Hydrological Transport Modeling: Bridging Theory, Simulation, and Decision Support

LAHORE UNIVERSITY OF MANAGEMENT SCIENCES DEPARTMENT OF PHYSICS Hydrological Transport Modeling: Bridging Theory, Simulation, and Decision Support A thesis submitted in partial fulfillment of the requirements for the degree of Bachelor of Science in Physics Submitted by: Ahmed Saeed Supervised by: Dr. Sabieh Anwar Co-supervised by: Dr. Talha Manzoor Lahore, Pakistan May, 2025 Abstract This work explores hydrological transport processes across scales by weaving together fundamental theory, numerical simulation, and field-informed application. In Part I, we review catchment water balance concepts and subsurface flow dynamics, drawing on Darcy’s Law and Richards’ equation to establish a conceptual foundation. Part II develops two complementary tools: a semi-distributed tank model in Vensim for simulating rainfall–runoff and reservoir operations, and a modular Python implementation of Richards’ equation that serves as a virtual laboratory for examining wetting-front behavior and mass conservation. In Part III, these methods are applied to the Namal Reservoir catchment in Pakistan, where the Vensim model is calibrated and validated using hydrometric data from the Centre for Water Informatics at LUMS. Additionally, a clear-acrylic soil column test bed is designed to calibrate capacitive soil moisture sensors under controlled conditions while visualizing infiltration phenomena. Finally, we consider the Representative Elementary Watershed (REW) framework as a promising direction for upscaling and incorporating process memory. 1 Acknowledgments First and foremost, I am deeply indebted to my supervisors , Dr. Talha and Dr Sabieh, whose patient guidance, detailed meetings, and steadfast oversight turned the potential chaos of this exploratory research into a coherent journey. This work simply would not have been possible without their help and encouragement. I am forever grateful to my family: to my father for his guidance and support over the years that made all of this possible; to my mother for her unwavering reassurance when times were tough and home was far; and to my sister for always being proud of me and cheering me on. I am thankful to Dr. Rizwan Khalid for giving me the bravery to embrace the unknown and for teaching me that academic paths need not follow a one-size-fits-all model. To Dr. Faryad, thank you for showing me that a true scientist extends kindness to everyone, especially to themselves. To Dr Safee Ullah Chaudhary, for his infectious desire to do research to create good for the little time we have in this world. A special note of gratitude to my roommate, Essa, for the cafeteria runs when I was overwhelmed with deadlines, and for his help and support. To Hasan and Jarri, thank you for keeping me connected to life beyond the walls of the School of Science and Engineering. To the dwellers of the senior room, Saad Hasan, Jibrael, Alaynah, and others, thank you for your friendship and advice. I would like to thank the Physics Department at LUMS for providing me with all of the resources I needed, and the Centre for Water Informatics at LUMS for housing a confused physics student who did not know where to put his tools to work. And finally, to my best friend Adan, for standing by me throughout every step of this process and for knowing me: before, during, and after it was all over. 2 “Water can flow, or creep, or drip, or crash”, “Be water, my friend.” – From a 1971 episode of the American television series “Longstreet” — uttered by Bruce Lee. In Chinese Taoism, water represents power and flexibility in the face of obstacles. Contents I Fundamentals 6 1 Introduction 7 1.1 Overview ....................................... 7 1.2 Thesis Structure .................................... 7 2 Hydrological Principles: Catchment Water Balance and Subsurface Flow 9 2.1 Catchment Model Water Balance ........................... 9 2.2 Sub-surface ...................................... 10 II Models 13 3 Tank-Based Watershed Modeling: Concepts and Implementation 14 3.1 Watershed Model Overview ............................. 14 3.1.1 Significance of Watershed Modeling ..................... 14 3.1.2 Conceptual Basis of the Tank Model ..................... 15 3.1.3 Storage Units: Stocks ............................. 15 3.1.4 Flow Pathways: Water Movement ...................... 16 3.1.5 General Development Framework ...................... 16 3.1.6 Impacts and Applications ........................... 17 4 Computational Modeling of Hydrological Dynamics 18 4.1 Introduction and Workflow Overview ........................ 18 4.2 Soil Hydraulic Property Curves ............................ 19 4.3 Solver Function Implementation ........................... 19 4.4 Numerical Experiment Setup ............................. 21 4.5 Static Vertical Profiles ................................ 