DRILLING MORE WELLS OR DOING PUMPING TEST: INVESTIGATING THE RELATIVE VALUE OF WATER HEAD AND CONDUCTIVITY MEASUREMENTS IN REDUCING INVERSE GROUNDWATER MODELING ERROR

ABSTRACT

Given limited resources, an investigator interested in inversely estimating the groundwater conductivity field can either invest in hydraulic head (H) measurements (e.g., drill a well) or conductivity (K) estimates (e.g., conduct pumping tests). They result in different information content and cost. While advanced stochastic methods exist to estimate the worth of data, there is no first-order answer that enables fast decision making. Here we empirically determine the worth of H or K data, in terms of reducing calibrated K error, as a function of data density, recharge rates, and boundary conditions. We found that normalized K error can be well approximated by a smooth function of heterogeneity-normalized H and K data densities. Stepwise polynomial fitting suggests including only the first-order terms and a mild interaction term. The relative information content ratio of H to K for the next data point (R_n) suggests the worth of K is always more than that of H, and could be as higher than 10 when K data is sparse. R_n is a function of data densities and K correlation length, thus the decision must consider the amount of presently available data and the heterogeneity of the field. We suggest investment decisions can be charted by following a straight line on which the information content per unit cost ratio is separated to be less- and greater-than-1 values.

         

Table of Contents

List of Figures ………………………………………………………………………………………………… vi

List of Tables ……………………………………………………………………………………………….. viii

Acknowledgements …………………………………………………………………………………………. ix

Chapter 1 Introduction ……………………………………………………………………………………… 1

1.1 Study Motivation ……………………………………………………………………………………. 1

1.2 Overview on groundwater model calibration ……………………………………………… 4

1.3 Overview on study of data worth to groundwater modeling …………………………. 7

1.4 Research objectives ……………………………………………………………………………….. 10

1.5 Terminology …………………………………………………………………………………………. 11

Chapter 2 Methodology ………………………………………………………………………………….. 13

2.1 Introduction ………………………………………………………………………………………….. 13

2.2 Methodology of parameter estimation in MODLFOW and PEST ……………….. 14

2.2.1 Conceptual model identification ……………………………………………………….. 14

2.2.2 Groundwater flow process ……………………………………………………………….. 15

2.2.3 Parameter estimation ……………………………………………………………………….. 17

2.2.4 Implementation of Gauss-Marquardt-Levenberg approach …………………… 20

2.2.5 Regularization ………………………………………………………………………………… 21

2.3 Process model development ……………………………………………………………………. 23

2.3.1 Synthetic model construction ……………………………………………………………. 23

2.3.2 Observation abstraction and parameterization …………………………………….. 25

2.3.3 Termination criteria of inverse modeling iterations ……………………………… 26

2.4 Dimensionless control variables for data densities and calibration errors ……… 27

2.5 Calibration errors to different recharge and boundary condition ………………….. 31

2.6 Experimental design and multi-variable polynomial curve fitting ……………….. 33

2.7 Relative data worth and economic analysis ………………………………………………. 34

Chapter 3 Result and Discussion ……………………………………………………………………… 36

3.1 Verification of the dimensionless numbers ……………………………………………….. 36

3.2 Impact of recharge and boundary condition on model calibration errors ………. 37

3.3 Errors as a function of effective data densities ………………………………………….. 39

3.4 Multi-variable polynomial curve fitting ……………………………………………………. 41

3.5 Economic worth of data acquisition against model calibration error reduction 46

Chapter 4 Conclusions and Limitations …………………………………………………………….. 51

Appendix A: Random log(K) Fields and Evaluations with Variograms …………………. 53

Appendix B: MATLAB Function to Plot 𝑅_𝑛 and

References …………………………………………………………………………………………………….. 58

Chapter 1 Introduction

1.1 Study Motivation

As a vital component of the global hydrological cycle, groundwater plays a crucial role in maintaining life and developing human society. Excluding the water locked in polar ice, about 89% of fresh water is stored in subsurface, and more than 1.5 billion people are living on groundwater supply, globally (Alley, 2002; Koundouri, 2004).

