Analysis of Measurement Uncertainty of Natural Source Zone Depletion Rate of Hydrocarbons as a Function of the Uncertainty of Subsurface Temperature

Abstract

Environmental monitoring and remediation of hydrocarbon contamination in soil, particularly through passive methods, has become a critical area of focus for oil companies adapting to current environmental standards. Natural source zone depletion (NSZD) is a passive remediation process in which the degradation of petroleum hydrocarbons by microorganisms produces measurable thermal effects on the subsurface. Accurate estimation of the NSZD rate is heavily dependent on the precision of temperature measurements, as small uncertainties in temperature can cause significant variations in the estimated rates. Despite growing interest in using subsurface temperature data for NSZD monitoring, there is a lack of studies addressing the impact of temperature measurement uncertainty on the reliability of depletion rate estimates. This paper proposes a Monte Carlo method approach to assess the propagation of temperature measurement uncertainties through the NSZD rate estimation process. By simulating different uncertainty scenarios, this work defines acceptable limits for temperature measurement errors to ensure accurate and representative NSZD rate calculations. For the analyzed case study, it was determined that the relationship between uncertainties was nearly linear, with a slope of 52.5 L m⁻² year⁻¹ in the estimated NSZD rate for each degree Celsius of uncertainty in the temperature measurements.

1. Introduction

Environmental, social, and corporate governance (ESG) has gained increasing prominence in recent years, requiring companies to adopt more sustainable practices in all industries [1,2,3]. The oil extraction and processing sectors, in particular, face significant challenges due to the risk of unintentional spills of petroleum products during transportation, storage, or use [4]. These incidents can cause substantial contamination, which affects the long-term quality of soils and groundwater worldwide [5]. To align with ESG principles, oil and gas companies are increasingly focused on preventing environmental disasters and implementing effective and well-defined remediation procedures when spills occur. Remediation methods generally fall into two categories: active and passive. Active remediation involves direct intervention to remove contaminants from the soil, employing techniques such as soil excavation, chemical treatments, or bioremediation. Passive remediation, on the other hand, is based on continuous monitoring to assess if natural decomposition of hydrocarbons is sufficient to reduce contamination levels to acceptable values within a reasonable time frame [6]. Although it is common to use an active remediation method soon after the contamination event, passive methods have proven to be viable alternatives to treat the residual free phase [7]. An example of such methods is the natural source zone depletion (NSZD), which can be implemented alone or in conjunction with active remediation methods, depending on site-specific risk management needs [8]. This dual approach reflects the growing recognition of the need to balance immediate environmental recovery with cost-effectiveness and long-term sustainability. By adopting these strategies, companies can effectively address their environmental responsibilities in accordance with ESG goals. However, a great challenge associated with passive methods is the high cost of monitoring, as spills often occur in isolated locations, and analyses typically require laboratory processing.

Early studies exploring the use of heat production to estimate reaction rates in subsurface environments are discussed in [9,10]. Based on these findings, it is known that the hydrocarbon degradation process in contaminated soil can generate measurable effects on soil temperature, causing changes in the subsurface temperature profile [11,12,13,14]. These effects can be effectively measured using thermal monitoring techniques, which provide long-term measurements of biogenic heat and can be used to assess long-term NSZD rates. These rates are defined as the rate of bulk mass depletion of the fuel in the free phase, also known as light non-aqueous phase liquid (LNAPL), typically expressed as the volume of LNAPL degraded per unit area per year. There is limited information on the magnitude of the temperature increase in the subsurface caused by the biogenic heat generated during NSZD for different sites, but previous studies indicate that this increase is in the order of 2 °C. For example, the authors in [11] observed that the subsurface temperature within an LNAPL-affected region was 2.0 °C to 2.5 °C higher compared to a nearby site with similar soil characteristics but unaffected by LNAPL contamination. In addition, a temperature difference ranging from 1.2 °C to 2.9 °C between the affected zone and the background was reported in [12].

