Balance De Materia

Reservoir Recovery Techniques 2010

Material Balance Equations

Department of Petroleum Engineering and Applied Geophysics Professor Jon Kleppe

Norwegian University of Science and Technology August 24, 2010

1

Material Balance Equations

To illustrate the simplest possible model we can have for analysis of reservoir behavior, we will start with

derivation of so-called Material Balance Equations. This type of model excludes fluid flow inside the reservoir,

and considers fluid and rock expansion/compression effects only, in addition, of course, to fluid injection and

production. First, let us define the symbols used in the material balance equations:

Symbols used in material balance equations

Bg Formation volume factor for gas (res.vol./st.vol.)

Bo Formation volume factor for oil (res.vol./st.vol.)

Bw Formation volume factor for water (res.vol./st.vol.)

Cr Pore compressibility (pressure-1)

Cw Water compressibility (pressure-1)

ΔP P2 − P1

Gi Cumulative gas injected (st.vol.)

Gp Cumulative gas produced (st.vol.)

m Initial gas cap size (res.vol. of gas cap)/(res.vol. of oil zone)

N Original oil in place (st.vol.)

Np Cumulative oil produced (st.vol.)

P Pressure

Rp Cumulative producing gas-oil ratio (st.vol./st.vol) = Gp / N p

Rso Solution gas-oil ratio (st.vol. gas/st.vol. oil)

Sg Gas saturation

So Oil saturation

Sw Water saturation

T Temperature

Vb Bulk volume (res.vol.)

Vp Pore volume (res.vol.)

We Cumulative aquifer influx (st.vol.)

Wi Cumulative water injected (st.vol.)

Wp Cumulative water produced (st.vol.)

ρ Density (mass/vol.)

φ Porosity

Then, the Black Oil fluid phase behavior is illustrated by the following figures:

Fluid phase behavior parameters (Black Oil)

Bo Rso

P P P P

B g Bw

Reservoir Recovery Techniques 2010

Material Balance Equations

Department of Petroleum Engineering and Applied Geophysics Professor Jon Kleppe

Norwegian University of Science and Technology August 24, 2010

2

Oil density: ρ

ρ ρ

o

oS gS so

o

R

B

=

+

Water compressibility: C

V

V

w P

w

w

= −( )( )T 1 ∂

∂

Water volume change: Bw2 Bw 1e B 1 c P

c P

w1 w

= − wΔ ≈ ( − Δ )

Finally, we need to quantify the behavior of the pores during pressure change in the reservoir. The rock

compressibility used in the following is the pore compressibility, and assumes that the bulk volume of the rock

itself does not change.

Pore volume behavior

Rock compressibility: C

r P T = ( 1)( )

φ

∂φ

∂

Porosity change: φ w2 φw1 φ

c P

e w1 1 cr P = rΔ ≈ ( + Δ )

The material balance equations are based on simple mass balances of the fluids in the reservoir, and may in

words be formulated as follows:

Principle of material conservation

Amount of fluids present

in the reservoir initially

(st. vol.)

Amount of

fluids produced

(st. vol.)

Amount of fluids remaining

in the reservoir finally

(st. vol.)

⎧

⎨ ⎪

⎩ ⎪

⎫

⎬ ⎪

⎭ ⎪

−

⎧

⎨ ⎪

⎩ ⎪

⎫

⎬ ⎪

⎭ ⎪

=

⎧

⎨ ⎪

⎩ ⎪

⎫

⎬ ⎪

⎭ ⎪

We will define our reservoir system in terms of a simple block diagram, with an initial reservoir stage before

production/injection starts, and a final stage at which time we would like to determine pressure and/or

production.

Block diagram of reservoir

Gas

Oil

Water

Initial stage (1)

Gas

Oil

Water

Final stage (2)

oil production: Np

gas production: RpNp

water production: Wp

aquifer influx: We

gas injection: Gi

water injection: Wi

Reservoir Recovery Techniques 2010

Material Balance Equations

Department of Petroleum Engineering and Applied Geophysics Professor

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