# Applying Petroleum the Pressure Buildup Well Test Procedure on Thermal Response Test—A Novel Method for Analyzing Temperature Recovery Period

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Literature Overview

## 3. Theoretical Background

#### 3.1. Solutions of the Diffusivity Equations for the Case of Infinite Medium and Line Source Well

#### 3.2. Application of Horner’s Pressure Build-Up Method in Applied Thermogeology and TRT

_{p}is the duration of the constant production rate before the well shut-in, while ∆t is the time following the well shut-in. A typical Horner’s method procedure is to analyze the data and identify where the curve becomes linear on a semi-log graph, as this is the period of specific interest, from which the slope of the curve, m, can then be determined. This factor describes the pressure change (or temperature change during TRT) over one logarithmic cycle of time. The curve from semi-log paper can be arbitrary and logically divided into three regions: Early Time Region (ETR), Middle Time Region (MRT), and Late Time Region (LTR). In petroleum well testing, ETR is affected by altered (damaged) permeability of the near-well zone and the afterflow effect from the reservoir into the wellbore after shut-in. In thermogeology, and particularly TRT analysis, ETR describes the time necessary to achieve a semi-steady heat flow state from the borehole heat exchanger to the surrounding ground. Like in petroleum well testing, initial data distortion in the form of unsteady-state heat flow occurs under influence of borehole thermal resistances; this is primarily because of the low conductivity of pipe material and grout properties, as described in Section 3.4.

_{p}.

_{p}, as required by Horner’s method (Figure 2), the pressure drop at time Δt can be obtained by the principle of superposition as:

_{p}+ Δt) + (pressure drop caused by rate change (−Q) for the time Δt).

_{wf}can be replaced with static pressure p

_{ws}at a certain time Δt after closing the well; therefore, according to Horner’s method, the corresponding equation for pressure build-up becomes:

_{ws}observed during a shut-in period is plotted versus the logarithm of (t

_{p}+ Δt)/Δt, a straight line should be obtained (Figure 1).

_{p}, when turning off the heaters and changing the heat rate to (−q) for the time t

_{p}+ Δt, temperature recovery (equal to pressure buildup test and fall-off test) (Figure 2 quadrant b1, b2 and d1, d2) can be derived as:

#### 3.3. Application of Petroleum Engineering Derivation Curves in TRT Interpretation

#### 3.4. Interpretation of Borehole Thermal Resistance Analogous to Well Testing Skin Effect

- Thermal resistance due to the advective heat transport in the pipes and between the pipes (R
_{ff}) - Thermal resistance due to pipe wall material and grout transition (R
_{fig}, R_{fog}) - Thermal resistance due to grout–soil exchange (R
_{gs})

_{p}, for a pressure drawdown test or extraction of heat from the ground:

_{p}, it could be approximated that (t

_{p}+ Δt)/t

_{p}= 1. Rearranging Equation (25) and arbitrarily choosing a value of 1 h for Δt, and replacing T

_{i}= T

_{1h}, it could be rewritten [21]:

_{1h}is not strictly set, and changing the value of ∆t in the equation to values other than 1 h would merely change the constant 0.351 in Equation (27).

_{p}as the duration of the first period.

## 4. Experimental Site Setup

## 5. Results and Discussion

^{SDR11}/D32

^{SDR11}), the volume flow and velocity, the viscosity and density of water, and the pipe roughness, there was a fully developed turbulent regime in both the annular space (Re = 7100) and the column pipe (Re = 23,000).

^{2}/d, depending on the moisture content. As a reasonable estimation, we assumed a value of 0.050 m

^{2}/d for further analysis. If the appearance of SS-state is deduced from the standardized equation by Mogensen presented in Section 3.3, then the corresponding time would be 7.5 h for a borehole diameter of 110 mm. To achieve maximum accuracy and nullify transient effects, the ground thermal conductivities were derived from intervals of 15–96 h for all test conditions in the TRT period and recovery period.

_{avg}vs. t

_{p}was plotted for the Semi-Steady State or SSS interval. As stated by Equation (10), since a semi-log plot is used, there is the need to determine the change of temperature for a one-log cycle of time (m), and then calculate the effective ground conductivity. The same result could be obtained by using the standardized principle of plotting T

_{avg}vs. ln(t

_{p}) on a normal graph, and then using Equation (9) and the slope of the line κ.