Different temperature transducers have been used in the experiments reported in the literature to estimate the NSZD rate, but thermistors [11,15] and thermocouples [13,16,17] are the most widely used options. In [14], it is stated that thermistors are the preferred choice for short-term estimation of the NSZD rate, whereas thermocouples are the preferred choice for year-long measurements. In addition, resistance temperature detectors have been used for NSZD studies in [18]. Given that the phenomenon of interest does not manifest as a large change in subsurface temperature values, it is quite important to correctly measure the temperature values, and the measurement uncertainty of typical temperature transducers can pose a challenge to correctly estimate the NSZD rate. The authors in [11] used a mesh with three Thermochron iButton transducers, obtaining an uncertainty of ±0.5 °C in temperature measurement. The authors in [13] used thermocouples coated with polytetrafluoroethylene and conducted laboratory tests, which estimated that the combination of the thermocouple and the acquisition system achieved measurement errors on the order of ±0.1 °C. It is important not to generalize this value. It can only be obtained for specific temperature ranges after careful calibration. The standardized measurement uncertainty for good quality thermocouples is around ±0.5 °C [19]. Additionally, the uncertainty of the temperature transducer used for cold junction compensation must also be considered. Some studies report the measurement uncertainty of the temperature transducers used in their setups. However, no study was found that estimates acceptable uncertainty limits for ground temperature measurements. These limits are important to ensure that the NSZD rate estimates remain representative.

The aim of this work is to propose a method to evaluate the impact of temperature measurement uncertainties on the uncertainty of the NSZD rate estimate. Although the approach is applicable to any kind of temperature transducer, thermistors were used in the case study of this paper. The proposed method uses the Monte Carlo method (MCM) to propagate the probability distributions of the subsurface temperature measurements, as well as the probability distributions of the temperature transducer positions, to obtain the corresponding distribution of the NSZD rate considering an analytic nonlinear model.

The remainder of this paper is structured as follows. Section 2 introduces the mathematical model for calculating the rate of depletion of hydrocarbons. Section 3 details the monitoring station used in this study. Section 4 outlines the method for assessing the uncertainty associated with temperature measurement. Section 5 covers the analysis of uncertainty, presenting the algorithm developed and its results. Finally, Section 6 provides the conclusions of this paper.

2. Mathematical Model for Calculating the Depletion Rate of Hydrocarbons

Traditional methods for estimating the NSZD rate involve calculating the flux of carbon dioxide (CO₂) and methane (CH₄) gases in the source zone. These estimates can be influenced by environmental and climatic factors, such as microbial respiration in the soil, moisture levels, rainfall, and fluctuations in atmospheric pressure [16]. To address the limitations associated with conventional approaches, alternative methods have been developed to estimate NSZD rates by calculating biodegradation in the source zone through the measurement of subsurface temperature profiles.

In the aquifer region, where the contaminant plume resides, heat is generated through the degradation of organic matter under anaerobic conditions, characterizing the methanogenesis zone [11,12,16]. In this area, hydrocarbons are transformed through methanogenesis, producing heat, CO₂, and CH₄. These gases rise to the upper layer, where they mix with oxygen infiltrating from the atmosphere into the soil, leading to the oxidation of CH₄, a process that also releases heat. Finally, the gases pass through the aerobic transport zone before being dispersed into the atmosphere. The magnitude of the gas flux is depicted by the size of the arrows in Figure 1, where the gray color is used to represent the flux of CO₂, light blue is used for CH₄, and dark blue is used for oxygen (O₂).

Based on the concept of an increase in temperature caused by the decomposition of petroleum hydrocarbons, the author in [16] developed a method with a mathematical model capable of estimating the rates of NSZD. This method involves measuring soil temperature at different depths and forming a vector with the temperature data of the affected zone. In order to identify the temperature variation caused solely by hydrocarbon depletion, any external effects must be mitigated. In view of this, a temperature vector with the same depths should be obtained at a nearby site with similar soil characteristics, but in a background region not affected by the contaminant source. Thus, it is possible to isolate the variations caused by hydrocarbon depletion. This is done by separating them from those caused by external factors, such as solar irradiation on the surface and variations in ambient temperature. The result is a corrected temperature change that takes into account just the changes caused by the decomposition of organic matter. On the basis of the corrected temperature values, it is possible to estimate the energy generated by the decomposition, which can then be converted into an NSZD rate ($R_{\mathrm{NSZD}}$). Figure 1 illustrates the modeled process, where the red crosses denote the corrected temperature points.

