# The Principle and Downhole Testing of Water Injection Exploration in Depleted Reservoirs
## 1 Introduction
For instance, the typical formula for permeability of smooth fractures [14] is as follows:  
\[K_{f}=\frac{gb^{2}}{12v_{w}} \tag{1}\]  
In the formula, \(g\) represents a constant; \(b\) stands for fracture width; and \(v_{w}\) represents the dynamic viscosity coefficient of water, which is also a constant. Additionally, the -fracture width \(b\) is influenced by both groundwater pressure and in situ stress. Building upon previous research [15; 16; 17], the combined effect of groundwater pressure and lateral stress is referred to as the equivalent normal stress. The equivalent normal stress can be simply expressed as  
\[\sigma_{ne}=P_{c}-P_{f} \tag{2}\]  
\(P_{c}\) represents the vertical fracture pressure, and \(P_{f}\) stands for internal water pressure. It is only when the equivalent normal stress exceeds zero that the permeability of fractures significantly increases [18; 19]. Furthermore, considering the three-dimensional stress effects, research by [PERSON] and [PERSON] [20] indicates that the expression for the effective normal stress (\(\sigma_{n}\)) on fractures is as follows:  
\[\sigma_{n}=\sigma_{2}+\mu(\sigma_{1}+\sigma_{3})-\beta P \tag{3}\]  
In the equation, \(\sigma_{2}\) represents the stress perpendicular to the crack, while \(\sigma_{1}\) and \(\sigma_{3}\) are the stresses parallel to the crack. The pore pressure within the crack is denoted as \(P\). \(\mu\) denotes [PERSON]'s ratio, and the coefficient \(\beta\) represents the ratio of the connected area within the crack to the total area. Based on the above equation, the formula for the permeability coefficient of fractures considering the influence of three-dimensional stress and pore pressure can be derived. Similar formulas and patterns can also be found in other studies [21]. \[K_{f}=\frac{gb^{2}}{12v_{w}}e^{\frac{3\sigma_{n}}{K_{n}}}=\frac{gb^{2}}{12v_{w }}e^{\frac{3(\beta P-|\sigma_{n}+|\sigma_{3}|)}{K_{n}}} \tag{4}\]  
From this equation, it can be observed that when \(\sigma_{n}=0\), the permeability coefficient remains in its original state, and when \(\sigma_{n}\) is less than 0, the permeability coefficient will significantly increase. In coal and rock layers near the goal, due to the adjacent side of the goal being in an unloading state, the horizontal stress perpendicular to the goal will sharply decrease or even become zero. On the other hand, according to the relevant literature [22], the distribution of coal pillars near the goal includes fractured zones, plastic zones, elastic zones, and original rock stress zones, with vertical stress reduction observed in fractured and plastic zones. Moreover, with time passing, the stress near the goaf side will further decrease. The literature [23] also suggests that the goaf side is in a stress reduction zone. Therefore, both horizontal and vertical stresses in the coal and rock layers near the goaf are reduced. When injecting water near the goaf, under the same water pressure conditions, the effective stress \(\sigma_{n}\) will significantly increase, leading to a significant increase in the permeability coefficient. Utilizing this principle, we can employ water injection methods to explore and conduct advanced exploration for depleted areas or similar structural bodies. The literature [24] also suggests that the extent of the voided area and damage characteristics can be determined by measuring water injection during drilling. During tunneling, boreholes are constructed and water injection tests are performed. When significant anomalies in water pressure, water volume, and permeability are observed during the water injection test, it indicates the presence of depleted areas within a certain range. Furthermore, in the coal wall close to the goaf, there are also disturbances and damages caused by mining activities. In the disturbed zone, fractures develop, and during water injection, sudden changes in water pressure and water volume may occur. Underground Testing of Water Injection Advanced Exploration Method for Depleted Areas Based on Stress-Seepage Coupling Principle