A generalized non-Darcian model for packer tests considering groundwater level and borehole inclination
Introduction
The packer test, also known as constant head test, pressure test or Lugeon test, is an important tool to characterize the permeability or hydro-mechanical properties of soil and rock aquifers, and has been widely used in the fields of hydrogeology, civil engineering and petroleum engineering (Lugeon, 1933; Foyo et al., 2005; Wang et al., 2015; Huang et al., 2016; Ishii, 2020; Shahbazi et al., 2020). Compared with other well tests, e.g., constant rate test (Theis, 1935; Liu et al., 2017), slug test (Bouwer and Rice, 1976; Audouin and Bodin, 2007) and recovery test (Banton and Bangoy, 1996; Zhu and Yeh, 2006), the packer test has a number of important advantages: (1) packer test only requires measuring the outflow rate from a borehole section separated by packers under a known constant pressure (Houlsby, 1976; Quinn et al., 2011), which is less expensive and more convenient than pumping test (Zangar, 1953; Ferris et al., 1962); (2) it is applicable for various formations from highly permeable gravel beds (Yamada et al., 2005) to low-permeability rocks (Hamm et al., 2007); (3) it can be used to access the variability of local permeability along a borehole as it intersects various hydrogeological units (Yihdego, 2017); and (4) its enhanced version, the high-pressure packer test (HPPT, where the maximum injection pressure is much higher than 1 MPa), can be applied to examine the pressure-sensitive permeability of fractured rocks (Cappa et al., 2006; Derode et al., 2013; Huang et al., 2018; Zhou et al., 2019a).
Data interpretation of a packer test relies mostly on Darcy's law-based models, for permeability estimation of the tested formation (Hvorslev, 1951; Zangar, 1953). Nonlinearity, however, frequently manifests in the injection pressure (P) versus volumetric flow rate (Q) curves (Quinn et al., 2011; Chen et al., 2020), especially for HPPTs (Chen et al., 2015a; Zhou et al., 2018). This is attributed to non-negligible inertial losses induced by increasing water pressure and hydraulic gradient (Bear, 1979; Chen et al., 2015a, Chen et al., 2015b; Zhou et al., 2019b). Non-Darcian interpretative models based on Izbash's (1931) or Forchheimer's (1901) law were therefore developed for estimating the aquifer properties from the nonlinear P − Q curves (e.g., Yamada et al., 2005; Quinn et al., 2011; Chen et al., 2015b). The development of all the above models involves two assumptions: the borehole is vertical and the groundwater table is infinitely far from the test interval. These assumptions are not always realistic in the field conditions. For instance, non-vertical boreholes or wells have been increasingly utilized (Zhan and Zlotnik, 2002; Onur et al., 2004; Tsou et al., 2010).
Horizontal or slanted boreholes may be used under the following conditions: (1) a vertical borehole may damage weak geological structures or artificial permanent structures in the formation; (2) the tested aquifer is of small thickness (Zhan and Zlotnik, 2002); (3) permeability anisotropy needs to be estimated (Yihdego, 2017); and (4) the drilling has multiple purposes, e.g., for measuring both the permeability and in-situ stresses (Chen et al., 2015a). Some analytical solutions considering the inclination angle of wells are available in the literature (Zhan and Cao, 2000; Tsou et al., 2010; Hazlett and Babu, 2014), but not targeted for data interpretation of packer test. On the other hand, Zangar (1953) clarified the influence of groundwater level on permeability estimation in Darcian flow regime by proposing two interpretative models for packer test performed above and below groundwater level, respectively. It remains unclear, however, whether the negligence of groundwater level and borehole inclination, as commonly done previously, induces a significant error for data interpretation of pack tests in slanted boreholes where non-Darcian flow occurs. Recalling that the data interpretation of packer tests in the vadose zone is complicated by the effect of unsaturated flow (Stephens and Neuman, 1982; Philip, 1985), we focus on the tests in the saturated zone.
The goal of this study is to quantify the effects of groundwater level and borehole inclination on packer tests where non-Darcian flow occurs. A Forchheimer's law-based interpretative model is proposed using image theory and velocity integration method, and validated by numerical simulations. Parametric and error analyses are conducted to reveal the conditions under which the previously developed models (e.g., Chen et al., 2015b) apply within negligible errors. The proposed model is finally applied to data interpretation of the packer tests performed in fractured sedimentary rocks in Qiongzhong County, Hainan Province, China, where the influences of borehole inclination and non-Darcian flow are evaluated.
Section snippets
Procedure and type curve of packer test
As shown in Fig. 1, the equipment of packer test usually consists of a downhole component for conducting the injection and an above-ground component for monitoring the procedure and recording the data (Chen et al., 2015a; Quinn et al., 2016). During testing, water is injected by a piston pump into the isolated test section via the pipe system after inflation of the two packers, and the test lasts for a period of time until the injection pressure becomes stabilized. A complete packer test
Numerical validation
The assumptions and the derivation procedure (image method and velocity integration) may cause errors to the analytical model. To validate the rationality of the assumptions and the accuracy of the proposed model, numerical simulations with finite element method are performed in a cylindrical model of rock (400 m in diameter) that contains a test interval, as shown in Fig. 5. The bottom surface of the model is located in the plane of z = 0, and the top surface is set at z = 200 m. The initial
Application to field data
The proposed model is applied to data interpretation of the high-pressure packer tests (HPPTs) performed in the sedimentary rocks of Cretaceous Lumuwan group (K1Lm) around an underground cavern system for a pumped-storage power station in Qiongzhong County, Hainan Province, China (Fig. 11a). The test site, equipment and procedures have been detailed in Chen et al., 2015a, Chen et al., 2015b. As shown in Fig. 11b, the test boreholes ZK129–1 (33.4 m in length) and ZK129–2 (45.2 m in length) were
Discussion
As a guidance to data interpretation of packer tests, Fig. 16 shows the relationships among the proposed model, Chen et al.'s (2015b) model, Zangar (1953) and Hvorslev (1951). The proposed model in this study is the most general one, and it immediately reduces to other models for cD ≫ 1 and/or β = 0. From Fig. 10, Fig. 16, the applicability of these models is summarized as follows:
In Darcian flow regime where the flow rate Q increases linearly with the pressure head H, the proposed model with β
Conclusions
This study presented a generalized non-Darcian model for data interpretation of packer tests by taking into account the effects of groundwater level and borehole inclination. The flow was assumed to follow the Forchheimer's law, and the model was developed by means of image theory and velocity integration. The proposed model becomes identical with Zangar's (1953) model for vertical borehole in laminar flow condition. Numerical verifications and parametric analyses show that the effects of
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
The financial supports from the National Natural Science Foundation of China (No. 51925906) and the National Key R&D Program of China (No. 2018YFC0407001) are gratefully acknowledged.
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