The Essential Guide to Analytical Problem-Solving in Applied Sciences This textbook offers a wealth of analytical solutions to engineering and applied sciences problems, with a primary focus on heat transfer, groundwater flow, and contaminant transport in shallow subsurface environments. It emphasizes a deeper understanding of the mathematical framework underlying physical processes, inspiring an appreciation for the power and elegance of analytical modeling. Whether you are a student, instructor, researcher, or professional, this resource delivers the clarity and tools necessary to address ambiguities in existing models and unlock the methods to solving complex boundary value problems.

This paper introduces an analytical model analyzing the effect of groundwater flow on heat transfer in an infinite conductive-convective porous domain representing shallow geothermal systems with arbitrarily configured cylindrical heat sources. The model is formulated based on the moving source concept and solved based on the spectral analysis method and the superposition principle. Compared to models based on the Green's function and the Laplace transform, the proposed spectral model has a simpler formulation, computationally efficient and easy to implement in computer codes. It can handle random time-dependent thermal loads and any arbitrarily configured grid distribution. The verification and numerical examples demonstrate the computational capabilities of the model, and show how the groundwater flow can play an important role in the thermal interaction between heat sources. They also feature how to make use of the direction of groundwater flow to avoid undesirable thermal interaction between neighboring installations, rapid depletion of energy sources and unfair mining of geothermal energy.

This paper introduces a spectral model for a moving cylindrical heat source in an infinite conductive-convective domain. This physical process occurs in many engineering and technological applications including heat conduction-convection in ground source heat pump systems, where the borehole heat exchangers likely go through layers with groundwater flow. The governing heat equation is solved for Dirichlet and Neumann boundary conditions using the fast Fourier transform for the time domain, and the Fourier series for the spatial domain. A closed form solution based on the modified Bessel functions is obtained for the Dirichlet boundary condition and an integral form for the Neumann boundary condition. Limiting cases of the moving cylindrical heat source to represent a moving line heat source are also derived. Compared to solutions based on the Green's function and the Laplace transform, the spectral model has a simpler form, applicable to complicated time-variant input signals, valid for a wide range of physical parameters and easy to implement in computer codes. The model is verified against the existing infinite line heat source model and a finite element model.

This paper introduces a thermo-hydro-mechanical finite element model for energy piles subjected to cyclic thermal loading. We address four particular features pertaining to the physics of energy piles: three-dimensionality, embedded heat exchangers, soil constitutive modeling and pile–soil interface. The model is designed to capture the strong coupling between all important physical and thermomechanical processes occurring in a concrete pile embedding U-tubes heat exchangers and surrounded by a saturated soil mass. It encompasses solid and fluid compressibility, fluid and heat flow, thermoplastic deformation of soil, buoyancy, phase change, volume change, pore expansion, melting point depression, cryogenic suction and permeability reduction due to ice formation. The model is distinct from existing energy pile models in at least two features: (1) it can simulate the detailed convection-conduction heat flow in the heat exchanger and the associated unsymmetrical thermal interactions with concrete and soil mass; and (2) it can simulate cyclic freezing and thawing in the system and the associated changes in physical and mechanical properties of the soil mass that likely lead to thermoplasticity and deterioration of pile shaft resistance. The performance of the model is demonstrated through a numerical experiment addressing all its features.

This paper presents a new parameter identification (PI) algorithm for estimating effective and detailed thermal parameters of ground source heat pump systems using data obtained from the well-known thermal response test. The PI comprises an iterative scheme coupling a semi-analytical forward model to an inverse model. The forward model is formulated based on the spectral element method to simulate transient 3D heat flow in ground source heat pump (GSHP) systems, and the inverse model is formulated based on the interior-point optimization method to minimize the system objective function. Compared to existing interpretation tools for the thermal response test, the proposed PI algorithm has several advanced features, including: it can handle fluctuating heat pump power and inlet temperatures; interpret data obtained from multiple heat injection or extraction signals; produce accurate backcalculation for short and long duration experiments; and handle multilayer systems. The PI algorithm is tested against synthesized data, using a wide range of random noise, and versus an available laboratory experiment. The computational results show that the PI algorithm is accurate, stable and exhibiting relatively high convergence rate.

