BAlance de materia para pozos de gas

1. Technology Services
Reservoir & Well Performance
Material Balance:
The Forgotten
Reservoir
Engineering Tool
John McMullan

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Understanding the Oil
And Gas Reservoir
Using Material Balance
Lafayette, Louisiana
October 21, 2004

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http://www.cgrpttc.lsu.edu/products/matbal/

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Possible Talk Titles
Š Material Balance: The Forgotten Reservoir
Engineering Tool
Š Are Traditional Material Balance Calculations
Obsolete?
Š Material Balance: Obsolete in 2005?
Š Material Balance: A Quaint Reservoir Engineering
Tool from the Past
Š Material Balance, Why Bother?

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Material Balance
Š Can Provide an estimate of initial HC in place
– independent of geological interpretation
– can be used to verify volumetric estimates
Š Determines the degree of aquifer influence
– understanding of the “drive mechanism”
– estimate recovery factor
Š Estimate of recoverable reserves

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Uses of Material Balance
Š As a precursor to reservoir simulation
Š Identify undrained hydrocarbons
Š Can be used as a forecasting tool in certain situations
Š Can be used to help evaluate operating strategies
such as new wells, accelerated rate, compression
Š In some cases can be used to screen for enhanced
recovery

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Gas Material Balance
G ≡ initial gas
volume - SCF
Bgi – bbl/SCF Vgi = G × Bgi
Vgi ≡ initial gas
volume - bbls

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Gas Material Balance
G × Bgi
Bg is a function
of new pressure
Expansion (bbls)
= Vg - Vgi
= G (Bg – Bgi)
}
Vg = G × Bg

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Gas Material Balance
Expansion (bbls)
= Vg - Vgi
= G (Bg – Bgi)
}
Expansion must
equal production:
Gp Bg = G (Bg – Bgi)
Fix piston
Bleed off Gp SCF
of gas until
pressure equals
the same as before.

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Gas Material Balance
Vti = Vgi + Vwi (bbls)
Vgi = Vti (1-Sw) = G Bgi
Vti = G Bgi / (1-Sw)
Vwi = Vti Sw = G Bgi Sw / (1-Sw)
Vgi = G × Bgi
Vwi = G Bgi Sw / (1-Sw)

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Gas Material Balance
The change in water volume can be found:
∆Vwi = Vwi cw ∆p
Since
Vwi = Vti Sw = G Bgi Sw / (1-Sw)
Substituting:
∆Vwi = G Bgi cw ∆p Sw / (1-Sw)
(the expansion of water with a drop in pressure)

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Gas Material Balance
G × Bgi
Expansion (bbls)
= G (Bg – Bgi) +
G Bgi cw ∆p Sw / (1-Sw)
}
Vwi =
G Bgi Sw / (1-Sw)
Vg = G × Bg

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Gas Material Balance
As before,
Expansion must equal production:
G (Bg – Bgi) + G Bgi cw ∆p Sw / (1-Sw)
= Gp Bg + WpBw
Fix piston
Bleed off Gp SCF
of gas, Wp water until
pressure equals
the same as before.

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Expansion (bbls)
= G (Bg – Bgi)
+
G Bgi cw ∆p Sw / (1-Sw) + We
}
Suppose while the pressure drops, we
inject We reservoir barrels of water.
Expansion must equal production:
G (Bg – Bgi) + G Bgi cw ∆p Sw / (1-Sw) + We
= Gp Bg + WpBw
Gas Material Balance
We

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Gas Material Balance
Finally, consider the possibility that the actual
initial pore volume will reduce as the pressure falls:
∆Vti = Vti cf ∆p
Recall,
Vti = G Bgi / (1-Sw)
Substituting:
∆Vti = cf ∆p G Bgi / (1-Sw)
This loss in original volume results in
an additional amount of expansion from
the original volume.

