Aquifer Model

The presence of an aquifer into both gas and oil reservoirs can highly change the pressure drop and the fluid production during a well life time. There are different models to describe the water influx into a reservoir. Some models are simple and have a lot limitation and others a more complex and handle any type of cases, however, usually engineers need to provide more information / characteristics about the aquifer.

Small Pot

This is the simplest aquifer model. It is time independent, and just account for simple aquifer material balance, as shown above:

where:

- initial volume of water in the aquifer;

- water enchroament, also called water influx for simplicity

- total aquifer compressibility;

- initial aquifer/reservoir pressure;

- current reservoir pressure (pressure at the original oil (or gas) water contact);

This model is only applicable to very small aquifers.

Fetkovitch Method

Fetkovitch method uses the same equations and principles of the flow oil from a reservoir into a well, however it is applied for a flow of aquifer into a hydrocarbon reservoir. The flow equation is given by:

where:

- productivity index of the aquifer;

- average pressure of the aquifer;

For a simple material balance of the aquifer, we can write the water influx as:

Rearrange the above equation, we can express the aquifer average pressure as following:

where:

- maximum influx of a sealed aquifer;

The derivative of the aquifer average pressure in relation to time is given by:

Separating the variables of the above equation and integrating from zero to any time, we can express the water influx as:

The cumulative water influx can be expressed by:

The above equation gives the water influx as function of time for a pressure drop constant at the reservoir-aquifer contact, .

In the real life, the pressure drop is not constant at the reservoir-aquifer contact, therefore, the above equation is not direct applicable. Fetchovitch showed a technique to solve this problem without applying superposition. He divided the water influx in n time intervals , and he estimated the water influx during each period as:

where:

- average pressure of the aquifer at the final of period ;

- average pressure at the reservoir-aquifer contact during the time interval ;

The above pressure can be calculated as following:

The productivity index of the aquifer is determined by its generic form allowing any aquifer geometry, as:

where:

- area of aquifer;

- Dietz (1965) shape factor;

- Euler's exponential constant;

For productivity index of the aquifer to be expressed in field units , we use the following equation:

where:

The cumulative water influx was validated against the calculation present in Dake, 1998, Chapter 9, Example 9.3 (Pg. 329 - 332), as shown in figure below:

Reference

1. Dake, L.P., 1998. Fundamentals of Reservoir Engineering, seventeenth impression. Developments in Petroleum Science, 8.