Previous 2D Models Research Articles

3D Simulation of Externally Driven Reconnection

A 3D magnetohydrodynamic simulation is presented. The essential conclusions obtained by the previous 2D model, such as the slow shock formation and the strong plasma jet generation, are reconfirmed. In addition, several new findings pertinent to three dimensionality are obtained. Among them, particularly interesting and important is the generation of field aligned currents at the slow shock associated with a local interruption of the neutral sheet current. It is also interesting to observe the generation of super–magnetosonic flows with the Mach number of 2.

Three-Dimensional Simulation of Hydraulic Fracturing

Summary A comprehensive three-dimensional (3D) simulator of hydraulic fracturing has been developed. The model formulation couples 3D, two-phase flow in the reservoir with a 3D fracture model in a vertical plane, proppant transport, and heat transfer. The numerical implementation is in finite differences that use several integrated grid systems. The model is computationally efficient and greatly increases the realism of modeling compared with two-dimensional (2D) simulators. The new features important for applications include treatment of proppant transport and closure process, simulation of fracture growth after shut-in, and in particular the vertical fracture propagation, which gives the model the capability to predict fracturing pressure increase or decrease. predict fracturing pressure increase or decrease. Examples show the sensitivity of the model to design parameters and properties of confining strata, and give parameters and properties of confining strata, and give comparisons between simulations that use different models of fracture geometry, Introduction Modeling of hydraulic fracturing has progressed recently, spurred by the increased importance of well stimulation and other fracturing applications in the field. The historical background and survey of the field is included in Refs. 1 and 2. Developments in computation of fracture geometry are discussed in Ref. 3, which presents a comprehensive 3D model of fracture geometry that is practically oriented and efficient. practically oriented and efficient. The approach taken is the same as in the previous 2D model described in Refs. 1 and 2--i.e., simultaneous modeling of the fracturing process, reservoir fluid flow and heat transfer, and proppant transport. The 2D model includes much of the physics necessary for realistic simulation of fracturing treatments, especially in low-permeability reservoirs, as discussed in Ref. 2. However, it has a weakness (common with other existing industrial models) of treating the fracture geometry as 2D--i.e., with separately determined fracture height. The model described here represents a major advance because it determines simultaneous lateral and vertical fracture evolution coupled with leakoff and subsequent production from a 3D reservoir. A new concept of a production from a 3D reservoir. A new concept of a pseudo-3D hydraulic (P3DH) fracture geometry has pseudo-3D hydraulic (P3DH) fracture geometry has been developed and integrated within the overall model. The resulting scheme for calculation of fracture geometry is described in detail in Ref, 3. It is computationally very fast and, when combined with efficient integration of the components of the overall model, results in a practical 3D simulation tool. This paper describes the formulation and integration of the overall model and deals in detail with some of its components that have not been described before. The treatment of the physics of filtration, fracturing fluid rheology, reservoir flow, etc. are not repeated here. Although the model can treat both vertical and horizontal fractures, we deal here only with fractures extending in a plane perpendicular to the reservoir bed. An example of an application including horizontal (pancake-type) fracture is found in Ref. 4. The physical and numerical formulation of the model is presented in the first two sections with the exception of the proppant transport and fracture-closure model. These parts of the system are coupled only weakly and for parts of the system are coupled only weakly and for clarity are dealt with in a separate section. The capabilities of the model are illustrated by examples of fracturing treatment simulation using both 2- and 3D fracture propagation. propagation. Model Formulation The model formulation consists of a set of equations describing (1) the 3D, two-phase flow in the reservoir, (2) two-phase non-Newtonian flow in the fracture, (3) heat transfer in the fracture and in the reservoir, including overburden losses, (4) lateral and vertical propagation of the fracture (resulting in a general fracture propagation of the fracture (resulting in a general fracture shape), (5) proppant transport, and (6) wellbore hydraulics. Additional relations describe the PVT and rheology behavior of reservoir and fracturing fluids, the physics of filtration on fracture walls, interaction of solid physics of filtration on fracture walls, interaction of solid stress and fluid flow, etc. The general physical configuration of the model is shown in Fig. 1. The direction of the lateral propagation of the fracture can be specified by giving the path of the leading edge--which can be independent of the bedding plane of the reservoir--and its height. Given this constraint, the fracture can then propagate arbitrarily through the reservoir and the adjoining propagate arbitrarily through the reservoir and the adjoining strata, as dictated by their elastic properties and confining stress. As a special case, the geometry model can be used without fracture height growth with generalized Perkins-Kern (PK) or Christianovich-Geertsma-DeKlerk (CGD) Perkins-Kern (PK) or Christianovich-Geertsma-DeKlerk (CGD) type geometries (Fig. 2). Fluid Flow and Heat Transfer. The fractured well is assumed to be at the origin of the x-y, plane and the fracture propagates along the x-z plane. JPT P. 1177