Transient heat transfer in a horizontal well in hot dry rock – Model, solution, and response surfaces for practical inputs

Highlights

• Geothermal energy system analysis of heat transfer (HT) to a horizontal well.

• FEA solution of equations governing TH in the rock mass and working fluid.

• Identified range of the 3 dimensionless parameters for physically meaningful inputs.

• Developed Response Surface Models (ROMs) for fluid Temperature vs time (T-t).

• Demonstrate that ROMs accurately & rapidly evaluate T-t histories.

Abstract

This article presents a solution to the geothermal problem of transient heat production from hot dry rocks using a horizontal well. Dimensionless forms of the governing equations are derived, including conduction in the rock, convection between the wellbore rock and the fluid, and advection and conduction in the fluid along the well. Ten model material and geometric parameters are reduced to three dimensionless parameters: $\alpha =4\frac{L}{D}\mathit{St}$ is the ratio of the rate of heat storage in the fluid to the rate the heat convection to the fluid from the rock, $\beta =\frac{1}{\mathit{Pe}}$ is the ratio of the rate of conductive versus advective heat transfer in the fluid, and $\gamma =2\frac{L}{D}\mathit{Bi}$ is the ratio of heat convection to the rock from the fluid to the rate of heat depletion in the rock. An axisymmetric finite element method (FEM) program is developed and yields the solution for the temperature of the rock mass and fluid over time. For physically meaningful inputs, analysis indicates that the effect of $\beta$ is negligible, that combinations of very large $\alpha$ and very low $\gamma$ (and vice versa) do not occur, and that all other things being equal, increasing $\alpha$ or decreasing $\gamma$ leads to higher fluid outlet temperatures. System response at different times and dimensionless temperature are captured in contour plots for values of $\alpha$ and $\gamma$ spanning practical injection rates, well geometries and fluid and rock properties. Power law response surface models are fitted using model outputs at discrete intervals and provide a means to accurately and rapidly compute the temperature-time histories of practical geothermal systems.

Introduction

Switching from fossil fuels to renewable and low carbon energy sources can play an important role in addressing potential environmental challenges. Geothermal heat could be a source of steady, clean, carbon-free, reliable and sustainable energy.

Geothermal heat can be used for electricity generation and for direct use; this can result in greater thermodynamic efficiency. The estimated amount of available heat in the earth varies using different methodologies; although the predicted amount varies based on the definitions and the methods used, all agree that it is a huge number [1], [2], [3], [4], [5]. The total worldwide installed capacity of geothermal electricity generation has increased from 0.2 GW in 1950 to 12.7 GW in 2015 [6], [7]. Geothermal heat could provide 1400 TWh of electrical energy by 2050, about 3.5% of global electricity demand, avoiding about 800 Mt of CO₂ emissions from fossil fuels per year [7]. Geothermal energy use has grown rapidly in many countries such as the United States, the Philippines, Indonesia, Turkey, New Zealand, and Iceland [8]. Direct power generation from dry and wet steam sources has increased to more than 82 countries from 28 in 1995 [9], [10]. Geothermal energy is already a viable source of power; in addition, direct geothermal heat use is increasing in industry and direct habitat heating.

Unlike traditional geothermal resources (i.e., hot, shallow, steam-generating fluids), which are locally distributed,Hot Dry Rocks (HDRs), without fluids and flow paths, are widely available across the world at various depths and present a remarkable potential resource for geothermal energy. By far the largest portion of geothermal resources in the world are in HDRs at accessible depths, approximately 10 million quads ($1\mathit{quad}=1.06\times {10}^{18}\text{J}$) [11]. Until recently, the existence of a high temperature source at a drillable depth (<2–3 km), the presence of porous and permeable rock, and the availability of sufficient volumes of hot fluid have been the three intertwined and necessary basic parameters of commercializable geothermal power resources. Although areas with high thermal gradient anomalies are found in many HDR regions, the absence of sufficient natural fluid, low-permeability rock, and great depth present challenges to the extraction of heat from HDRs [12].

