Reservoir Estimates

Reservoir Estimates
One of the important functions of the reservoir engineer is the periodic
calculation of reservoir oil and gas in place and the recovery anticipated under
the prevailing reservoir drive mechanisms.
Reverse estimation methods are usually categorized into three types:
1. Analogy
2. Volumetric methods
3.performance based techniques
 Material balance calculations
 Decline curve analysis
 Pressure transient analysis

Numerical simulation techniques.
1.Analogy:
During this period , before any wells are drilled on the property , any estimates
will be of a very general nature based on experience from similar pools or
wells in the same area. i.e by analogy.
2. Volumetric methods
The volumetric methods involve a determination of the bulk reservoir rock
volume , average porosity , fluid saturations , formation volume factors from
which the total reservoir hydrocarbon volume is calculated.
Recoverable reserves are then estimated by application of a suitable recovery
factor and the formation /surface volume factor for the produced fluid.
Recoverable oil =
Where :
Vb x Ø x (1-Sw) x R.F.
Bo
Vb is the bulk reservoir volume
Ø
is the fractional porosity
(1-Sw) is the hydrocarbon saturation
R.F. is the recovery factor
Bo is the oil formation volume factor
A recovery factor is approximated considering :
- laboratory measurement of oil displacement in cores
So –Sor
So
-type of displacement mechanism involved
-correlation of sweep efficiency based on a similar reservoir
Reservoir estimates are needed at various stages of a project
1)Geophysical exploration stages
Same order –of- magnitude estimate of the reserves which a structure might
contain is necessary to rank various projects probability of economic success
prior to making bids or possibly to relinquish undrilled acreage.
The first estimate is based on the volume of the structure determined from
seismic maps supplemented by information on local geological trends which
may indicate the thickness of porous beds which may be encountered.
By applying the common range of rock parameters , porosity (7 to 30%) ,
water saturation (8 to 40%) and recovery factor (10 to 50%) a possible range
of reserves that the structure might contain is estimated.
2)Exploration stage
With the drilling of a discovery well the uncertainty of encountering
hydrocarbons is removed , and measured values for porosity , and water
saturation became available for the section of pay traversed. Assuming that
well log data corroborate the prior seismic data , now only the contour of the
hydrocarbon /water contact (O.W.C.) is required to make reasonable estimate
for this stage.
3)Field development stage:
As new wells are drilled the volume and geometrical distribution of the
reservoir become even more accurately defined as well as the average
reservoir porosity and saturation values .
On the other hand , fluid withdrawals and injections into the reservoir and the
corresponding changes in fluid interfaces must be accounted for as the
inventory of reserves is continuously upgraded .
Whether the accounting of hydrocarbon reserves is made by computer or
manually , the procedures are the same in principle. Obviously , the trend in
reservoir studies is toward numerical simulation on which not only the static
inventory of reserves is kept , but which can predict the future behaviour of a
field.
Calculation of the reserve
The gross reservoir rock volume enclosed by the structure above the
hydrocarbon /water contact is calculated in the following steps:
1)
A net isopach map , giving the contour of equal thickness of pay with
the water contact assigned zero elevation contour is the most convenient
basis for rock volume calculations.
2)
The area within each contour is determined by planimetering , and a
plot prepared of area contained in each contour versus depth:
3) The gross rock volume is A dh which may be found by planimetering
again or by application of a numerical integration rule . In the Schlumberger
field studies approach the volumetric reservoir distribution is calculated
numerically and plotted by machine as isopach and isovolume maps.
V =h
An + A n+1 + An A n+1
3
- Trapezoidal formula  V = h
An +An+1
2
Vb = h Ao +2A1 +2A 2 …2A n-1 +A n + t av x An
2
- Pyramidal formula
