## 3 Method
### Methods for Recapping Shale Free Gas Content
#### 3.2.3 The Method of Paleo-Pressure Recovery  
This paper utilizes a pressure simulation equation to reconstruct paleo-pressure. A new set of pressure development equations specifically suited for the properties of shale gas reservoirs were developed. The mass conservation theorem governs the migration of shale gas, and is expressed through the continuity equation:  
\[\left\{ \begin{array}{c}\frac{d(\rho\varnothing)}{dt}=\
abla\big{(}\rho\!\! \cdot\!\varnothing v^{\dagger}\pm\rho\!\!\cdot\!\varnothing v^{\rightarrow} \big{)}+\rho\!\!\cdot\!q_{\rm g}\\ \qquad\varnothing v^{\dagger}=D\frac{\mu}{P}\!\!\cdot\!\frac{1}{\mu}\
abla P^{ \dagger}\\ \qquad\varnothing v^{\rightarrow}=K\!\!\cdot\!\frac{1}{\mu}\
abla P^{ \rightarrow}\\ \qquad\rho=\frac{P\ M}{RT}\end{array} \right. \tag{6}\]  
The relationship between porosity and shale overlying pressure and formation fluid pressure is as follows:  
\[\frac{d\varnothing}{dt}=-\beta_{\rm s}(1-\varnothing)\frac{d(S-P)}{dt} \tag{7}\]  
According to the basic theory of fluid mechanics, the ideal state equation of compressible gas is:  
\[\frac{1}{\rho}\frac{d\rho}{dt}=\frac{1}{P}\frac{dP}{dt}-\frac{1}{T}\frac{dT}{dt} \tag{8}\]  
Substituting Equations (7) and (8) into Equation (6):  
\[\left(\frac{\varnothing}{P}+\beta_{\rm s}-\varnothing\beta_{\rm s}\right)\frac {dP}{dt}=\frac{1}{\rho}\
abla\left(\rho\frac{D}{P}\
abla P^{\dagger}\pm\rho \frac{K}{\mu}\
abla P^{\rightarrow}\right)+(1-\varnothing)\beta_{\rm s}\frac{ dS}{dt}+\varnothing\frac{1}{T}\frac{dT}{dt}+q_{\rm g} \tag{9}\]  
In the formula: \(\varnothing\)--porosity of the reservoir, dimensionless; \(P\)--gas reservoir pressure, MPa; \(\
abla P^{\dagger}\)--vertical cap pressure gradient, MPa/m; \(\
abla P^{\rightarrow}\)--reservoir lateral pressure gradient, MPa/m; \(T\)--formation temperature, \({\rm K}\); \(\beta_{\rm s}\)--isothermal compression coefficient of shale skeleton, MPa\({}^{-1}\); \(t\)--time, Ma; \(\rho\)--underground density, kg/m\({}^{3}\); \(D\)--cap diffusivity, m\({}^{2}\)/s; \(K\)--Reservoir bedding permeability, m\({}^{2}\); \(\mu\)--shale gas viscosity, MPa\(\cdot\)s; \(M\)--molarmass of methane, g/mol; \(S\)--overlying formation pressure, MPa; and \(q_{\rm g}\)--rate of gas volume formation per unit volume, m\({}^{3}\)/Ma. Formula (9) represents the contribution of four factors to pressure formation in shale gas reservoirs: vertical and lateral migration, compaction, temperature, and gas generation rate. The left side of the equation shows the change in pressure over time. Depending on the stage of shale tectonic uplift, there are two alternative versions of Formula (9) for a given computation. The stratum's occurrence was mild due to the nearby tectonic units not having formed prior to the most recent uplift. The pressure between the adjacent tectonic units was roughly equal, so the reservoir's lateral pressure gradient (\(\
abla P^{\rightarrow}\)) can be disregarded. The pressure gradient of the vertical cap (\(\
abla P_{\rm p}^{\uparrow}\)) is determined by the gas reservoir pressure, cap thickness, cap bottom, and hydrostatic pressure at the top of the cap. The pressure evolution equation is used to determine the gas reservoir pressure at the base of the cap based on the time-depth relationship derived from the stratum's burial history data, as well as the pressure gradient, geothermal gradient, and pressure gradient of the static rock in the study area. The gas yield curve with maturity (\(q_{\rm g}\)) was obtained from thermal simulation experiments on hydrocarbon generation in shale [2]. The \"time-gas generation rate\" relationship is then calculated using the shale's thermal maturity history and other intermediate characteristics, such as organic matter maturity.