21 4.6 Time-Series Diagnostics ............................... 23 4.7 Animation of Transient Wetting Front ........................ 24 4.8 Interpretation and Pedagogical Value ......................... 25 4 III Application 26 5 Decision-Support Hydrological Modeling for Namal Reservoir Operations 27 5.1 Hydrological Modeling of the Namal Valley: A Comprehensive Case Study . . . . 27 5.1.1 Historical Context of the Namal Valley ................... 27 5.1.2 Project Objectives and Rationale ....................... 28 5.1.3 Methodology: Model Development ..................... 29 5.1.4 Outcomes and Implications .......................... 33 6 Design of a Controlled Calibration Environment 35 6.1 Soil Moisture Sensor Calibration Test Bed ...................... 35 6.1.1 Introduction .................................. 35 6.1.2 Design of the Test Bed ............................ 36 6.1.3 Calibration Procedure ............................ 38 6.1.4 Visualization of Hydrological Phenomena .................. 40 6.1.5 Implementation and Preliminary Results ................... 41 6.1.6 Conclusion .................................. 42 7 Discussions 44 7.1 Drivers for a New Paradigm ............................. 44 7.2 The REW Conceptual Framework .......................... 44 7.3 Comparison with Classical Models .......................... 45 7.4 Implications for Practice ............................... 45 7.5 Conclusion ...................................... 46 5 Part I Fundamentals 6 Chapter 1 Introduction 1.1 Overview Water resource management poses significant challenges in semi-arid regions, particularly in countries such as Pakistan, where population growth, climate variability, and competing agricultural demands strain existing infrastructure. The need to develop and calibrate models at various scales to understand the flow of water is increasingly vital to our survival, to ensure that we utilize it to its best purpose. This exploratory research follows a three-pronged structure, progressing from fundamental hydrological theory to model development and real-world application. The theoretical foundation addresses the dynamics of the water balance in the catchment and the mechanisms of subsurface flow, drawing on Darcy’s law and the Richards equation. Building on this, a semi-distributed tank model is developed to simulate hydrological processes, using field data provided by the Centre for Water Informatics and Technology (WIT) for model validation and strategy development. This is complemented by numerical simulations that model the physical processes described by Darcy [1856] and Richards [1931]. 1.2 Thesis Structure The document progresses systematically from theoretical foundations to practical implementations: •Part I: –Establishes core hydrological principles. –Covers catchment water balance equations. –Explores subsurface flow mechanisms using the Darcy-Richards framework. •Part II: 7 –Details technical implementations: ∗Development of a Vensim-based watershed model. ∗Creation of a Python-based program to simulate concepts from Part I. •Part III: –Presents the Namal Reservoir case study: ∗Analysis of hydrological behavior. ∗Model calibration and validation results. ∗Test bed design and calibration applications. This structure progressively builds understanding from basic principles to complex system interactions, culminating in practical tools for water resource managers. The integrated approach: •Addresses technical hydrological challenges through multi-scale modeling. •Considers socio-ecological implications via conflict-aware operational frameworks Sivapalan et al. [2012]. 8 3.1.2 Conceptual Basis of the Tank Model The tank model offers a framework for simulating hydrological processes by conceptualizing a watershed as a series of storage units, or tanks, interconnected by flow pathways. Each tank represents a distinct component of the hydrological cycle, such as surface water, soil moisture, groundwater, or reservoir storage. The movement of water between these tanks is governed by flows, which are driven by physical principles and empirical relationships, such as rainfall input, runoff generation, infiltration into the soil, and controlled reservoir outflows. As a semi-distributed model, the tank model balances computational efficiency with spatial detail, dividing the watershed into functional zones to capture variability in hydrological behavior without requiring the intensive data demands of fully distributed models. This approach is particularly advantageous in data-scarce regions, where empirical relationships can be calibrated to limited observations. Implemented in system dynamics software, such as Vensim, the tank model leverages feedback loops and dynamic interactions to simulate both natural processes and human interventions, such as reservoir management. Its flexibility allows it to adapt to a wide range of catchment characteristics, making it a versatile tool for hydrological analysis. Figure 3.1: Conceptual schematic of a tank model, illustrating storage units (e.g., surface water, soil moisture, groundwater, reservoir) and their interconnecting flows (e.g., rainfall, runoff, infiltration, outflow). 