Technology developments have led to a relatively new concept for heat energy extraction from HDRs, known as an Enhanced Geothermal System (EGS), where the rock mass properties are enhanced by multiple well hydraulic stimulation (hydrofracturing or hydroshearing) with a circulating fluid provided to access the heat. The heat exchange volume and area may be further enhanced by longer multiple wellbores, including inclined or horizontal wellbores. Although no full-scale commercial projects greater than 10 MW are yet built, the EGS approach may be able to overcome the challenges discussed above. EGS is far less efficient than shallow (<2–3 km) steam sources in terms of power generation and is certain to remain so, but it is far more widely applicable geographically.

A new approach, known as Closed Loop Geothermal System (CLGS), was proposed to produce heat from both hydrothermal systems and HDRs using horizontal wells [13]. The same approach is currently (2020) being tested at full-scale, in Canada. In this approach, horizontal wellbores connect an injection well to a production well, forming a U-shape closed loop. The horizontal wellbores must be long enough to give sufficient time for the injected fluid to be heated to an appropriate temperature. CLGS is isolated from the environment and has the advantage of significantly reducing risks in green-house gas (GHG) emissions, seismicity, water loss, and aquifer contamination.Heat production from CLGS was assessed by Song et al., [14] using numerical methods; however, the boundary condition on the well wall for updating the rock and fluid temperature is simply assumed to be a function of temperature gradient, which can have a significant impact on the outlet fluid temperature over time. The study is also limited to specific well geometry characteristics and rock and fluid properties. There is a need for models with fewer assumptions and a need for solutions to these models for a wide range of practical system and rock properties.

Ground Heat Exchangers (GHEs) are another type of closed-loop systems, widely used at shallow depths for space heating and industrial applications. GHEs have been extensively studied with various analytical and semi-analytical solutions [15], [16], [17], [18], [19], [20], [21], [22]. Although GHEs are types of closed-loop systems and related to CLGS, GHEs are sufficiently different from CLGS that models and solutions used for GHEs are not directly applicable to CLGS. Closed-loop GHEs are commonly classified in two categories: U-pipes, comprising a pair of straight pipes, connected with a U-turn at the bottom [23]; and concentric or coaxial pipes, as simple as one straight pipe inside a bigger diameter pipe or as more complex configurations, such as the cylindrical heat source [24], the ring-coil source [25], the helical-line source [26], [27], [28], and the spiral source [29], [30]. Since the upward and downward flowing pipes are very close together, there is heat transfer between these pipes, affecting the overall system efficiency. While the CLGS systems considered here are also U-shaped, flow within the well is one way and the vertical segments of the well are separated by a long (+1 km) horizontal section. Thus, the interaction between vertical segments is negligible and so the models and the solutions available for GHE are not directly appropriate for CLGS. Furthermore, many of the analytical solutions for GHEs are valid for a specific time and space scale; these studies also use empirical constants, which need to be determined by experimental and computational data [31]. In yet other studies, the models used only evaluated the rock temperature, while assuming constant heat flow rate, constant temperature, or temperature gradient at the wellbore [24], [32]. Thus, while there are numerous existing models for GHEs, they are not directly applicable to the system of interest.

As geothermal systems become more important, many efforts have been made to evaluate the heat transfer process in different conditions and configurations. However, only a few studies have evaluated CLGS so far for specific rock and system properties. Hence, a comprehensive study is needed to evaluate the heat production of CLGS for a wide range of practical well and rock and fluid characteristics. This study develops a semi-analytical solution to evaluate the output fluid temperature of a uniform diameter horizontal wellbore for a wide range of system characteristics (such as well diameter and length, injection flow rate, and rock and fluid properties) over time (years). Dimensionless forms of the governing partial differential equations (PDEs) are derived to evaluate the impact of first-order parameters and better understand their impact on the output fluid temperature. Then, a finite element method (FEM) program is developed to evaluate the advection-diffusion heat transfer process from the rock mass to the fluid flowing along the horizontal wellbore. The Response Surface Method (RSM) and power laws are used to present the relationship between output fluid temperatures and times as a function of dimensionless parameters. The power law models provide an empirical approximate solution for the mathematical model and can be used in a way similar to analytical solutions. The empirical solution has the same advantages over FEA as analytical solution – ease and expediency of evaluations, as is often required for optimization of systems.