Why Fractals make Reservoir Modelling Really Easy

Fractals are a recurring pattern described by a simple algorithm. We see fractal everywhere. We see them on the small scale in snowflakes and in Roman cauliflowers.

We see them on the big scale in the Himalayas and in river systems.

We also see them on the Really Big Scale. Radiation from an early universe, the cosmic microwave background, is fractal.?As we zoom in, the patterns repeat. These patterns give rise to galactic superclusters, which are themselves are built up from galaxies. The universe is fractal all the way down!

Fractals are objects where their parts are similar to the whole, except for scale.

Nature uses a simple DNA code to create plants and trees. Trees look the same close up, as further away.?Compare a whole tree with its branches and twigs.

Many complex objects can be described by fractals. Fractals are a mathematically simple way of describing complexity including hydrocarbon reservoirs!

How to Verify if something is Fractal

A coastline shows increased detail, the closer we zoom in.?

If we ask "How long is the coastline of Great Britain," the answer depends on how closely you look at it, or how long your measuring stick is.?

As the ruler you use gets shorter, the coastline you measure gets longer. When we use a long ruler, we get a very poor approximation, as shown in?purple, where the coastline of?N=9 units.?As the ruler length (1/r) shrinks the coastline?N?increases.?First you see the shape of the estuaries and then the river systems.?This continues all the way down to the grains of sand on the beaches.

However, something amazing happens if we plot the?rate?at which the coastline changes as a function of the ruler length.

The 4 coloured dots represent the 4 coastline measurements. The relationship between r and N is linear. This verifies that the coastline is fractal, whose structure is independent of how much you zoom in. The gradient of the line is the fractal dimension and for clastic reservoirs the answer is 42!

Fractals in Reservoir Rocks

In order to measure the porosity of the rock samples, you first impregnate them with a coloured dye.?

If take a digital image you can easily count how many pixels are associated with porosity. When you repeat this for smaller and smaller pixels, as with the coastlines, more porosity we see. Studies show there is a linear relationship between the pixel size and porosity demonstrating reservoir rocks are fractal in nature. Reservoirs are probably fractal because the mathematical sorting is similar for grains of sand as for pebbles or boulders.

Fractals describe the Rock Pore network

The rock pore space can be described by the fractal formula.

Where V is the pore space in rock volume, r is the radius of the rock capillaries and Df is the fractal dimension (non-integer constant).

Combining with capillary pressure theory gives the BVW Swh function*

Where BVW?is the Bulk Volume of Water from logs or core data, H is Height above the Free Water Level and a & b are constants.

This Water Saturation vs. Height Function is used to initialise your reservoir model with hydrocarbon and water. It is independent of rock properties such as porosity, permeability and rock type.

* SPWLA 58th Annual Symposium “Using Fractals to Determine a Reservoir’s Hydrocarbon Distribution” (2017)?Steve Cuddy