Groundwater resource is being increasingly used in agriculture, industry and commercial facilities as a subsequence of the rapid blossom of pumping technologies (Konikow and Kendy, 2005). During the year of 2000, a rate of worldwide groundwater abstraction of 734 (±82) km³a^-1 was estimated (Wada et al. 2010). While concentrating on the United States, the USGS conducted a survey to nation-wide water use in 2010: about 355 billion gallons of water in total are customed per day, 22.33% of which are from fresh and saline groundwater resource (Maupin et al., 2010). McGuire et al. (2000) reported a typical example of High Plain where groundwater storage has reduced by 6% of predevelopment cumulation during the 20^th century. Morever, Wada et al. (2010) drawn the conclusion that groundwater depletion has been increased to 283 (±40) km³a^-1 in 2000 from 126 (±32) km³a^-1 in 1960 based on a numerical model.

Excessive withdraw of groundwater can result in extensive damages like degradation in water quality, ground subsidence, higher cost to maintain wells yields and devastating impact to relevant ecosystem etc. (Konikow and Kendy, 2005; Wada et al., 2010), therefore, the management of groundwater has attracted much concern of people during the past decades; as depletion has extended dramatically all around the world (Alley, 2006). However, estimation of large-scale groundwater condition is an issue with challenges due to complexities of groundwater property, various investigation system with different spatiotemporal scales and personal research objectives (Reilly et al.,

2008). Furthermore, there is no such regular arrangement to integrate groundwater information on the federal level, hence the researchers mostly rely on the local resources. Despite the varying of temporal and spatial scales inherent in the defective condition above and comparing materials available all across the country, still, people have drawn a sketch picture (Figure 1) of present national groundwater condition regarding water level declines and pointed out further direction to better assessment and management of groundwater resource (Dennehy et al., 2015; Reilly et al., 2008).

As a response to the explosion of groundwater withdraws in the last half century, methods of studying groundwater have made significant progress. Based on field investigations, USGS has established a database of groundwater-related data throughout 50 states for more than 100 years, This database is the primary resource for nationwide groundwater managers and policymakers (USGS FS-058-95). With the basis of geophysical principles, electronic and magnetic methods are commonly used to study aquifer structure and subsurface flow regarding containments (Karlik and Kaya, 2001; Wattanasen and Elming, 2008).

 

Figure 1. Water-level declines. Red regions indicate areas in excess of 500 square miles that have water-level in excess of 40 feet in at least one confined aquifer since predevelopment, or in excess of 25 feet of decline in unconfined aquifers since predevelopment. Blue dots are wells in the USGS National Water Information System database where the measured water-level difference over time is equal to or greater than 40 feet (Reilly et al. 2008).

During the last several decades, computer-based numerical models have been increasingly used in groundwater studies. As stated by Gallagher and Doherty (2007),

“computer models are often used to predict the behavior of environmental systems”. Computer-based simulation and modeling have been demonstrated to be effective approaches to solving groundwater management problems. Like most other hydrological models, groundwater models are typically operated both conceptually and numerically. However, conceptual models are highly simplified against realistic systems, in addition that the phase of parameterization is hardly determined to sufficiently accounts the heterogeneity of domain aquifer. Mathematical algorithm solving the governing equations may not accurately infer the values of parameter either. All these matters can subsequently introduce errors to the models thus weakening the confidence with which one can rely on them (Christensen and Doherty, 2008; Muleta and Nicklow, 2005; Tonkin and Doherty, 2009).  In addition to such a discrepancy between model outputs against historical field records, uncertainty is another intrinsic feature of models. Quantitative knowledge of uncertainty in a spatial perspective can help modelers assess the areas where the existing models are more to deliver inaccurate results.

Error and uncertainty pertaining to both parameter and prediction are commonly regarded as essential criteria when evaluating the performance of a computer model.  Data collection is the foundation of a model with deterministic impact on error and uncertainty quantification. Incomplete or inaccurate data can cause large misfit between model outputs and data records. Fallacious locations where data being collected can leave areas with uncertainty to be uninformed where predictions will be biased. On the other hand, it is never practicable to ceaselessly investigate the site with limited resource. Overly considered data collection can needlessly waste millions of dollars given the fact that a single water sample for contamination test may require thousands of dollars (James and Gorelick, 1994). In one word, tradeoff decision must be made to improve the model performance by appropriately adding data information and to ensure such investigation are financially feasible.

