The present fluid phases are water, oil and gas, and they are represented by the subscript a = w,o,g respectively.
Notice that in petroleum engineering the spatial co-ordinate system is right-hand-oriented with z-axis pointing downward.
The flow equation then becomes Leverett also pointed out that the capillary pressure shows significant hysteresis effects.
During 1951-1970 commercial computers entered the scene of scientific and engineering calculations and model simulations.
But the progress in upscaling was slow, and it was not until 1990-2000 that directional dependency of relative permeability and need for tensor representation was clearly demonstrated,[4][5] even though at least one capable method[6] was developed already in 1975.
One such upscaling case is a slanted reservoir where the water (and gas) will segregate vertically relative to the oil in addition to the horizontal motion.
If you inject water updip (or gas downdip) for a period of time, it will give rise to different relative permeability curves in the x+ and x- directions.
It can be represented by an IF-statement in the software code, and it occurs in some commercial reservoir simulators.
The process (or rather sequence of processes) may be due to a backup plan for field recovery, or the injected fluid may flow to another reservoir rock formation due to an unexpected open part of a fault or a non-sealing cement behind casing of the injection well.
The flow equation in component form (using summation convention) is The Darcy velocity
is not the velocity of a fluid particle, but the volumetric flux (frequently represented by the symbol
(or short but inaccurately called pore velocity) is related to Darcy velocity by the relation The volumetric flux is an intensive quantity, so it is not good at describing how much fluid is coming per time.
The preferred variable to understand this is the extensive quantity called volumetric flow rate which tells us how much fluid is coming out of (or going into) a given area per time, and it is related to Darcy velocity by the relation We notice that the volumetric flow rate
is a scalar quantity and that the direction is taken care of by the normal vector of the surface (area) and the volumetric flux (Darcy velocity).
A version of the multiphase flow equation, before it is discretized and used in reservoir simulators, is thus In expanded (component) form it becomes The (initial) hydrostatic pressure at a depth (or level) z above (or below) a reference depth z0 is calculated by When calculations of hydrostatic pressure are executed, one normally does not apply a phase subscript, but switch formula / quantity according to what phase is observed at the actual depth, but we have included the phase subscript here for clarity and consistency.
However, when calculations of hydrostatic pressure are executed one may use an acceleration of gravity that varies with depth in order to increase accuracy.
Such high accuracy is not needed in reservoir simulations so acceleration of gravity is treated as a constant in this discussion.
The initial pressure in the reservoir model is calculated using the formula for (initial) overburden pressure which is In order to simplify the terms within the parenthesis of the flow equation, we can introduce a flow potential called the
-potential, pronounced psi-potential, which is defined by It consists of two terms which are absolute pressure and gravity head.
A reservoir may consists of several flow units that are separated by tight shale layers.
Before we start the conversion, we notice that both the original (single phase) the flow equation of Darcy and the generalized (or extended) multiphase flow equations of Muskat et al. are using reservoir velocity (volume flux), volume rate and densities.
First, we take the flux version of the equation and rewrite it as We want to place the composite conversion factor together with the permeability parameter.
The task here is to have a gravity term that is consistent with the applied units ("H-units") for the pressure gradient.