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Gas General Material
Balance Equation
G (Bg – Bgi) + G Bgi cw ∆p Sw / (1-Sw) + We + cf ∆p G Bgi / (1-Sw)
= Gp Bg + WpBw
Gas
Production
Water
Production
+
=
Gas
Expansion
Water
Expansion
Water
Influx
Formation
Expansion
+
+
+
Š Note that all terms are a function of pressure
Š Equation can not be directly solved
Š An iterative approach is required for solution

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Oil General Material
Balance Equation
N (Bt – Bti) + N m Bti (Bg - Bgi ) + (N Bti + N m Bti ) cw ∆p Sw / (1-Sw)
+ cf ∆p (N Bti + N m Bti ) / (1-Sw) + We + WI BwI + GI BgI
= Np Bt + Np (Rp – Rsoi) Bg + WpBw
Bgi
Oil
Expansion
Water
Expansion
+ +
Gas Cap
Expansion
Water
Influx
Formation
Expansion
+ + Water
Injection
+ Gas
Injection
+
Free Gas
Production
Water
Production
+
+
Oil & Dissolved
Gas Production
=

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Gas Material Balance as a
Straight Line
G (Bg – Bgi) + Bgi cw Sw + cf ∆p = Gp Bg + WpBw - We
1-Sw
⎡
⎢
⎣
⎤
⎥
⎦
⎩
⎧
⎨
⎫
⎬
⎭
Xg ≡ (Bg – Bgi) + Bgi cw Sw + cf ∆p
1-Sw
⎡
⎢
⎣
⎤
⎥
⎦
Yg ≡ Gp Bg + WpBw - We
Yg = G Xg

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Yg - rb
Xg – rb/SCF
m = G
Yg - rb
Xg – rb/SCF
m = G
Gas Material Balance as a
Straight Line

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Water Influx
Š Aquifers come in all shapes and sizes
– Aquifers can be extremely large relative to the reservoir
size, even infinite “acting”.
– Aquifers can be small, even neglected.
– Aquifer productivity can be either high or low (relative to the
withdrawal rates from the reservoir).
Š Aquifers can be hydraulically connected to more than
one reservoir.
Š Aquifers can even be connected to the surface.

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Steady-State and Semisteady-
state Aquifer Models
dWe
dt
= qaq = k’ × (Paq – P)
qaq = instantaneous aquifer flow rate (rb/day)
k’ = aquifer influx constant (rb/day/psi)
Paq = average aquifer pressure (psi)
P = average reservoir pressure (psi)

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Steady-State and Semisteady-
state Aquifer Models
k’ is similar to the “Productivity Index” often used
to describe an individual well’s productivity.
Using a similar definition:
qaq = Jaq × (Paq – Pr)

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Steady-State and Semisteady-
state Aquifer Models
If the pore volume (Vaq), compressibilities, and average
aquifer pressure are known, the total water influx at
any point in time can be estimated by:
We = Vaq × cavg × (Pi – Paq)
Recall: Cavg = SoCo + SwCw + SgSg + Cf
So for aquifers, Cavg = Cw + Cf

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Steady-State and Semisteady-
state Aquifer Models
Define a term Wei that we will refer to as the
“maximum encroachable water”:
Wei = Vaq × cavg × (Pi – Paq)
0
Wei = Vaq × cavg × Pi
In words, this is the volume of water that will flow
from an aquifer if it’s pressure is lowered to zero.

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Steady-State and Semisteady-
state Aquifer Models
The average aquifer pressure at any point in time
can then be estimated by:
Paq = Pi × (1 – We / Wei)
qaq = Jaq × (Paq – Pr)
This equation, along with the previously shown equation
below, form the basis for steady-state and semisteady-
state aquifer models.

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Steady-State and Semisteady-
state Aquifer Models
Š “Pot” aquifer
– defined as an aquifer where the aquifer and reservoir pressure
remain (nearly) equal as the reservoir depletes
– this implies a small aquifer with high productivity
Š “Schilthuis” steady-state aquifer
– aquifer is extremely large and consequently, the aquifer pressure
can be assumed to remain constant
Š “Fetkovitch” semisteady-state aquifer
– aquifer rate and pressure are assumed to both change with time as
described by the previous equations

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Pot Aquifer
Definition: P = Paq
Recall:
Paq = Pi × (1 – We / Wei)
Then:
P = Pi × (1 – We / Wei)
Solving for We yields:
We = Wei × (1 – P / Pi)
We can be directly substituted into any of
the material balance equations.

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Steady-state Aquifer
Definition: Paq = Pi
Recall:
qaq = Jaq × (Paq – P)
Then:
qaq = Jaq × (Pi – P)
(∆We )n = Jaq × Pi –
⎡
⎣ ⎦
⎤
Pn + Pn-1
2
× ∆t

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Fetkovich Analytical
Aquifer
Recall “encroachable” water was defined by:
W ei = V aq × P i × c aq
Aquifer
Pore
Volume
Initial
Pressure
Aquifer
Compressibility
Aquifer pressure at any point in time is given by:
P aq = P i × (1 - W e / W ei )
Cumulative Water Influx

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Fetkovich Analytical
Aquifer
Finally, the influx rate at any point in time is given by:
q aq = J aq × ( P aq - P res ) =
Aquifer
Productivity
Index
Reservoir
Pressure
dWe
dt
Note: A large J and small W ei models a pot aquifer.
A large W ei models an infinite aquifer.