_{avg}is plotted vs. (t

_{p}+ Δt)/Δt, as shown by Figure 6c. Effective ground thermal conductivity is then derived from Equation (17) by knowing the log slope m, just like in the case for a standard TRT period. When the temperature recovery line is extended until (t

_{p}+ Δt)/Δt = 1, initial conditions—i.e., undisturbed ground temperature—are reached. Figure 8 shows the entire Horner’s procedure for all ten heat step conditions. It is important to note that testing times of 15–96 h were used in this analysis, in order to nullify transient effects, just like in the case of TRT. When extending the data trendline until (t

_{p}+ Δt)/Δt = 1, it can be seen that the initial temperature value ranges between 14.0 °C and 15.2 °C. As mentioned before, the reason for this effect is the shallow installation of the coaxial system with a final depth of 35 m, while solar energy perturbation penetrates to a depth of 10 m. Therefore, the initial condition values are somewhat dependent on the climate and season in which the measurements are taking place.

_{air}

_{.}Since the air temperature was recorded for the entire ~2000 h of measurement, the hypothesis is that climate interference would be zero if the air temperature were the same value as the current EST of the borehole fluid. The higher the temperature difference between the air temperature and the EST of the borehole fluid at any moment, the higher the interference would be on the TRT ground conductivity analysis, since a certain heat transfer would take place between the environment and the surface collector pipes and equipment.

_{horner avg}= 2.13 W/m °C), as seen in Figure 9.

^{2}/d as expected for clays, would only change the resistivity/skin term. Inverse analysis could also be conducted, by calculating the thermal resistance term based on a known well geometry and material thermodynamic characteristics, and determining the soil thermal diffusivity term using ILS equations. However, practical geometry and the completion of the borehole heat exchanger is often different from the one assumed in this project. This problem is often related to misaligned pipes in the wellbore (especially in inclined coaxial systems) and the quality of the grouting (especially perfect adherence of grout and pipe). Soil diffusivity could be more precisely determined by laboratory measurements of undisturbed soil samples, by determining density, specific heat capacity and conductivity.

## 6. Conclusions

## Supplementary Materials

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Nomenclature

B | formation volume factor (m^{3}/m^{3}) |

c | specific heat capacity (kJ/kg °C) |

c_{t} | compressibility of the rock (kJ/kg °C) |

e | Euler number (2.7183) |

Ei | exponential integral |

k | permeability (m^{2}) |

m | slope of the line in log chart |

p(r, t) | pressure in the function of time and radius (Pa) |

p_{i} | initial pressure (Pa) |

p_{wf} | bottom hole flow pressure (Pa) |

p_{ws} | bottom hole static pressure (Pa) |

Q | rate of production (m^{3}) |

q′ | heat power per meter of borehole (W/m) |

r | radius around line source (m) |

r_{w} | wellbore radius (m) |

R_{b} | equivalent borehole resistance (m °C/W) |

s | skin factor, dimensionless |

t | time (h) |

t_{p} | duration of constant production rate (h) |

T(r, t) | temperature in function of radius and time (°C) |

T_{i} | initial borehole temperature (°C) |

T_{est} | entering source temperature (°C) |

T_{ext} | temperature during extraction of the heat from the ground (°C) |

T_{rej} | temperature during rejection of the heat to the ground (°C) |

u | integral parameter |

α | thermal diffusivity (m^{2}/h) |

Δp_{skin} | pressure drop due to skin effect (Pa) |

Δt | shut-in time (h) |

ΔT_{skin} | temperature drop/rise due to skin effect (°C) |

Φ | porosity, fraction |

γ | Euler’s constant (0.5772) |

η | hydraulic diffusivity factor (m^{2}/h) |

κ | slope of the line |

λ | thermal conductivity of ground (W/m °C) |

μ | viscosity (Pa s) |

ρ | density of the ground (kg/m^{3}) |

SS | Steady-state |

SSS | Semi-steady state |

TRT | Thermal Response Test |

USS | Unsteady-state |

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**Figure 2.**Display of corresponding behavior of pressure/temperature during different types of well tests/thermal response tests.

**Figure 4.**Components of borehole thermal resistance for (

**a**) the CXA arrangement; and (

**b**) the CXC arrangement.

**Figure 5.**Test site setup: (

**a**) Borehole drilling equipment; (

**b**) Inserting coaxial borehole heat exchanger; (

**c**) Heat exchanger shaft; (

**d**) TRT equipment; (

**e**) Schematics of testing site; (

**f**) Lithology description.

**Figure 6.**Example of two-step TRT analysis carried out for each heat power condition (CXC 71 W/m presented here); (

**a**) Recorded borehole and air temperatures and unit heat power; (

**b**) Determination of conductivity from TRT period; (

**c**) Determination of conductivity by Horner’s method from recovery period.

**Figure 8.**Horner’s semi-log method for determining ground thermal conductivity applied to a TRT recovery period for each heat power step and flow regime.

