a note on the error analysis of time compression approximations the accuracy of the time compression analysis tca is analyzed by comparison with a numerical solution both the standard tca and a new modified tca are considered for a power law diffusivity and constant surface flux as expected the error of the approximations decreases with increasing power and the error of the modified tca is about half the error of the standard tca in a second part the errors of the two tcas are measured using a simple analytical solution instead of a numerical solution it is shown that the conclusions remain the same for the analytical and numerical solutions the advantage of using the analytical solution is to obtain simple analytical expressions showing the influence of parameters this is done to estimate the maximum error of both tcas a practical estimate of the errors can be obtained from equations which only require knowledge of the soil water diffusivity it appears that for real soils the errors of the tca are always 1 and thus are a very reliable tool for practical problems although not studied systematically it also appears that gravity effects reduce the errors of the tca so that the error obtained in the absence of gravity provides a conservative estimate when gravity is present

analytical solution for one dimensional drainage burgers and simplified forms solutions are developed for burgers equation and for the surface water content subject to one dimensional drainage these are found as limiting cases of richards equation using diffusivity and conductivity functions from fujita as extended by broadbridge and white the solution from burgers equation can generally be expressed as a function of only a dimensionless time depth and initial reduced water content for the more strongly nonlinear case initially the surface water content decreases in proportion to the square root of time for large times the surface water content approaches the residual water content inversely with the square root of time

scope for further analytical solutions for constant flux infiltration into a semi infinite soil profile or redistribution in a finite soil profile we attempt to generate new solutions for the moisture content form of the one dimensional richards 1931 equation using the lisle 1992 equivalence mapping this mapping is used as no more general set of transformations exists for mapping the one dimensional richards equation into itself starting from a given solution the mapping has the potential to generate an infinite number of new solutions for a series of nonlinear diffusivity and hydraulic conductivity functions we first seek new analytical solutions satisfying richards equation subject to a constant flux surface boundary condition for a semi infinite dry soil starting with the burgers model the first iteration produces an existing solution while subsequent iterations are shown to endlessly reproduce this same solution next we briefly consider the problem of redistribution in a finite length soil in this case lisle s equivalence mapping is generalized to account for arbitrary initial conditions as was the case for infiltration however it is found that new analytical solutions are not generated using the equivalence mapping although existing solutions are recovered

integrable forms of the one dimensional flow equation for unsaturated heterogeneous porous media the equation for the horizontal transport of a liquid in an unsaturated scale heterogeneous porous medium is t x x c x x e x x c e d a systematic search for lie backlund symmetries leads to the requirement that c a b 2 as in the homogeneous 1 case more generally e may be 1 mx 1 3 2 c or exp mx 3c 2 in these cases the transport equation may be linearized and solved exactly examples of more complicated heterogeneous extensions are presented for the integrable nonlinear diffusion equations and for burgers equation

reciprocal transformations of two component hyperbolic systems and their invariants abstract criteria for the equivalence under reciprocal transformations of two component hyperbolic systems are established in terms of reciprocal invariants these results are applied to a 1 1 dimensional isentropic gasdynamics system to determine reciprocal equivalence to the canonical system with a polytropic gas law the four parameter model pressure density relations for which this reciprocal equivalence holds are parameterized in terms of the lagrangian signal speed

analytical decomposition of the nonlinear unsaturated flow equation an analytical procedure for the solution of the nonlinear unsaturated flow equation is proposed the horizontal and vertical flow equations subject to constant boundary conditions and continuous wetting are approximated using analytical decomposition series the results are verified with the philip 1955 and the parlange 1971 classical solutions and with experimental data a convergence theorem with proof and discussion on the numerical accuracy of decomposition series are included applications to the modeling of infiltration subject to hysteresis in the soil water functional relationships soil heterogeneity and time variable point rainfall are also discussed other potential applications in water management models under more realistic conditions are possible the methodology offers advantages over the common small perturbation solutions the possibility to study nonstationarity in the random quantities statistical separability from the physics of the problem i e independence between the system parameters and the input quantities rather than forced through closure approximations and the construction of an analytical series solution that converges uniformly to the true nonlinear solution finally because of stability and computational economy with respect to linearized numerical solutions an analytical solution appears promising

