Thermal-Hydrodynamic Simulation of Unsteady Drilling Fluid Intrusion and Mud Cake Evolution in Porous Reservoirs Based on CFD

During the drilling process, the positive pressure difference between the drilling fluid column and the formation pore pressure leads to the continuous invasion of drilling fluid into the surrounding porous reservoir formations [1, 2]. As this fluid intrudes, suspended solid particles in the drilling fluid deposit onto the wellbore surface, gradually forming a low-permeability mud cake layer. This mud cake plays a pivotal role in regulating both hydraulic and thermal behaviors at the fluid–formation interface. Its formation and structural evolution exert a critical influence not only on the penetration depth of the fluid but also on the associated heat transfer mechanisms. The permeability of the formation directly impacts both the extent of fluid intrusion and the properties of the mud cake itself [3, 4]. In high-permeability formations, rapid mud cake development results in a compact structure that restricts both fluid flow and thermal migration. In contrast, low-permeability formations hinder effective particle deposition, leading to a loosely structured mud cake with limited capacity to obstruct intrusion or modulate thermal exchange. This divergence highlights the need for a systematic investigation into the interplay between mud cake evolution, multiphase seepage, and heat transfer dynamics in varying geological settings.

To date, predictions of drilling fluid intrusion depth have been approached primarily through physical simulation and numerical modeling. Traditional filtration experiments, such as API filtration loss tests, are based on static or steady-state assumptions. While these offer operational guidance, they fail to capture the inherently transient, heat-sensitive nature of downhole fluid behavior. In reality, drilling fluid invasion is a multiphase and unsteady process where thermal effects and flow dynamics co-evolve. The temperature gradient between the high-temperature formation and the comparatively cooler drilling fluid alters viscosity, permeability, and local pressure distribution, thereby influencing fluid penetration. Moreover, the deposition and compaction of solids continuously modify both the hydraulic and thermal conductivities near the wellbore, forming a dynamic boundary layer with evolving thermophysical properties.

Some researchers have addressed these complexities through innovative experimental designs. For example, a radial sand disk model was developed to visualize resistivity changes during multiphase intrusion [5], while others have constructed core-holder systems with integrated pressure and temperature controls to simulate coupled filtration under realistic wellbore conditions [6, 7]. These setups have contributed to a better understanding of the evolving saturation and thermal profiles. On the numerical side, studies have shown that radial filtration loss correlates positively with exposure time and pressure differential, and negatively with fluid viscosity—factors all strongly influenced by temperature. Building on two-phase flow theory and ion diffusion principles, a predictive model was proposed that accounts for reservoir-specific characteristics under variable thermodynamic conditions [8].

Efforts to characterize dynamic mud cake formation have also advanced. Sharma et al. [9] introduced a filtration-based mathematical model using loss coefficients and formation damage indicators to characterize the interplay between invasion and near-wellbore alteration. More recent models integrate the conservation of mass and radial flow resistance to simulate the coupled movement of fluid and solids, with validation from laboratory tests such as API filtration and dynamic core plugging experiments [10]. The modeling landscape has progressed from one-dimensional linear approximations [11] to more comprehensive three-dimensional CFD frameworks [12]. However, many of these models simplify the process to single-phase flow and overlook the temperature-dependent behaviors of mud cake and drilling fluids. When fluid properties differ significantly or thermal effects dominate, single-phase assumptions lead to inaccuracies. Thus, two-phase flow models incorporating heat transport offer a more robust and realistic basis for predicting dynamic invasion behavior under varying reservoir and thermal conditions.

Despite these developments, several key issues remain unresolved. Notably, the formation mechanisms and structural features of mud cake vary significantly across reservoirs, and the impact of these variations on both fluid intrusion depth and local thermal field distribution is still not well understood. To address this, the present study constructs a CFD-based multiphase seepage model that couples fluid flow with the thermal and structural evolution of the mud cake. By incorporating time-dependent permeability and thickness parameters of the mud cake, and solving the two-phase Darcy flow equations alongside the conservation of energy and mass, this study simulates how mud cake development governs the depth and thermal spread of fluid invasion across formations of differing permeability. Comparative analysis of static and dynamic mud cake models under various geological and thermal conditions is performed to identify their respective domains of applicability and critical threshold behaviors. The results aim to enhance the accuracy of drilling fluid invasion prediction while offering a thermally informed framework for assessing near-wellbore formation damage and designing adaptive fluid strategies in complex, temperature-sensitive drilling environments.