3.1.3 Storage Units: Stocks The storage units, or stocks, in a tank model represent the primary reservoirs where water resides within the watershed. Surface water encapsulates the accumulation of precipitation on the ground, 15 serving as the initial point for runoff or infiltration. Soil moisture accounts for water stored within the soil matrix, governed by properties such as porosity and hydraulic conductivity. Groundwater represents water in subsurface aquifers, contributing to long-term storage and baseflow. Finally, reservoir storage corresponds to water collected in lakes or dams, managed for purposes such as irrigation or flood mitigation. Each stock is characterized by its capacity, determined by physical attributes like soil depth or reservoir volume, and initialized with baseline conditions that are refined during model calibration. 3.1.4 Flow Pathways: Water Movement The flow pathways in a tank model describe the transfer of water between stocks, driven by hydrological processes. Rainfall serves as the primary input, delivering water to the surface water stock. Runoff transports surface water to streams or reservoirs, influenced by factors such as surface slope and land cover. Infiltration facilitates the movement of water from the surface to the soil moisture stock, modulated by soil saturation levels. Percolation governs the downward transfer of water from soil to groundwater, occurring most readily when the soil is fully saturated. Outflows from reservoirs, whether controlled by human management or natural overflow, regulate water release to downstream systems. These flows are typically parameterized using mathematical equations or lookup functions to capture nonlinear relationships, such as the dependence of infiltration rates on soil moisture. 3.1.5 General Development Framework The development of a tank model follows a systematic and iterative process to ensure scientific rigor and practical applicability. Initially, a conceptual model is formulated, identifying the key hydrological processes to be represented and defining the stocks and flows that encapsulate them. This step draws on expert knowledge and literature to establish a theoretical framework. Next, relevant data—such as precipitation records, soil characteristics, and water level measurements—are collected to inform parameter values and support model validation. In regions with limited data, empirical relationships or generalized values from similar catchments may be employed. The model is then implemented in system dynamics software, where stocks are represented as storage units and flows as dynamic pathways, with causal relationships and feedback loops defined to capture system behavior. Parameters, such as runoff coefficients or soil porosity, are assigned based on data or assumptions, often using lookup functions for complex processes. Calibration involves adjusting these parameters to align model outputs with observed data, such as streamflow or reservoir levels, while validation confirms the model’s accuracy across diverse scenarios. Sensitivity analysis is conducted to identify parameters with significant influence on model performance, guiding further refinement and data collection efforts. 16 Figure 3.2: Schematic of flow pathways in a tank model, depicting the inflows and outflows of a single tank. 3.1.6 Impacts and Applications The tank model offers a straightforward framework for managing water resources. Because each “tank” (stock) and its associated flows are explicitly represented, it’s easy to spot and trace any implausible behavior—e.g. when the ending volume in a tank exceeds its capacity or when total water in the system grows without an external input, you can immediately follow the chain of flows and identify the culprit. Beyond debugging, the model supports basic forecasting: by adjusting inputs (rainfall, withdrawals, etc.) you can explore how the system responds under heavy-rain or drought scenarios. That in turn can inform design and operation decisions—such as sizing spillways or timing reservoir releases to balance flood protection against irrigation needs. Finally, because the stocks and flows are transparent, the tank model can help frame discussions around allocation trade-offs (upstream vs. downstream abstractions, environmental demands, etc.) on a common, easy-to-understand footing. This clarity makes it a useful decision-support tool for routine planning and for facilitating stakeholder dialogue. 