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Aquifer Productivity Index
Source: Applied Petroleum Reservoir Engineering
Craft, Hawkins, and Terry

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Fetkovich Analytical
Aquifer
Through algebraic manipulation and integration, the water
influx for a constant drop in pressure for a time “t” becomes:
We = ( pi – p ) (1 - e-J pi t / Wei)
Wei
pi
Fetkovich showed that this equation can be applied in a
difference form without the need for superposition to
model a system with a continuously falling pressure:
∆Wen = ( pn-1 – pRn ) (1 - e -J pi ∆tn / Wei)
Wei
pi

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Aquifer Boundary Pressure

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Some Comments
Regarding Aquifer Models
Š Steady-state and semisteady-state models are often
thought to be less accurate than unsteady-state models
(like the often used Hurst and van Everdingen model)
Š None of the analytical aquifer models directly consider
the growing water invaded zone and its impact on
aquifer productivity or MB (see SPE papers by Al-
Hashim & Bass and Lutes et al)
Š Truly “rigorous” treatment of aquifer influx requires
reservoir simulation

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Material Balance Calculations
Two Approaches
Š “Traditional” XY Plot
– Uses observed pressures and production in GMBE and
aquifer model to calculate N or G
– Some iteration may be required to estimate gas cap volume,
aquifer properties, etc.
– Sparse pressure data, erratic production rates a problem
Š Alternate method
– Uses observed production, aquifer model, and assumed
values of N or G in GMBE to calculate pressure
– Iterate until calculated and observed pressures agree
– Excellent for investigating sensitivities

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Gas Example No. 1
Š Moderate Water Drive
Š Project Based on 60-100 BCF Volumetric
Estimate
Š Unconstrained MB Analysis Suggested
Reservoir Was Depleted
Š Subsequent Well Was Dry Hole

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Gas Example No. 2
Š Weak Water Drive
Š P/Z Suggested 860 BCF
Š Downdip Water Production Suggested
Limited Water Influx

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Gas Example No. 2 (cont.)
Š Material Balance 725 BCF
Š HC Pore Volume 450 MMBbls
Š Water Influx 100 MMBbls

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Gas Example No. 3
Š Strong Water Drive
Š “Classic” Rate Sensitive Reservoir
Š Used in Field to Establish Production
Priority

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Gas Example No. 4
Š Example of “Pot” Aquifer
Š Observe the Sensitivity to Formation
Compressibility (49.7 to 58.5 BCF)

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Oil Example No. 1
Š Initially Undersaturated, Moderate Water Drive
Š Observe the Extreme Sensitivity to Formation
Compressibility (11 to 30 MMBO)
Note Culled Points

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Oil Example No. 2
Š Initially Undersaturated, Weak Water Drive
Š MB Analysis Reveals Pressure Behavior That Could
Not Be Matched
Š Subsequent Simulation Study Was a Failure
Š Anomalous Behavior Was Later Determined to be the
Result of Several Casing Leaks

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Oil Example No. 3
Š Small Offshore, Above BP Pressure
Š N = 35 MMBO from “XY” Plot
Š N of 66.5 MMBO Agrees with
Volumetric Estimate of 65 MMBO
Š Dominated by Aquifer Influx
From Dake, Exercise 3.4
The Practice of Reservoir Engineering

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Oil Example No. 4
Š Venezuela, Water-drive, Water Injection, Gas-cap
Expansion, Solution Gas Drive
Š N = 27 MMBO with “XY” Plot and Fixed Gas Cap Size
Š N = 34 MMBO Allowing Gas Cap Size to Vary
Note Culled Points
From Havelana and Odeh, JPT, July 1964

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Material Balance Compared
To Reservoir Simulation
Š Reservoir Simulation
– expensive
– time consuming
– requires geologic
description
– driven with single
phase
– ability to forecast
– determines location
and distribution of
unswept HC’s
Š Material Balance
– cheap
– fast
– independent of
geology
– uses production of all
phases in calculations
– limited forecasting
– can determine the
existence of unswept
HC’s

46. Technology Services
Reservoir & Well Performance
Material Balance:
The Forgotten
Reservoir
Engineering Tool
John McMullan