Comments on “Steady- and transient-state inversion in hydrogeology by successive flux estimation” by P. Pasquier and D. Marcotte

Pasquier and Marcotte [Pasquier P, Marcotte D. Steady- and transient-state inversion in hydrogeology by successive flux estimation. Adv Wat Res 2006;29:1934–52] propose some modifications to the Comparison Model Method (CMM), in order to apply it to transient 3D ground water flow data for conductivity identification. We present some remarks on that paper to improve the comprehension of the basic features of the CMM and of the real value of the novelties introduced by Pasquier and Marcotte.     

Modeling water resources of a highly irrigated alluvial plain (Italy): calibrating soil and groundwater models

Modern and effective water management in large alluvial plains that have intensive agricultural activity requires the integrated modeling of soil and groundwater. The models should be complex enough to properly simulate several, often non-linear, processes, but simple enough to be effectively calibrated with the available data. An operative, practical approach to calibration is proposed, based on three main aspects. First, the coupling of two models built on well-validated algorithms, to simulate (1) the irrigation system and the soil water balance in the unsaturated zone and (2) the groundwater flow. Second, the solution of the inverse problem of groundwater hydrology with the comparison model method to calibrate the model. Third, the use of appropriate criteria and cross-checks (comparison of the calibration results and of the model outputs with hydraulic and hydrogeological data) to choose the final parameter sets that warrant the physical coherence of the model. The approach has been tested by application to a large and intensively irrigated alluvial basin in northern Italy.     

Is the forward problem of ground water hydrology always well posed?

Complex aquifer systems are often modeled with quasi-three-dimensional models, which consider two-dimensional horizontal flow in the aquifers and one-dimensional vertical flow through aquitards. When the aquifer system consists of a phreatic aquifer and one or more semiconfined aquifers connected by aquitards, the discrete model consists of a nonlinear system of algebraic equations, because the transmissivity of the phreatic aquifer depends on the phreatic head. If the water extraction is very high, the phreatic aquifer can be depleted and the equations of the model must be modified accordingly. There are not simple and general criteria to state if the phreatic aquifer is depleted before solving the system of equations. Therefore, the iterative procedures (e.g., relaxation methods), used to find the solution to the forward problem, must handle these particular conditions and can suffer several problems of convergence. These problems can be caused by the choice of the initial head values or of the relaxation coefficient of the iterative algorithms; however, they can also be caused by the nonexistence or nonuniqueness of the solution to the system of nonlinear equations. The study of existence and uniqueness of the general problem is very difficult and, therefore, we consider a simplified problem, for which the discrete model can be handled analytically. The results of the numerical experiments show that the solution to the forward problem can be nonunique. Only for some cases it is possible to invoke physical arguments to eliminate tentative solutions.     

Discrete stability of the Differential System Method evaluated with geostatistical techniques

The Differential System Method (DSM) permits identification of the physical parameters of finite-difference groundwater flow models in a confined aquifer when piezometric head and source terms are known at each point of the finite-difference lattice for at least two independent flow situations for which the hydraulic gradients are not parallel. Since piezometric head data are usually few and sparse, interpolation of the measured data onto a regular grid can be performed with geostatistical techniques. We apply kriging to the sparse data of a synthetic aquifer to evaluate the stability of the DSM with respect to uncorrelated measurement errors and interpolation errors. The numerical results show that the DSM is stable.     

Discrete stability of the Differential System Method evaluated with geostatistical techniques

The Differential System Method (DSM) permits identification of the physical parameters of finite-difference groundwater flow models in a confined aquifer when piezometric head and source terms are known at each point of the finite-difference lattice for at least two independent flow situations for which the hydraulic gradients are not parallel. Since piezometric head data are usually few and sparse, interpolation of the measured data onto a regular grid can be performed with geostatistical techniques. We apply kriging to the sparse data of a synthetic aquifer to evaluate the stability of the DSM with respect to uncorrelated measurement errors and interpolation errors. The numerical results show that the DSM is stable.     

A technique to test finite difference schemes to model some geophysical processes in a geological structure

A preliminary problem to solve in the set-up of a mathematical model simulating a geophysical process is the choice of a suitable discrete scheme to approximate the governing differential equations. This paper presents a simple technique to test finite difference schemes used in the modeling of geophysical processes occurring in a geological structure. This technique consists in generating analytical solutions similar to the ones characterizing a geophysical process, given general information on some relevant parameters. Useful information for the choice of the discrete scheme to employ in the mathematical model simulating the original geophysical process can be obtained from the comparison between the analytical solution and the approximated numerical solutions generated by means of different discrete schemes. Two classes of numerical examples approximating the differential equation that governs the steady state earth's heat flow have been treated using three different finite differences schemes. The first class of examples deals with media whose phenomenological parameters vary as continuous space functions; the second class, instead, deals with media whose phenomenological parameters vary as discontinuous space functions. The finite difference schemes that have been utilized are: Centered Finite Difference Scheme (CDS), Arithmetic Mean Scheme (AMS), and Harmonic Mean Scheme (HMS).The numerical simulations showed that: the CDS may yield physically inconsistent solutions if the lattice internodal distance is too large, but in case of phenomenological parameters varying as a continuous function, this pitfall can be avoided increasing the lattice node refinement. In case of phenomenological parameters varying as a discontinuous function, instead, the CDS may yield physically inconsistent solutions for any lattice-node refinement. The HMS produced good results for both classes of examples showing to be a scheme suitable to model situations like these.