In the first stage of applying the model, an energy balance is made for the release of the LNAPL. In the second step, heat flows and generating sources are considered. Finally, the model processes the temperature differences between the affected zone and the reference in the background region for measurements taken at the same depth. In addition, for the model to be solved, two considerations are necessary: the behavior of the soil in the observed region is similar to that of the region used as a reference and the thermal conductivity of the soil is homogeneous. The temperature at different depths is the sum of the heat generated by the decomposition of NSZD and other sources, such as the effects on the soil surface and the effects of groundwater [20]. Assuming that the temperature at the background is affected by the same factors except for the presence of the heat source from the decomposition of the LNAPL, the temperature variation caused by NSZD at depth z and time instant i can be approximated by:

$$T_{{\mathrm{NSZD}}_{z,i}}=T_{{\mathrm{imp}}_{z,i}}-T_{{\mathrm{r}}_{z,i}};$$

(1)

where: $T_{{\mathrm{imp}}_{z,i}}$ is the temperature observed in the impacted region at depth z and time instant i; and $T_{{\mathrm{r}}_{z,i}}$ is the temperature in the background region at the same depths and time instant. The energy generated by the decomposition of the NSZD can be obtained for any instant of time i after solving a balance equation for the control volume illustrated in Figure 2 and assuming that heat exchanged in the horizontal direction can be neglected [21]:

$${\dot{E}}_{z_{\inf },i}-{\dot{E}}_{z_{\sup },i}+{\dot{E}}_{\mathrm{NSZD},i}={\displaystyle \frac{d{E}_{\mathrm{sto},i}}{dt}};$$

(2)

where: ${\dot{E}}_{z_{\inf },i}$ is the energy flux at the bottom of the control volume; ${\dot{E}}_{z_{\sup },i}$ is the energy flux at the top of the control volume; ${\dot{E}}_{\mathrm{NSZD},i}$ is the vertically integrated energy produced through NSZD over the height of the control volume; $E_{\mathrm{sto},i}$ is the vertically integrated stored energy over the height of the control volume; and t is time.

The conductive energy fluxes at the top and bottom of the control volume can be calculated using Fourier’s law as:

$${\dot{E}}_{z_{\inf },i}=-k_{\mathrm{sat}}{\left.{\displaystyle \frac{d{T}_{{\mathrm{NSZD}}_{z,i}}}{dz}}\right|}_{z=z_{\inf }};$$

(3)

$${\dot{E}}_{z_{\sup },i}=-k_{\mathrm{unsat}}{\left.{\displaystyle \frac{d{T}_{{\mathrm{NSZD}}_{z,i}}}{dz}}\right|}_{z=z_{\sup }};$$

(4)

where: $k_{\mathrm{sat}}$ is the thermal conductivity in the saturated region; $k_{\mathrm{unsat}}$ is the thermal conductivity of the unsaturated region; and $z_{\sup }$ and $z_{\inf }$ denote the depths corresponding to the upper and lower limits of the evaluated control volume, respectively. The change in energy stored within the control volume with respect to time, ${\displaystyle \frac{d{E}_{{\mathrm{sto}}_i}}{dt}}$, can be approximated using a finite difference:

$${\displaystyle \frac{d{E}_{{\mathrm{sto}}_i}}{dt}}\approx {\displaystyle \frac{E_{\mathrm{sto},i}-E_{\mathrm{sto},i-1}}{t_i-t_{i-1}}};$$

