## 5 Results  
The developed mathematical model of pressure distribution in a structurally complex reservoir is implemented in the Python programming 3.12 environment. The input influence is the pressure levels at the location points of producing wells, formed in the form of selected spatial modes with certain numbers (spatial frequencies):  
\[p(i_{z},j_{z},f_{z},t)=U(x,y), \tag{14}\]  
where \(U(x,y)=A+A\cdot\cos\left(\frac{\pi x}{L_{x}}x\right)\cos\left(\frac{\pi J}{L_{ y}}y\right)\); \(A\)--input amplitude; \(\xi\) and \(\gamma\)--numbers of spatial modes in \(x\) and \(y\), respectively; \(L_{x}\) and \(L_{y}\)--object dimensions along X and Y axes, respectively; \(p\)--target pressure; \(i_{z},j_{z},f_{z}\)--wellbore coordinates of the \(z\)-th wellbore in the deposit. This study considers modelling a section of a gas-bearing horizon containing nine wells. At the output of the system, we determine the level distribution in the given (control) points of the reservoir. The points at the bottom of the producing wells will be considered as control points. Investigate the behaviour of the model when an input disturbance is applied to it. In Equation (12), assume the amplitude of the input influence \(A=40\) MPa. Spatial modes \(\xi_{1}=1,\xi_{2}=3\) and \(\gamma_{1}=1,\gamma_{2}=3\). By applying influences on the model for each pair of modes, we obtain the following transient plots (Figures 3 and 4). Figure 3: Graph of the transient process by mode 1. In all graphs in this chapter, the solid blue line represents the well transient in Block 6. By analysing the transient graph of mode 1 and mode 3 shown in Figures 3 and 4, we can conclude that the transfer function (TF) of the object \(W_{i}(G,s)\) for each mode can be represented as an aperiodic link of the first order with a delay:  
\[W_{i}(G,s)=\frac{k_{i}(G)}{T_{i}(G)s+1}e^{-\tau_{i}}, \tag{15}\]  
where \(T_{i}(G)\)--time constant of the object by \(i\)-th mode: \(k_{i}(G)\)--gain of the object by \(i\)-th mode; \(\tau_{i}\)--delay time by \(i\)-th mode; \(s\)--Laplace operator; \(G\)--generalised coordinate; \(i=1,3\). The gain is determined by the formula  
\[k_{i}(G)=\frac{y_{equ,i}}{x_{in,j}}, \tag{16}\]  
where \(y_{equ}\)--steady-state transient value and \(x_{in}\)--system input. According to the results of modal modelling of the distributed object, the following results were obtained \(y_{equ,1}~{}=~{}46.7\) MPa and \(y_{equ,3}~{}=~{}43.7\) MPa. The input impact is calculated by the formula:  
\[U_{i}(x,y)=A_{i}+A_{i}\cdot\cos(\frac{\pi\xi_{i}}{L_{x}}x)\cos(\frac{\pi\tau_ {Ti}}{L_{y}}y). \tag{17}\]  
Substituting all values into Formula (15), we obtain the values of the input influence: \(U_{1}(x,y)=48.96\) MPa and \(U_{3}(x,y)=~{}40\) MPa. The obtained values are used in Formula (14), and we obtain the gain of TF:  
\[k_{1}(G)=\frac{46.7}{48.96}=0.95,\] \[k_{3}(G)=\frac{43.7}{40}=1.09.\]  
We determine the time constant \(T_{1}(G)\) and the delay time \(\tau_{1}\) by approximating the transient graph (Figure 3). Figure 4: Graph of the transient process by mode 3. According to the graph in Figure 5, determine the value of \(T_{1}(G)=8\) h, \(\tau=1\). Thus, the TF of the object by mode 1 will take the following form:  
\[W_{1}(G,s)=\frac{0.95}{8s+1}e^{-s}. \tag{18}\]  
According to the graph in Figure 6, determine the value \(T_{3}(G)=6\) h, \(\tau_{3}=1\). Thus, the TF of the object by mode 3 will take the following form:  
\[W_{3}(G,s)=\frac{1.09}{6s+1}e^{-s}. \tag{19}\]  
Having obtained the TF of the object by modes 1 and 3, let us proceed to the synthesis of a distributed control system; namely, we develop a distributed high-precision controller (DHC), the TF of which has the following form [52; 57]:  
Figure 3:0  
Figure 3:1 Approximation of the time constant by mode 1 (the blue line indicates pressure dynamics; the black line denotes the approximation of the time constant \(T_{1}\)).