17 Chapter 4 Computational Modeling of Hydrological Dynamics 4.1 Introduction and Workflow Overview This chapter reproduces how Richards’ equation is implemented as a modular Python notebook, transforming the nonlinear partial differential equation into a reproducible computational experiment as demonstrated by Ireson [2025] . The workflow is divided into clear stages that mirror the logical progression of a physical soil column test bed: •1. Library imports and configuration. Soil–water property routines (van Genuchten retention, capacity, and conductivity) are loaded alongside numerical libraries (NumPy for arrays, SciPy’s odeint for integration, and Matplotlib for visualization). •2. Definition of the solver function. The core subroutine, RichardsModel, computes the right-hand side of Richards’ equation in finite-difference form, combining Darcy flux calculations with local mass continuity. •3. Experiment parameters. Grid resolution, boundary conditions, and initial hydrostatic profile are specified in one place, enabling rapid parameter sweeps without rewriting code. •4. Numerical integration. Matric potential profiles, 𝜓(𝑧, 𝑡), are obtained by integrating the system of ODEs over time. •5. Post-processing and diagnostics. The raw output is converted into volumetric water content, total soil storage, change in storage, and boundary fluxes, which are then visualized. This structure promotes clarity, reproducibility, and ease of extension, just as a well-designed laboratory protocol would. 18 4.2 Soil Hydraulic Property Curves Before simulating infiltration, the model evaluates the soil’s constitutive functions. Figure 4.1 shows the graphs for three fundamental relationships: •𝜃(𝜓): volumetric water content versus pressure head. •𝐶(𝜓)=𝑑𝜃/𝑑𝜓: differential water capacity highlighting the inflection point where moisture release is most sensitive to suction changes. •𝐾(𝜓): unsaturated hydraulic conductivity exhibiting strong nonlinearity near saturation. These curves define the soil’s response to changing matric potential and form the backbone of both the numerical model and the forthcoming physical test bed. 4.3 Solver Function Implementation The Python function RichardsModel encapsulates the finite-difference scheme for one-dimensional, variably saturated flow. It proceeds by: - Computing water capacity 𝐶(𝜓)and node conductivity 𝐾(𝜓). - Applying boundary conditions: a prescribed infiltration flux or head at the soil surface and free drainage or fixed head at the base. - Calculating interface fluxes via Darcy’s law: 𝑞𝑖+1/2=−𝐾𝑖+1/2𝜓𝑖+1−𝜓𝑖 Δ𝑧+1, where the “+1” accounts for gravitational head. - Enforcing continuity to derive each cell’s temporal derivative: 𝑑𝜓𝑖 𝑑𝑡 =−𝑞𝑖+1/2−𝑞𝑖−1/2 Δ𝑧 𝐶(𝜓𝑖). Below is the core routine, shown as a LaTeX-formatted code listing: 1def RichardsModel(psi, t, dz, n, p, vg, qTop, qBot, psiTop, psiBot): 2# Differential capacity 3C = vg.CFun(psi, p) 4# Interface flux array 5q = np.zeros(n+1) 6 7# Surface boundary 8if qTop is None: 9Ksurf = vg.KFun([psiTop], p)[0] 10 q[n] = -Ksurf*((psiTop - psi[n-1])/(2*dz) + 1) 11 else: 19 Figure 4.1: van Genuchten soil hydraulic property curves: retention (top), differential capacity (middle), and conductivity (bottom) as functions of pressure head. 20 12 q[n] = qTop 13 14 # Base boundary logic omitted for brevity 15 16 # Internal fluxes 17 Kn = vg.KFun(psi, p) 18 Kmid = 0.5*(Kn[:-1] + Kn[1:]) 19 slope = (psi[1:] - psi[:-1]) / dz + 1.0 20 q[1:-1] = -Kmid * slope 21 22 # Continuity 23 dpsidt = -(q[1:] - q[:-1]) / (dz * C) 24 return dpsidt This function, together with odeint, forms the numerical solver for Richards’ equation. 4.4 Numerical Experiment Setup For a typical infiltration simulation, the following configuration is used: •Soil column depth: 5 m, discretized into 50 cells (Δ𝑧=0.1 m). •Initial condition: hydrostatic suction given by 𝜓0(𝑧)=−𝑧. •Boundary forcing: constant infiltration flux of 0.01 m/day at the top; free drainage at the bottom. •Simulation period: 0 to 10 days, with 101 equally spaced time points (0.1-day interval). These choices ensure that the advancing wetting front is resolved while maintaining solver stability. 4.5 Static Vertical Profiles Vertical slices of pressure head and moisture content at selected times illustrate the progression of infiltration and redistribution. Figure 4.2 is a placeholder for these plots, showing the initial hydrostatic profile, the emergence of a sharp inflection at mid-infiltration, and the approach to a near-uniform state under sustained flux. These snapshots reveal the nonlinear behavior of the wetting front and the strong coupling between matric potential and moisture content. 21 Figure 4.2: Vertical profiles of matric potential (left) and volumetric water content (right) at initial, mid-infiltration, and near-steady-state times. 22 4.6 Time-Series Diagnostics To complete the water balance analysis, the model computes the total change in soil storage and compares it with infiltration and discharge fluxes. Figure 4.3 is a placeholder for the resulting time-series plot, which confirms mass conservation as infiltration increases storage until discharge catches up, eventually reaching steady state. Figure 4.3: Time series of change in storage (blue), infiltration (orange), and discharge (green) over the 10-day simulation. This diagnostic plot highlights the temporal interplay between storage dynamics and boundary fluxes, reinforcing the conceptual catchment water balance. 