A continuous‐time inverse operator for groundwater and contaminant transport modeling: Deterministic case

The value of time-dependent data in the calibration of numerical models of groundwater flow has been recently highlighted in the literature. In this paper the background for a new development in the inverse problem is placed, and a new operator for inversion of the spatially discretized diffusion equation is introduced. The motivation for the development of an inverse problem solution procedure which uses information in transient data is illustrated with a focused review of recent literature. Such an approach which implicitly incorporates the continuous-time dependency of the flow model is then presented. This approach is based upon the new concept of inverting the model after the (for example, finite element) spatial discretization but before the (typically) finite difference temporal discretization of the model, not afterward, as has been done in the past. The Laplace transform is used to replace discretization in time of the system of ordinary differential equations to discretization in the Laplace variable of the algebraic system in Laplace space. The existence of the deterministic solution in the Laplace transform space does not require knowledge of the eigenvalues of the diffusion matrix but only the specification of the Laplace variables near zero. A simple numerical example from the literature is used to generate a set of three simplified test cases, upon which the proposed method is tested. The implications of the theoretical investigation and the results of the numerical experiments are discussed with a view toward prospects of practical implementation.

[1]  Analytic solution of spatially discretized groundwater flow equations , 1973 .

[2]  G. Marsily,et al.  An Automatic Solution for the Inverse Problem , 1971 .

[3]  S. Yakowitz,et al.  Instability in aquifer identification: Theory and case studies , 1980 .

[4]  S. P. Neuman.  Calibration of distributed parameter groundwater flow models viewed as a multiple‐objective decision process under uncertainty , 1973 .

[5]  J. Bredehoeft,et al.  Computer model of two-dimensional solute transport and dispersion in ground water , 1978 .

[6]  C. S. Simmons.  A stochastic‐convective transport representation of dispersion in one‐dimensional porous media systems , 1982 .

[7]  S. P. Neuman,et al.  Estimation of Aquifer Parameters Under Transient and Steady State Conditions: 1. Maximum Likelihood Method Incorporating Prior Information , 1986 .

[8]  A. D. Kraus,et al.  Matrices for engineers , 1987 .

[9]  S. P. Neuman,et al.  Estimation of Aquifer Parameters Under Transient and Steady State Conditions: 3. Application to Synthetic and Field Data , 1986 .

[10]  The Problem of Complex Eigensystems in the Semianalytical Solution For Advancement of Time in Solute Transport Simulations: A New Method Using Real Arithmetic , 1986 .

[11]  Some algorithms for parameter estimation in water resources systems , 1983 .

[12]  M. Mariño,et al.  The inverse problem for confined aquifer flow: Identification and estimation with extensions , 1987 .

[13]  S. Nakagiri,et al.  Identifiability of Spatially-Varying and Constant Parameters in Distributed Systems of Parabolic Type , 1977 .

[14]  A. Cheng,et al.  An efficient laplace transform solution for multiaquifer systems , 1989 .

[15]  S. P. Neuman,et al.  Estimation of aquifer parameters under transient and steady-state conditions: 2 , 1986 .

[16]  John H. Cushman,et al.  Development of stochastic partial differential equations for subsurface hydrology , 1987 .

[17]  R. Bellman,et al.  A NUMERICAL INVERSION OF THE LAPLACE TRANSFORM , 1963 .

[18]  G. Chavent.  Identification of functional parameters in partial differential equations , 1974 .

[19]  Andrés Sahuquillo,et al.  An eigenvalue numerical technique for solving unsteady linear groundwater models continuously in time , 1983 .

[20]  John H. Cushman,et al.  On unifying the concepts of scale, instrumentation, and stochastics in the development of multiphase transport theory , 1984 .

[21]  John H. Cushman,et al.  On Measurement, Scale, and Scaling , 1986 .

[22]  R. Goodson,et al.  A definition and some results for distributed system observability , 1970 .

[23]  W. Yeh.  Review of Parameter Identification Procedures in Groundwater Hydrology: The Inverse Problem , 1986 .

[24]  S. P. Neuman.  A statistical approach to the inverse problem of aquifer hydrology: 3. Improved solution method and added perspective , 1980 .

[25]  G. Yeh,et al.  An eigenvalue solution continuous in time to the spatially discretized solute transport equation in steady groundwater flow. , 1984 .

[26]  J. Bear.  Dynamics of Fluids in Porous Media , 1975 .

[27]  R. Willis.  A planning model for the management of groundwater quality , 1979 .