The earliest research on (additional) data collection worth can be ascended to 1970s (Davis et al., 1972; Gates and Kisiel, 1974; James and Gorelick, 1994) followed by remarkable amount of others’ work generally focusing on either numerically exposing the uncertainty associated in model itself or relating uncertainty analysis to groundwater management problems. The latter always involves monetary consideration of measurement costs with various site investigation scenarios. The conclusion of cost-effective data collection scheme is drawn with respect to the reliability of the model, which is determined by uncertainty in model calibration and prediction. However, previous work in literature rarely considered model calibration errors in terms of not only measurements, but parameters as well. It lacks an integrated economic concept relating data collection cost to both error and uncertainty quantification. Moreover, early work generally makes determinations of optimal measurement scenario among various alternatives within the aquifer under study. On the other hand, studies such as how well particular field investigation strategy can serve simulations of various domains can further complete people understanding on data sufficiency.

1.2 Overview on groundwater model calibration

In the following section, we will briefly introduce some previous work on groundwater modeling as well as associated error and uncertainty analysis. Thereafter we will display the current stage of research on data worth in improving the modeling quality and simultaneously to control data measuring budget. Before continuing, one is strongly recommended to glance over the “Terminology” at the end of present chapter in order to get familiarized with the terms that are particularly used in groundwater modeling research.

A number of computer-based models have been released and demonstrated to be “satisfying” in representing realistic aquifer system. One of the most popular applications of modeling is to simulate subsurface flow with chemical contamination. For instance, GFLOW (Yager and Neville, 2002) and STANMOD (Feinstein and Guo, 2004) uses analytical methods to simulate subsurface flow; Model Viewer (Zhen, 2004) and Environmental Insite (Tonkin and  Becker, 2005) visualize the model-and-user interface; other tools like GIS and Microsoft Excel are representatively used in ModTech (Pint and Li, 2006) and Jiao and Leung’s work (2003) as supplement to provide better match with modeler’s needs, just to name a few. Another computer program suite, MODFLOW (firstly named as Modular Model), was originally developed by U.S. Geologic Survey (USGS) in the 1980s together with the first documentation of McDonald and Harbaugh (1983) and has become one of the most widely used groundwater flow models since 1990s. MODFLOW-2000 (Harbaugh et al., 2000) was then developed to integrate groundwater flow transport and parameter estimation to transition the program from package-based to process-based (Mcdonald and Harbaugh, 2003). The applications of MODFLOW (-2000) during the last two decades are rarely exclusive, but always associated with other programs like PEST, specific cases will be introduced in later reviews.

As stated earlier, data source is the basis of a computer model and can crucially but also partially determine model error and uncertainty. Often, parameters constituting aquifer properties cannot be readily measured, let alone an extensive relevant network. This is primarily due to the vast financial and technical requirements of such measurements. Inverse modeling is a more advanced approach to curve fitting of observations that is used by van Genuchten et al. (1991) under the assistance of nonlinear least-squares optimization scheme to estimate the parameters all over the domain (Vrugt et al, 2001). Abundant work has been done to expose and reduce error and uncertainty within parameter estimation process.

Spatial characterization of hydrogeological condition within the model domain can provide more prior information to calibration especially when using zones, each of which contains uniform hydraulic property and parameter value, to parameterize the model. Zonation is implemented before actual parameter estimation process based on modeler’s knowledge on the area of study, which can hardly embrace the complexity of study field at adequate or even just satisfactory level (Doherty 2003a). Pilot point method has been widely used as an effective alternative to parameterize the model with capability to generate a smooth variation of the parameter while also reflect the heterogeneity of studied domain (Doherty, 2003; Lavenue and de Marsily, 2001; Ramarao et al, 1995).

As attempts are being made to sufficiently reflect the spatial heterogeneity of some aquifer properties like hydraulic conductivity, one can rapidly increase the number of pilot points as parameters. Such a matter can lead to substantial non-uniqueness in parameter estimation, where a variety of parameter sets can fit the model with similar model-to-measurement discrepancies, especially when the number of parameters exceeds the number of observations. Regularization methodology has occupied a commonplace in treating such a drawback of pilot point by introducing “modeler preference” of parameter value distribution in inverse modeling (Doherty, 2003; Tonkin and Doherty, 2005).  Its advanced variant, namely, adaptive regularization is demonstrated to be more efficient in stabilizing parameter values in (Doherty and Skahill, 2006).