23 4.7 Animation of Transient Wetting Front To capture the continuous evolution of matric potential and moisture content through the column, we expanded on Ireson [2025]’s work by using matplotlib.animation.FuncAnimation. The update routine simply replaces the data for two pre-initialized line objects—one for 𝜓, one for 𝜃—at each frame index: 1def update(frame): 2line_psi.set_data(psi[frame,:], z) 3line_theta.set_data(theta[frame,:], z) 4ax_psi.set_title(f"Time = {t[frame]:.2f} days") 5return line_psi, line_theta 6 7from matplotlib.animation import PillowWriter 8anim = FuncAnimation(fig, update, 9frames=range(t.size-1), 10 interval=200, # ms per frame 11 blit=True) 12 anim.save('richards_animation.gif', 13 writer=PillowWriter(fps=20)) A representative frame from the resulting GIF is shown in Figure 4.4. It highlights the rapid advance of the wetting front and its gradual slowdown as deeper soil layers saturate. Figure 4.4: Snapshot from the animated profile showing the transient wetting-front propagation over ten days. 24 Figure 5.2: Simplified schematic of the Vensim tank model for the Namal Valley, illustrating stocks and flows. in lake elevation was attributable to the event. The storm event was programmed as a lookup function in Vensim, as shown in Figure 5.3, to drive the model’s rainfall input [Centre for Water Informatics and Technology, Lahore University of Management Sciences,2024]. Satellite-based ERA5 data were initially evaluated but discarded due to inconsistencies, such as missing time lags between inflows and precipitation. Soil properties, including porosity (0.5) and thickness (0.1 m), were derived from literature on the valley’s sandy/loamy soil, supplemented by data from the Pakistan Council of Research in Water Resources. Model Implementation in Vensim The model was constructed in Vensim’s graphical interface, with stocks represented as boxes and flows as arrows. Causal relationships and feedback loops were defined to capture dynamic interactions, such as the feedback between soil moisture and infiltration rates. Vensim’s simulation engine solved the system of differential equations governing stock accumulation and flow rates, using a time step of 10 minutes to align with sensor data resolution. The model’s structure was informed by a perceptual model, based on knowledge of the Namal Valley’s hydrology, and refined into a conceptual model incorporating physical processes like mass conservation and empirical relationships like runoff coefficients. The procedural model was coded in Vensim, with parameters tuned during calibration. 31 Figure 5.3: Lookup function for the July 12, 2021, storm event in Vensim. Calibration and Validation The model was calibrated using the July 12, 2021, storm event, comparing simulated lake elevation to observed data from WIT’s sensors. Initial runs revealed a discrepancy: the model predicted a 3-inch rise in lake elevation, while the observed rise was 3 feet. This was attributed to excessive infiltration and percolation, which were adjusted by refining the lookup functions for infiltration and percolation rates. The runoff coefficient was fine-tuned to 0.4 to better match observed inflows. Validation was performed using additional storm events from WIT’s dataset, ensuring the model could replicate lake elevation patterns under varying conditions. Sensitivity analysis identified infiltration and runoff coefficients as critical parameters, guiding further data collection efforts. Assumptions and Limitations The model incorporates several assumptions to simplify the complex dynamics of the Namal Valley: 1. Soil parameters, such as porosity, hydraulic conductivity, and thickness, are assumed constant across the catchment, neglecting variations due to soil type or land use. 2. Precipitation is assumed to be evenly distributed in the model, despite using Thiessen polygons for data processing to account for spatial heterogeneity. 3. Evapotranspiration and evaporation are excluded to focus on rainfall-runoff dynamics, potentially underestimating water losses. 32 Component Value/Description Rainfall Storm Event ×0.001 (m) Storm Event Lookup for July 12, 2021 Runoff Surface Water ×0.4 Infiltration Lookup based on soil moisture Percolation Lookup based on soil moisture Inflow Runoff with 8-hour delay Gate Outflow Max(0, 0.1 ×(Lake Elevation −Gate Threshold)) Soil Capacity Basin Area ×0.1 m ×0.5 Porosity 0.5 Soil Thickness 0.1 m Hydraulic Conductivity Lookup based on soil moisture Table 5.1: Key model components and their parameter values. 4. The catchment is treated as homogeneous for some processes, limiting the model’s ability to capture fine-scale spatial variability. 