Revisiting the Fundamental Analytical Solutions of Heat and Mass Transfer: The Kernel of Multirate and Multidimensional Diffusion

Abstract

There are two types of analytical solutions of temperature/concentration in and heat/mass transfer through boundaries of regularly shaped 1‐D, 2‐D, and 3‐D blocks. These infinite‐series solutions with either error functions or exponentials exhibit highly irregular but complementary convergence at different dimensionless times, td. In this paper, approximate solutions were developed by combining the error‐function‐series solutions for early times and the exponential‐series solutions for late times and by using time partitioning at the switchover time, td0. The combined solutions contain either the leading term of both series for normal‐accuracy approximations (with less than 0.003 relative error) or the first two terms for high‐accuracy approximations (with less than 10−7 relative error) for 1‐D isotropic (spheres, cylinders, slabs) and 2‐D/3‐D rectangular blocks (squares, cubes, rectangles, and rectangular parallelepipeds). This rapid and uniform convergence for rectangular blocks was achieved by employing the same time partitioning with individual dimensionless times for different directions and the product of their combined 1‐D slab solutions. The switchover dimensionless time was determined to minimize the maximum approximation errors. Furthermore, the analytical solutions of first‐order heat/mass flux for 2‐D/3‐D rectangular blocks were derived for normal‐accuracy approximations. These flux equations contain the early‐time solution with a three‐term polynomial in td and the late‐time solution with the limited‐term exponentials for rectangular blocks. The heat/mass flux equations and the combined temperature/concentration solutions form the ultimate kernel for fast simulations of multirate and multidimensional heat/mass transfer in porous/fractured media with millions of low‐permeability blocks of varying shapes and sizes.

Revisiting the Analytical Solutions of Heat Transport in Fractured Reservoirs Using a Generalized Multirate Memory Function

“…The average matrix temperature, T av m t ð Þ , in the REV can be written (Zhou, Oldenburg, Rutqvist, & Birkholzer, 2017;Zhou, Oldenburg, Spangler, & Birkholzer, 2017) as follows:…”

“…We also use a local memory function, g(t), and its Laplace transform, g*(s), to represent the conductive heat flux through fracture-matrix interfaces per unit temperature change in fractures. A new memory function, g*(s), is developed in section 3.2 using the diffusive flux equation developed recently (Zhou, Oldenburg, Rutqvist, & Birkholzer, 2017;Zhou, Oldenburg, Spangler, & Birkholzer, 2017). With G * 0 x; s ð Þ and g*(s) available, the new dualcontinuum solutions for heat transport in fractured reservoirs are developed using G *…”

“…For a single matrix block of regular shape (sphere, cylinder, slab, square, rectangle, cube, and rectangular parallelepiped), we use the unified-form diffusive flux equation with the first-order approximation (Zhou, Oldenburg, Rutqvist, & Birkholzer, 2017;Zhou, Oldenburg, Spangler, & Birkholzer, 2017):…”

“…The TRAT code was developed based on the Fortran code in Moench (1995) and was extended for pressure propagation (Zhou et al, 2009). The SHPALib also includes the subroutines for unified-form diffusive flux equations for various matrix blocks and intrablock solutions of solute, heat, and pressure transport (Zhou, Oldenburg, Rutqvist, & Birkholzer, 2017;Zhou, Oldenburg, Spangler, & Birkholzer, 2017).…”

“…A representative elementary volume (REV) of the fractured reservoir is conceptualized to contain multiple rectangular matrix blocks with different sizes, aspect ratios, properties, and volume fractions that are bounded by multiple well-mixed orthogonal fractures. A generalized multirate memory function, g * (s), is developed using the diffusive flux equation (Zhou, Oldenburg, Rutqvist, & Birkholzer, 2017;Zhou, Oldenburg, Spangler, & Birkholzer, 2017) to account for multirate conductive heat exchange between fractures and matrix blocks per unit matrix volume of a REV. The suite of analytical solutions is developed in the form:…”