**Figure 9.**Results of the ground thermal conductivity values obtained for classic TRT and recovery period as a function of variable test time 36–96 h.

**Figure 10.**Ground temperature rise at a certain radius from the borehole for five different TRT heat steps.

**Figure 11.**Example of fitting measured EST data with the infinite line source equation and two different values of thermal conductivity (the first obtained from the TRT period, and the second from the recovery period) for a case of 71 W/m and CXC setup.

**Figure 12.**Results of sum of squares of differences analysis: (

**a**) SUMXMY2 between ILS fitted curve and EST measured data; (

**b**) SUMXMY2 between air temperature and EST measured data.

CXC | CXA | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

TRT Heat Step, Mean | 70.9 | 61.2 | 54.5 | 42.5 | 35.3 | 70.8 | 60.6 | 53.9 | 43.0 | 35.5 |

Standard Error | 0.031 | 0.024 | 0.015 | 0.019 | 0.016 | 0.028 | 0.023 | 0.026 | 0.022 | 0.013 |

Median | 71.0 | 61.4 | 54.5 | 42.6 | 35.3 | 70.8 | 60.7 | 54.1 | 43.2 | 35.5 |

Mode | 72.1 | 61.6 | 54.4 | 43.1 | 34.7 | 69.8 | 60.4 | 54.2 | 43.5 | 35.7 |

Standard Deviation | 1.06 | 0.81 | 0.53 | 0.64 | 0.56 | 0.96 | 0.80 | 0.88 | 0.73 | 0.45 |

Sample Variance | 1.13 | 0.65 | 0.28 | 0.41 | 0.31 | 0.91 | 0.64 | 0.77 | 0.54 | 0.20 |

Kurtosis | −0.73 | −0.29 | −0.26 | −0.67 | −0.92 | −0.17 | −0.29 | −0.70 | −0.64 | −0.56 |

Skewness | −0.32 | −0.65 | −0.12 | −0.33 | −0.08 | −0.39 | −0.54 | −0.38 | −0.49 | −0.49 |

Range | 4.77 | 4.41 | 2.64 | 2.94 | 2.35 | 5.33 | 3.97 | 3.97 | 3.19 | 2.24 |

Minimum | 68.3 | 58.3 | 53.0 | 40.7 | 34.0 | 67.7 | 58.3 | 51.8 | 41.2 | 34.1 |

Maximum | 73.1 | 62.7 | 55.7 | 43.7 | 36.3 | 73.0 | 62.2 | 55.7 | 44.4 | 36.4 |

Sum | 81,499 | 70,671 | 62,709 | 48,874 | 40,579 | 81,443 | 69,752 | 61,999 | 49,536 | 40,813 |

Count | 1150 | 1154 | 1151 | 1150 | 1151 | 1151 | 1151 | 1151 | 1151 | 1151 |

Confidence Level (95.0%) | 0.06 | 0.05 | 0.03 | 0.04 | 0.03 | 0.06 | 0.05 | 0.05 | 0.04 | 0.03 |

CXC | CXA | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

Mean, TRT Heat Step | 70.9 | 61.2 | 54.5 | 42.5 | 35.3 | 70.8 | 60.6 | 53.9 | 43.0 | 35.5 |

Air Temperature, Mean | 10.2 | 9.1 | −1.2 | −1.1 | 21.0 | 14.4 | 12.3 | 14.0 | 16.2 | 19.8 |

Standard Error | 0.30 | 0.29 | 0.25 | 0.31 | 0.29 | 0.24 | 0.36 | 0.29 | 0.34 | 0.24 |

Median | 10.7 | 8.7 | −1.3 | −0.5 | 20.2 | 13.8 | 13 | 13.2 | 16 | 19.5 |

Mode | 6.0 | 13.9 | −0.1 | 0.4 | 18.0 | 12.4 | 4.4 | 11.1 | 18.8 | 21.0 |

Standard Deviation | 4.12 | 4.04 | 3.47 | 4.28 | 4.06 | 3.35 | 5.07 | 4.04 | 4.78 | 3.27 |

Sample Variance | 16.9 | 16.3 | 12.0 | 18.3 | 16.5 | 11.2 | 25.7 | 16.3 | 22.9 | 10.7 |

Kurtosis | −0.71 | −1.13 | 0.77 | −0.42 | −0.71 | −0.34 | −0.95 | −0.32 | −0.63 | −0.63 |

