Review of pseudo-three-dimensional modeling approaches in hydraulic fracturing

In the field of hydraulic fracture modeling, the pseudo-three-dimensional (P3D) approach is an efficient and practical computational tool serving as a compromise between two-dimensional and planar three-dimensional models. This review discusses the P3D modeling approach from its early developmental stage in the 1980s to the present. The evolution of P3D modeling is drawn over time based on the major differences in the governing formulation and assumptions considered by each model. The problems of equilibrium height growth and vertical viscous fluid resistance (i.e., non-equilibrium height growth) emphasize the primary differences among these models. Besides, the P3D-based complex fracture network models for shale oil and gas reservoirs accounting for the interaction between preexisting natural fractures and induced hydraulic fractures are discussed. Finally, in the application section, several simulations are reported to demonstrate the validation of the P3D numerical algorithm by comparing it with the Perkins–Kern–Nordgren (PKN) large and small asymptotic solutions, as well as the effect of time-dependent variable injection rates on the hydraulic fracture propagation. The results showed a good matching between P3D and PKN solutions and a significant effect of the wellbore variable injection rate on the evolution of the fracture length.

Quantitative Study on Bed Load Proppant Transport during Slickwater Hydraulic Fracturing

Bed load proppant transport is a significant phenomenon during slickwater hydraulic fracturing. However, the mechanism of bed load proppant transport is still unclear. In the present study, the proppant transport process during slickwater hydraulic fracturing was simulated with a coupled computational fluid dynamics- (CFD-) discrete element method (DEM) model, and the mechanism of the bed load proppant transport was analyzed. A model for calculating the mass flux of the bed load layer was proposed and verified with experimental results from the literature. The results show that bed load migration is an essential mechanism of proppant transport. When the shear force of fluid acting on the surface of the sand bank reaches the critical Shields number, the proppant in the upper layer of the sand bank begins to migrate in the form of bed load. The movement of the bed load layer increases the time for the sand bank to reach the equilibrium height. In addition, the mass flux of the bed load layer significantly affects the equilibrium height of the sand bank. The mass flux of the bed load layer decreases, and the equilibrium height increases as the proppant density, proppant diameter, or rolling friction coefficient and static friction coefficient of the proppant increase, but the mass flux of the bed load layer increases, and the equilibrium height decreases as the fluid viscosity increases.

Numerical Investigation on Proppant–Water Mixture Transport in Slot under High Reynolds Number Conditions

Water hydraulic fracturing involves pumping low viscosity fluid and proppant mixture into the artificial fracture under a high pumping rate. In that high Reynolds number conditions (HRNCs, Re > 2000), the turbulence effect is one of the key factors affecting proppant transportation and placement. In this paper, a Eulerian multiphase model was used to simulate the proppant particle transport in a parallel slot under HRNCs. Turbulence effects in high pumping rates and frictional stress among the proppant particles were taken into consideration, and the Johnson-Jackson wall boundary conditions were used to describe the particle-wall interaction. The numerical simulation result was validated with laboratory-scale slot experiment results. The simulation results demonstrate that the pattern of the proppant bank is significantly affected by the vortex near the wellbore, and the whole proppant transport process can be divided into four stages under HRNCs. Furthermore, the proppant placement structure and the equilibrium height of proppant dune under HRNCs are comprehensively discussed by a parametrical study, including injection position, velocity, proppant density, concentration, and diameter. As the injection position changes from the lower one to the top one, the unpropped area near the entrance decrease by 7.1 times, and the equilibrium height for the primary dune increase by 5.3%. As the velocity of the slurry jet increases from 2 m/s to 5 m/s (Re = 2000–5000), the vortex becomes stronger, so the non-propped area near the inlet increase by 5.3 times, and the equilibrium height decrease by 5.2%. The change of proppant properties does not significantly change the vortex; however, the equilibrium height is affected by the high-speed flush. Thus, the conventional equilibrium height prediction correlation is not suitable for the HRNCs. Therefore, a modified bi-power law prediction correlation was proposed based on the simulation data, which can be used to accurately predict the equilibrium height of the proppant bank under HRNCs (mean deviation = 3.8%).