Among literature model calibration has also been implemented using tools other than PEST. For example, to study optimal data network, Thompson et al. (2013) integrate an established surface and groundwater model with a subsurface model to create a unified MODFLOW-SUEFACT model, which is later to be calibrated with some data that is not always emphasized by modelers. Another method named, Meta-Heuristic, is used to calibrate groundwater models. It is demonstrated to be valid with the illustration of a case study on Ghaen plain, Iran by criticizing sum of squared deviation (SSD) and the absolute value of deviation (SAN) between simulated and observed hydraulic head values (Haddad et al., 2013).

1.3 Overview on study of data worth to groundwater modeling

Previous work has been done seeking optimal measurement strategy to serve groundwater management decisions and mostly focus on subsurface environment remediation. James and Gorelick (1994) studied the cost-effectiveness of sampling based on Bayesian data worth framework to solve a hypothetical contamination problem where the pollutant plume is uncertain in both location and extent. The cost-effectiveness here is defined as the cost of potential remediation program reduced by data information is greater than that to obtain them. Under multiple assumptions, say, two-dimensional steady state, the number of 6 is deemed to be optimum for the studied case, such the number is also particularly sensitive to the variance of hydraulic conductivity of studied aquifer (see Figure 2). Similar work can be found in (Freeze, et al, 1992), Risk-cost-benefit objective function (Freeze et al., 1990) is used to facilitate tradeoff between the cost of data collection and expected value of risk reduction brought by the data. A synthetic landfill leachate issue is illustrated, where the risk depends on the uncertainty associated with aquitard continuity as well as hydraulic conductivity. Adjusting the objective to Bayesian theory, data worth pertaining to reducing the uncertainty of both aspects above is calculated, which is thus the tolerant cost to gain a data sample.

Data collection strategy has also been studied under other various approaches help to solve groundwater problems. Tucciarelli and Pinder (1991) employed a chance-constrained stochastic technique to find best number and locations of additional measurements that can result in minimum summation cost of data acquisition and groundwater reclamation. One similar chance-constrained model is

 

(b)

 

Figure 2. (a) Sampling cost, remediation cost total cost, and number of remaining plume realizations versus number of samples collected in example sampling program for preselected plume. (b) Sensitivity of average optimum number of samples, prior remediation cost, average optimum remediation cost, and average optimum total cost to variance of log hydraulic conductivity (James and Gorelick, 1994).

coupled with an integer-programing sampling network design model in Wagner (1999) to optimize pumping and sampling strategy. In a synthetic aquifer remedial case design, the former model identifies least cost of possible pumping plan while the latter model identifies optimal sampling network in reducing simulation model uncertainty.

What is worth raising in Wagner’s work is that he compared the contribution of multiple data types to reduce model uncertainty, which will thus save groundwater remediation cost. Conclusively, the worth of measuring hydraulic conductivity is much greater than that of aquifer state investigation (i.e. to measure hydraulic head and pollutant concentration).

The hydraulic conductivity (K) of groundwater aquifers is an important parameter for either contaminate fate and transport studies or evaluations of groundwater resources (Freeze and Cherry 1979). While large-scale estimates of groundwater conductivities become increasingly available, e.g., (Gleeson et al. 2014), there are still many districts in the world, e.g., California desert in the US, where high-quality K data is too scarce for groundwater modeling purposes. K can be inferred from pumping tests (Marsily 1986) or lithological estimates (Hördt et al. 2007; Hyndman and Gorelick 1996). Typically, K values are not randomly distributed spatially, but are auto-correlated in space (Rehfeldt et al. 1992). Therefore, one K measurement not only informs us about the K values at the site, but also gives us some information about the adjacent region. On the other hand, K can also be estimated inversely through calibrating a suitable groundwater flow model to observed hydraulic head (H) using estimated recharge. Often they make use of available K data points and geostatistical model to constrain the inversion process. This inversion is possible because, given an accurate estimate of recharge, H carries information about K.

More efforts can be found in literature studying data worth on groundwater model development. For example, Vrugt et al (2001) developed a Parameter Identification Method based on Localization of Information (PIMLI) to serve uniqueness of parameter estimation. With a synthetic model where there are artificially known values of parameters, a conclusive figure is obtained where specific subsets of data are separated out containing the most data information on one of the parameters, which means, each value of parameters is sensitive most to one of the data subsets.