5. The basin area is assumed constant, ignoring changes due to sedimentation, erosion, or human activities. These assumptions introduce uncertainties, addressed through calibration and validation. Future refinements could incorporate spatial variability and additional processes to enhance accuracy. 5.1.4 Outcomes and Implications The model successfully reproduced the shape and time lag of lake elevation changes for the July 12, 2021, storm event, though initial discrepancies in magnitude highlighted the need for refined infiltration and percolation parameters. It provides insights into how rainfall translates into reservoir inflows, enabling predictions of water availability under various scenarios, such as monsoon floods or drought, and helps educate crucial decisions of water allocation in the most optimal way. The model’s socio-hydrological components, particularly the gate outflow, allow simulation of management decisions, informing strategies to balance upstream flood control and downstream irrigation needs. By elucidating upstream-downstream dynamics, the model supports conflict resolution and equitable water allocation, addressing long-standing tensions in the valley. Its scalability makes it applicable to other rainfed catchments, contributing to global efforts in sustainable water management. In conclusion, the hydrological modeling of the Namal Valley represents a significant step toward sustainable water management in a complex socio-ecological system. By combining physical rigor with empirical flexibility, the tank model offers a robust framework for addressing local challenges while contributing to broader hydrological science [Dingman,2015a]. 33 Figure 5.4: Comparison of simulated and observed lake elevation for the July 12, 2021, storm event. 34 Chapter 6 Design of a Controlled Calibration Environment 6.1 Soil Moisture Sensor Calibration Test Bed Soil moisture is a critical parameter in hydrological research, agricultural management, and environmental monitoring, influencing water availability, crop productivity, and ecosystem dynamics. At the Centre for Water Informatics and Technology (WIT) at Lahore University of Management Sciences, capacitive soil moisture sensors are developed to provide precise measurements of soil moisture content. Accurate calibration of these sensors is essential to ensure reliable performance across diverse soil types and moisture conditions. This chapter presents the design and functionality of a soil moisture sensor calibration test bed, specifically engineered to calibrate WIT’s sensors while also serving as an educational platform for visualizing fundamental hydrological phenomena, such as Darcy’s law and Richards’ equation. By enabling observation of the wetting front propagation, the test bed offers valuable insights into water movement through porous media, enhancing both sensor accuracy and pedagogical outcomes. 6.1.1 Introduction Soil moisture measurements are pivotal for optimizing irrigation, predicting hydrological processes, and managing water resources, particularly in regions facing water scarcity, such as Pakistan. Capacitive soil moisture sensors, like those developed at WIT, measure changes in the dielectric constant of soil to estimate moisture content, offering a non-invasive and efficient method for monitoring. However, the accuracy of these sensors depends on robust calibration to account for variations in soil composition, texture, and environmental conditions. The current calibration method for WIT’s sensors involves measuring capacitance in air (representing dry conditions) and water (representing full saturation), assuming a linear relationship between these extremes and soil moisture levels. While straightforward, this approach oversim35 plifies the complex interactions between soil, water, and sensor response, potentially leading to inaccuracies in field applications. A more precise calibration method is required, one that involves controlled hydration of soil samples with measured water volumes to establish a direct correlation between sensor readings and actual moisture content. The soil moisture sensor calibration test bed described herein addresses this need by providing a controlled environment for incremental soil hydration and gravimetric moisture measurement. Beyond calibration, the test bed serves as an educational tool, enabling visualization of key hydrological principles, including Darcy’s law, which governs fluid flow through porous media, and Richards’ equation, which describes water movement in unsaturated soils. The clear acrylic construction of the test bed allows for direct observation of the wetting front, enhancing understanding of infiltration and water redistribution processes. This dual-purpose apparatus thus contributes to both practical sensor development and theoretical hydrological education. 