3.Performance based techniques
 Material balance calculations
In many cases , porosity , saturations , and reservoir bulk volume are not
known with any reasonable accuracy , and emphasis on volumetric
calculations for reserve estimates is not advisable. The material balance is a
useful auxiliary tool for confirming reservoir estimates. The material balance
equation allow dependable estimates of the initial hydrocarbons in place as
well as prediction of the future reservoir performance.
The material balance equation (MBE) relates the volumes of fluids withdrawn
and encroached to the resulting reservoir pressures. Its principal utility ,
however , lies in predicting reservoir behavior and not in the estimation of
initial hydrocarbon in place.
The generalized material balance equation:
The general form of the oil material balance , first presented by Schilthuis is
derived as a volume balance which equates :
(a)
cumulative observed production , expressed as an underground
withdrawal , to
(b)
the expansion of the fluids in the reservoir resulting from a finite
pressure drop.
For the general case, the material balance has the form:
underground fluid
expansion of oil zone
expansion of any reduction of
withdrawal of oil , = (includes liquid oil+dissolved gas)+ gas cap
+ HCPV due to
,water, gas(Bbls)
(Bbls)
(Bbls)
connate water
expansion and
rock compaction
and water
encroachment
(Bbls)
Symbols:
cumulative oil production, Np ,STB
cumulative gas production ,Gp ,SCF
cumulative water production , Wp ,STB
original oil in place ,N ,STB
original gas in place ,G ,SCF
cumulative water influx ,res. bbl , We
m = gas cap size , G X Bgi
oil zone size , N X Boi
Bo,Bg , Bw , Bt are PVT properties of the different reservoir , s fluids
Bt = Bo+(Rsi-Rs)Bg
Ri = GOR = Kg/Ko x Mo/Mg x Bo/Bg +Rs (instantaneous gas oil ratio)
Average GOR ,Rp = Cumulative gas production
= Gp
SCF
Np
STB
Cumulative oil production
Bgc = gas cap Bg
Bgs = solution gas Bg
Gpc = cumulative gas production from gas cap
Gps = cumulative gas production from solution gas.
Derivation of the oil MBE
Let us define the following quantities:
N= the initial oil in place(STB) = VBO  (*1-Swi) /Boi
(!) Gas cap expansion = New gas cap volume - original gas cap volume
New gas cap volume = (G-Gpc) B gc
Original gas cap volume = G x Bgci
(1)Gas cap expansion = (G-Gpc) B gc - G x Bgci
(2) Remaining release gas = original soluble gas –remaining soluble gas –
- cumulative produced gas.
= [N Rsi -(N-Np)Rs -Gps] Bgs
(3) Remaining oil volume = (N-Np) Bo
(4) Net water influx
= (W e -W p Bw)
(5) Rock and water expansion is neglected in the presence of gas.
Gas Cap
Gas Cap
Gas cap expansion
Release gas
Oil
Remaining Oil
(N Boi )
Connate water
Pi
Connate Water
P
Rock & water
expansion
Net water influx
Aquifer ( water bearing zone)
Aquifer
( water bearing zone)
Condition I
Condition II
Pressure = Pi
NP = Zero
GP = Zero
WP = Zero
P<Pi
NP = +ive
GP = +ive
WP = +ive
Now, by equating the initial conditions , to the final conditions resulting from a
finite pressure drop.
N Boi = (N-NP) Bo + [(G-GPC) Bgc -G Bgci] +[NRsi –(N-NP)Rs–GPS ]Bgs+(We -WPBw)
NBo – NPBo + G Bgc- GPC Bgc - G Bgci + N RSi Bgs- N RS Bgs+ NP Rs Bgs -
N Boi =
-GPS Bgs + (We-WP Bw)
NBoi-NBo-NRsi Bgs +NRsBgs=-NPBo+NPRsBgs-GPSBgs+GBgc-GPcBgc-GBgci+(We-WPBw)
N[Boi-Bo-RsiBgs+RsBgs]=Np[-Bo+RsBgs]-Gps Bgs+GBgc-Gpc Bgc-GBgci+(We-WpBw)
But :
Bgc
= Bgs = Bg
and
Bgci = Bgsi Bgi
N[Boi-Bo+(-Rsi+Rs)Bg] = Np[-Bo+RsBg]-Gps Bg+GBg –GpcBg-GBgi+(We-WpBw)
Np[Bo-RsBg]+(Gps+Gpc)Bg =N[Bo-Boi+(Rsi-Rs)Bg]+ G(Bg-Bgi)+(We-WpBw)
But :
(Gpc+Gps)Bg = GpBg
Therefore ,
and Rp = Gp/Np and Bt = Bo+(Rsi-Rs)Bg
Np [ Bo - RsBg]+GpBg =N[(Bo-Boi)+(Rsi-Rs)Bg]+ G(Bg-Bgi)+(We-WpBw)
By adding and subtracting RsiBg -RsiBg
Np[Bo-(Rsi-Rs)Bg]-RsiBg]+GpBg =N[(Bo-Boi)+(Rsi-Rs)Bg]+ G(Bg-Bgi)+(WeWpBw)
Np[Bt-RsiBg]+Np Rp Bg =N[(Bo-Boi)+(Rsi-Rs)Bg]+ G(Bg-Bgi)+(We-WpBw)
Np [Bt – Rsi Bg + Rp Bg] = N [(Bo - Boi)+(Rsi-Rs)Bg] + G(Bg-Bgi) +(We-WpBw)
And finally :
Np [Bt + (Rp - Rsi) Bg] = N (Bt – Bti ) + mNBoi(Bg - Bgi) + (We-WpBw) …..(1)
Bgi
Where :
m = GBgi
Nboi
, G = mNBoi
Bgi
N (Bt – Bti ) +
Np [Bt + (Rp - Rsi) Bg] =
Cumulative oil
withdrawal
Depletion Drive
mechanism
mNBoi(Bg - Bgi)
Bgi
Gas cap drive
mechanism
+ (We-WpBw)
Water drive
mechanism