(5)

where: $t_i$ is the current instant of time and $t_{i-1}$ is the time of the previous sample. In addition, $E_{{\mathrm{sto}}_i}$ is defined by:

$$E_{{\mathrm{sto}}_i}=C_{\mathrm{unsat}}{\int }_{z_{\sup }}^{z_{\mathrm{wt}}}{T}_{{\mathrm{NSZD}}_{z,i}}dz+C_{\mathrm{sat}}{\int }_{z_{\mathrm{wt}}}^{z_{\inf }}{T}_{{\mathrm{NSZD}}_{z,i}}dz;$$

(6)

where: $C_{\mathrm{sat}}$ is the heat capacity of saturated soil; $C_{\mathrm{unsat}}$ is the heat capacity of unsaturated soil and $z_{\mathrm{wt}}$ is the height of the water table.

After substituting (3) to (6) in (2) and isolating ${\dot{E}}_{\mathrm{NSZD}}$, the NSZD rate on a volumetric base (volume per time per unit area) can be calculated as [21]:

$$R_{\mathrm{NSZD},i}=-{\displaystyle \frac{{\dot{E}}_{\mathrm{NSZD},i}}{\Delta H_{\mathrm{decane}}{\rho }_{\mathrm{decane}}}};$$

(7)

where: $R_{\mathrm{NSZD},i}$ indicates the rate of NSZD per unit area at time instant i; $\Delta H_{\mathrm{decane}}$ is the enthalpy of the hydrocarbon degradation reaction, with an approximate value of −6797.1 kJ mol⁻¹ obtained from an average constituent (decane in this case); and ${\rho }_{\mathrm{decane}}$ is the molar concentration of the LNAPL constituents, which in this case was approximated by the value for decane, 5.14 mol L⁻¹ [22].

3. System for Thermal Monitoring and Quantification of the Mass Depletion of Petroleum Hydrocarbons

Thermal monitoring to estimate the NSZD rate is a relatively recent approach, so not many tools have been reported in the literature for this purpose. Published temperature-based approaches involve either continuously measuring vertical temperature profiles in both a background well and areas impacted by LNAPL [11,12,21], or focus solely on areas impacted by LNAPL without the need to measure the background [13].

The system used in this study to remotely monitor hydrocarbon depletion in regions affected by oil spills is designed to operate autonomously, powered by a battery recharged by a solar panel. The measured data are stored in a datalogger, which communicates wirelessly with a computer in a station inside a laboratory. Field data are processed and made available online via a custom application developed specifically for this purpose.

The NSZD estimation model contains various inputs, which come from data from transducers installed in both contaminated and background areas. In addition to the vertically distributed temperature transducers in the soil, the system for thermal monitoring includes the following: soil moisture transducer to determine the heat transfer properties of the soil; a transducer to determine the water table level, since the distribution of contaminants in the soil can migrate with the variation in level and the water table level is important to determine the thermal properties of the surrounding soil; a transducer to measure relative humidity and air temperature; a transducer to monitor the incident solar radiation in the region; a transducer to measure rainfall in order to characterize the conditions of the external environment. Figure 3 illustrates the designed system.

The temperature transducers used in the system are Measurement Specialties Micro-BetaCHIP 10K3MCD1 thermistors [23] in a voltage divider configuration with a precision 10 kΩ resistor. The final setup was calibrated at 10 points around the temperature values of interest using a Wika CTH6500 thermometer [24] as standard. The repeatability observed in the calibration process was ±0.04 °C, which combined with the accuracy of the standard (±0.03 °C) and the inherent error of the Steinhart-Hart equation (±0.02 °C) resulted in a measurement uncertainty of ±0.1 °C for temperature measurements from 10.0 °C to 40.0 °C. This temperature range corresponds to the expected subsurface temperatures in the region under analysis.

