In previously published work, we have analyzed transient injection of water from a growing vertical hydrofracture into a low-permeability compressible rock of uniform properties, filled with a fluid of identical mobility. Here we extend the prior analysis1
to water injection into a layered rock initially filled with a fluid of different mobility. We then develop a new control model of water injection from a growing hydrofracture into a layered formation. Based on the new model, we design an optimal injection controller that manages the rate of water injection in accordance with the hydrofracture growth and the formation properties. As we have already demonstrated, maintaining the rate of water injection into low-permeability rock above a reasonable minimum inevitably leads to hydrofracture growth if flow in a uniform formation is transient. The same conclusion holds true for transient flow in layered formation. Analysis of field water injection rates and wellhead injection pressures leads us to conclude that direct links between injectors and producers can be established at early stages of waterflood, especially if the injection policy is aggressive. On one hand, injection into a low-permeability rock is slow and there is a temptation to increase the injection pressure. On the other hand, such an increase may lead to irrecoverable reservoir damage: fracturing of the formation and water channeling from the injectors to the producers. Such channeling may be caused by thin highly permeable reservoir layers, which may conduct a substantial part of injected water. Considering these field observations, we expand our earlier model. Specifically, we consider a vertical hydrofracture in contact with a multi-layered reservoir where some layers have high permeability and they, therefore, quickly establish steady state flow from an injector to a neighboring producer.
The main part of this paper is devoted to the development of an optimal injection controller for purely transient flow and for mixed transient/steady-state flow in a layered formation. The objective of the controller is to maintain the prescribed injection rate in the presence of hydrofracture growth. Such a controller will be essential in our proposed automated system of field-wide waterflood surveillance and control. We design optimal injection controllers using methods of optimal control theory. The history of injection pressure and cumulative injection, along with estimates of the hydrofracture size are the controller input data. By analyzing these inputs, the controller outputs an optimal injection pressure for each injector. When designing the controller, we keep in mind that it can be used either off-line as a smart advisor, or on-line in a fully automated regime.
We demonstrate that the optimal injection pressure depends not only on the instantaneous measurements, but it is determined by the whole history of injection and growth of the hydrofracture. Because our controller is process-based, the dynamics of the actual injection rate and the pressure can be used to estimate an effective area of the hydrofracture. The latter can be passed to the controller as one of the input parameters. Finally, a comparison of the estimated fracture area with independent measurements leads to an estimate of the fraction of injected water that flows directly to the neighboring producers due to channeling or thief-layers.