Inverse methods in hydrogeology: Evolution and recent trends

Highlights

• Track the evolution of the inverse methods over the last decades in hydrogeology.

• Discuss issues encountered in the inverse problem, e.g., handling multiGaussianity, and considering prior statistics.

• Reveal the development motivation of the inverse methods and their recent trends.

Abstract

Parameter identification is an essential step in constructing a groundwater model. The process of recognizing model parameter values by conditioning on observed data of the state variable is referred to as the inverse problem. A series of inverse methods has been proposed to solve the inverse problem, ranging from trial-and-error manual calibration to the current complex automatic data assimilation algorithms. This paper does not attempt to be another overview paper on inverse models, but rather to analyze and track the evolution of the inverse methods over the last decades, mostly within the realm of hydrogeology, revealing their transformation, motivation and recent trends. Issues confronted by the inverse problem, such as dealing with multiGaussianity and whether or not to preserve the prior statistics are discussed.

Introduction

Mathematical modeling of subsurface flow and mass transport is needed, for instance, for groundwater resources management or for contaminant remediation. The forward model requires specification of a variety of parameters, such as, hydraulic conductivity, storativity and sources or sinks together with initial and boundary conditions. However, in practice, it is impossible to characterize the model exhaustively from sparse data because of the complex hydrogeological environment; for this reason, inverse modeling is a valuable tool to improve characterization. Inverse models are used to identify input parameters at unsampled locations by incorporating observed model responses, e.g., hydraulic conductivities are derived based on hydraulic head and/or solute concentration data. Deriving model parameters from model state observations is common in many other disciplines, such as petroleum engineering, meteorology and oceanography. This work mostly focuses on inverse methods used in hydrogeology.

The forward problem involves predicting model states, e.g., hydraulic head, drawdown and solute concentration, based on a prior model parameterization. Combining mass conservation and Darcy’s laws, the forward groundwater flow model in an incompressible or slightly compressible saturated aquifer can be written as [1]

$$\nabla ·(K\nabla h)=S_s\frac{\partial h}{\partial t}+Q$$

subject to initial and boundary conditions, where $\nabla ·$ is the divergence operator $\left(\frac{\partial }{\partial x}+\frac{\partial }{\partial y}+\frac{\partial }{\partial z}\right)$, $\nabla$ is the gradient operator ${\left(\frac{\partial }{\partial x}\text{,}\frac{\partial }{\partial y}\text{,}\frac{\partial }{\partial z}\right)}^T\text{,}K$ is hydraulic conductivity (L T⁻¹), h is hydraulic head (L), $S_s$ is specific storage (L⁻¹), t is time (T), and Q is source or sink (T⁻¹). The differential equation governing non-reactive transport in the subsurface is:

$$\varphi \frac{\partial C}{\partial t}=-\nabla ·\left(\mathit{qC}\right)+\nabla ·\left(\varphi \mathrm{D}\nabla C\right)$$

subject to initial and boundary conditions, where C is the concentration of solute in the liquid phase (M L⁻³), $\varphi$ is porosity (–), $\mathrm{D}$ is the local hydrodynamic dispersion tensor (L² T⁻¹) usually defined as $D_i={\alpha }_i|q|+D_m$ where ${\alpha }_i$ refers to the longitudinal and transverse dispersivities (L) and $D_m$ is the molecular diffusion coefficient (L² T⁻¹), and q is the Darcy velocity (L T⁻¹) given by Darcy’s law as $q=-K\nabla h$.

The inverse problem aims at determining the unknown model parameters by making use of the observed state data. In the early days of groundwater modeling, it was common to start with a prior guess of the model parameters, run the forward model to obtain the simulated states, and then enter in a manual loop iteratively modifying the parameters, and then running the forward model, until observed and simulated values were close enough so as to accept the model parameter distribution as a good representation of the aquifer. This “trial and error” method falls into the scope of “indirect methods” as opposed to the “direct methods” which do not require multiple runs of the forward model to derive the model parameters [2] as will be discussed below.

Sagar et al. [3] classified the inverse problem into five types according to the unknowns, i.e., model parameters, initial conditions, boundary conditions, sources or sinks and a mixture of the above. Most documented inverse methods fall into the first type, that is, they try to identify model parameters, which contribute largely to the model uncertainty due to the inherent heterogeneity of aquifer properties. Parameter identification is of importance considering the fact that no reliable predictions can be acquired without a good characterization of model parameters. Parameter identification is a broad concept here including not only the property values within facies but the facies distribution, or in other words, geologic features. The effect of geologic uncertainty in groundwater modeling is examined, for instance, by He et al. [4] in a real case study. Furthermore, data scarcity deteriorates the characterization of the model parameters and raises the uncertainty. Besides estimating aquifer parameters, the inverse methods also play a critical role in assessment of uncertainty for the predictions. Furthermore, the inverse problem might be used as a guide for data collection and the design of an observation network. The reader is referred to Poeter and Hill [5] who discussed the benefits of inverse modeling in depth. In this work we are mainly concerned with the uncertainty introduced by the unknown model parameters and thus the inverse methods that are used to characterize these parameters.