lie symmetry classification analysis for nonlinear coupled diffusion the group properties of the 1 1 dimensional matrix diffusion equation of the form with respect to point symmetries are given it is shown that in particular cases the group properties of this equation are similar to those of the corresponding scalar equation namely the lie algebra is extended for power and exponential functions with additional extensions when powers of and 2 exist however a specific form of is shown to exist for cases admitting both infinite and finite groups the result obtained is used to construct new group invariant solutions for impulsive boundary inputs using two similarity variables a reduction is applied to the coupled diffusion of temperature and volumetric moisture content in porous media under periodic boundary conditions a perturbation method is employed to obtain an explicit solution for the case when can be expressed exponentially in terms of the similarity variables

null generalized helices in lorentz minkowski spaces we obtain a lancret type theorem for null generalized helices in lorentz minkowski spaces l n i nl 3 we find that the only null generalized helices are the ordinary null helices however in l 5 we have to consider two types of null generalized helices according to whether the axis is non null or null in both cases we obtain the solutions to the natural equations problem

the depth of invariant rings and cohomology abstract let g be a finite group acting linearly on a vector space v over a field k of positive characteristic p and let p g be a sylow p subgroup ellingsrud and skjelbred compositio math 41 1980 233 244 proved the lower bound depth k v g min dim v p 2 dim v for the depth of the invariant ring with equality if g is a cyclic p group let us call the pair g v flat if equality holds in the above in this paper we use cohomological methods to obtain information about the depth of invariant rings and in particular to study the question of flatness for g of order not divisible by p2 it ensues that g v is flat if and only if h1 g k v 0 or dim vp 2 dim v we obtain a formul a for the depth of the invariant ring in the case that g permutes a basis of v and has order not divisible by p2 in this situation g v is usually not flat moreover we introduce the notion of visible flatness of pairs g v and prove that this implies flatness for example the groups sl2 q so3 q su3 q sz q and r q with many interesting representations in defining characteristic are visibly flat in particular if g sl2 q and v is the space of binary forms of degree n or a direct sum of such spaces then g v is flat for all q pr with the exception of a finite number of primes p along the way we obtain results about the buchsbaum property of invariant rings and about the depth of the cohomology modules hi g k v we also determine the support of the positive cohomology h g k v as a module over k v g in the appendix the visibly flat pairs g v of a group with bn pair and an irreducible representation are classified

local geometric invariants of integrable evolution equations the integrable hierarchy of commuting vector fields for the localized induction equation of 3d hydrodynamics and its associated recursion operator are used to generate families of integrable evolution equations which preserve local geometric invariants of the evolving curve or swept out surface

an analytical solution to richards equation for time varying infiltration an analytical solution is developed for richards equation for time varying infiltration the infiltration rates must be specified as piecewise constants with time the answer is expressed as a sum of two terms the first is a function of the instantaneous infiltration and the second an integral which accounts for the water distribution within the profile for the previous infiltration history the solutions assume specific forms for the soil water diffusivity and hydraulic functions used earlier by broadbridge and white 1988

extension of the fujita solution to air and water movement in soils for specific soil properties the fundamental solution of fujita 1952b is extended to include the effect of air movement on horizontal absorption this exact solution is then used to determine the accuracy and range of applicability of two approximations to the sorptivity with air effects previously presented by parlange et al 1982 and sander and parlange 1984 the optimal sorptivity of parlange et al has very good accuracy while the surface water content s is near saturation but decreases in accuracy as s decreases the approximate sorptivity of sander and parlange however proves to be extremely accurate for all s with a maximum error of only 0 5 sander and parlange additionally give an approximation to the water profile which is also shown by comparison with the exact solution to have a maximum error of only 0 5 the aim of this paper is primarily of a theoretical nature in that we use the exact solution to check the accuracy of the approximate solutions as stated above alternatively the solution may be used for checking the accuracy of numerical models

second order accurate monotone finite volume scheme for richards equation in this work we perform a theoretical and numerical analysis of richards equation for certain types of nonlinearities we provide explicit analytical solutions these solutions are used to show that conventional unconditionally monotone finite volume schemes have only first order accuracy we derive necessary and sufficient conditions for the monotonicity of finite volume discretizations and use these conditions to construct a monotone finite volume discretization accurate to second order

