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Material Balances

Material balances are typically mass balances, based on the Law of Conservation of Mass. This can be explained with this statement:
"total mass input" = "total mass output".
The general balance equation, a form of the previous statement, is:

Input is the mass that enters through the system boundaries.
Generation is the mass that is produced within the system.
Output is the mass the exits the system boundaries.
Consumption is that consumed within the system.
Accumulation is the buildup within the system.
Here is a basic schematic of a system:

```                  _____________
|           |
----------------->|  Process  |----------------->
flow rate in    |   Unit    |   flow rate out
q(in), mass/time  |___________| q(out), mass/time

```
Material balances are heavily used in Chemical Engineering and Biomedical Engineering to determine flow rates and ideality of a system, and can also be applied to other fields as in population models.

Here is a simple example population problem solved with a material balance:

Every year 30,000 people move into East Newark, 70,000 people move out, 20,000 are born, and 17,000 die. Using a material balance, determine how much the City's population changes each year (accumulation).

Solution:
Let P stand for people.
Using the equation above,

```                  ______________
|             |
----------------->|    East     |----------------->
30,000 P/yr    |   Newark:   |   70,000 P/yr
|+20,000 P/yr |
|-17,000 P/yr |
|_____________|

Input       + Generation -  Output -      Consumption = Accumulation
30,000 P/yr + 20,000 P/yr - 70,000 P/yr - 17,000 P/yr = Accumulation
Accumulation = -37,000 P/yr
```
This means that 37,000 people leave East Newark each year.

Material balance is used in petroleum engineering to estimate the amount of available fluid (usually natural gas) in a reservoir (also known as a pool).

The simplest material balances are just simple plots with obvious relationships. The most commonly used plot for a gas material balance is a 'P/Z Plot'. The name, however, is misleading. This type of plot has cumulative gas production on the X axis, and (pressure / compressability factor) on the Y axis. For each pressure test done on a natural gas well, a point is plotted at the cumulative production of that well to date, and the pressure yielded from the test divided by the compressability factor of the gas (which is determined from a number of gas properties, such as the CO2 concentration, the gas gravity, and the average reservoir temperature). When this has been done for all the wells believed to be producing from the same gas reservoir, a linear relationship is usually easily visible. A least squares best fit line is usually fit over the data, and where the best fit line intersects the X axis (when the pressure is 0), that reading from the X axis is your IGIP (initial gas in place). However, in the case of a connected reservoir or a waterdrive reservoir, the pressure in your reservoir is not constant with time, and therefore time must be factored into the material balance. You can use time to determine the artificial pressure of a reservoir over a number of years, and plotting that line on a P/Z plot will reveal your /actual/ P/Z plot (which, in the case of a simple reservoir, will follow your best-fit line exactly, if done right).

Pressure-Transient Analysis, Rate-Transient Analysis, Decline Analysis, and Volumetric calculations are other popular methods for determining the gas in place in a reservoir. They'll all tend to give different but similar results for any good set of data.

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