Abstract

Starting from the original concept proposed by Carter, Howard and Fast, this paper reviews the description of fracturing fluid leakoff in view of modeling flow in porous media. It is shown how various linear leakoff models have been developed and why a new, radial leakoff concept is necessary for high-permeability fracturing, where the injection time is commensurable to the response time of the reservoir. Using Laplace space methods the radial leakoff law is calculated and compared to linear leakoff. For comparison purposes a calibration test executed in high-permeability formation is interpreted using several approaches, namely: linear leakoff + bulk leakoff coefficient; filtercake resistance + linear flow in the formation and finally, filtercake resistance + radial flow in the formation.

Introduction

The polymer content of the fracturing fluid is partly intended to impede the loss of fluid. The phenomenon is envisioned as a continuous build-up of a thin layer (the filtercake) which manifests a resistance to flow through the fracture face. For one of the latest reviews see McGowen and Vitthal. During fracturing, the actual leakoff is determined by a coupled system, of which the filtercake is only one element. The other two important elements are the region invaded by the polymer and/or filtrate and the bulk reservoir itself, containing the original (slightly compressible) reservoir fluid. This work concentrates on the aspect of fluid leakoff which is connected with the bulk reservoir. The methods used are borrowed from the literature on flow in porous media.

As usual, we assume that the two wings of a vertical fracture are identical. For modeling purposes we will deal only with one wing. All our variables, including injection rate, i, injected volume, Vi, fracture volume, V refer to one wing. (If we want to refer to total injection rate, we write 2i.) By the fracture surface, A we mean the area of one face of one wing. All these variables may refer to a given time, t during the treatment. It is important to make a clear distinction between the values of the above variables at any time, t, and at the end of pumping, i.e, at time te. We will use the subscript e if we wish to emphasize that a given value corresponds to the end of pumping. Figure 1 shows the basic notation on an example of radial fracture. Fluid efficiency, is defined as the fraction of the fluid remaining in the fracture: = V/Vi. As any other state variable it might vary with time. The average width, w, is defined by the relation V = Aw. The difference of injected and contained volume is the lost volume. The leakoff rate, qL defined here as the volume leaving one wing in unit time and can be calculated from an appropriate leakoff model.

Often we assume that the fracture is contained in the permeable layer. Then the whole fracture surface takes part in the leakoff process. If we know the height of the permeable layer, hp, we can be more rigorous in taking into account only the actual leakoff surface. Figure 2 shows how we calculate the ratio of the leakoff surface to the total surface for radial geometry. The ratio, rp, is unity for a fracture contained perfectly in the permeable layer and is less than unity if the fracture grows out from the permeable layer. In the case of rectangular fracture shape rp is the ratio of the "net" height to the "gross" height. The factor is easily incorporated into the derivations, but in the following we do not show rp to increase the readability of the equations.

Previous work
Carter Leakoff Model

A fruitful approximation dating back to Carter, Howard and Fast considers the combined effect of the different phenomena as a material property. According to this concept, the leakoff velocity, uL, is given by the Carter equation: (1)

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