reciprocal transformations in 2 1 dimensions reciprocal backlund transformations are introduced for nonlinear integro differential equations in 2 1 dimensions invariance under such transformations is investigated

time to ponding comparison of analytic quasi analytic and approximate predictions an analytic expression for time to ponding is introduced using the nonlinear model of broadbridge and white 1987 the hydraulic properties of this model can encompass properties ranging from those of a highly nonlinear green ampt like soil to those satisfying the weakly nonlinear burgers equation because of its versatility this analytic solution is used as a benchmark against which extant analytic quasi analytic and approximate expressions are compared time to ponding is parameterized here in terms of the readily measured field properties sorptivity and hydraulic conductivity in the limit of green ampt like properties the analytic solution reduces exactly to the parlange and smith 1976 approximation a similar functional dependence of time to ponding on rainfall rate is found from quasi analytic approximations based on this a modified approximation is suggested which should give time to ponding for most soils to within 10 some existing approximations are found to have unacceptable deviations from the analytic solution and their continued use appears unwarranted finally we address the field problem of predicting time to ponding at any antecedent water content given sorptivity measured at only one initial water content

exact solutions for water infiltration with an arbitrary surface flux or nonlinear solute adsorption several authors have now used the backlund transformation to linearize richard s equation and so obtain analytical solutions for infiltration and redistribution processes for the linearizable class of diffusivities and hydraulic conductivities after transformation a convection diffusion equation is obtained to date all the analytical results making use of the backlund transform have been based on a specified constant flux that is assumed to exist at the surface boundary we generalize the previous work by removing this restriction and obtain a general solution for an arbitrary temporal variation in surface flux the solution contains a function that is specified by a volterra integral equation in practice the volterra equation must be evaluated numerically this task is accomplished using a relatively straightforward algorithm the analytical results for infiltration can be applied to the process of solute transport subject to a nonlinear adsorption isotherm since these two phenomena are equivalent under another mapping our new analytical results therefore also apply to this class of solute transport processes the general form of the adsorption isotherm for which exact results can be obtained is derived this isotherm appears capable of representing a relatively large family of physically relevant cases although this topic remains open for future investigation

constant rate rainfall infiltration a versatile nonlinear model 1 analytic solution analytic solutions are presented for a nonlinear diffusion convection model describing constant rate rainfall infiltration in uniform soils and other porous materials the model is based on the darcy buckingham approach to unsaturated water flow and assumes simple functional forms for the soil water diffusivity d and hydraulic conductivity k which depend on a single free parameter c and readily measured soil hydraulic properties these d and k yield physically reasonable analytic moisture characteristics the relation between this model and other models which give analytic solutions is explored as c the model reduces to the weakly nonlinear burgers equation which has been applied in certain field situations at the other end of the range as c 1 the model approaches a green ampt like model a wide range of realistic soil hydraulic properties is encompassed by varying the c parameter the general features of the analytic solutions are illustrated for selected c values gradual and steep wetting profiles develop during rainfall aspects seen in the laboratory and field in addition the time dependent surface water content and surface water pressure potential are presented explicitly a simple traveling wave approximation is given which agrees closely with the exact solution at comparatively early infiltration times

infiltration models for semi infinite soil profiles 1 infiltration models for hydrological applications are developed for rational forms of the soil hydraulic conductivity and moisture retention functions the models are based on the traveling and kinematic wave solutions of the governing equation that proved to be highly accurate in specific flow conditions the approximate solutions describe the one dimensional infiltration into a semi infinite medium for uniform initial conditions under either a time dependent flux or a constant head boundary condition they capture the profile development from early to late time accurately due to the proper derivation of the wetting front speed in terms of the time varying surface boundary potential and flux rate algebraic expressions of theoretical and practical importance are derived in terms of well defined soil water parameters and include the effect of the antecedent moisture content in a specific fashion practical results include simple expressions for the surface moisture variation for variable rainfall patterns time to ponding expressions and infiltration equations that account explicitly for the initial condition formulas for the depth of the wetting zone and algebraic equations for parameter estimation of the soil hydraulic parameters by inverse analysis