and this is the generalized material balance equation for combination drive
reservoir neglecting the rock and connate water expansion.
N ( B t – B t I ) = D.D.I
Np[Bt+(Rp-Rsi)Bg]
m N Boi/ Bgi (Bg - Bgi) = GCDI
BgiNp [Bt + (Rp - Rsi) Bg]
( We- Wp Bw) =
Np[Bt+(Rp-Rsi)Bg]
WDI
Driving index of any mechanism:
The driving index of any mechanism represents the fractional contribution of
the total oil withdrawal produced by that mechanism.
Driving
Index
GCDI
DDI
WDI
Time or Np or Pressure
It is clear from the previous figure that the DDI and GCDI decrease both of
them with time.
This is because that both of the gas cap gas and solution gas are limited
volume wise and pressure wise. However , the water aquifer is a huge water
bearing zone (unlimited volume in case of infinite aquifer), which means that
the volume of water available in the aquifer is very great.
Normally , the reservoir pressure in the aquifer will remain more or less
constant , while the reservoir pressure is decreasing with time.
Therefore, the pressure draw down between the aquifer and the reservoir
increases with time which means increasing in water influx and consequently
an increasing water drive index(WDI).
Rock and water expansion:
Definitions:
1.
Compressibility :
The change in volume per unit volume per unit pressure.
2.
Compressibility factor (z):
It is a measure of the deviation of natural gas with respect to ideal gas.
C = -  V
.
1
V
P
Note that formation compressibility = pore volume compressibility.
Cf = -  V
.
1
V
(Pi-P)
Now , let us consider the rock expansion = Cf Vp (Pi-P)
…….(A)
Saturation = fluid volume
Pore volume
Soi = (1-swi) =
OOIP =
Vp
NBoi
Vp
And thus:
Vp =
(NBoi )
(1-Swi)
substitute in (A)
Rock expansion = Cf
(NBoi ) (Pi-P)
(1-Swi)
Wate expansion :
Cw = -  Vw
.
1
Vw
 Vw =
Vw Cw
SW = Vw
(Pi-P)
then : Vw = Vp. Sw
Vp
(Pi-P)
Therefore
Water expansion =
 Vw = Cw x Vp
Sw(Pi-P)
Water expansion =
 Vw = Cw x Sw
NBoi
1-Swi
also ,
(Pi-P)
Therefore ,
Rock & water expansion = Cf
(NBoi ) (Pi-P) +
(1-Swi)
+ Cw x Sw ( NBoi ) (Pi-P)
(1-Swi )
= (Cf +CwSw) ( NBoi ) (Pi-P)
(1-Swi )
The most general form of Material Balance Equation is:
Np [Bt + (Rp - Rsi) Bg] =
N (Bt – Bti ) +
mNBoi(Bg - Bgi)
Bgi
+ (We-WpBw)
+ (Cf +CwSw) (NBoi ) (Pi-P) ……(2)
(1-Swi )
Case (1): Water drive reservoir:
a)
Below the bubble point pressure:
The driving mechanisms involved are :
1.Water drive mechanism
2.Depletion drive mechanism
The material balance equation is :
Np [Bt + (Rp - Rsi) Bg] =
b)
N (Bt – Bti ) + (We-WpBw) …………………..(3)
Above the bubble point pressure:
The driving mechanisms involved are :
1.Water drive mechanism
2.Depletion drive mechanism and
3.Rock and water expansion mechanism.
The material balance equation is :
Np Bo = N (Bo – Boi )
(Since
Rp = Rsi
+ ( We –Wp Bw ) + (Cf +CwSw) (NBoi ) (Pi-P )......(4)
(1-Swi )
= Rs = Constant) .
Effective oil compressibility :
Co = -  Vo
Vo
Bo - Boi
=
1
P
Co Boi
=
Bo-Boi
Boi
1
(Pi-P)
(Pi-P)
Substitute this value in equation (4):
Np Bo = N Co Boi (Pi-P) + ( We –Wp Bw ) + (Cf +CwSw) (NBoi ) (Pi-P ) ..(5)
(1-Swi )
+
Np Bo + Wp Bw = N Boi (Pi-P)
Cf +CwSw
(1-Swi )
Np Bo + Wp Bw = N Boi (Pi-P)
Cf +CwSw +Co So
(1-Swi )
Np Bo + Wp Bw = N Boi Ce (Pi-P) +
Co
We
+ We ………..(6)
+ We …....…..(7)
…………………….(8)
Where :
Ce = Cf + Cw Sw +So So = effective oil compressibility.
1-Swi
Case (2): Gas cap drive reservoir:
The driving mechanisms involved are :
1.Gas cap drive mechanism and
2.Depletion (solution gas) drive mechanism
Np [Bt + (Rp - Rsi) Bg] =
N (Bt – Bti ) +
mNBoi (Bg - Bgi)
Bgi
Case (3): Depletion drive reservoir:
a) Below the bubble point pressure:
The driving mechanism involved is :
Depletion (solution gas) drive mechanism only.
Np [Bt + (Rp - Rsi) Bg] =
c)
N (Bt – Bti )
Above the bubble point pressure (Under-saturated reservoir) :
The driving mechanism involved is :
Rock and fluid expansion only.
The material balance equation is:
Np Bo
= N(Bo-Boi) +
Cf +CwSw +Co So
(1-Swi )
Bo-Boi = Co Boi (Pi –P ) , finally:
Np Bo
=
N Boi Ce P
NBoi
( Pi - P )
+ We
If there is water production :, the equation form becomes :
Np Bo + Wp Bw
=
N Boi Ce P
This is the material balance equation for depletion drive reservoir (DDR)
producing
above the bubble point pressure (under- saturated reservoir) ,
taking into account the rock and water compressibilities ( expansion
