenerate parabolic problem is approximated in a domain shared in two homogeneous parts each of them being characterized by its relative permeability and capillary curves functions of the phase saturations we first give a weak form of the conservation equations on the whole domain with a new general expression of the conditions at the interface between the two regions we then propose a finite volume scheme for the approximation of the solution which is shown to converge to a weak solution in one two or three dimensional domains we conclude with some numerical tests