material balance equations · material balance equations ... equation 3: gas material balance...

TPG4150 Reservoir Recovery Techniques 2003Material Balance Equations

Department of Petroleum Engineering and Applied Geophysics Professor Jon KleppeNorwegian University of Science and Technology August 13, 2003

1

Material Balance Equations

To illustrate the simplest possible model we can have for analysis of reservoir behavior, we will start withderivation 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 andproduction. First, let us define the symbols used in the material balance equations:

Symbols used in material balance equations

BgFormation volume factor for gas (res.vol./st.vol.)

BoFormation volume factor for oil (res.vol./st.vol.)

BwFormation volume factor for water (res.vol./st.vol.)

Cr Pore compressibility (pressure-1)Cw Water compressibility (pressure-1)DP P P2 1-

GiCumulative gas injected (st.vol.)

GpCumulative 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.)N p

Cumulative oil produced (st.vol.)

P PressureRp Cumulative producing gas-oil ratio (st.vol./st.vol) = G Np p/

RsoSolution gas-oil ratio (st.vol. gas/st.vol. oil)

SgGas saturation

SoOil saturation

SwWater saturation

T TemperatureVb

Bulk volume (res.vol.)

VpPore volume (res.vol.)

WeCumulative aquifer influx (st.vol.)

WiCumulative water injected (st.vol.)

WpCumulative water produced (st.vol.)

r Density (mass/vol.)f 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

BwB g



2

Oil density: rr r

oo S g S so

o

R

B=

+

Water compressibility: CV

V

Pww

wT= -( )( )

1 ∂

∂

Water volume change: B B e B 1 c Pw2 w 1c P

w1 ww= ª -- D D( )

Finally, we need to quantify the behavior of the pores during pressure change in the reservoir. The rockcompressibility used in the following is the pore compressibility, and assumes that the bulk volume of the rockitself does not change.

Pore volume behavior

Rock compressibility: CPr T= ( )( )

1f

∂f∂

Porosity change: f f fw2 w1c P

w1 re 1 c Pr= ª +D D( )

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

Ï

ÌÔ

ÓÔ

¸

˝Ô

˛Ô

-Ï

ÌÔ

ÓÔ

¸

˝Ô

˛Ô

=Ï

ÌÔ

ÓÔ

¸

˝Ô

˛Ô

We will define our reservoir system in terms of a simple block diagram, with an initial reservoir stage beforeproduction/injection starts, and a final stage at which time we would like to determine pressure and/orproduction.

Block diagram of reservoir

Gas

Oil

Water

Initial stage (1)

Gas

Oil

Water

Final stage (2)

oil production: Npgas production: RpNpwater production: Wp

aquifer influx: We

gas injection: Giwater injection: Wi



3

The two stages on the block diagram are reflected in the fluid phase behavior plots as follows:

Initial and final fluid conditions

B

B

P

P

o Rso

P

wg

P

| |

|

||

|

|

|

1

11

1

2 2

22

B

Note: If a gas cap is present ini-tially, then the initial pressure isequal to the bubble point pressure

Now, we will apply the above material balance equation to the three fluids involved, oil, gas and water:

Equation 1: Oil material balance

Oil present

in the reservoir

initially

(st. vol. )

Oil

produced

(st. vol. )

Oil remaining

in the reservoir

finally

(st. vol. )

Ï

ÌÔÔ

ÓÔÔ

¸

˝ÔÔ

˛ÔÔ

-Ï

ÌÔ

ÓÔ

¸

˝Ô

˛Ô

=

Ï

ÌÔÔ

ÓÔÔ

¸

˝ÔÔ

˛ÔÔ

orN - N = V S /Bp p o o2 2 2

yielding

SN N B

Vop o 2

p 22 =

-( )

Equation 2: Water material balance

Water present

in the reservoir

initially

(st. vol. )

Water

produced

(st. vol. )

Water

injected

(st. vol. )

Aquifer

influx

(st. vol. )

Water remaining

in the reservoir

finally

(st. vol. )

Ï

ÌÔÔ

ÓÔÔ

¸

˝ÔÔ

˛ÔÔ

-Ï

ÌÔ

ÓÔ

¸

˝Ô

˛Ô

+Ï

ÌÔ

ÓÔ

¸

˝Ô

˛Ô

+Ï

ÌÔ

ÓÔ

¸

˝Ô

˛Ô

=

Ï

ÌÔÔ

ÓÔÔ

¸

˝ÔÔ

˛ÔÔ

orV S /B - W + W + W = V S /Bp w w p i e p w w1 1 1 2 2 2

yielding

( ) ( )S 1 m NBS

S1

BW W W

B

Vw2 o 1w1

w1 w1i e p

w2

p 2

= +-

Ê

ËÁ

ˆ

¯˜Ê

ËÁ

ˆ

¯˜ + + -

È

ÎÍÍ

˘

˚˙˙1



4

Equation 3: Gas material balance

Solution gas

present in the

reservoir initially

(st. vol.)

Free gas

present in the

reservoir initially

(st. vol. )

Gas

produced

(st. vol. )

Gas

injected

(st. vol. )

Solution gas

present in the

reservoir finally

(st. vol. )

Free gas

present in the

reservoir finally

(st. vol. )

Ï

ÌÔÔ

ÓÔÔ

¸

˝ÔÔ

˛ÔÔ

+

Ï

ÌÔÔ

ÓÔÔ

¸

˝ÔÔ

˛ÔÔ

-Ï

ÌÔ

ÓÔ

¸

˝Ô

˛Ô

+Ï

ÌÔ

ÓÔ

¸

˝Ô

˛Ô

=

Ï

ÌÔÔ

ÓÔÔ

¸

˝ÔÔ

˛ÔÔ

+

Ï

ÌÔÔ

ÓÔÔ

¸

˝ÔÔ

˛ÔÔ

orNR + mNB /B - R N = (N-N )R + V S /Bso o g p p p so p g g1 1 2 2 2 2 2

yielding

†

Sg2 = N (Rso1 - Rso2) + m( Bo1Bg2

)È

Î Í Í

˘

˚ ˙ ˙ - N p (Rp - Rso2) + Gi

Ï Ì Ô

Ó Ô

¸ ˝ Ô

˛ Ô (Bg2

Vp2)

In addition to these three fluid balances, we have the following relationships for fluid saturations and porevolume change:

Equation 4: Sum of saturations

S S S 1.0o w g+ + =

Equation 5: Pore volume change

V =V ( +c P)p p r2 1 1 D



5

By combining the 5 equations above, and grouping terms, we obtain the material balance relationships, asshown below:

THE COMPLETE BLACK OIL MATERIAL BALANCE EQUATION:

( ) ( )F N E mE E W W B G Bo g f w i e w2 i g 2= + + + + +,

where

production terms are

( )[ ]F N B R R B W Bp o 2 p so2 g 2 p w2= + - +

oil and solution gas expansion terms are

( ) ( )E B B R R Bo o 2 o 1 so1 so2 g 2= - + -

gas cap expansion terms are

E BB

B1g o 1

g 2

g 1

= -Ê

ËÁÁ

ˆ

¯˜̃

and rock and water compression/expansion terms are

( )E 1 m BC C S

1 SPf w o 1

r w w1

w1, = - +

+

-D

material balance equations · material balance equations ... equation 3: gas material balance...

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