mechanism ). However, the rock and water compressibilities are small relative
to the oil compressibility and thus , the rock and water expansion mechanism
can be cancelled. Therefore , the equation will be as follows:
Np Bo =
N (Bo - Boi )
Np Bo =
N Bo - N Boi
and therefore :
(N – Np ) = N Boi
this is the simplest form of the material balance equation which represents
a depletion drive reservoir (DDR) producing above the bubble point pressure
(under- saturated reservoir) neglecting the rock and water expansion
mechanism.
The last equation can be driven simply by considering the initial and
remaining oil in-place only ( of course , in addition to the connate water).
Pressure = Pi
Original Oil
in-Place
Remaining
oil
N Boi
(N-Np) Bo
Connate water
11
Pressure = P < Pi
Connate water
Reservoir performance curves
Typical performance curves of primary recovery mechanisms:
It is very rare to find a reservoir driven by single mechanism. In most cases ,
the reservoirs are driven by more than one mechanism.
For example, if you have a water drive reservoir producing above bubble point
pressure, the mechanisms will be water influx and fluid and rock expansion.
Once the reservoir pressure introduce to become lower than bubble point
pressure , the driving mechanism of the same reservoir will be water drive and
solution gas drive.
Sharp reduction in pressure
GCDR
P
Fluid &Rock Expansion
Mechanism
DDR
WDR
0
Np/N or R.F
GOR
1.0
Bg is very high
GCDR
Pb
WDR
F& Rock Expansion
0.0
DDR
NP/N
or
R.F
1.0
The following table represents the different recovery mechanisms and the
recovery factors associated with each of them:
Reservoir mechanism
Ultimate recovery factor,%
Fluid &rock expansion(F&R Exp.M)
5% of the OOIP
Depletion drive mechanism(D.D.M)
15-30% of the OOIP
Gas cap drive mechanism(G.C.D.M)
40-50% of the OOIP
Water drive mechanism (W.D.M.)
50% or more of the OOIP
1- For water drive reservoirs:
A rapid pressure reduction is occurred and then the water influx can
compensate this reduction causes gradual pressure reduction.
Gas oil ratio ( GOR) : Almost constant = solubility of gas
2-For Gas cap drive reservoirs:
The reduction in reservoir pressure is slow due to the high expansibility of the
gas , then reduction of the reservoir pressure is compensated by the gas cap
expansion .
Near the end of the reservoir life , reservoir pressure is considerably decline ,
the gas cap gas invades the oil zone and you can,t avoid the production of the
gas cap gas . thus you will concern a rapid increase in GOR , which is
corresponding to rapid decline in the reservoir pressure.
3-For depletion drive reservoirs:
The reservoir pressure declines slowly at the beginning and then rapidly.
The GOR remains more or less constant for the period from the initial
reservoir pressure up to the Pb ( bubble point pressure). Then it decreases for
a short period and then increases again to reach a maximum value after
which it starts to decrease again. To explain this, it is well known that the gas
solubility above the bubble point pressure is constant and therefore the
surface GOR is constant as well.
Once the pressure is reduced below bubble point pressure , the solution gas
start to release from oil , however , this gas can not be move (immobile) until
its saturation exceeds the critical gas saturation. So , the observed surface
GOR in this short period ( from Pb
to Sgc) decreases.
After that , the released free gas becomes mobile and can be produced to be
added to the solution gas which results in increasing the GOR.
Near the end of the reservoir life , the reservoir pressure declines to a low
value which means an increase in the gas formation volume factor (Bg) .
since the surface gas oil ratio ,GOR is expressed as :
GOR = K g . o .o + Rs
Kg
g
g
Thus , surface GOR starts to decreases due to increasing
g
Rock and fluid expansion :
As for the rock and fluids expansion which dominates for a very short period
of reservoir life , the reservoir pressure declines very rapidly while thesurface
GOR remains constant.
How to determine the reservoir driving mechanism:
The worst conditions is the rock and fluid expansion only.
This exists in the case of DDR above Pb.
Np o +
N
W p w
Np o +
=
=
N oi Ce  P
W p w
oi Ce  P
i.e
Y
Y
=
Constant.