The water table level transducer is a Campbell Scientific CS456 [25], based on differential pressure measurement. The measurement range is 0.00 m to 2.00 m and the accuracy is ±0.1%. The transducer accuracy was combined with the positioning accuracy and then expanded considering a coverage probability of 95.45%, resulting in ±10 mm (±0.01 m). The groundwater level data used in this study oscillate between 8.50 m to 9.05 m. In addition, the pollutant is located at an approximate depth of 8 m. Since the variation in subsurface temperature due to ambient temperature changes is practically minimal below 8 m [21], the positioning uncertainty of the temperature transducers has no significant impact on the measurement results. In addition, methanogenesis is an anaerobic process, so the depth of the LNAPL does not affect the released heat.

4. Method for Propagating Uncertainties Applied to NSZD Results

The rate of hydrocarbon decomposition is obtained by identifying and separating the amount of heat generated by the decomposition of LNAPL from the energy that crosses the boundaries of the control volume. In general, in cases where a measurand Y is not obtained directly, but from other N quantities, $X_1,X_2,\cdots ,X_N$, considered as input variables, by means of a functional relationship f, the representation $Y=f\left(X_1,X_2,\cdots ,X_N\right)$ is usual. In these cases, an estimate y of the measurand Y is obtained by $y=f\left(x_1,x_2,\cdots ,x_N\right)$, where $x_i$ is the expected value of $X_i$. In the case of this work, measurand Y is the NSZD rate and the input variables are the temperatures in the affected region, the reference temperatures in the background region; the observed depths; the height of the groundwater; and the characteristics of the soil.

Determining the uncertainty of y using the classic uncertainty propagation method requires calculating the sensitivity coefficients of the input quantities, which are derived from the partial derivatives of the output quantity in relation to each input variable [26]. For some cases, such as those in which the mathematical model of the measurement is highly non-linear, the evaluation of the uncertainty of y be carried out using a numerical method [27]. This involves performing random sampling of the probability distributions of the model input variables and applying the random values in the input of the mathematical model of a measurement. This alternative approach makes it possible to evaluate the distribution of the uncertainty associated with the measurement by combining an appropriate number of simulated random samples. MCM is a practical approach recommended to extract random samples from probability density functions (PDFs) [27,28]. In implementing the technique, the PDF for the variable $X_i$ is denoted by $g_{x_1}\left({\xi }_i\right)$, where ${\xi }_i$ is a variable representing the possible values of $X_i$, and M vectors $x_{r,1},x_{r,2},\cdots ,x_{r,N},r=1,\cdots ,M$ are generated. Using f, M estimates $y_r=f\left(x_{r,1},x_{r,2},\cdots ,x_{r,N}\right)$ are obtained. By sorting the values of $y_r$ in increasing order, an output vector G of size M is obtained, from which the cumulative distribution function $G_y\left(\eta \right)$ can be approximated, where $\eta$ denotes the possible values of the measurand Y. The quality of the approximation of $G_y\left(\eta \right)$ strongly depends on the size of the discrete representation G and is decisive in obtaining the statistical parameters of interest, such as the mean and standard deviation, which are associated with the estimated value $\widehat{y}$ and its standard uncertainty $u\left(\widehat{y}\right)$, respectively. According to [27], good approximations are generally obtained from a million simulations ($M={10}^6$). In this study, the NSZD rate is the measurand y, so each $y_r$ of the MCM will be denoted as $R_{\mathrm{NSZD},r}$, and the estimate of the NSZD rate, ${\overline{R}}_{\mathrm{NSZD}}$, can be obtained by:

$${\overline{R}}_{\mathrm{NSZD}}={\displaystyle \frac{1}{M}}\sum _{r=1}^M{R}_{\mathrm{NSZD},r}.$$

(8)

The standard uncertainty obtained from the M Monte Carlo simulations, $u\left({\overline{R}}_{\mathrm{NSZD}}\right)$, can be obtained by:

$$u\left({\overline{R}}_{\mathrm{NSZD}}\right)=\sqrt{{\displaystyle \frac{1}{M-1}}\sum _{r=1}^M{\left(R_{\mathrm{NSZD},r}-{\overline{R}}_{\mathrm{NSZD}}\right)}^2}.$$

(9)

This alternative approach also makes it easier to estimate the expanded uncertainty of ${\overline{R}}_{\mathrm{NSZD}}$, $U\left({\overline{R}}_{\mathrm{NSZD}}\right)$, given a coverage probability p, even in cases where the distribution representing the possible measurement values is not normal. Given that the equation that relates subsurface soil temperature to hydrocarbon depletion rate is complex, the MCM for propagating distributions is the most suitable tool for assessing uncertainties based on the variability of input data.