Since formation permeability plays an important role in the penetration depth of drilling fluid and the physical parameters of mud cake formation, this paper divides the common reservoirs into five types of reservoirs, including ultra-high permeability, high permeability, medium permeability, low permeability and ultra-low permeability, according to the conventional oil and gas reservoir classification standard and the permeability as the dividing condition, and the permeability range is shown in Table 1.

Table 1. Reservoir division criteria

During the over-equilibrium drilling process, the drilling fluid invades the reservoir driven by the positive pressure difference, triggering the redistribution and migration of the oil-water phases in the near-well zone, forming a coupled two-phase seepage system composed of well bore, dynamic mud cake and reservoir. In order to accurately describe this dynamic process, a mathematical model needs to be established to couple the dynamic filtration loss of drilling fluid in the well bore with the seepage process of oil and water in the reservoir. Based on the actual physical mechanism, the model is based on the following basic assumptions: (1) there are only oil and water two-phase fluids in the reservoir; (2) The multiphase seepage in the strata around the well follows Darcy's law; (3) Ignore the influence of capillary force.

3.1 Mathematical model of drilling fluid intrusion formation without considering mud cake conditions

In the theoretical modeling of drilling fluid intrusion into the formation, if the formation and influence of mud cake are ignored, it can be assumed that the drilling fluid is always in direct contact with the well wall, and the effective pressure difference acting on the formation interface is constant, which is equal to the difference between the drilling fluid column pressure and the formation pore pressure. Under these conditions, the drilling fluid intrusion process can be simplified to single-phase seepage in the porous medium, and its flow behavior conforms to Darcy's law, which can be expressed as [18]:

$V_1=-\frac{K}{\mu} \nabla P$

where, in $V_1$ the Darcy penetration velocity, m/s; $K$ is the formation permeability, mD; $\mu$ is the dynamic viscosity of the fluid, Pa·s; $\nabla P$ is the driving force, MPa.

Under the condition of ignoring the source sink term, the flow of drilling fluid in the porous medium follows the law of conservation of mass, and its continuity equation is:

$\frac{\partial}{\partial \mathrm{t}}\left(\varphi_f \rho_1\right)+\nabla \cdot\left(\rho_1 V_1\right)=0$

where, in $t$ time, s; $\rho_1$ is the density of the fluid, kg/m³.

Substituting Darcy's law into the continuity equation gives the basic seepage differential equation describing pressure diffusion:

$\frac{\partial}{\partial \mathrm{t}}\left(\varepsilon_s \rho_1\right)=\nabla \cdot \frac{\rho_1 \kappa}{\mu}(\nabla P)$

For the multiphase seepage situation, the two-phase Daxi flow model is used to describe the interaction between drilling fluid and formation fluids (such as oil phase). The effective viscosity of each phase was characterized by the coupling of relative permeability and saturation. The mixture viscosity model can be expressed as [19]:

$\bar{\mu}=\frac{\bar{\rho}}{\frac{k_{r 1} \rho_1}{\mu_1}+\frac{+k_{r w} \rho_w}{\mu_w}}$

where, in $k_{r 1}$ relative permeability of oil; $k_{r w}$ is the relative permeability of water;

Mixture equivalent viscosity model:

$\frac{1}{\bar{\mu}}=S_1 \frac{k_{r 1}}{\mu_i}+S_w \frac{k_{r w}}{\mu_w}$

In the porous media multiphase flow module, the fluid volume fraction inside the porous medium lacks the relevant control conditions. For simulating the migration and diffusion of drilling fluid intrusion in porous media, the two-phase Darcy Law module has ideal and uniform results due to the control of relevant conditions [20].

$S_1+S_w=1$

where, in $S_1$ integral number of drilling fluid; $S_w$ is the oil volume fraction of the formation.

