# Research on Grouting Dynamic Monitoring Based on Borehole-Tunnel Joint Resistivity Method
## 3 Resistivity Monitoring
### 3D Resistivity Inversion Algorithm  
The acquired monitoring data undergo three-dimensional resistivity inversion imaging. The forward modeling component employs the finite volume method to discretize partial differential equations (PDEs), solved using a preconditioned biconjugate gradient stabilized (BiCGSTAB) algorithm [25]. The governing PDE for the forward problem is expressed as:  
\[\
abla\cdot(-\sigma\
abla\emptyset)=\mathrm{I}(\delta(\mathrm{r}-\mathrm{r}_{ \mathrm{s}^{+}})-\delta(\mathrm{r}-\mathrm{r}_{\mathrm{s}^{-}})) \tag{1}\]  
where \(\sigma\) represents the medium's conductivity, \(\emptyset\) is the electric potential, \(\mathrm{I}\) denotes the input current, and \(\mathrm{r}_{\mathrm{s}^{+}}\), \(\mathrm{r}_{\mathrm{s}^{-}}\) are the positions of the positive and negative current sources, respectively. The Dirac delta function \(\delta(\mathrm{r}-\mathrm{r}_{\mathrm{s}})\) is defined as:  
\[\delta(\mathrm{r}-\mathrm{r}_{\mathrm{s}})= \begin{cases}0&(r\
eq\mathrm{r}_{ \mathrm{s}})\\ 1&(r=\mathrm{r}_{\mathrm{s}})\end{cases}  \tag{2}\]  
Rewriting Equation (1) in matrix form yields:  
\[(\mathrm{DS}(\sigma)\mathrm{G})\mathrm{u}=\mathrm{A}(\sigma)\mathrm{u}= \mathrm{q} \tag{3}\]  
Here, \(\mathrm{D}\) and \(\mathrm{G}\) represent the three-dimensional divergence and gradient operators, respectively. \(\mathrm{S}(\sigma)\) is a diagonal matrix containing conductivity values, \(\mathrm{u}\) is the potential vector, \(\mathrm{A}(\sigma)\) is the forward operator, and \(\mathrm{q}\) is the source vector encoding the positions of the positive and negative current sources. The potential \(\mathrm{u}\) encompasses the electric potential  
Figure 5: Schematic cross-section of borehole electrode string. values at every point within the entire 3D domain. However, the inversion process requires only a subset of \(\mathrm{u}\). To extract the necessary data points for inversion, a projection matrix \(\mathrm{Q}\) is defined, leading to the final formulation of the forward problem:  
\[\mathrm{d}=\mathrm{Qu}=\mathrm{QA}^{-1}(\sigma)\mathrm{q} \tag{4}\]  
For the inversion process, a quasi-Gauss-Newton method optimizes model fitting, with model updates computed via a preconditioned conjugate gradient (PCG) algorithm. Tikhonov regularization incorporates prior information to stabilize the ill-posed problem [26; 27]. The objective function is formulated as:  
\[\mathcal{O}(\mathrm{m})=\frac{1}{2}\|\mathrm{d}(\mathrm{m})-\mathrm{d}_{ \mathrm{obs}}\|^{2}+\frac{\beta}{2}\left\|W\Big{(}\mathrm{m}-\mathrm{m}_{ref} \right\|^{2} \tag{5}\]  
Here, \(\mathrm{d}_{\mathrm{obs}}\) represents the measured data, specifically the observed voltage divided by the input current. \(\mathrm{d}(\mathrm{m})\) denotes the simulated data obtained by forward modeling a given conductivity model. The second term in the equation is the Tikhonov regularization term, which incorporates prior information about the solution. The parameter \(\beta\) acts as the regularization parameter, balancing the influence of data misfit and model regularization during minimization. \(\mathrm{W}\) is the model regularization matrix, \(\mathrm{m}\) is the inversion model parameterized as \(\mathrm{ln}(\sigma)\) (where is conductivity), and \(\mathrm{m}_{ref}\) is the reference model. The first term of the objective function ensures that the inverted model matches the observed data, while the second term imposes prior constraints (e.g., shape, structural continuity) on the final model. The database of the monitoring system establishes an automated interaction with the processing software, enabling fully autonomous processing and interpretation without manual intervention. For 3D inversion using the processing software, users must input mesh generation parameters and an initial model. The grid size is typically determined by the electrode spacing, while the initial model is preferably a homogeneous model constructed from surrounding rock resistivity.