Skewness | 0.00 | 0.05 | 0.51 | −0.29 | 0.45 | 0.43 | −0.24 | 0.58 | −0.21 | 0.37 |

Range | 18.2 | 16 | 19.6 | 19.6 | 16.2 | 15.4 | 20 | 17 | 19.9 | 14 |

Minimum | 1.8 | 0.8 | −9.3 | −10.1 | 13.9 | 7.8 | 1.8 | 7.4 | 4.9 | 14.1 |

Maximum | 20.0 | 16.8 | 10.3 | 9.5 | 30.1 | 23.2 | 21.8 | 24.4 | 24.8 | 28.1 |

Sum | 1977 | 1758 | −234 | −207 | 4062 | 2788 | 2374 | 2710 | 3124 | 3819 |

Count | 193 | 193 | 193 | 193 | 193 | 193 | 193 | 193 | 193 | 193 |

Confidence Level (95.0%) | 0.58 | 0.57 | 0.49 | 0.61 | 0.58 | 0.48 | 0.72 | 0.57 | 0.68 | 0.46 |

Flow Setup | CXC | CXA | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

TRT Heat Step, Mean | 70.9 | 61.2 | 54.5 | 42.5 | 35.3 | 70.8 | 60.6 | 53.9 | 43.0 | 35.5 |

Thermal conductivity @96 h, W/m °C during TRT period | 2.62 | 2.38 | 2.72 | 2.65 | 2.44 | 2.50 | 2.47 | 2.46 | 2.47 | 2.40 |

Thermal conductivity @96 h, W/m °C during Recovery period | 2.25 | 2.26 | 2.05 | 2.04 | 2.04 | 2.11 | 2.19 | 2.08 | 2.24 | 2.05 |

Initial temperature, TRT circulation, °C | 14.6 | 14.7 | 14.8 | 14.6 | 14.8 | 14.7 | 14.7 | 14.7 | 14.7 | 14.7 |

EST after 96 h of TRT, °C | 36.0 | 33.7 | 32.0 | 28.1 | 26.6 | 37.8 | 33.9 | 31.9 | 28.2 | 26.5 |

EST after 96 h of Recovery period, °C | 15.7 | 15.9 | 15.9 | 15.6 | 16.2 | 16.7 | 16.3 | 16.0 | 15.7 | 16.1 |

Initial temperature, Horner, °C | 13.9 | 14.4 | 14.5 | 14.5 | 15.2 | 14.8 | 14.7 | 14.5 | 14.7 | 15.1 |

Borehole resistance, Horner, m °C/W | 0.138 | 0.138 | 0.127 | 0.126 | 0.129 | 0.135 | 0.133 | 0.131 | 0.135 | 0.128 |

m slope, Horner, °C/log cycle | 5.6 | 5.0 | 4.9 | 3.8 | 3.2 | 6.2 | 5.1 | 4.7 | 3.5 | 3.2 |

skin factor, Horner, °C | 2.0 | 2.0 | 1.6 | 1.6 | 1.7 | 1.8 | 1.8 | 1.7 | 1.9 | 1.7 |

T1 h, Horner, °C | 25.1 | 24.2 | 24.2 | 22.0 | 21.5 | 27.0 | 24.8 | 24.0 | 21.7 | 21.4 |

Tp, Horner, °C | 36.0 | 33.6 | 32.0 | 28.1 | 26.6 | 37.7 | 33.9 | 31.9 | 28.2 | 26.5 |

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**MDPI and ACS Style**

Kurevija, T.; Strpić, K.; Koščak-Kolin, S.
Applying Petroleum the Pressure Buildup Well Test Procedure on Thermal Response Test—A Novel Method for Analyzing Temperature Recovery Period. *Energies* **2018**, *11*, 366.
https://doi.org/10.3390/en11020366

**AMA Style**

Kurevija T, Strpić K, Koščak-Kolin S.
Applying Petroleum the Pressure Buildup Well Test Procedure on Thermal Response Test—A Novel Method for Analyzing Temperature Recovery Period. *Energies*. 2018; 11(2):366.
https://doi.org/10.3390/en11020366

**Chicago/Turabian Style**

Kurevija, Tomislav, Kristina Strpić, and Sonja Koščak-Kolin.
2018. "Applying Petroleum the Pressure Buildup Well Test Procedure on Thermal Response Test—A Novel Method for Analyzing Temperature Recovery Period" *Energies* 11, no. 2: 366.
https://doi.org/10.3390/en11020366