This paper introduces a thermo-hydro-mechanical computational model for freezing and thawing in porous media domains, with focus on freezing and thawing in soil. The model is formulated based on the averaging theory and discretized using a mixed discretization scheme, where the standard and extended finite element methods are simultaneously employed. It is capable of capturing the strong coupling between all important phenomena and processes occurring during relatively high freezing-thawing rates in porous media. Solid and fluid compressibility, buoyancy, phase change, thermomechanical behavior, water volume change, pores expansion, cryogenic suction, melting point depression and water migration to the freezing zone are all considered in the model. The cryogenic suction, in particular, is central to the occurrence of many of these phenomena and processes, and thus treated as a primary state variable, and discretized using the partition of unity method to make sure that it can be captured accurately. The paper presents detailed formulation of the governing equations and the numerical discretization. Verification and numerical examples are given to demonstrate the accuracy and computational capability of the model in describing the behavior of a soil mass subjected to boundary conditions resembling those occurring in the vicinity of an energy pile. The numerical examples show that the model is effectively mesh-independent and can simulate all important phenomena using relatively coarse meshes.

A semi-analytical model for simulating transient conductive-convective heat flow in a three-dimensional shallow geothermal system consisting of multiple borehole heat exchangers (BHE) embedded in a multilayer soil mass is introduced. The model is formulated in three steps, starting from an axial symmetric system and ending in a 3D multilayer, multiple BHE system. In step 1, the model is formulated as a single BHE embedded in an axial symmetric homogeneous soil layer, and the governing heat equations are solved analytically using the fast Fourier transform, the eigenfunction expansion and the modified Bessel function. In step 2, the model is extended to incorporate multiple layers using the spectral element method. And in step 3, the model is extended to incorporate multiple borehole heat exchangers using a superposition technique suitable for Dirichlet boundary conditions. The ensuing computational model solves detailed three-dimensional heat flow using minimal CPU time and capacity. The number of the required spectral elements is equal to the number of soil layers embedded in which any number of borehole heat exchangers with any layout configuration. A verification example illustrating the model accuracy and numerical examples illustrating its computational capabilities are given. Despite the apparent rigor of the proposed model, its high accuracy and computational efficiency make it suitable for engineering practice.

In this paper, we introduce a fully coupled thermo-hydrodynamic-mechanical computational model for multiphase flow in a deformable porous solid, exhibiting crack propagation due to fluid dynamics, with focus on CO₂ geosequestration. The geometry is described by a matrix domain, a fracture domain, and a matrix-fracture domain. The fluid flow in the matrix domain is governed by Darcy's law and that in the crack is governed by the Navier-Stokes equations. At the matrix-fracture domain, the fluid flow is governed by a leakage term derived from Darcy's law. Upon crack propagation, the conservation of mass and energy of the crack fluid is constrained by the isentropic process. We utilize the representative elementary volume-averaging theory to formulate the mathematical model of the porous matrix, and the drift flux model to formulate the fluid dynamics in the fracture. The numerical solution is conducted using a mixed finite element discretization scheme. The standard Galerkin finite element method is utilized to discretize the diffusive dominant field equations, and the extended finite element method is utilized to discretize the crack propagation, and the fluid leakage at the boundaries between layers of different physical properties. A numerical example is given to demonstrate the computational capability of the model. It shows that the model, despite the relatively large number of degrees of freedom of different physical nature per node, is computationally efficient, and geometry and effectively mesh independent.

This paper introduces a semi-analytical model for the simulation of transient heat transfer with friction heat gain in a single U-tube geothermal borehole heat exchanger subjected to an arbitrary heat flux signal. The friction effect appears as a nonhomogeneous term in the governing equations, which constitutes a set of coupled partial differential equations describing heat flow in the three components of the borehole; pipe-in, pipe-out and grout. We utilize the spectral analysis for discretizing the time domain, and the eigenfunction expansion for discretizing the spatial domain to solve the governing initial and boundary value problem. The proposed model combines the exactness of the analytical methods with an important extent of generality in describing the geometry and boundary conditions of the numerical methods. The model is verified analytically against a simplified one-dimensional solution. A numerical example is given to illustrate the effect of friction on heat transfer in the borehole heat exchanger for different fluid velocities and viscosities. The analysis shows; for the geometry, materials fluid velocities and viscosities, typically utilized in shallow geothermal systems; the friction is not really significant. However, the main advantage of this work is on the solution technique that can be useful for many other applications, including fluid flow in narrow pipes, high fluid velocities, high fluid viscosities, and pipes made of composite materials and of complex geometry. Also, the method can be useful for solving other nonhomogeneous coupled partial differential equations.