on the depth of invariant rings of infinite groups abstract let k be an algebraically closed field for a finitely generated graded commutative k algebra r let cmdef r dim r depth r denote the cohen macaulay defect of r let g be a linear algebraic group over k that is reductive but not linearly reductive we show that there exists a faithful rational representation v of g which we will give explicitly such that cmdef k v k g k 2 for all k n

null w slant helices in e13 abstract in this paper we give the necessary and sufficient conditions for null curves in e 1 3 to be w slant helix in terms of their curvature functions mainly throughout this paper relationships between the null w slant helices and their pseudo spherical images are obtained furthermore some illustrative examples of the null w slant helices and their pseudo spherical indicatrices in e 1 3 are demonstrated

on the depth of the invariants of the symmetric power representations of sl2 fp abstract we study the depth of the ring of invariants of sl 2 f p acting on the n th symmetric power of the natural two dimensional representation for n p these symmetric power representations are the irreducible representations of sl 2 f p over f p we prove that when the greatest common divisor of p 1 and n is less than or equal to 2 the depth of the ring of invariants is 3 we also prove that the depth is 3 for n 3 p 7 and n 4 p 5 however for n 3 p 7 the depth is 4 and for n 4 p 5 the depth is 5 in these two exceptional cases the ring of invariants is cohen macaulay

inverse transformations and the reduction of nonlinear dirichlet problems involutory transformations are introduced which allow the solution of a class on nonlinear dirichlet problems the application of coupled involutory and backlund transformations is also described

on the construction of a functional solution method for the infiltration in porous media problems abstract in this paper we consider a transient vertical one dimensional flow problem of water in unsaturated porous media modelled by the non linear richards equation constitutive relations of van genuchten will be employed and pade approximants are used to represent the hydraulic capacity and conduction in a simplified fashion from the proposed methodology a construction of a functional solution is presented with the objective to define an initialisation of a recursive scheme in the spirit of adomian decomposition this solution is optimised and evaluated using the governing equation for a self consistency test the results are presented for some soil types and its related soil parameters that are reported in the literature

clean two and three dimensional analytical solutions of richards equation for testing numerical solvers 1 this technical note derives clean analytical solutions of richards equation for three dimensional unsaturated groundwater flow clean means that the boundary conditions and steady state solutions are closed form expressions and the transient solutions have relatively simple additional fourier series terms two dimensional versions of these solutions are also given the primary purpose for the solutions is to test linear and nonlinear solvers in finite difference volume element computer programs for accuracy and scalability using architectures ranging from pcs to parallel high performance computers this derivation starts from the quasi linear assumption of relative hydraulic conductivity varying exponentially with pressure head and the separate approximation that relative hydraulic conductivity varies linearly with moisture content this allows a transformation to be used to create a linear partial differential equation separation of variables and fourier series are then used to obtain the final solution physically reasonable material properties are also used a total of four solutions are given in this technical note steady state and transient solutions for two different boundary conditions of the sample problem

analytical model of infiltration under constant concentration boundary conditions 1 known integrable models for 1d flow in unsaturated soil have a rescaled soil water diffusivity that is either constant or proportional to c c 1 c 2 where is the degree of saturation and c 1 is constant with a wider more realistic range of hydraulic conductivity functions than has been used in this context before a formal series solution is developed for infiltration subject to constant concentration boundary conditions a readily programmed iteration algorithm applicable for any value of c is used to construct many coefficients of the infiltration series without requiring any numerical integration in particular for either c 1 small or 1 c small several infiltration series coefficients are constructed as formal power series in c 1 or in 1 c for which we construct a number of terms explicitly in the limit as the diffusivity approaches a delta function the infiltration coefficients are obtained in simpler closed form all but the sorptivity depend on the form of the conductivity function

