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

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

random walk in markovian environment we prove a quenched central limit theorem for random walks with bounded increments in a randomly evolving environment on z d we assume that the transition probabilities of the walk depend not too strongly on the environment and that the evolution of the environment is markovian with strong spatial and temporal mixing properties

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

on homogenization of time dependent random flows we study a diffusion with a random time dependent drift we prove the invariance principle when the spectral measure of the drift satisfies a certain integrability condition this result generalizes the results of 13 7

canonical rational equivalence of intersections of divisors one way to define an operation in intersection theory is to define a map on the group of algebraic cycles together with a map on the group of rational equivalences which commutes with the boundary operation assuming the maps commute with smooth pullback the extension of the operation to the setting of algebraic stacks is automatic the goal of section 2 of this paper is to present the operation of intersecting with a principal cartier divisor in this light we then show how this operation lets us obtain a rational equivalence which is fundamental to intersection theory a one dimensional family of cycles on an algebraic variety always admits a unique limiting cycle but a family of cycles over the punctured affine plane may yield different limiting cycles if one approaches the origin from different directions an important step in the historical development of intersection theory was realizing how to prove that any two such limiting cycles are rationally equivalent the results of section 2 yield as a corollary a new explicit formula for this rational equivalence another important rational equivalence in intersection theory is the one that is used to demonstrate commutativity of gysin maps associated to regularly embedded subschemes in section 3 we exhibit a two dimensional family of cycles such that the cycles we obtain from specializing in two different ways are precisely the ones we need to show to be rationally equivalent to obtain the commutativity result our explicit rational equivalence respects smooth pullback and hence the generalization to stacks is automatic this simplifies intersection theory on deligne mumford stacks as in 8 where construction of such a rational equivalence fills the most difficult section of that important paper since our rational equivalence arises by considering families of cycles on a larger total space we are able to deduce section 4 that the rational equivalence is invariant under a certain naturally arising group action the key observation is that we can manipulate the situation so that the group action extends to the total space this equivariance result is used but appears with mistaken proof in 2 where an important new tool of modern intersection theory the theory of virtual fundamental classes is developed acknowledgement the author would like to thank s bloch w fulton t graber and r pandharipande for helpful advice and the organizers and staff of the mittagleffler institute for hospitality during the 1996 97 program in algebraic geometry

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

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

cycle groups for artin stacks we construct an algebraic homology functor for artin stacks of finite type over a field and we develop intersection theoretic properties

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

central limit theorem for additive functionals of reversible markov processes and applications to simple exclusions we prove a functional central limit theorem for additive functionals of stationary reversible ergodic markov chains under virtually no assumptions other than the necessary ones we use these results to study the asymptotic behavior of a tagged particle in an infinite particle system performing simple excluded random walk

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

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

the intrinsic normal cone let x be an algebraic stack in the sense of deligne mumford we construct a purely 0 dimensional algebraic stack over x in the sense of artin the intrinsic normal cone frak c x the notion of perfect obstruction theory for x is introduced and it is shown how to construct given a perfect obstruction theory for x a pure dimensional virtual fundamental class in the chow group of x we then prove some properties of such classes both in the absolute and in the relative context via a deformation theory interpretation of obstruction theories we prove that several kinds of moduli spaces carry a natural obstruction theory and sometimes a perfect one

random walks in quenched i i d space time random environment are always a s diffusive we consider a general model of discrete time random walk x t on the lattice 1 in a random environment t x t x 1 with i i d components t x previous results on the a s validity of the central limit theorem for the quenched model required a small stochasticity condition in this paper we show that the result holds provided only that an obvious non degeneracy condition is met the proof is based on the analysis of a suitable generating function which allows to estimate l 2 norms by contour integrals

convection diffusion equation with space time ergodic random flow we prove the homogenization of convection diffusion in a time dependent ergodic incompressible random flow which has a bounded stream matrix and a constant mean drift we also prove two variational formulas for the effective diffusivity as a consequence we obtain both upper and lower bounds on the effective diffusivity

a degeneration formula of gw invariants this is the sequel to the paper 23 in this paper we construct the virtual moduli cycles of the degeneration of the moduli of stable morphisms constructed in 23 we also construct the virtual moduli cycles of the moduli of relative stable morphisms of a pair of a smooth divisor in a smooth variety based on these we prove a degeneration formula of the gromov witten invariants

the central limit theorem for markov chains started at a point the aim of this paper is to prove a central limit theorem and an invariance principle for an additive functional of an ergodic markov chain on a general state space with respect to the law of the chain started at a point no irreducibility assumption nor mixing conditions are imposed the only assumption bears on the growth of the l 2 norms of the ergodic sums for the function generating the additive functional which must be with the result holds almost surely with respect to the invariant probability of the chain

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

torus localization and wall crossing for cosection localized virtual cycles abstract the theory of virtual fundamental class defines important invariants such as the gromov witten and the donaldson thomas invariants it has been generalized to the cosection localized virtual cycle which has applications in seiberg witten fan jarvis ruan witten and other invariants in this paper we prove the formulas of virtual pullback torus localization and wall crossing for cosection localized virtual cycles

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