## 3 Field Case Study of Pumping Test
### Geological and Numerical Models
#### 3.5.1 Two-dimensional Radial Flowing Model  
The sedimentary layers in this region are quite stable and almost flat over a few tens of kilometers. Thus, it is valuable to make a rough estimate of some key parameters using simple models for which theoretical solutions are available. For this end, a two-dimensional (2D) radial flowing model of an infinite and isotropic horizontal layer was tested. Under the assumption that [PERSON]'s law holds, the theoretical solution of the pore pressure change is given by [39]  
\[\Delta P(t,r)=Q\frac{\eta}{4\pi kH}\Gamma\left(0,\frac{r^{2}}{4 Dt}\right) \tag{12}\]  
\[D=\frac{k}{\eta S_{a}},S_{a}=\phi\left(\beta_{f1}+\beta_{pv}\right) \tag{13}\]  
In Equations (12) and (13), \(t\) (s) is time since the start of pumping, \(r\) (m) is the distance from pumping well. \(Q\)(m\({}^{3}\)/s) is the pumping rate, \(H\) (m) and \(K\) (m\({}^{2}\)) are thickness and permeability of the layer, \(D\) (m\({}^{2}\)/s) is hydraulic diffusivity, \(\eta\) (Pa\(\cdot\)s) is dynamic viscosity of water, \(S_{a}\) (Pa\({}^{-1}\)) is unconstrained specific storage coefficient, \(\beta_{\#}\) (Pa\({}^{-1}\)) and \(\beta_{pv}\) (Pa\({}^{-1}\)) are compressibility of the fluid and pores, respectively.  
In the present problem, all parameters except \(K\) and \(\beta_{pv}\) can be estimated based on the borehole data. In this study history matching approach was applied to estimate the values of \(K\) and \(\beta_{pv}\). First, sensitivity tests were performed to determine if \(K\) and \(\beta_{pv}\) are in a tradeoff relationship. As shown in Figure 6, the predicted water level response during pumping and after shutdown is sensitive to both parameters. Thus, the permeability of the layer and the compressibility of the pores, which are poorly known and show large uncertainty in many operation sites, can be determined from the observed water level data. The parameters \(K\) and \(\beta_{pv}\) can be determined by fitting the model predictions to the observed data using a grid-searching algorithm.  
The water level change in the upper Chonan formation during the initial pumping period from Well-3 and Well-4 were then considered. It was found that, for a single well pumping test, the water level change observed at Well-2, located 630 m away from Well-3 (Figure 2), can be represented fairly well using a single-layer 2D radial flowing model (Figure 7a). For multiple tests, when a single set of \(K\) and \(\beta_{pv}\) values were used, the observed data were not well represented (Figure 7b). However, when using different sets of \(K\) and \(\beta_{pv}\) values, the fitting results greatly improved (Figure 7c), indicating lateral inhomogeneity in the physical properties. \(K\) and \(\beta_{pv}\) values estimated in such a manner may be considered as a type of \"mean\" or \"equivalent\" value and can be used as initial values in the coupled model.  
A rough estimate of the bulk modulus of the Chonan formation can also be obtained from the maximum change in the water level and the maximum fiber strain observed during the pumping test. Under drained conditions, the volumetric strain \(\varepsilon_{v}\) caused by a change of pore pressure \(\Delta P\) is correlated with the bulk modulus as  
\[\Delta P=K\varepsilon_{v}=K\left(\varepsilon_{zz}+\varepsilon_{xx}+ \varepsilon_{yy}\right)\approx Kc\varepsilon_{zz} \tag{14}\]  
where \(c\) is a constant ranging from 1 to 3.