Effective Parameters of Hydrogeological Models

Although hydrogeological conditions can be of interest per se, most hydrogeological investigations are of applied nature, and their results are used in decision-making that may carry large ecological and financial risks. For example, when developing a reservoir project, the developers have to evaluate possible losses of water from the reservoir, the stability of the dam, and how adjacent soils and rocks could be affected by different project decisions. Hydrogeological investigations related to the use of an aquifer for water supply should not only conclude that the usage is possible. The developers must also have estimates on how long and with what intensity the aquifer can be exploited by a well or group of wells. The developers of a landfill project must know whether the landfill can cause contamination of the aquifer below and, if so, whether and when the contaminant plume will reach water supply wells and the concentration of the pollutant at the wells. The developers of an irrigation project need to know to what extent and how fast the water table rise should be expected, what consequences are possible, how to deal with them effectively, etc.

The situation with the deterministic approach to predictive simulations is transparent. It can provide evaluations of the uncertainty of the simulation results in some typical circumstances for which engineering experience exists. These evaluations are of statistical nature. They are based on observed successes and failures of decisions made based on results of the corresponding simulations. However, if such experience does not exist, the engineering approach fails to provide provable estimates for the uncertainty of the simulation results. The situation seems more complicated with the geostatistical approach.

To predict responses of geological objects on man-made or natural impacts by applying mathematical methods, i.e., by solving differential or integral equations, the pertinent properties of the geological objects should be assigned continuously, that is, at each point of the objects and at each instant of the period of simulation, if the properties vary in time, besides maybe countable sets of points, i.e., isolated in space and time points. The boundary and initial conditions must be known in the same way. Unfortunately, only an infinitesimal part of the required geological information is available from direct observations and measurements. This information gap must be filled, and geological models have to do the job. They are a tool for interpolation and extrapolation of the sparse available data to all points of the geological objects of interest.

One of the effects of the phenomenon of problem dependence of model identification is that the model parameters effective in a given formulation of the model identification may not be, and often are not, effective in the coupled predictive simulations. The reason is that the coupled predictive problem differs from the model identification problem in many respects. They can have different impacts, boundaries and boundary conditions, nature of simulating fields (water tables and streamlines, as in the example discussed in Sect.​ 4.​4), quality criteria of simulation, and monitoring networks on which quality is evaluated. Even the mathematical models applied are often different (steady-state filtration in calibration versus transient in predictive simulations). The goal of model identification must be to provide the model parameters effective in predictive simulations, not just in calibration. The concept of transforming mechanisms introduced below is focused on providing the model parameters effective in predictive simulation.

Let us consider one-dimensional steady-state underground flow in an unconfined aquifer on a horizontal base with constant recharge N to a fully penetrating trench at X ₀ = 0 (Fig. 6.1). The aquifer is piecewise homogeneous. Its hydraulic conductivity changes at locations X ₁, X ₂, and X ₃, taking within the intervals [X ₀, X ₁], (X ₁, X ₂], (X ₂, X ₃], and (X ₃, X ₄] the values K ₁, K ₂, K ₃, and K ₄. Recharge N = 0.0001 m/day, and X ₁ = 25, X ₂ = 50, X ₃ = 75, and X ₄ = 100 m.

Linear transforming mechanisms are rare in practical applications. Even the mechanisms presented in Sect.​ 6.​2 were obtained by linearization of nonlinear mechanisms. Mathematical descriptions of nonlinear mechanisms, and their inferences and applications are considerably more complicated. However, it is still possible to find simple examples for illustrations.

In the examples of Chaps. 6 and 7 the transforming mechanisms were obtained analytically. Such a direct approach can be cumbersome and even not available in many situations. The two-level modeling introduced below is more universal and seems to be more practical.

As mentioned in Chap.​ 4, the term “inverse problem” is not a synonym for the terms “model identification,” “model calibration,” “historical matching,” or “site-specific validation.” These relate to evaluating the effective characteristics for a given simulation model, which is usually an optimization problem. The goal of the inverse problem is to estimate the actual properties of geological objects using available observations on natural geological phenomena or on responses on manmade impacts.

Contemporary computational techniques permit simulation of any predictive problem based on up-to-date hydrogeological theories and concepts. The real issue is the reliability of the simulation results, their uncertainty. Geological objects and their properties are not known in full, and how the inaccuracy of a simulation model can affect the simulation results is unknown.