Mechanisms Controlling Fracture-Height Growth in Layered Media

Abstract

Summary. This paper considers the mechanisms that affect the lateral and vertical extension of a hydraulically induced fracture. These include natural mechanisms-such as contrasts in stresses, moduli, toughness, and leakoff-and those induced by leakoff and fluid heterogeneities. Treatment trends during the injection and closure are analyzed with a state-of-the-art 3D fracture model. The effect of the vertical fracture penetration on conductivity and water coning and the parameters influencing them are then quantified. Introduction Factors influencing the vertical propagation of a hydraulic fracture have been discussed in the literature. They generally fall into three categories: natural factors, artificial (or desired) effects, and response of the formation to fluid invasion. The first category refers to in-situ conditions that control the propagation and include stress contrasts, in-situ stress gradients, elastic (or plastic) behavior, tough-ness, discontinuities and natural fractures, fracturing-fluid leakoff, and heat-transfer properties. The second category-injection characteristics (flow rate and rheology), proppant placement, and temperature profiles-depends on injection and has not been effectively exploited. Finally, when a reservoir medium is invaded by fracturing-fluid filtrate, backstresses may be induced, and poroelastic and thermoelastic effects may have a significant effect, especially in oil reservoirs. Characteristic fracture geometries and treating pressures are presented for those different effects. The effect of natural fractures and geological discontinuities has been dis-cussed elsewhere; this paper focuses on an ideally isotropic layered system (Fig. 1). The usual problem encountered in field practice is to overcome adverse containment conditions (small stress contrasts between layers). Controlling height growth has led to small injection rates or small viscosities to avoid exceeding a critical pressure that would lead to out-of-zone propagation. Techniques have been (revised to control height growth artificially, and the conditions for those treat-ments to be effective are discussed. he need to monitor (or even to control) fracture growth prompted the need to develop fracturing-pressure interpretation techniques. As early as 1979, Nolte proposed a technique that allowed fracture parameters to be determined from the interpretation of pressure behavior after injection. Nolte and Smith also gave guidelines for determining the mode of fracture propagation from the excess pressure evolution (leakoff and containment). Note, however, that the pressure-decline analyses developed by these authors assumes a constant relation between pressure and fracture volume ("compliance"), and it is important to determine the validity of such an assumption for a formation with varying properties (elastic properties, stresses, and leakoff). Comparisons between closure times predicted by Nolte's theory and those obtained with a numerical simulator for layered media are presented. Finally, the problem of controlling vertical fracture growth is generally motivated by the need to prevent possible water or gas breakthrough for adjacent layers and to optimize the area drained by the fracture. An implicit multiphase simulator is used here to simulate the first category of effects, while the second category of effects can be addressed with classic pressure-transient analysis techniques. From an operational point of view, the problem is to determine the layer properties (through minifractures and pressure-decline analysis), to determine (for production) the optimum fracture position with respect to layers (to ensure the most effective fracture) and to water- and gas-bearing formations, and to design the treatment to achieve the desired fracture geometry (eventually through artificial height-growth-control techniques). Modeling Fracture Propagation Through Layers Models and simulators have become increasingly popular for treatment design, sensitivity analyses of selected treatment variables or formation properties (owing to advances in numerical analysis) and reduction in computing-time cost. Most models are based on the assumptions of linear elastic isotropic behavior of the layers; boundary-integral techniques are rapidly becoming the dominant numerical approach because of their inherent property to discretize only the fracture (or discontinuity) surface. instead of the whole formation, with classic finite elements. Theoretically, 3D models allow simulation of the effect of any continuous lateral and vertical stress distribution, which may be inferred from logging tools or microhydraulic fracturing tests. Three types of boundary-integral techniques have been developed to simulate fracture propagation. The first one is based on a variational (finite-element) formulation of the boundary integrals. Introduced by Clifton and Abou Sayed for planar fractures, it was subsequently generalized for out-of-plane growth. The two other techniques-a surface integral scheme and the displacement discontinuity method -use double-layer formulations for the equations, considering the displacement discontinuity (or fracture width) as the unknown of the formulation. A significant advantage in these three schemes is that multiple fractures can be modeled with the same formulation, because those fractures are considered to be discontinuity surfaces, including the interaction between a hydraulic fracture and a natural preexisting fracture, the interaction between simultaneously created fractures (in deviated wells), or slippage at frictional interfaces. Major numerical problems with the development of such simulators generally result from the strong pressure gradients prevailing near the fracture tip and the use of a leading edge to lump such variations. Inherent in the use of boundary integrals, contrasts in elastic moduli generally require a discretization of the boundary between the layers, significantly increasing the computing requirements. More problems are the result of 2D fluid, proppant, and heat flow in the fracture. Pseudo-3D height-growth (P3DH) models are used more effectively for treatment design, sensitivity analysis, and pressure matching. Cleary described the relation between the different variables in terms of shape functions, relating the evolution of different variables along the fracture. Those shape factors generally are determined with sophisticated numerical modules, but they may eventually be stored in data bases with different levels of accuracy, thus speeding up the simulation process. Cleary et al. considered such an approach in a lumped P3DH model, and Settari described a discrete simulator that provided realistic geometries for aspect ratios as low as one (Touboul et al. made a comparison with a fully 3D model). The present study uses the general 3D technique described by Touboul et al. for uncontained fractures and large stress gradients and includes a model based on the discrete formulation to simulate the mechanisms of impedance for height growth. At least an extra order of magnitude in computational power (in a planar mode) is required to run the 3D model. SPEPE P. 142^