# Investigation on Interference Test for Wells Connected by a Large Fracture
## 3 Model and Solution for the Interference Test  
### Mathematical Model and Pressure Transient Solution  
A simplified physical model considers two wells A and B connected by a large fracture. Well A is an active well, and it shuts in after a period of production time \(t_{p}\). Well B is an observation well, in which a precision pressure gauge is set up at the bottom-hole and kept closed. When the production time is short and the matrix of the reservoir is tight, well A and B influence each other mainly by the flow in large fractures, and the impact of flow in the matrix can be ignored. The fracture is presumed to run through the formation vertically. Therefore, the flow in the fracture can be treated as one-dimensional flow. The original of the coordinate system is set at Well A, and the right direction of axis \(x\) is along the fracture from Well A to Well B (Figure 2). The latter analysis assumes that the activation signal does not reach the end of the fracture which is treated as an infinite condition. The one-dimensional equation of flow in porous media is [39]  
\[\frac{\partial^{2}p}{\partial x^{2}}=\frac{1}{\eta}\frac{\partial p}{\partial t} \tag{1}\]  
where \(p\) is the reservoir pressure (Pa), \(x\) is the coordinate along fracture (m), \(t\) is the time (s) and \(\eta\) is the pressure conductivity coefficient (m\({}^{2}\)/s). Considering the pressure transient when Well A produces with a production rate \(q\), boundary conditions and initial conditions are  
\[p(t=0)=p_{i} \tag{2}\]  
\[p(x=\infty)=p_{i} \tag{3}\]  
\[\frac{\partial p}{\partial x}\bigg{|}_{x=0}=\frac{qB\mu}{2 bhK} \tag{4}\]  
where \(b\) is the fracture width (m), \(h\) is the reservoir thickness (m), \(K\) is the permeability of fracture (m\({}^{2}\)), \(B\) is the volumetric ratio of fluid, \(\mu\) is the viscosity (Pa-s), \(p_{i}\) is the original pressure of reservoir, (Pa). Figure 1: Active well production rate and observation well pressure of interference test. Defining the dimensionless quantities  
\[p_{D}=\frac{2 Kh}{q\mu B}(p_{i}-p),\ x_{D}=\frac{x}{b},\ t_{D}=\frac{\eta t}{b^{2}} \tag{5}\]  
The governing Equation (1) becomes Equation (6) [39]  
\[\frac{\partial^{2}p_{D}}{\partial x_{D}^{2}}=\frac{\partial p_{D}}{\partial t_ {D}} \tag{6}\]  
The initial conditions and boundary condition Equations (2)-(4) can be written as Equations (7)-(9)  
\[p_{D}(t_{D}=0)=0, \tag{7}\]  
\[p_{D}(x_{D}=\infty)=0, \tag{8}\]  
\[\left.\frac{\partial p_{D}}{\partial x_{D}}\right|_{x_{D}=0}=-1. \tag{9}\]  
Given the above equations, the following similarity transformation is performed according to [PERSON] et al. [39]. \[p(t=0)=p_{i} \tag{2}\]0  
\[p(t=0)=p_{i} \tag{2}\]1  
Then  
\[p(t=0)=p_{i} \tag{2}\]2  
\[p(t=0)=p_{i} \tag{2}\]3  
\[p(t=0)=p_{i} \tag{2}\]4  
Therefore, the governing equation becomes  
\[p(t=0)=p_{i} \tag{2}\]5  
and the boundary conditions and initial conditions are written as the following  
\[p(t=0)=p_{i} \tag{2}\]6  
\[p(t=0)=p_{i} \tag{2}\]7  
A special solution of Equation (15) is  
\[p(t=0)=p_{i} \tag{2}\]8  
\[p(t=0)=p_{i} \tag{2}\]9 Schematic diagram of wells connected by a large fracture. Assuming another special solution that is linearly independent with \(U_{1}\) is  
\[p(x=\infty)=p_{i} \tag{3}\]0  
Substituting Equation (19) into the control differential Equation (15)  
\[p(x=\infty)=p_{i} \tag{3}\]1  
Solving Equation (20), the following equation is derived  
\[p(x=\infty)=p_{i} \tag{3}\]2  
Therefore, another special solution is  
\[p(x=\infty)=p_{i} \tag{3}\]3  
The general solution is  
\[p(x=\infty)=p_{i} \tag{3}\]4  
where \(c_{1}\) and \(c_{2}\) are constant, and \(erf\) denotes the error function.