Material balance is a mathematic way to express mass conservation in a reservoir and a simple key principle is “what reservoir is produced must be replaced by other mass.”

**Volume Produced = Volume Replaced**

**Volume Produced** comes from Gas Production, Oil Production, and Water Production.

**Volume Replaced** comes from volume expansion, water in flux and water/gas injection.

Figure 1 shows the relationship of the material balance.

Let’s take a look at each component of equation.

**Volume Produced**

Gas Production (rb) = Np (Rp – Rs) Bg

Oil Production (rb) = Np Bo

Water Production (rb) = Wp Bw

Total Production = Np [Bo + (Rp – Rs) Bg ] + Wp Bw

**Volume Expansion**

** ****Oil Expansion**

Oil expansion consists of two parts. The first one is only oil expansion and the second one is solution gas expansion that happens when reservoir pressure is below a bubble point.

Oil expansion (rb) = N (Bo – Boi)

Solution gas expansion (rb) = N (Rsi – Rs) Bg

**Total oil expansion (rb) = N [ (Bo – Boi) + (Rsi – Rs) Bg ]**

**Rock and Connate Water Expansion**

Rock expansion (rb) = Vp ×Cf ×ΔP

Connate water expansion (rb) = Vp ×Swc× Cw ×ΔP

Total rock and connate water expansion (rb) = Vp ×(Cf + Swc× Cw)× ΔP

Pore Volume (Vp) = [ N× (1 + m) ×Boi ]÷ (1 – Swc)

**Initial Gas Cap Expansion**

Initial gas cap volume (rb) = m ×N ×Boi

Current gas cap volume (rb) = m ×N ×Boi ×(Bg ÷ Bgi)

Gas cap expansion (rb) = Current gas cap volume – Initial gas cap volume

**Gas cap expansion (rb)** **= m ×N ×Boi ×[(Bg ÷ Bgi)-1]**

**Expansion Term**

**Oil expansion term (Eo)**

Eo = (Bo – Boi) + (Rsi – Rs) **×**Bg

** Rock and connate water expansion term (Ef,w)**

**Gas cap expansion term (Eg)**

Eg = Boi **×**[(Bg ÷ Bgi)-1]

With all expansion terms, total expansion can be mathematically expressed like this:

Total expansion (rb) = N (Eo + Efw + m Eg)

**Aquifer Influx and Injection**

** **Aquifer influx volume (rb) = We

Water injection volume (rb) = Wi**×**Bw

Gas injection volume (rb) = Gi**×**Bg

All relationships can be put in a material balance equation based on the following concept:

The equation above is simplified by using an expansion term.

The equation above can be rearranged like this:

Simplify a term like this:

# F = N×Et + We

Where;

F = Net fluid production (Volume Produced – Volume Injection)

N = Oil in place

Et = Total expansion term (oil, water, gas cap and rock)

We = water influx

This simplified equation was presented by Havlena + Odeh (1963). With this equation form, graphical plots can be easily made and material balance evaluation is donemore easily and accurately.

**#1 form:**

The graphic can be drawn based on #1 form. The line should be a horizontal line and an intersection at y-axis is N (oil in place). Deviation from the horizontal line indicates adding or losing enery (Figure 2).

**#2 form:**

This is the form of a simplified material balance equation.

A slope of the curve is 1 and the intersection on the y-axis is N. Deviation from the straight line indicates extra energy in or energy out (Figure 3).

**Nomenclatures**

N = oil initially in place (STOIIP) in reservoir (stb)

Np = cumulative oil production (stb)

Boi = oil volume factor at initial reservoir pressure (rb/stb)

Bo = oil volume factor at current reservoir pressure (rb/stb)

Rsi = solution GOR at initial reservoir pressure (scf/stb)

Rs = solution GOR at current reservoir pressure (scf/stb)

Rp = cumulative produced gas oil ratio (scf/stb)

G = gas volume initially in place (GIIP) in reservoir (scf)

m = ratio of initial gas cap volume to initial oil volume (rb/rb)

Bgi = gas volume factor at initial reservoir pressure (rb/scf)

Bg = gas volume factor at current reservoir pressure (rb/scf)

Swc = connate water saturation (fraction or %)

Cw = water compressibility (1/psi)

Cf = formation (rock) compressibility (1/psi)

Wp = cumulative water production (stb)

We= cumulative water influx from aquifer (rb)

Bw = water volume factor at initial reservoir pressure (rb/stb)

Wi = cumulative water injection (stb)

Gi = cumulative gas injection (scf)

Gp = cumulative gas production (scf)

Eg = gas expansion term (rb/stb)

Eo = oil expansion term (rb/stb)

Efw = formation and connate water expansion term (rb/stb)

**References**

Abhijit Y. Dandekar, 2013. Petroleum Reservoir Rock and Fluid Properties, Second Edition. 2 Edition. CRC Press.

L.P. Dake, 1983. Fundamentals of Reservoir Engineering, Volume 8 (Developments in Petroleum Science). New impression Edition. Elsevier Science.

Tarek Ahmed PhD PE, 2011. Advanced Reservoir Management and Engineering, Second Edition. 2 Edition. Gulf Professional Publishing.