series solution for richards equation under concentration boundary conditions and uniform initial conditions a series solution which expresses the moisture profiles and surface fluxes during infiltration and exfiltration is derived the solution applies for conditions of fixed surface moisture content and uniform initial moisture content within an unbounded homogeneous nonhysteretic soil column the series derivation is similar to j r philip s but with a transformed time variable the series converges for infiltration problems involving soils which exhibit steep increase in capillary pressure near saturation under these conditions the solution captures the transition from early square root of time behavior to the longtime traveling wave solution without the use of approximate joining techniques the soil water retention models for which convergence occurs include the widely applied brooks corey and van genuchten models with minor restrictions on the shape parameters of the latter other cases examined include infiltration and exfiltration assuming power law linear quadratic and constant dependence of diffusivity and conductivity on moisture content for these cases the convergence limits were finite the solution is particularly well suited for infiltration into sandy soils for which the transition between short time and longtime behavior occurs relatively early

application of linked backlund transformations to nonlinear boundary value problems a class of nonlinear boundary value problems is reduced to linear canonical form by a combination of backlund transformations

an eulerian lagrangian approach with an adaptively corrected method of characteristics to simulate variably saturated water flow a relatively simple method of characteristics is developed to simulat e one dimensional variably saturated water flow the method uses the eulerian lagrangian approach to separate the governing flow equation into convection and diffusion parts which are solved with the method of characteristics and the conventional finite element method respectively the method of characteristics combines a single step reverse particle tracking technique with a correction strategy to ensure accurate mass balances the correction process is implemented by weighing the calculated convective contribution to the pressure head at each node with the pressure head values of two upstream nodes using an adaptive weighing factor a the value of a is automatically adjusted by considering the global mass balance at each time step numerical experiments for ponded infiltration are presented to illustrate the scheme s performance for situations involving highly nonlinear soil hydraulic properties and extremely dry initial conditions results indicate that the proposed method is mass conservative virtually oscillation free and computationally quite efficient the method is especially effective for simulating highly nonlinear flow scenarios for which traditional finite difference and finite element numerical methods often fail to converge

parameter estimation for an infiltration problem abstract the burgers equation is considered as a mathematical model for nonhysteretic infiltration in nonswelling soil under appropriate physical conditions for this nonlinear partial differential equation the modal approximation scheme for estimating parameters such as initial water distribution and precipitation evaporation history is introduced the function space parameter estimation convergence property of this scheme is proved the idea is to change the nonlinear problem to a linear one by an appropriate change of variables also numerical simulations are performed

the quasilinear approximation for unsaturated porous media flow this paper gives a review of the quasilinear approximation an exponential relationship between hydraulic conductivity and moisture potential and the uses this has been put to since the time of its creation one of the major attractions of this approximation is that under steady regimes the highly nonlinear richards equation governing unsaturated flow can be linearized this simplification has enabled analytic and semianalytic solutions to be found for many unsaturated flow problems these are reviewed and the current research areas in which the quasilinear approximation is being used are discussed

infiltration models for soil profiles bounded by a water table 1 approximate solutions of richards equation are developed for rational forms of the soil hydraulic conductivity and moisture retention functions using a perturbation expansion method a novel approach is devised in the perturbation technique whereby a priori introduced parameters are estimated in an optimal context to ensure the accuracy of the perturbation solution at all times and for all flow conditions the perturbation model describes the one dimensional nonlinear nonsteady infiltration for arbitrary initial conditions under either a time dependent surface flux or a constant head boundary condition into either a semi infinite or a finite medium bounded from below by a shallow water table the optimal perturbation approach proved to be an effective method by which closed form linear solutions can be adapted to nonlinear soils through proper scaling factors the results show that the first order linear term of the optimal perturbation solution is sufficiently accurate for practical applications especially for shallow water table conditions they offer the means to analyze the effect of nonuniform initial conditions and the influence of the water table on the infiltration process practical results in terms of well defined soil water parameters and the antecedent moisture content include algebraic expressions for the surface moisture content variation under variable rainfall conditions time to ponding expressions and infiltration equations for deep and shallow water table conditions and a prediction formula for the water table recharge rate

exact solutions of the richards equation with nonlinear plant root extraction the richards equation commonly used to model water flow in unsaturated soils is highly nonlinear thus making it very challenging to solve analytically for situations meaningful in practical applications the inclusion of realistic forms of root water uptake rates in this equation adds complications in deriving exact solutions this study provides for the first time analytical solutions of the richards equation with a sink term nonlinearly dependent on soil water content these solutions are applied to irrigation furrows using cartesian coordinates and irrigation from a circular plate in cylindrical coordinates

