Modern Physics

Pressure gradient: what it is and how it is calculated

The pressure gradient consists of variations or differences in pressure in a given direction, which can occur within or at the limits of a fluid. In turn, pressure is the force per unit area exerted by a fluid (liquid or gas) on the walls or on the edge that contains it.

For example, in a pool filled with water, there is a positive pressure gradient in the vertical downward direction because pressure increases with depth. With every meter (or centimeter, foot, inch) of depth, the pressure increases linearly.

However, at all points located on the same level, the pressure is the same. Therefore, in a pool, the pressure gradient is zero (zero) in the horizontal direction.

In the oil industry, the pressure gradient is very important. If the pressure at the bottom of the borehole is greater than at the surface, the oil will easily exit. Otherwise, the pressure difference must be created artificially, by pumping or steam injection.

Fluids and their interesting properties

A fluid is any material whose molecular structure allows it to flow. The bonds that hold the fluid’s molecules together are not as strong as they are with solids. This allows them to withstand the least resistance to pull, and therefore to the flow.

This circumstance can be observed by observing that solids maintain a fixed form, while fluids, as already mentioned, adopt to a greater or lesser degree that of the container that contains them.

Gases and liquids are considered fluids because they behave that way. A gas expands completely to fill the volume of the container.

Liquids, on the other hand, do not reach as much as they have a certain volume. The difference is that liquids can be considered incompressible , while gases are not.

Under pressure, a gas compresses and adapts easily, taking up all the available volume. When pressure increases, its volume decreases. In the case of a liquid, its density – given by the ratio of its mass to its volume – remains constant over a wide range of pressure and temperature.

This last sizing is important as, in reality, almost any substance can behave like a fluid under certain conditions of extreme temperature and pressure.

Within the earth where conditions can be considered extreme, rocks that would otherwise be solid on the surface melt into magma and can flow to the surface in the form of lava.

pressure calculation

To find the pressure exerted by a column of water or any other fluid on the floor of the container, the fluid will be considered to have the following characteristics:

  • Its density is constant
  • it’s incompressible
  • It is in static equilibrium conditions (rest)

A column of fluid in these conditions exerts a force on the bottom of the container that contains it. This force is equivalent to its weight W :

W = mg

Now, the density of the fluid, which, as explained above, is the ratio of its mass m and its volume V , is:

ρ = m / V

Density is usually measured in kilograms / cubic meters (kg / m 3 ) or pounds per gallon (ppg)

Substituting the density expression in the weight equation, it becomes:

W = ρVg

The hydrostatic pressure P is defined as the ratio between the force exerted perpendicularly on a surface and its area A:

Pressure = Force / Area

When substituting column fluid volume V = base area x column height = Az, the pressure equation is:

Pressure gradient: what it is and how it is calculated 2

Pressure is a scalar quantity whose units in the international measurement system are Newton / meter 2 or Pascal (Pa). The British units of the system are widely used, especially in the oil industry: pounds per square inch (psi).

The above equation shows that denser liquids will exert greater pressure. And that the pressure is greater, while the surface on which it is exerted is smaller.

By substituting column fluid volume V = base area x column height = Az, the pressure equation is simplified:

Pressure gradient: what it is and how it is calculated 3

The above equation shows that denser liquids will exert greater pressure. And that the pressure is greater, while the surface on which it is exerted is smaller.

How to calculate pressure gradient?

The equation P = ρgz indicates that the pressure P of the fluid column increases linearly with depth z. Therefore, a pressure variation ΔP will be related to a depth variation Δz as follows:

ΔP = ρgΔz

Define a new quantity called fluid specific weight γ, given by:

γ = ρg

Specific weight comes in units of Newton / volume or N / m 3 . Thus, the equation for the pressure variation is:

ΔP = γ Δz

Which is rewritten as:

Pressure gradient: what it is and how it is calculated 4

This is the pressure gradient. We now see that, under static conditions, the pressure gradient of the fluid is constant and equivalent to its specific gravity.

Pressure gradient units are the same as specific weight, but can be rewritten as Pascal / meter in the International System. It is now possible to visualize the interpretation of the gradient as the change in pressure per unit of length as defined at the beginning.

The specific weight of water at a temperature of 20 °C is 9.8 kgPascal / m or 9800 Pa / m. It means that:

“For every meter that descends into the water column, the pressure increases by 9800 Pa”

Density Conversion Factor

English system units are widely used in the oil industry. In this system, the pressure gradient units are psi / ft or psi / ft. Other convenient units are bar / subway. For density, pound per gallon or ppg is widely used.

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The density and specific gravity values ​​of any fluid have been experimentally determined for various conditions of temperature and pressure. They are available in the stock tables.

To find the numerical value of the pressure gradient between different systems of units, conversion factors that bring the density directly to the gradient must be used.

The 0.052 conversion factor is used in the petroleum industry to move from a density in ppg to a pressure gradient in psi / ft. In this way, the pressure gradient is calculated as follows:

GP = conversion factor x density = 0.052 x density ppgFor example, for fresh water, the pressure gradient is 0.433 psi / ft. The value 0.052 is deduced using a cube whose side measures 1 foot . To fill this bucket, 7.48 gallons of some fluid is needed.

If the density of this fluid is 1 ppg , the cube’s total weight is 7.48 lb/ft and its specific gravity is 7.48 lb/ft 3 .

Now in 1 foot 2 there are 144 square inches, so in 1 foot 3 there will be 144 square inches for every foot of length. Dividing 7.48 / 144 = 0.051944, which is approximately 0.052.

For example, if you have a fluid whose density is 13.3 ppg, the pressure gradient would be: 13.3 x 0.052 psi / ft = 0.6916 psi / ft.

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