X

Past performance:
1-production data
2-pressure data
and
from production commencement so far.
3-PVT data.
Gas cap or We
X
X

Y


X

X
Correct assumption
O
O


O

O
O
O
Lost production or
Thief zone
P
Where :
Y = Apparent N =
Np o +
W p w
oi Ce  P
The driving mechanism of any reservoir can be determined as follows :(1) plot the past performance (GOR & pressure Vs Np/N ) and compare
the same with the typical performance curves of the different driving
mechanisms to guess which mechanism
(2) to check if your guessing is correct or not ,do the following:
(3) assume the lowest efficient driving mechanism i.e (that is to say) a fluid
& rock expansion only (D.D.R above Pb ) i.e. under saturated DDR
(4) The M.B. equation for this reservoir can be written as:
NPBo +WPBW= N BOICe
 P
(5) Use the past performance data of your reservoir (P, NP. and PVT
data)
(6) Use the past performance to solve the M.B. equation at different time
intervals and plot apparent N versus pressure .
If this plot represent a horizontal line , this means that your assumption is
good and your driving mechanism is really fluid & rock expansion
mechanism.
(7)If not , there are two possibilities :
a) The apparent (N) increases with pressure . This indicate that there is an
other external force rather than DDR mechanism which might be gas cap
or water influx (We) or both . To check which of which go to the logs.
b)The apparent (N) is decreases , this means that your reservoir
connected to a thief zone .
General material balance can be also derived as follows :
EXPASNSION TERMS
Oil :
N(o – oI)
Gas cap :
m N oi
gc-gci
gci
Liberated gas (liberated .solutionn gas)
water :
Vp Sw (w-wi) = N oi ( 1+m ) Sw Cw  P
wi
1-Sw
Vp (1-) . m-mi
rock:
: N (Rsi –Rs) gs