5. Calculating the Impact of Temperature Measurement Uncertainty on NSZD Estimates

The following simplifications were considered to assess the impact of temperature uncertainties on the estimation of NSZD decay levels:

• the characteristics of the soil in the monitored region and in the background region are considered the same [21];
• the uncertainties of thermal conductivity in the saturated and unsaturated regions are zero;
• the uncertainty in the enthalpy of the hydrocarbon degradation reaction is zero;
• the uncertainty associated with the depth of the temperature transducers is ±0.5 mm and the uncertainty of groundwater level measurements is ±10 mm, as detailed in Section 3;
• the temperature data used in this study were acquired over one month of measurements, a time period in which the drift of the thermistors can be neglected.

The method presented is general, while numerical results refer to a specific case study. The method remains applicable regardless of the scenario under analysis. It can evaluate the effects of components omitted in this analysis, although this is not the primary focus. This study primarily investigates the influence of temperature measurements on the NSZD rate. The findings aim to support the selection of suitable transducers for specific applications.

Applying the propagation of the uncertainty distribution through the model presented in (7), it was possible to assess the impact of the temperature uncertainty on the estimates of the NSZD rate. All sources were assumed to have uniform distributions. Algorithm 1 was implemented, where:

• $t_{\mathrm{v}}$ is the temperature array (containing measured temperatures);
• $u_{\mathrm{v}}$ is the temperature uncertainty array (uniform distribution bounds for each temperature);
• $d_{\mathrm{v}}$ is the array with the depth of each temperature transducer;
• $u_{\mathrm{dv}}$ is the depth uncertainty array (uniform distribution bounds for each depth);
• $w_{\mathrm{v}}$ is the array with the level of the water table;
• $u_{\mathrm{wv}}$ is the water table level uncertainty array (uniform distribution bounds);
• $i_{\mathrm{v}}$ is the array with the other parameters necessary to calculate the NSZD rate;
• $l_{\mathrm{b}}$ and $u_{\mathrm{b}}$ are the lower and upper bounds of the uncertainty associated with the temperature, respectively;
• $l_{\mathrm{db}}$ and $u_{\mathrm{db}}$ are the lower and upper bounds of the uncertainty associated with the depth;
• $l_{\mathrm{wb}}$ and $u_{\mathrm{wb}}$ are the lower and upper bounds associated with the water table level;
• $t_{\mathrm{adjusted}}$, $d_{\mathrm{adjusted}}$, and $w_{\mathrm{adjusted}}$ are arrays of adjusted temperature, depth, and water table levels, respectively, after adding random values from their uncertainty distributions;
• $R_{{\mathrm{NSZD}}_{\mathrm{v}}}$ is the array that contains the NSZD rate estimates for the adjusted temperatures and depths;
• ${\overline{R}}_{\mathrm{NSZD}}$ is the mean value of the NSZD rate estimates; and $C_{\mathrm{interval}}$ represents the confidence interval bounds considering a coverage probability of 95.45%.

Figure 4 shows the histogram and the results obtained for the particular case described in Section 3, with temperature uncertainty of ±0.1 °C. In this case, for a 95.45% confidence interval, the estimated value of the NSZD rate would be 5.2 ± 2.7 L m⁻² year⁻¹. The values and standard uncertainties used in the simulation are provided in

Table 1. Parameters and uncertainties used in the simulation.