The worth of different types of data to characterize aquifer heterogeneity and to reduce uncertainty in parameter estimation is focused by Fu and Jaime

Gómez-Hernández (2009) with a new blocking Markov chain Monter Carlo algorithm. The conclusion was outlined as log-conductivity data can reduce the spatial distribution uncertainty of conductivity more than that of the piezometric head, whereas measurement of water head works reversely. And the combination of both data types can generally act better in model uncertainty reduction. To name a few of other works that focus on data worth to groundwater model calibration, say, Christiansen et al. (2011); Thompson et al. ( 2013); Lubczynski and Gurwin (2005); Bakr and Butler (2004).

1.4 Research objectives

Either H and K data helps reduce the uncertainty about the K field, i.e., they both have mutual information (Cover and Thomas 1991) with the K field. However, they vary in the information content they carry and their cost. Field-collected data can be very expensive, and the cost scenarios can vary greatly from place to place and from time to time. Traditionally, H data points can be read from existing wells or new wells. A K data point, if estimated using pumping test, requires more efforts and time. Therefore, we assume that a K data point costs more than an H data point. More recently, data collection effort can take the form of securing and interpreting existing well records and pumping records. Even in this scenario, a K data point can take more effort to obtain, and it indeed tends to be sparser. Therefore, given limited resources, which could be time and budget, an investigator needs to decide on the type of data is the most worthwhile to pursue. Thus, from a practical point of view, there is a need for understanding the relative worth of H and K data, condensed into the question “should we obtain an H or K data point next?”. However, searching through literature we found that there are no clear, simple answers to this inquiry.

The goal of this paper specifically includes:

1. To produce the first-order estimate of the relative worth of H and K data points on calibrated K error reduction, and to identify its main controlling factors.
2. To examine if calibrated K error can be described by a function of recharge, boundary conditions, and relative data densities of H and K.

• To seek methods reducing the dimensionality of the problem by non-dimensionalization, and verify the effectiveness of the dimensionless variables.

1. To comment on the best strategy in directing investment in H and K data collection honoring the economic costs of data collections

1.5 Terminology

A number of technical terms are frequently used in the field of groundwater modeling. To avoid confusion, the specific definition of each is qualitatively introduced presently. Mathematical specifications of several can be found in Chapter 3.

1. Conceptual model vs. numerical model: Conceptual model is a simplified, conceptual representation of characteristics of groundwater system or aquifer, and always concerns all available information related, like geology, boundary and initial conditions, discharge and recharge, hydrological source, etc. Numerical model is to mathematically solve the physical governing equations with numerical methods such as discretization or finite difference approach. ii. Parameter: In groundwater system there is a set of physical properties configuring general condition and behavior of aquifer that are relatively consistent with time but always spatially distributed, those properties are called parameters. Popularly concerned parameters in groundwater modeling include conductivity, recharge, thickness of aquifer and storage coefficient etc. The process using limited known parameter values to determine overall model inputs throughout temporal and spatial scale of modeling is called parameterization.

• Objective function: A mathematical function used to quantitatively calculate misfit between measurement and model prediction.

1. Calibration: The process that modelers refine the simulation to better represent realistic circumstance and obtain the best agreement of measured data and model output (i.e. the minimum value of objective function). The term is equally referred as parameter estimation.
2. Insensitivity: Inclusion of sensitivity in model calibration is used to derive model generation from parameter. Insensitivity occurs when field observation does not contain sufficient information to promote successful parameter estimation.
3. Non-uniqueness: More than often in hydrologic modeling, multiple combinations of parameter values can provide modeler with similar calibration results. This is always led by the heterogeneity of simulated system and simplification of the model.

• Regularization: The effort made to achieve uniqueness of the solution, which is exact parameter value obtained from calibration.
• Pilot points: A set of scattering points each of which is assigned with hydraulic property values, which are further used as parameters in model calibration.

DRILLING MORE WELLS OR DOING PUMPING TEST: INVESTIGATING THE RELATIVE VALUE OF WATER HEAD AND CONDUCTIVITY MEASUREMENTS IN REDUCING INVERSE GROUNDWATER MODELING ERROR

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