Demystifying the G-function
Diagnostic fracture injection tests (DFITs) are commonly interpreted by plotting pressure versus a function of time called the G-function. The G-function seems rather arcane and is often misunderstood. However, it has a simple, intuitive meaning. In this post, I will try to demystify the G-function.
The G-function was presented by Ken Nolte in his paper “Determination of fracture parameters from fracturing pressure decline” in 1979. This paper is the foundation of DFIT analysis.
In short, the G-function is derived so that the cumulative volume of fluid leaked off from the fracture after shut-in is linearly proportional to the G-function. In other words, at a G-time of 4, twice of much fluid has leaked off after shut-in as at G-time of 2. The G-function is derived based on several assumptions. One important assumption is that the fluid pressure in the fracture is constant during shut-in. Obviously, the fluid pressure is not constant. It decreases over time after shut-in. But during the early shut-in period, it is approximately valid to assume that fracture pressure is constant.
Mathematically, what is the G-function? You can refer to Nolte’s paper or other references to find the exact mathematical form of the G-function. There are several possible forms of the G-function, depending on your assumptions about the rate of fracture propagation. Fortunately, it doesn’t make a huge difference which form of the G-function that you use.
Except at early time, the G-function is approximately proportional to the square root of shut-in time. If you are doing a quantitative analysis, you should use the full form of the G-function. But if you are looking for an intuitive idea of what the G-function is, it’s a pretty good approximation to just think of it as the square root of shut-in time.
Why is it useful to define a number that is proportional to leakoff volume after shut-in? Let’s do a bit of calculus (sorry). From the chain rule, the derivative of pressure with respect to the G-function is:
where Vf is the volume of fluid in the fracture and P is the pressure in the fracture. This equation expresses the idea that the pressure decline rate is equal to the fracture stiffness (more generally, the equivalent stiffness including the effect of wellbore storage) times the derivative of leakoff volume with respect to G-time. As long as the fracture walls are not touching, the stiffness is approximately constant. Because G-time is derived to be proportional to leakoff volume during shut-in, the derivative of leakoff volume with respect to G-time is constant. Therefore, for an “ideal” DFIT, a plot of pressure versus G-time should make a straight line. A critical insight is that the slope of this straight line could be used quantitatively to infer the leakoff coefficient, or equivalently, the formation permeability.
As will be addressed in a future post, many processes can cause deviation from the ideal straight line. Remember, the G-function is derived assuming that leakoff rate can be calculated from an assumption of constant pressure in the fracture. Once the fracture pressure decreases sufficiently, this assumption is no longer valid, and the curve deviates from a straight line. Fracture closure (the walls come into contact) affects both the leakoff rate and the stiffness. A variety of other processes have been proposed as possibly causing ”non-ideal” changes in either stiffness or leakoff rate.
”Pressure transient analysis” is a general term for the use of pressure trends to infer properties in the subsurface. The earliest work on pressure transient analysis dates back to the work of Charles Theis in 1935. Nolte’s innovation was to show how to infer formation properties from the pressure transient after creation of a hydraulic fracture. Since 1979, Nolte and many others have extended these ideas and expanded what can be learned from fracturing tests.