# Research Status and Development Direction of Formation Damage Prediction and Diagnosis Technologies
## 3 Laboratory Tests and Numerical Approach Method
### Mathematical Modeling Method  
The mathematical model is a scientific or engineering model constructed by mathematical logic methods and mathematical language. It can yield quantitative or qualitative outcomes but must be reflected in quantitative form. Therefore, the mathematical model is a strong tool that can be used for interpretation. However, since mathematical models are usually abstracted from complex real-world problems and are too idealized, some calculation parameters are often difficult to obtain. [PERSON] et al. [74] compares the measured and predicted data and points out the minimal error in Figure 5.  
In 1952, [PERSON] et al. [75] and [PERSON] [76] predicted the saturation index method for inorganic scaling. Although this was relatively simple and easy to use, it did not consider the impacts of pressure, temperature, ionic strength, and other factors, which ultimately affected the validity of the application. [PERSON] [77] presented a theoretical-empirical model to describe the filtration loss process of compressible mud-cake based on [PERSON]'s law and some dynamic and static filtration loss experiments. In addition, the relationship between the accumulation of drilling fluid filtrate and time was also developed for medium- and high-porosity formations. The methods of [PERSON] and [PERSON] are too complicated and extremely inconvenient to use [78; 79]. In 1994, [PERSON] and [PERSON] improved the prediction model based on the Amaefble theory after learning from the experience of many previous scholars, and finally introduced a more refined prediction model, the Oddo-Tomson saturation index method [80]. In 1994, [PERSON] et al. proposed using the water phase trapping index to predict the damage extent due to water phase trapping [81]. Based on the bulk volume of water (BVW) method, the influence of  
Figure 4: Schematic diagram of the CT scan of the rock sample [66]. permeability, relative permeability curve form, pore structure (capillary pressure curve form), intrusion profile, formation pressure (the maximum return pressure difference that may be provided), and other factors relating to water-phase trap damage were considered. However, in practical applications, parameters such as relative permeability correction factor, intrusion profile correction factor, and formation pressure correction factor are more difficult to obtain and are often omitted in favor of their simple basic forms for prediction, which also limits the application of this method:  
\[APT_{i}=0.25[\lg(K_{a})]+2.2(S_{wi})-RP_{a}-IP_{a}+PR_{a} \tag{5}\]  
where \(APT_{i}\) is the water-phase trapping index, dimensionless; \(K_{a}\) is the average gas-measured permeability of the formation, \(10^{-3}\)\(\upmu\)m\({}^{2}\); \(S_{wi}\) is the initial water saturation, dimensionless; \(RP_{a}\) is the relative permeability correction factor, dimensionless; \(IP_{a}\) is the intrusion profile correction factor, dimensionless; and \(PR_{a}\) is the formation pressure correction factor, dimensionless. In 1994, [PERSON] et al. [82] and [PERSON] et al. [83] also combined laboratory-based experimental data with wellsite evaluation and used large amounts of experimental data to establish several mathematical models to represent each damage factor separately. Then, they calculated several parameters, such as damage zone radius, post-damage permeability, and epithermal coefficient, under different damage factors. Finally, they obtained the proportion of each damage factor in the total damage degree according to the formation damage superposition principle.