6.1.2 Design of the Test Bed The soil moisture sensor calibration test bed facilitates precise control over soil hydration and sensor placement. The test bed consists of a one-meter-high rectangular tank with a 0.3 m ×0.3 m opening at the top, constructed from clear acrylic to enable visual observation of soil moisture dynamics. A showerhead at the top simulates rainfall, ensuring uniform water distribution across the soil surface. At the bottom, a drainage layer—comprising gravel or a high-density polyethylene (HDPE) drainage mat—prevents waterlogging and maintains soil integrity. The apparatus includes several key components, labeled A through H, as illustrated in Figures 6.1 through 6.7. Key Components of the Test Bed The test bed comprises the following components, each designed to fulfill a specific function in the calibration and visualization process: A) Collection Water Tank: Located at the base of the main soil tank, this tank collects percolated water. It is open-topped and structurally reinforced to support the full system’s weight. The tank includes volume markings for precise water measurement (Figure 6.3). B) Soil Moisture Probe Entry Point: A slanted, 30-degree opening enables the secure insertion of moisture probes while minimizing soil displacement. The opening is dimensioned to ensure a tight fit, reducing leakage (Figures 6.4 and 6.5). C) Showerhead: A 12-inch ×12-inch showerhead uniformly distributes water across the soil surface. The adjustable flow rate simulates different rainfall intensities, modeled by: 𝑣𝑙=0.09 ×𝑥 where 𝑥is the rainfall rate (mm/hr), and 𝑣𝑙is the volume (liters/hour). Example rates: light rain (¡2.5 mm/hr, 0.224 L/hr), moderate rain (2.5–7.6 mm/hr, 0.45 L/hr), heavy rain (¿7.6 mm/hr, 0.684 L/hr). Excess outlets can be sealed to localize water application. 36 Figure 6.1: Disassembled view of the test bed apparatus. D) Drainage Layer: A gravel bed or HDPE drainage mat (e.g., HDPE Drainage Cell Mat Board from https://www.alibaba.com) is placed at the tank base. It ensures efficient water outflow and prevents waterlogging, maintaining soil structure (Figure 6.6). E) Drainage Outlet: Positioned at the tank’s lowest point with a slight base gradient, this outlet ensures full drainage and minimizes pooling or blockages. F) Drip Logger (Optional): A drip logger can track percolation in real-time by counting individual water droplets, enabling advanced analysis of infiltration and solute transport dynamics. G) Main Soil Tank: Made of clear acrylic for visualizing infiltration fronts. The panels are chemically bonded to handle soil and water loads without leaking (Figure 6.7). H) Volumetric Water Tank: With a 45-liter minimum capacity, this tank supports hydration of a 90-liter soil column (0.3 m ×0.3 m ×1 m). A 12V DC submersible pump (e.g., available at https://hallroadlahore.pk) delivers water through the showerhead and can pump to heights of 1.3 meters. Additionally, stress gauges (e.g., coin-sized sensors from the Physics Lab at LUMS) can be installed beneath the soil tank to provide real-time weight data. Once calibrated, these can monitor changes in moisture retention and bulk density. Consultation with Dr. Sabieh Anwar is recommended for implementation. 37 Figure 6.2: Assembled 3D view of the test bed (drawn to scale). The CAD design file for the test bed is available at: https://www.tinkercad.com/things/ j22VTP2KQ79/edit?returnTo=%2Fdashboard&sharecode=TQeXbQ8JCOy0TWjkDyw34dI_GsB9MaLg17_ pKPijfBw 6.1.3 Calibration Procedure The calibration of WIT’s capacitive soil moisture sensors using the test bed follows a systematic procedure to ensure accurate and reliable measurements: 1. Soil Preparation: Fill the main soil tank with a known mass of soil representative of field conditions (e.g., sandy or loamy soil common in the Namal Valley). Pack the soil uniformly to avoid preferential flow paths. 2. Initial Moisture Measurement: Insert the soil moisture sensors at multiple depths (e.g., 10 cm, 20 cm, and 30 cm) through the slanted probe entry points. Record the initial capacitance readings when the soil is completely dry. 3. Controlled Hydration: Add a measured volume of water using the volumetric tank and pump system through a showerhead to simulate rainfall. Begin with small increments (e.g., 5% of soil volume) and gradually increase to achieve specific moisture levels. Adjust the showerhead’s flow rate to simulate light, moderate, or heavy rainfall. 38 Figure 6.3: Percolation collection tank. 4. Equilibration: Allow the soil to equilibrate for 30 minutes to 1 hour after each hydration step to ensure uniform water distribution throughout the soil column. 5. Sensor Readings: Record the capacitance readings from the sensors at each moisture level, noting the depth and position of each sensor. 