This paper introduces a computational model for transient high enthalpy fluid flow through geothermal wellbores. The drift-flux model is utilized to formulate the physical behavior of fluid, and the constitutive relationships are described using relevant equations of state and empirical relationships. The governing equations are solved using the finite element method. All important physical phenomena and processes occurring along the wellbore, including buoyancy, phase change, compressibility, thermal interaction, wall friction and slip between phases are considered. Airlifting of water and air, initially existing in the wellbore before production, is also considered. During airlifting and early stages of production, two fluids exist along the wellbore: airlifted water-dry air fluid, and reservoir water-vapor fluid; giving rise to a discontinuity in thermodynamic properties between the two fluids. The discontinuity is modeled using the level-set method. Two numerical examples illustrating the computational capability and accuracy of the model are presented. The physical phenomena occurring during airlifting and production along the wellbore are highlighted.

This paper introduces a multidomain-staggered technique for coupling multiphase flow in a porous medium, dominated by the Darcy laminar flow, with multiphase flow in a wellbore, dominated by the Navier Stokes viscous, compressible flow. The Darcy flow in the porous medium is formulated using the averaging theory, and the Navier Stokes flow in the wellbore is formulated using the drift-flux model. The governing equations are discretized using a mixed discretization finite element scheme, in which the partition of unity finite element method, the level set method and the standard Galerkin finite element method are combined in an integrated numerical scheme. A multidomain technique is utilized to uncouple the physical system into two subdomains, coupled back by enforcing flow constraints at their interaction boundaries. The resulting system of equations is solved using an iterative staggered technique and a multiple time-stepping scheme. This combination between the multidomain technique and the staggered-multiple time-stepping technique enables the use of different mathematical and numerical formulations for the two subdomains, and facilitates the implementation of a standard finite element computer code. The proposed model is tailored to simulate sequestered CO₂ leakage through heterogeneous geological formation layers and abandoned wellbores. A numerical example describing different leakage scenarios is given to demonstrate the computational capability of the model. The numerical results are compared to those obtained from a commercial simulator.

In this paper, we introduce analytical solutions for transient heat conduction in an infinite solid mass subjected to a varying single or multiple cylindrical heat sources. The solutions are formulated for two types of boundary conditions: a time-dependent Neumann boundary condition, and a time-dependent Dirichlet boundary condition. We solve the initial and boundary value problem for a single heat source using the modified Bessel function, for the spatial domain, and the fast Fourier transform, for the temporal domain. For multiple heat sources, we apply directly the superposition principle for the Neumann boundary condition, but for the Dirichlet boundary condition, we conduct an analytical coupling, which allows for the exact thermal interaction between all involved heat sources. The heat sources can exhibit different time-dependent signals, and can have any distribution in space. The solutions are verified against the analytical solution given by Carslaw and Jaeger for a constant Neumann boundary condition, and the finite element solution for both types of boundary conditions. Compared to these two solutions, the proposed solutions are exact at all radial distances, highly elegant, robust and easy to implement.

This paper introduces a semi-analytical model based on the spectral analysis method for the simulation of transient conductive-convective heat flow in an axisymmetric shallow geothermal system consisting of a double U-tube borehole heat exchanger embedded in a soil mass. The proposed model combines the exactness of the analytical methods with an important extent of generality in describing the geometry and boundary conditions of the numerical methods. It calculates the temperature distribution in all involved borehole heat exchanger components and the surrounding soil mass using the fast Fourier transform, for the time domain; and the complex Fourier and Fourier-Bessel series, for the spatial domain. Numerical examples illustrating the model capability to reconstruct thermal response test data together with parametric analysis are given. The CPU time for calculating temperature distributions in all involved components, pipe-in, pipe-out, grout, and soil, using 16,384 FFT samples, for the time domain, and 100 Fourier-Bessel series samples, for the spatial domain, was in the order of 3s in a normal PC. The model can be utilized for forward calculations of heat flow in a double U-tube geothermal heat pump system, and can be included in inverse calculations for parameter identification of shallow geothermal systems.