= N oi (1+m) .Cf 
mi
P
1-Sw
VOIDAGE TERMS
Npo
OIL :
Librated gas (librated solution gas) : (Gps- NpRs) gs
Gas cap :
Gpc gs
Water :
–W e + Wpw
Injection :
–Gi gi –Wi w
Equating the expansion terms to the voidage terms , and solving for the
original oil in place , we get
General M.B Equation for oil in – place :
N = Np o + [ Gps – NpRs ] gs +Gpc gc+ W pw – W e -Gi gi –W i w
(o-oi)+(Rsi-Rs) gs+moi(gc-gci)+ oi(1+m)(CwSw+Cf) P
gci
(1-Swi )
IF no distinction is made between solution gas , gas cap gas and injection
gas , the general material balance equation can be written as:
N = Np o + [ Gp– NpRs ] g – ( W p +Wp w ) -Gi g – W i w
(o-oi)+(Rsi-Rs) g+moi(g-gi)+ oi (1+m)(CwSw+Cf) P
gi
Notice that :
(1-Swi )
G= m Noi
gi
Although this equation is general and includes all terms yet special cases
may be considered in which certain of the terms become zero and allow
considerable simplifications . some of these special cases are given in the
following pages. When there is no gas injection or water injection , the
principal unknowns in the MB equation are N , m and We.
any one of these unknowns can be calculated if the other two are
satisfactorily known.
also, it is some times possible to solve simultaneously to find two of these
unknowns.
one of the important uses of the MB equation is predicting the effect of
production rate and/or injection rates (for gas or water)on the reservoir
pressure.
Special Cases of the MB Equation:
1-Under saturated oil reservoir above the bubble point pressure:
In this case :
m
= zero (no gas cap )
the producing GOR R
= Rs
= constant , namely Rsi , so that
Gp = Np Rsi ,
Also , with no gas injection , Gi =
reduced to :
N = Np o + ( W p –Wi ) w - W e
(o-oi)+ oi (CwSw+Cf) P
(1-Swi )
or
N = Np o + ( Wp –Wi ) w - W e
oi Ce P
zero , the MB equation
where:
Co = Bo-Boi
.
1
P
Boi
Ce =
= Compressibility of single phase oil
Co So +Cw Sw +Cf
1 -Swi
Also , if Wp = zero
, Wi
=
zero and
We = zero,
Therefore ,
Np o
= N oi Ce P
2- Solution gas drive reservoir ( initially at bubble point pressure):
(under saturated oil reservoir below Pb):
In this case , Pi = Pb
,
m = zero
, We = zero , Wp = zero or
negligible , and Wi = zero , ( no water or gas injections) . Also , usually ,
rock and connate water expansions are negligible compared to gas and
hydrocarbons expansion , so that , Cf = zero and Cw = zero .
Hence the MB Equation reduces to:
N =
Np o + [ Gps – NpRs ] gs
(o-oi)+(Rsi-Rs) g
Where : N = the oil in place at the bubble point pressure.
It is customary to base solution gas drive calculations on an initial (N) value
representing the oil in place at the bubble point pressure Pb . Thus the
original oil in place at the initial pressure Pi would be this oil in place at the
bubble point pressure plus the cumulative oil production obtained until the Pb.
Is reached. (case 1 obtained before). That is :
Ni
=
Nbubble +
 N I-
b
Gas reservoirs
For these gas reservoirs, the reservoir fluid is always a single phase gas for
the life of the reservoir because in these systems the formation temperature is
greater than the cricondentherm.
Phase diagram :
These diagrams are drawn from PVT date:
P
single phase liquid
2
Retrograde condensation
 C
Bubble point line
1
Two phase region
1
Cricondentherm
Single phase (gas)
Dew point line
Temperature
Point C, the critical point: the point at which the liquid and gas are in
equilibrium or has the same properties.
Point 1, the cricondentherm : the maximum temperature where the two phase
or liquid may exist.
Point 2 the cricondenbar : the maximum pressure where the two phase
or gas may exist.
Point 3 retrograde condensation: at which , although the pressure is reduced
, the gas is condensed to the liquid state.
How you can confirm the type of reservoir:
According to the phase diagram: as shown in the attached figure:
C
B
A
P
C1
C B1
2
C2 A3
B2
A2
A1
Separator
Temperature
Reservoir A is a gas reservoir
A –A1
A – A2
Dry gas reservoir at the surface
A-A3
Wet gas reservoir
Reservoir B
B –B1
Gas reservoir
B- B2
Gas
Condensate reservoir
Reservoir C
C–C1
Under saturated reservoir
Contains no free gas
C- C2
Saturated reservoir
contains both
oil and free gas
Note that :
-the reservoir temperature is constant
-reservoir pressure declines with production
Determination of the original gas in place(OGIP):
As with oil reservoirs, there are two main ways to estimate the original gas in
place : the volumetric method and Material balance method.
1-Volumetric method :
this approach is used early in the life of the reservoir ( for instance , before
5% of the reserves have been produced).
The standard cubic feet of gas in a reservoir which has a gas pore volume
of Vg ( cu.ft) is simply :
G = Vg  g
Where: g =gas formation volume factor ( Scf/ft 3) .
As (g) changes with pressure , the gas in place also changes as pressure
declines.
Also, the gas pore volume (Vg) may also be changing owing to water influx
into the reservoir.
The simplest form of the equation for the volumetric or pore volume method
that assumes a homogenous , isotropic reservoir is:
G = 43,560 A h  (1-Sw)
gi
(1)
where :
G = original gas in place ,scf
43,560 = conversion factor , sq.feet per acre.
A
= reservoir productive area , acres
h
= net thickness , feet

