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Dynamic Data Analysis – v5.12.01 - © KAPPA 1988-2017
Chapte
r 2 – T heory- p14/743
Darcy’s Law states that the pressure drop between two points, close enough to consider all
parameters to be constant, will be:
proportional to the flowrate density (q/A)
proportional to the fluid viscosity (
)
inversely proportional to the reservoir permeability (k)
Darcy’s Law is a fundamental law in dynamic data analysis. It is used to model the flow in
several compartments in the reservoir:
At any point of the reservoir, it is one of the three equations that will be used to define the
diffusion equation (see next section).
When the well is flowing, it determines the pressure gradients at the sandface.
Close to reservoir boundaries it determines that the pressure gradient towards a no-flow
boundary is flat, or it allows the determination of the inflow from the pressure gradient.
Darcy’s law assumes a linear relation between the flow of fluid in one direction and the
pressure gradient, corrected for gravity, in the same direction. This assumes that the density
of flow is small enough to avoid turbulent behaviour.
When there is turbulence, a quadratic term is added and Darcy’s law is replaced by the
Forchheimer’s equation. We then speak about non-Darcy flow. In most cases, non-Darcy
problems will be solved with a numerical model.
2.A.2
The diffusivity equation
The diffusivity equation describes how, in an elementary piece of rock, the pressure will react
in time as a function of the local pressure gradient around this piece of rock.
There may be as many diffusivity equations as there are assumptions on what is happening
downhole. The basic theory in Dynamic Data Analysis uses the simplest possible diffusivity
equation, assuming the following:
The reservoir is homogeneous and isotropic.
The fluid is single-phase and only slightly compressible.
Gravity effects are ignored. If they were not the diffusivity equation would be written in
terms of potential and not pressure.
Darcy’s law applies.
Reservoir and fluid properties are independent of the pressure.
Under these conditions, the diffusivity equation is derived from the combination of:
(1)
The principle of conservation of mass
(2) Darcy’s law
(3)
Slightly compressible fluid equation
Some more complex versions of the diffusivity equation will have different components:
Darcy’s law may be replaced by the Forchheimer’s equation, and more complex PVT models
may be used: real gas diffusion, multiphase black oil correlations or an Equation of state.