Thermal lift effect in geothermal wells
The procedure presented here allows to estimate the value of the true depression and pressure that would be recorded at the production wellhead if the volumetric expansion of water associated with thermal heating of the borehole did not take place. Since heating of the borehole increases the volume of water inside casings, pressure recorded at the wellhead is always higher than if no thermal heating had occurred. In the case of sub-artesian wells, the water table in the pumped borehole is always higher compared to boreholes where the volumetric expansion of the extracted water is small or negligible. In both cases, the interpretation of wellhead pressure or water level during the hydrodynamic tests leads to an estimate of higher aquifer transmissivity than it actually is. Likewise – during the calibration of the numerical model based on monitoring data from production wells, this phenomenon should be taken into account. Forecasting operating conditions on the basis of incorrectly estimated filtration parameters can lead to unsustainable water resources management. Hence it is necessary to correct the pressure measured under such conditions and use the value of the so-called reduced wellhead pressure. Reduced wellhead pressure – i.e. wellhead pressure corrected for the volumetric expansion of water occurring during operation should be the basis for the interpretation of hydrodynamic tests and calibration of numerical models of geothermal reservoirs (Kawecki 1995, Bielec and Miecznik 2012). Figure 1 shows the temperature distribution in the well column under static (no flow) and dynamic conditions.
The hydrostatic pressure exerted by the water column is:
Equation 1
Since the pressure at the bottom of the well is independent of the temperature of the liquid inside it, the multiplication of the height of the liquid column and its density will be the same for the unheated hole as for the heated one:
Equation 2
Static conditions
Dynamic conditions
Fig. 1. Effect of borehole pumping on water temperature in the casing column (based on Kawecki 1995)
The average density of water in the borehole is as follows:
Equation 3
Although in general the density of water is a non-linear function of temperature, in the temperature range from 40 to 95°C (typical temperature of geothermal water in liquid systems) the approximation by a linear function can be considered sufficiently accurate. By omitting some transformations of Equation 3, one obtains a relation that says that the average density of the water column is the density of water for the average (weighted) temperature of the water column:
Equation 4
By inserting Equation 4 into Equation 2, the following relation is obtained:
Equation 5
The pressure measured at the bottom of the borehole under static conditions is as follows:
Equation 6
The bottomhole pressure during operation can be expressed in the form of Equation 7, which is analogous to Equation 6:
Equation 7
in a borehole is the difference between the static and dynamic bottom pressures: Equation 8
By substituting Equations 6 and 7 into Equation 8, one obtains a formula that allows to calculate the actual depression excluding the influence of the borehole thermal heating effect (also known as thermal lift) knowing only the static and dynamic wellhead pressure and the average water column temperature under static and dynamic conditions:
Equation 9
from the static wellhead pressure, taking into account the thermal lift effect during extraction, the so-called reduced wellhead pressure value is obtained
Equation 10
An example of the use of the presented calculation procedure on real data can be observed in Figure 2.
Bańska PGP-1
Fig. 2. Recorded wellhead pressure and calculated reduced wellhead pressure for the Bańska PGP-1 production well (data from 2001-2015). Source: Miecznik 2017, based on data provided by PEC Geotermia Podhalanska S.A.
Literature:
Bielec B., Miecznik M., 2012 – Efekt termiczny w obliczeniach przewodności hydraulicznej w otworach ujmujących wodę termalną. Technika Poszukiwań Geologicznych, Geotermia, Zrównoważony Rozwój, 51(2): 45–54.
Kawecki M., 1995 – Correction for temperature effect in the recovery of a pumped well. Ground Water, 33(6): 917–926.
Miecznik M., 2017 – Model zrównoważonej eksploatacji zbiornika wód geotermalnych w centralnej części Podhala do produkcji energii cieplnej i elektrycznej. Studia, Rozprawy, Monografie, Vol. 202, Wyd. IGSMiE PAN.
The project is financed under the Fund for Bilateral Relations through the European Economic Area Financial Mechanism (EEA FM) and the Norwegian Financial Mechanism (NFM) 2014-2021, programme “Environment, Energy and Climate Change”.
Project Operator: Ministry of Climate and Environment