A problem is properly posed if the solution exists uniquely and varies continuously as the input data changes smoothly. However, most of the inverse problems in hydrogeology are ill-posed and they cannot be solved unless certain assumptions and constraints are specified. Ill-posedness may give rise to three problems: non-uniqueness, non-existence and non-steadiness of the solutions, among which non-uniqueness is the most common. Non-uniqueness primarily stems from the fact that the number of parameters to be estimated exceeds that of the available observation data. Another reason is that the observations are sometimes not sensitive to the parameters to be identified; in other words, the information content of the observations is very limited. For instance, hydraulic heads close to the prescribed head boundaries are more influenced by the boundaries than by the nearby hydraulic conductivities (i.e., the hydraulic heads are not so sensitive to the conductivities), and on the contrary, the hydraulic heads close to the prescribed flux boundaries are determined to a large extent by the hydraulic conductivities nearby [6].

A series of suggestions have been proposed to alleviate the ill-posedness:

• Reduce the number of unknown parameters, e.g., using zonation, or collect more observation data so that the numbers of data and unknowns are balanced.

• Consider the prior information or some other type of constraint to restrict the space within which parameters may vary.

• Impose regularization terms to reduce fluctuations during the optimization iterations [2].

• Maximize the sensitivity of observations to model parameters, for instance, by designing properly the observation network.

• Minimize the nonlinearity in the model equation. Carrera and Neuman [6] argued that working with the logarithm of hydraulic conductivity reduces the degree of non-convexity during optimization. An alternative is to infer hydraulic conductivity using fluxes rather than heads as done by Ferraresi et al. [7], since the relationship between hydraulic conductivity and flux is linear (Darcy’s law) while the relationship between hydraulic conductivity and head is nonlinear.

Detailed discussions on this subject can be found in [2], [6], [8], [9] among others.

Besides the ill-posedness problem, computational burden is the second main hurdle for inverse problems [10]. There are several reasons for the high CPU time requirement. Since many inverse models are iterative, the forward model has to be run many times until an acceptable parameter distribution is obtained. The time needed to run the forward model grows exponentially with the degree of discretization and the level of heterogeneity. When multiple realizations are sought, CPU demand grows with the number of realizations. For those methods that use gradient minimization, the sensitivity matrices of model variables on parameters are computed, which is very time consuming. A few measures to reduce computational demand have been proposed, for instance, (a) certain kernel techniques render it possible to select representative realizations from a large ensemble so that the size of the ensemble can be reduced (e.g., [11], [12]); (b) upscaling can be performed prior to any forward simulation in order to reduce solution time (e.g., [13], [14], [15]); (c) use of parallel/distributed computing technologies (e.g., [16], [17]); (d) use of efficient surrogate models to reduce the number of unknowns that must be computed at each time step, i.e., reduced-order flow modeling (e.g., [18], [19], [20]).

The problem of scales is the third difficulty to be confronted in the application of the inverse method. Measurements from boreholes (made in situ or in the laboratory), local pumping tests, and regional aquifer estimates are the three common scales [21], [22] at which information is handled in aquifer modeling. As Emsellem and De Marsily [8] pointed out, “permeability is a parameter with no punctual value but with an average value in a region of a given size”. The support of the permeability measured from the field is normally smaller than the cell size of the numerical model. In practice, permeability should be upscaled to a scale consistent with that of the numerical model discretization, otherwise the forward model would be computationally very expensive. A variety of approaches to calculate the upscaled permeability or hydraulic conductivity are available (e.g., [23], [24], [25], [26], [27]). Besides, the scale inconsistency between field measurement support and numerical model discretization extends also to the observations, which can be obtained at different supports, too.

Despite all sorts of difficulties, many inverse methods have been proposed to solve the inverse problem. In the present paper we do not intend to review all current inverse methods, since several others have reviewed the topic from different points of view (e.g., [9], [28], [29], [30], [31]). But rather, we would like to analyze the evolution of the inverse models, from the simple trial-and-error approaches of yesterday to the sophisticated ensemble Kalman filters of today, pointing out the incremental improvements that happened in the way.

In the remainder of this paper we mainly focus on seven key topics, as follows:

• Section 2.1 discusses the direct method. Then, in the following sections, we focus on the indirect approaches.

• Section 2.2 shows a linear inverse method in which the groundwater flow model is solved by linearizing the partial differential equation under certain assumptions. Its shortcoming motivates development of the nonlinear inverse methods in which the forward problem is solved numerically. The inverse methods in the remaining sections all belong to the nonlinear type.

• Section 2.3 highlights the importance of considering uncertainty and introduces the inverse method based on Monte Carlo simulation in which multiple plausible realizations are used to represent the real system.

• Section 2.4 discusses sampling the posterior distribution rather than minimizing an objective function as the solution to the inverse problem.

• Section 2.5 focuses on integrating new observations sequentially without the need to reformulate the problem.

• Section 2.6 discusses whether the prior statistical structure of the model should be preserved through the inversion algorithm.

• Section 2.7 addresses the issue of multiGaussianity in inverse modeling and the difficulties to get away from it.

In each section, we will introduce a typical inverse method explaining its principle, implementation details, advantages and shortcomings. Recent trends of the inverse modeling are summarized in Section 3. The paper ends with some conclusions.