= porosity,%
Swi
= average water saturation ,% and
gi
= initial gas formation volume factor, cu.ft/scf.
The significance and derivation for gi is as follows:
(g) is used to signify gas formation volume factor which is equal to the
volume of gas at reservoir temperature and pressure divided by the
volume of the same amount of gas at standard conditions of temperature
and pressure. With this factor, we can relate gas reservoir volume to its
surface volume using the real gas law:
Z n R Tf
gi = Vreservoir =
Pi
= Zi Tf Psc
Vstandard
Zsc n R Tsc
Zsc Tsc P
Psc
gi = Psc . Tf Zi
cu.ft /scf
Pi Tsc
Normally , with field units , Tsc = 520 oR , Psc = 14.7 psia , Zsc =1.0
g = 0.0283 Z T
cu.ft/ scf
P
gi = 0.00504 Z T
bbl/scf
P
During the early life of the reservoir (such as zero , one or two wells drilled) ,
only one or two estimates for each of the parameters in equation (1) are
available . however, the geologist may have provide a structure map primarily
based on geophysical data from which a rough net- pay isopach can be
generated.
In this case , it would be suitable to use equation (1) together with estimates
of average values over the reservoir for all parameters except ( A) and (h) .
Net hydrocarbon (gas) , volume for this situation is determined by numerical
integration of the net pay isopach. This result in acre-ft is substituted into
the previous equation (1) in place of the product (A h) or VB.
The volumetric method makes use of subsurface and isopach maps based on
the data from electric logs , cores , and drill –stem and production tests .
- A subsurface contour map is a map showing lines connecting points of equal
elevations on the top of a marker bed , and it therefore , represents a map
showing geologic structure.
-A net isopach map is a map showing lines connecting points of equal net
formation thickness.
Two equations are used to estimate reservoir bulk volume (VB):
- Trapezoidal formula :
 VB = h ( An +
- Pyramidal formula :
2
 VB = h (( An +
3
An+1 )
An+1 ) + (An x
An+1)1/2
Calculation of unit recovery from volumetric gas reservoirs:
During the development period , VB is not known , so, it is better to place the
reservoir calculations on a unit basis (acre-ft).
Therefore ,
Connate water =
43,560  Sw
ft 3/ ac-ft
Reservoir gas volume = 43,560  (1- Sw)
Reservoir pore volume = 43,560 
ft 3/ ac-ft
ft 3/ ac-ft
Calculation of the initial gas in place (G) :
G= 43,560  (1- Sw) gi
Ga = 43,560  (1- Sw) ga
scf/ ac-ft
scf/ ac-ft
The unit recovery = the difference between the initial gas in-place and the
gas remaining at abandonment pressure.
Therefore , the unit recovery ,U.R = 43,560  (1- Sw) (gi - ga )
And the recovery factor , R.F.= ( G-Ga ) X100
G
scf/ ac-ft
=100 (gi - ga )
gi
%
Calculation of unit recovery from water drive gas reservoirs:
In many reservoirs under water drive , the pressure suffers an initial decline ,
after which water enters the reservoir at a rate equal the production and the
pressure stabilizes.
In this case , the stabilize pressure equals abandonment pressure.(Pb).
Then , under abandonment conditions ,
water volume = 43,560  (1- Swi)
,and
Reservoir gas volume = 43,560  Sgr
Surface units of gas = 43,560  Sgr x ga
The Unit recovery ,U.R= 43,560  [ (1-Swi) gi - Sgr x ga] scf/ac-ft
The Recovery factor ,R.F = 100 [ (1-Swi) gi - Sgr x ga]
%
(1- Swi) gi
Material balance equation of gas reservoirs
If enough production –pressure history is available for a gas reservoir , the
initial gas in –place (G) , the initial reservoir pressure (Pi) , and the gas
reserves can be calculated without knowing A , h,  , or Sw . This is
accomplished by forming a mass or mole balance on the gas. That is :
Moles produced = initial moles in – place =remaining moles
in equation form:
nf = ni
- np
np = ni - nf
applying the gas law ,
PV = n Z R T gives :
Psc Gp
=
Tsc Zst
PI VI
-
Ti Zi
Pf Vf
Tf
Zf
where :
vi = initial gas pore volume ,ft3
pf = final pressure
vf = final gas volume after producing Gp (separator) of gas
Vf
We =
=
Vi
- We
+ WpBw
cubic feet of water encroached into the reservoir
\Wp = cubic feet of water produced
Bw
= water formation volume factor , bbl/STB
Psc Gp
Tsc
=
PI VI
-
Pf (Vi - We+Wp Bw)
Ti Zi
Tf
Zi
Gp = Standard cubic feets (SCF) of produced gas measured at standard
temperature and pressure.
in case of volumetric reservoirs , there is no
water influx ( We) , Wp generally negligible , then :
Psc Gp
=
PI VI
Tsc
-
Pf Vi
Ti Zi
(2)
Tf Z
where :
Tf = formation temperature
Vi = reservoir gas volume
Pi = initial pressure and
P = reservoir pressure after producing scf (Gp ),
the reservoir gas volume can be put in units of scf by use of Bgi , that is
Vi = G Bgi
combining equation (2)
P
(3)
and (3) and solving for P gives
Z
=
Pi
-
Tf
Psc
x Gp
Tsc Bgi G
from which it is obvious that a plot of P/Z versus Gp will result a straight line
of slope (m) equals
m = Tf
Psc
and intercept at Gp = zero. of Pi/Zi
Tsc Bgi G
thus , both (G) and (Pi) can be obtained graphically.
the slope
m
=
 Gp
 (P/Z)
Equation (2) may be written in terms of gas volume factors gi and gf by
solving it for (Gp) :
Psc Gp
=
PI V I
Tsc
Gp
-
Pf (Vi - We+Wp Bw)
T Zi
=
Pi Tsc
T
Vi -
gi
=
Psc Zi T