6. Gravimetric Measurement: Extract small soil samples using a core sampler at each hydration level. Weigh the samples, then dry them in an oven at 105 C until a constant weight is achieved. Weigh the dry samples again and compute the gravimetric moisture content using the formula: Gravimetric Moisture Content (%) =Wet Soil Mass −Dry Soil Mass Dry Soil Mass ×100 7. Correlation and Calibration Curve: Plot the capacitance readings against the gravimetric moisture content to develop a calibration curve for each sensor. This enables the conversion of raw capacitance values into accurate soil moisture percentages. 8. Validation: Deploy the calibrated sensors under field conditions across various soil types and environmental scenarios to assess and confirm their accuracy, robustness, and reliability. This procedure ensures that WIT’s sensors provide precise soil moisture data, addressing the limitations of the previous air–water calibration method. 39 Figure 6.4: Cross-section of the probe entry point. 6.1.4 Visualization of Hydrological Phenomena Beyond calibration, the test bed serves as an educational platform for demonstrating core hydrological principles, thereby enhancing understanding of water movement in soils. Darcy’s law, which governs the flow of fluids through porous media, states that the flow rate is proportional to the hydraulic gradient and the medium’s permeability. Using the test bed, students can measure the hydraulic gradient by comparing water levels and assess flow velocity by observing the movement of water through the soil and its collection in the tank below. When equipped with a drip logger, the setup provides real-time data on flow rates, allowing for the empirical verification of Darcy’s law in a controlled environment [Dingman,2015a]. Richards’ equation describes water movement in unsaturated soils, accounting for both gravitational and capillary forces. By monitoring soil moisture at various depths over time using the calibrated sensors, the test bed provides a dynamic view of how water infiltrates and redistributes within the soil column. This enables a practical illustration of Richards’ equation and underscores the influence of soil properties on moisture behavior [?]. The propagation of the wetting front—the interface between wet and dry soil—is clearly visible through the transparent acrylic walls of the main tank. This allows direct observation of infiltration dynamics and highlights the effects of soil texture and porosity on the advancement of moisture. The use of dye tracers can further enhance visibility, making the progression of the wetting front even more apparent to students and researchers [?]. 40 Bibliography Keith J. Beven. Rainfall-Runoff Modelling: The Primer. John Wiley & Sons, 2nd edition, 2012. Centre for Water Informatics and Technology, Lahore University of Management Sciences. Hydrometric sensor network data for namal valley, 2024. Accessed online, specific URL pending. Henry Darcy. Les fontaines publiques de la ville de Dijon: exposition et application des principes ` a suivre et des formules ` a employer dans les questions de distribution d’eau. Victor Dalmont, Paris, 1856. In this work, Darcy formulated the law governing flow of fluids through porous media. G. K. Devia, B. P. Ganasri, and G. S. Dwarakish. A review on hydrological models. Aquatic Procedia, 4:1001–1007, 2015. S. Lawrence Dingman. Physical Hydrology. Waveland Press, 3rd edition, 2015a. S.L. Dingman. Physical Hydrology. Waveland Press, 3rd edition, 2015b. B. P. Ganasri and G. S. Dwarakish. Classification of hydrological models for flood prediction. Aquatic Procedia, 4:417–424, 2015. Andrew Ireson. Richardsequation. https://github.com/amireson/RichardsEquation, 2025. URL https://github.com/amireson/RichardsEquation. GitHub repository. Ali H. Kazmi and M. Qasim Jan. Geology and Tectonics of Pakistan. Graphic Publishers, 1997. Talha Manzoor, Hassam Arshad, Hasan Arshad Nasir, Muhammad Sheraz, and Malik Jahan Khan. Development of hydrometric sensor networks in formerly ungauged watersheds: Lessons from namal valley, mianwali. In 2022 IEEE International Geoscience and Remote Sensing Symposium (IGARSS), pages xxxxx–xxxxx, 2022. Grey S. Nearing, Frederik Kratzert, Alden K. Sampson, Craig S. Pelissier, Daniel Klotz, Jonathan M. Frame, Cristina Prieto, and Hoshin V. Gupta. What role does hydrological science play in the age of machine learning? Water Resources Research, 56(3):e2019WR026091, 2020. 47 Punjab Tourism Department. Namal lake and dam: Environmental and social impacts, 2023. Accessed online, specific URL pending. L.A. Richards. Capillary conduction of liquids through porous mediums. Physics, 1(5):318–333, 1931. doi: 10.1063/1.1745010. Murugesu Sivapalan, Hubert H. G. Savenije, and G¨unter Bl¨ oschl. Socio-hydrology: conceptualising human-flood interactions. Hydrology and Earth System Sciences, 16(8):2673–2685, 2012. Thorsten Wagener, Howard S. Wheater, and Hoshin V. Gupta. Rainfall-runoff modelling: a review of the current state of the art. Progress in Physical Geography, 28(1):1–29, 2004. 48