In a recent study a numerical model for Produced Water Re-injection (PWRI) under fracturing conditions was presented [1]. The model was a 2-dimensional analytical fracture growth model and hence only appropriate for constant-height fractures fully penetrating the injection layer, with a "square fracture" option allowing a first order estimate of radial fracture dimensions. In this paper we present a new 3-dimensional fracture growth model which permits the description of elliptical fractures in a multi-layer reservoir with fracture length, height up and height down all potentially growing at different rates.

The pressure field in the formation around a fully contained elliptical fracture will resemble that of a fracture in an unbounded reservoir (i.e. 3D solution), while far into the reservoir it will approach a 2D solution, with a gradual transition between the two limiting cases. Solutions to both of these limiting cases have been developed earlier [2,3]. Here we present a new method which approximates the gradual transition from 3D-elliptical symmetry to 2D-elliptical symmetry with an abrupt transition. In a multi-layer reservoir the model can be applied independently to all the layers if no crossflow between the layers is allowed.

A comparison is presented between the two models which demonstrates that the previous model predicts larger fracture sizes. This is particularly the case if the pressure drop over the filtercake is relatively small.

We also present a new approach to represent plugging within the fracture due to the build-up of filtercake resulting from suspended particulates in the produced water. This new approach allows for flow channels to develop within the filtercake in the fracture, rather than the approach used in the previous study which adopts a parallel-plate flow model and allows uniform growth of filter cake on fracture walls.

A number of PWRI field examples is also presented. The model is very suitable to assess the containment capacity of layered reservoirs, which is demonstrated using a field example of a large reservoir in the Middle East.

Comparisons with Prudhoe Bay data show that our model is not capable of predicting the virtually immediate response to changes in the water quality. The model still assumes that the filtercake properties are determined by the cumulative volume of solids pumped into the fo rmation. This needs to be reviewed and possibly extended to also include immediate effects.

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