group theoretic methods applied to burgers equation abstract in this study the group theoretic methods for calculating the solution of burgers equation with appropriate boundary and initial conditions is presented the application of a one parameter group reduces the number of independent variables by one and consequently the governing partial differential equation with the boundary and initial conditions to an ordinary differential equation with the appropriate corresponding conditions the obtained differential equation is solved analytically and the solution obtained in closed form for a specific choice of boundary condition

burgers equation a solution for soil water flow in a finite length burgers equation can be considered a special case of the one dimensional richards equation for vertical flow for which the soil water diffusivity is constant and the unsaturated hydraulic conductivity is proportional to the square of the reduced water content here a fourier series solution for burgers 1948 equation for a finite region is presented the solution is useful to modelers seeking a simplified representation of vertical water flow through the vadose zone and as an analytical solution for benchmarking which maintains many of the nonlinear characteristics of solutions to richards equation two example problems are presented in the first infiltration into a dry soil is modeled showing that burgers equation gives developing wetting front profiles similar to those observed for richards equation in the second example one step outflow from an 8 cm core is modeled comparisons between the analytical and finite difference solutions for burgers equation are presented to help verify the analytical solutions

finite volume integrated surface subsurface flow modeling on nonorthogonal grids in this paper we present an innovative finite volume surface subsurface integrated flow model on nonorthogonal grids the shallow water equation with diffusion wave approximation is used to formulate the surface flow system while the richards equation is used to formulate the saturated unsaturated subsurface flow system these two flow systems are discretized using a finite volume method and are then coupled by enforcing the continuity of pressure and flux at the surface subsurface interface which does not require unphysical parameters such as the interface permeability and thickness the numerical instability caused by enforcing the continuity of pressure and flux at the interface is resolved using a cell centered finite volume discretization the coupled systems are solved simultaneously by the newton iterative method a battery of benchmark analyses and laboratory experiments verify the proposed model s superior performance relative to existing models two numerical experiments over irregular terrain show that the nonorthogonal grids and diffusive wave approximation used in the proposed model accurately represent the interaction between surface and subsurface flows for irregular topographies in particular they capture the significant topographical effects on runoff discharges especially where gentle slopes are involved

some non local transformations between nonlinear diffusion equations generalized backlund transformations are applied to derive links between a large number of different types of nonlinear diffusion equations including many which are of physical significance some new exactly linearizable forms are determined

analytical solution to transient richards equation with realistic water profiles for vertical infiltration and parameter estimation a general analytical model for one dimensional transient vertical infiltration is presented the model is based on a combination of the brooks and corey soil water retention function and a generalized hydraulic conductivity function this leads to power law diffusivity and convective term for which the exponents are functions of the inverse of the pore size distribution index accordingly the proposed analytical solution covers many existing realistic models in the literature the general form of the analytical solution is simple and it expresses implicitly the depth as function of water content and time it can be used to model infiltration through semi infinite dry soils with prescribed water content or flux boundary conditions some mathematical expressions of practical importance are also derived the general form solution is useful for comparison between models validation of numerical solutions and for better understanding the effect of some hydraulic parameters based on the analytical expression a complete inverse procedure which allows the estimation of the hydraulic parameters from water content measurements is presented

non integrability of non linear diffusion convection equations in two spatial dimensions it is generally accepted that integrable partial differential time evolution equations possess lie backlund symmetries of arbitrarily high finite order it has already been established that in 1 1 dimensions the full class of nonlinear fokker planck convection diffusion equations having this property consists of the known integrable examples these are closely related either to the burgers equations or to the fokas yortsos rosen equation both of which have been applied to unsaturated flow in porous media here the author shows that higher order lie backlund symmetries do not exist for any examples of the nonlinear fokker planck equation in 1 2 one time and two space dimensions

reciprocal auto backlund transformations invariant backlund transformations of reciprocal type are derived for a class of n 1 th order conservation laws a new auto backlund transformation for the harry dym equation is presented as a special case of the analysis