In this model, the system of control equations for Darcy's law of two phases can be written as:

$\frac{\partial}{\partial t}\left(\varphi_0 c_g\right)+\nabla\left(c_g\left(-\frac{\kappa}{\mu} \nabla \mathrm{P}\right)\right)=\nabla \cdot c_g$

The initial conditions for the seepage field in the formation are defined by the original fluid pressure and saturation prior to drilling operations:

$\left\{\begin{array}{l}p_w=p_{w 0} \\ S_w=S_{w 0}\end{array}\right.$

Upon completion of drilling, the drilling fluid acts directly on the wellbore wall, where the internal pressure remains constant and equal to the wellbore pressure, while the fluid saturation also remains unchanged. Thus, the inner boundary condition of the formation is defined as:

$\left\{\begin{array}{l}p_w=\nabla P \\ S_w=S_{w b}\end{array}\right.$

For the outer boundary of the system, both the formation pressure and fluid saturation retain their original values. Therefore, the outer boundary condition of the formation is given by:

$\left\{\begin{array}{l}p_w=p_{w 0} \\ S_w=S_{w 0}\end{array}\right.$

3.2 Considering the drilling fluid intrusion formation model under mud cake conditions

Mud cake plays a key control role in the intrusion process of drilling fluid, and its formation is a dynamic evolution process accompanied by time-varying characteristics of porosity and permeability. The dynamic behavior of mud cake significantly affects the pressure distribution and filtration rate in the near well zone, which in turn affects the intrusion depth of drilling fluid and the degree of formation damage. During the filtration process, the porosity and permeability of the mud cake continued to change with time, and the permeability of the mud cake and the permeability of the thickness were affected. Therefore, it is very important to accurately characterize the intrusion depth of drilling fluid and the physical properties of the formation around the wellbore hole. The drilling mudcake prediction equation proposed by Zhao [21], as adopted in this study, derives its applicability from the underlying physical mechanisms of cross-flow filtration theory on which the model is based. The validity of this equation highly depends on the composition and stability of the drilling fluid system. Its core assumptions include a predictable particle size distribution and concentration of solid phases in the drilling fluid, as well as stability under shear, ensuring that particle deposition and compaction remain a mechanically-dominated process. As a result, the model demonstrates strong predictive performance for water-based or oil-based drilling fluids with sufficient solid content and stable properties. However, in formations rich in water-sensitive clay, physicochemical interactions such as hydration, swelling, and dispersion of the formation rocks may significantly interfere with the mechanically-dominated mudcake formation process. Under such conditions, additional modifications may be required to improve the predictive accuracy of the model. Based on mass conservation and particle deposition kinetics, the equation for the thickness of the mud cake is:

$L_{c(t)}=\sqrt{\left(k \frac{\Delta x_m}{k_m}\right)^2+2 \frac{\varepsilon_{s o}}{\varepsilon_s-\varepsilon_{s o}} \frac{k}{\mu} \Delta P \gamma t}-k \frac{\Delta x_m}{k_m}$

where, in $k$ is the permeability of mud cake, m²; $\Delta x_m$ is the thickness of the percolation medium, m; $k_m$ is the permeability of the permeability of the permeate medium, m²; $\varepsilon_{s o}$ is the solid phase content of drilling fluid; $\varepsilon_s$ is the solid phase content of the mud cake; $\mu$ is the viscosity of the liquid, Pa·s; $\Delta P$ is the filtration loss pressure difference, Pa; $\gamma$ is the possibility of spherical particles deposited on the surface of the mud cake; $t$ for time, s.

Dynamic permeability model:

$k=(0.197 \Delta P-1.09) L_{c(t)}-1.5 v+3.9$

where, in: L_c(t) is the thickness of the mud cake, mm; $V$ is the viscosity of the drilling fluid, m/s;

When the drilling fluid passes through the mud cake, there is a differential pressure loss, and when the drilling fluid touches the mud cake, the velocity equation at this time is:

$V_1=-\frac{K}{\mu} \nabla\left(P-P_s\right)$

The subsequent calculation can be brought into continue to calculate the depth of drilling fluid intrusion into the formation after considering the mud cake. However, some scholars simply consider the introduction of mud cake as the fixed value after equilibrium, so this paper analyzes the dynamic mud cake stability value as the third case.