## 1 Introduction
The authors found that the resolution in some tests was insufficient to verify the presence of a barrier for oocysts of _Cryptosporidium_. Here, the pressure decay test process was considered as a static process, which only took into account the test pressure at test completion, the maximum hydraulic head, the pore shape correction factor, and the material-specific contact angle. In gas-liquid fluidization systems, bubble dynamics plays a key role in dictating the transport phenomena, and the pressure affects the physical properties of the gas and liquid phases and hence the bubble behavior [9]. The dynamics of bubble flow is relatively complicated and is affected by many parameters, such as the defect diameter, gas flow rate, liquid height, wettability, and physical properties of the dispersed and continuous phases [10]. According to the bubble dynamics theory, the pressure decay test process, which involves gas entering wet microdefects and balancing the resistance, bubble formation and detachment, and then gas chamber pressure decay, would be a bubble dynamic process. The bubble dynamic behavior of membrane defects has a significant effect on the pressure decay characteristics and identification of membrane integrity. In addition, the membrane is well characterized with regard to its elastic properties, which are responsible for the gas pressure decay. Therefore, we can better understand the mechanisms in pressure decay tests from studies on the dynamic behaviors of bubbles and the air flow from the defect or breach on the membrane. Bubbling behaviors, including bubble formation, movement, and interaction, play an important role in the practical operation of many industry reactors. As a result, bubbling behavior has been a widespread topic of interest for a long time. [PERSON] et al. [11; 12] investigated the dynamic behavior of bubbles emerging from different micro-orifices on a plate in a stagnant and isothermal liquid and found that, with the increase in the height, the bubble velocity reduced first and then increased, while the bubble radius and aspect ratio appeared as a bifurcation structure. [PERSON] et al. [13] analyzed locally the dynamics of bubble growth and detachment from a rigid orifice and a flexible orifice and found that the differences in orifice nature and properties had a strong impact on the associated bubble formation phenomena. In a full-scale gas-liquid reactor, the different bubble generations was found to have a strong impact on the gas hold-up, the interfacial area, and the mass transfer. Furthermore, [PERSON] et al. [14] proposed an analytical model to describe the dynamics of bubble growth and its detachment process generated from a flexible rubber membrane orifice submerged in water, taking into account the rubber membrane features (elastic behavior and wettability). The predicted bubble diameters at detachment were in agreement with the experimental measurements, and the orifice coefficient, the elastic pressure, and the surface tension force appeared to be key parameters. The main characteristic considered nonspherical bubble formations from a flexible membrane sparger orifice. [PERSON] et al. [15] investigated the formation of bubbles in micro- and macroreactors and proposed a method to reveal the various mechanisms governing the formation from micro- to macroscale. Based on these previous studies, the objective of the present work was to study bubble formation on submerged defects on ultrafiltration membranes under exactly defined conditions and develop an analytical model to describe the pressure decay generated from membrane damage in the membrane gas chamber using bubble dynamic behaviors. The aim was to better understand the parameters that govern the pressure decay rate. By comparing the predicted pressure decay rates with the experimental data, this study presents the development and experimental validation of a theoretical model that describes changes in the pressure and the PDR in the membrane gas chamber during pressure decay tests.