error analyses of simplified unsaturated flow models under large uncertainty in hydraulic properties the functional relationships between soil hydraulic properties k and are needed for analyzing unsaturated water flow in soils and are generally of a nonlinear form these relationships estimated from field measured data usually show large variability because of measurement errors and natural spatial heterogeneity exhibited by many field soils a probabilistic approach is often employed to account for this uncertainty in parameters whereby the coefficients in the resulting partial differential equation are treated as random quantities usually characterized by their moments the objective of this analysis is to evaluate the errors resulting when simplified solutions based on simple forms of the hydraulic properties are used rather than nonlinear and numerically complicated models as an example of a parametric functional relationship for k we adopt the brooks corey formulation in which a single parameter is considered as random to emulate the spread encountered in practice for data that show a large enough spread and when one is interested in the mean behavior of k or simple flow models such as the green ampt model and the linear model are proposed as alternatives to the more accurate but numerically demanding richards equation to compare models scaled error definitions are introduced and are expressed as a percentage of the saturated hydraulic conductivity of the soil these time dependent errors are then computed under various boundary conditions in general the green ampt solution is found to yield smaller errors than the linear model when compared with solutions from richards equation it is concluded that the use of simplified models is justified when field data exhibit sufficient spread and a tight bound cannot be placed on the parameter estimates

the green ampt limit with reference to infiltration coefficients 1 recent progress with an analytic nonlinear model has provided the exact infiltration coefficients for realistic soil behaviors with nonsingular hydraulic functions as well as their exact delta function diffusivity limits after some correction and reinterpretation of the approximate analytical method the exactly solvable model validates some previously obtained approximate infiltration functions the green ampt infiltration function follows from a delta function diffusivity limit with a hydraulic conductivity that may be among other possibilities a linear function of water content just as a linear conductivity function is an overestimate for a realistic soil the second philip infiltration coefficient s1 in the green ampt infiltration function is too large due to conductivity being overestimated better agreement with experiment halving the value of s1 is obtained from the analytic nonlinear model with a limiting delta function diffusivity and a matching gardner exponential hydraulic conductivity function in general infiltration behavior is determined by the limiting forms of the diffusivity and conductivity relative to one another at the saturated water content or alternatively the relationship between the conductivity and soil moisture potential a new infiltration model demonstrates the possible range of s1 for physically valid limiting conductivity functions we show that in the delta function diffusivity limit the solution behaves as if the potential at the wet front were time dependent decreasing in magnitude from an initial value at the traditional green ampt level

on the exactly solvable equation s_t beta s gamma 2 s_x _x alpha beta s gamma 2 s_x occurring in two phase flow in porous media the symmetry approach of fokas j math phys 21 1980 pp 1318 1325 is used to show that the most general equation of the form s t g s s xx f s s x dg ds ne 0 which is exactly solvable is equivalent to the equation s t beta s gamma 2 s x x alpha beta s gamma 2 s x this equation is then mapped to the linear heat equation through a series of backlund transformations the above result is used to solve an initial boundary value problem modeling the process of waterflooding i e the displacement of oil initially in a porous reservoir by continuously injected water

linearized unsteady multidimensional infiltration the parameters in the linearized multidimensional unsteady equation for unsaturated flow in homogeneous soils can be chosen so that small time flow rates are given correctly with the quasi linear steady solution as the large time limit matched in this way the linearized equation offers insight into the time course of multidimensional infiltration the laplace transform reduces the equation and its governing conditions into forms similar to those for quasi linear steady flows the problems of unsteady infiltration from buried spherical and cylindrical cavities are solved in many cases the solutions are approximately reducible to the product of the steady solution and a function only of time and radial coordinate theorems show that approximate product solutions apply not only to these configurations but generally to multidimensional cavities of arbitrary shape multidimensional infiltration from finite supply surfaces involves two characteristic lengths and two characteristic times one set deriving from the capillarity gravity interaction and the other from the capillarity geometry interaction we explore the influence of the two characteristic times on the time scales of approach to steady moisture distribution near the supply surface and to steady discharge the analysis provides explanation of the relatively rapid approach to the steady state observed in three dimensional infiltration from small supply surfaces