Parameter	Value
$k_{\mathrm{sat}}$	2.29 W m−1 K−1
$k_{\mathrm{unsat}}$	1.10 W m−1 K−1
$C_{\mathrm{sat}}$	2.8 × 106 J m−3 K−1
$C_{\mathrm{unsat}}$	1.46 × 106 J m−3 K−1
$z_{\sup }$	0.1 m
$z_{\inf }$	20 m
$u_{\mathrm{w}}$	±0.01 m
$u_{\mathrm{t}}$	±0.1 °C
$u_{\mathrm{d}}$	±0.0005 m

, where: $u_{\mathrm{t}}$ represents the temperature transducer uncertainty, $u_{\mathrm{w}}$ is the uncertainty in groundwater level measurements, and $u_{\mathrm{d}}$ is the uncertainty associated with the depth of the temperature transducers. It is interesting to note that although rectangular distributions were considered for the model input data (uncertainties of the directly measured variables that act as the model input), the resulting uncertainty distribution has a shape that resembles a normal distribution. This behavior is explained by the central limit theorem, since none of the sources of input uncertainty is preponderant in relation to the others [29]. Although the results were obtained through simulation, the method presented here can be used to evaluate any scenario since the model used in the Monte Carlo simulation is the same as that used in real cases to evaluate natural remediation.

Algorithm 1: NSZD Estimation with Monte Carlo Temperature Uncertainty

It is also possible to use the proposed method to consider different theoretical scenarios in order to evaluate the effect of temperature uncertainty on the uncertainty of the NSZD rate. To do this, the uncertainties for the NSZD rate were theoretically obtained, considering that the temperature measurement chain has an uncertainty ranging from 0.01 °C to 1.00 °C, with a uniform distribution. The results of the uncertainty of the NSZD rate based on simulations with a step of 0.01 °C from 0.01 °C to 1.00 °C for the temperature uncertainties are presented in Figure 5. It can be observed that both variables have an almost linear relationship with an increase of 5.25 L m⁻² year⁻¹ in the uncertainty of the NSZD rate for each increase of 0.1 °C in the uncertainty of temperature measurements.

6. Conclusions

The method described in this paper allows establishing minimum requirements for specifying temperature measurement systems for thermal monitoring and quantification of the mass depletion of petroleum hydrocarbons. This capability has the potential to popularize the method as an alternative to active remediation techniques.

The proposed uncertainty evaluation method is based on the Monte Carlo simulation, which does not require the classification into Type A and Type B evaluations of uncertainty. The probability density functions assigned to all input quantities were primarily obtained from prior knowledge concerning the quantities based on information from the datasheets of the transducers used. In some cases in which prior knowledge was not available, such as in the placement of the temperature transducers, the standard deviation of independently obtained measurements was considered to define the probability density function of the input quantities. The combined approach provided a comprehensive analysis of measurement uncertainty, which improved the reliability of the results.

Simulations in a particular case study showed that the uncertainty of NSZD is directly related to the uncertainty of subsurface temperature measurements. Temperature uncertainty tends to be the most critical factor for the specification of the instrumentation system for indirect NSZD estimation, since the NSZD rate is evaluated from the temperature difference between the contaminated soil and a background reference. The proposed method enables the assessment of the propagation of uncertainty from the temperatures to the NSZD rate and establishes some minimum requirements so that hydrocarbon decay levels can be satisfactorily estimated. The Monte Carlo method proved to be a reliable alternative for this purpose. In the case study, it was noticed that the two variables exhibit an approximately linear relation, with the uncertainty of the NSZD rate increasing by 5.25 L m⁻² year⁻¹ for each 0.1 °C rise in the uncertainty associated with temperature measurements.

To improve the understanding of the impact of measurement uncertainties on the estimation of NSZD rate, the next steps of the study are to evaluate the results of the model considering the uncertainties of other quantities relevant to the estimation of NSZD rate, such as thermal conductivity and heat capacity of the soil, the composition of the LNAPL, and horizontal heat transfer. Evaluating the measurement uncertainties of the input quantities makes it possible to identify the reliability of the NSZD rate estimate, improving decision-making when it comes to specifying the instrumentation to be used in the field.