Numerical Simulation of Proppant Placement in Scaled Fracture Networks

While hydraulic fracturing is recognized as the most effective stimulation technique for unconventional reservoirs, the production enhancement is influenced by several factors including proppant placement inside the fractures. The goal of this work is to understand the proppant transport and its placement process in T-shaped fracture network through simulations. The proppant transport is studied numerically by coupling a computational fluid dynamic model for the base shear-thinning fluid and the discrete element methods for proppant particles. A scaling analysis has been performed to scale down the model from field scale to lab scale by deriving relevant dimensionless parameters. Different proppant size distributions and injection velocities are considered, as well as the friction and cohesion effects among particle and fracture surface. The simulation results show that in the primary fracture, the injected proppant could divide into three layers: the bottom sand bed zone, the middle rolling surface zone, and the top slurry flow zone. The total number of the proppants do not increase much after the dune reach an equilibrium height. The equilibrium height of sand dune in the minor fracture could be greater than the primary fracture, and the distribution of proppant dunes is symmetric. Two deposit mechanisms have also identified in the bypass fracture network: falling deposition and rolling deposition. Additionally, significant momentum changes due to the change in the flow direction at the intersection with natural fractures is identified as a potential factor in accelerating particle deposition.

Calibration of the Interaction Parameters between the Proppant and Fracture Wall and the Effects of These Parameters on Proppant Distribution

Saltation and reputation (creep) dominate proppant transport rather than suspension during slickwater fracturing, due to the low sand-carrying capacity of the slickwater. Thus, the interaction parameters between proppants and fracture walls, which affect saltation and reputation, play a more critical role in proppant transport. In this paper, a calibration method for the interaction parameters between proppants and walls is built. A three-dimensional coupled computational fluid dynamics–discrete element method (CFD–DEM) model is established to study the effects of the interaction parameters on proppant migration, considering the wall roughness and unevenly distributed diameters of proppants. The simulation results show that a lower static friction coefficient and rolling friction coefficient can result in a smaller equilibrium height of the sand bank and a smaller build angle and drawdown angle, which is beneficial for carrying the proppant to the distal end of the fracture. The wall roughness and the unevenly distributed diameter of the proppants increase the collision between proppant and proppant or the wall, whereas the interactions have little impact on the sandbank morphology, slightly increasing the equilibrium height of the sandbank.

Development of a New Mathematical Model To Quantitatively Evaluate Equilibrium Height of Proppant Bed in Hydraulic Fractures for Slickwater Treatment

Summary The proppant bed develops and its height grows until it reaches the critical velocity and equilibrium height. This paper proposes a comprehensive mathematical model to evaluate the equilibrium height for slickwater treatment. We use well-accepted published experimental data and models from other groups to validate our model. After that, we investigate the effects of proppant properties and fluid properties on the equilibrium height. This work can provide critical insights to optimize the design of proppant parameters in a hydraulic fracture. Meanwhile, this model can be incorporated into fracture-propagation simulators for simulating proppant transport.

Fractal imbibition in Koch's curve-like capillar tubes

Fractal geometry eects in capillary imbibition process are studied.Capillary rise analysis in Koch's curve-like tubes were be carried out withiterations i = 0; 1; 2; 3; 4; 5. The behaviour was characterized in function oftime, fractal geometry and height of capillary rise. An geometrical relationshipfor fractal dimension of ow tortuosity (dr) in porous media is obtained.The analytical model of Lucas-Washburn-Cai to describe the capillary rise byspontaneus imbibition in tubes with deterministic fractal geometry is adjusted.The equilibrium height time as function of fractal dimension of ow tortuosity incapillary tubes with tortuous path is also derived.