gf
scf/ft3
= Pf Tsc
Psc Zf T
Gp = 
gi
Vi = G

gi
Vi - 
gf (
Psc
Psc Zf T
, scf/ft3
Pi Tsc
Zf
Pf Tsc ( Vi – W e +W p Bw)
Zi T Psc

x Tsc
Vi –W e +W p  w)
Therefore , Gp = G - 
gf
( G – W e +W p  w)

Dividing by 
Gp

=

gf
Gp

G
=
g
g
G


Replace 
Gp 
-
G
gf
:
gf
gf
gi
We + Wp 
+
gi
1
- 1
gf

by
ft3/scf
= G (
w
gf
+
We - W p 
w
gi
- 
instead of scf/ft3 ,
gi
) + W e -W p 
w
therefore ,
production term = expansion term +Water influx term - water production term
For volumetric reservoir:
production volume = expansion volume
Gp 
g
= G (
gf
- 
gi
)
P/Z
Pi
Zi
Water drive
Volumetric Reservoir
Pa
Za
0
Reserve
G
P/Z Vs Gp graph for gas reservoir
Gp
The steady state radial flow equation
Well bore damage and improvement effects:
Ways to quantify damage or improvement :
1- skin factor (S):
The pressure drop at a damaged or improved well differs
from that at undamaged well by the additive amount :
 Ps = 141.2
q . S
k h
From Darcy, s equation :
Q = 0.00708 k h (pe -pwf )
 o ln re/rw
pe = pwf + 141.2 q  o ln re/rw
kh
pat r = pwf + 141.2 q  o ln r / rw
kh
if there is a skin ( altered zone) around the well :
Pe
P
Ka
P1
K
Pwf
Pwf1
R1
rw
Ra
Re
Re
P1 – Pwf =
141.2 q  o ln re/rw
K.h
P1 - Pwf1 = 141.2 q  o ln re/rw
Ka . h
Subtract equation (2) from equation (1)
Pwf – Pwf1 =
Pwf- Pwf1
141.2 q  o ln ra/rw  1 h
Ka
(1)
(2)
1 
K
= pressure drop due to skin . Multiply by K and divide by K
Gas reservoirs
For these gas reservoirs, the reservoir fluid is always a single phase gas for
the life of the reservoir because in these systems the formation temperature is
greater than the cricondentherm.
Phase diagram :
These diagrams are drawn from PVT date:
P
single phase liquid
2
Retrograde condensation
 C
Bubble point line
1
Two phase region
1
Cricondentherm
Single phase (gas)
Dew point line
Temperature
Point C, the critical point: the point at which the liquid and gas are in
equilibrium or has the same properties.
Point 1, the cricondentherm : the maximum temperature where the two phase
or liquid may exist.
Point 2 the cricondenbar : the maximum pressure where the two phase
or gas may exist.
Point 3 retrograde condensation: at which , although the pressure is reduced
, the gas is condensed to the liquid state.
How you can confirm the type of reservoir:
According to the phase diagram: as shown in the attached figure:
C
B
A
P
C1
C B1
2
C2 A3
B2
A2
A1
Separator
Temperature
Reservoir A is a gas reservoir
A –A1
A – A2
A-A3
Wet gas reservoir
Dry gas reservoir at the surface
Reservoir B
B –B1
Gas reservoir
B- B2
Gas
Condensate reservoir
Reservoir C
C–C1
Under saturated reservoir
Contains no free gas
C- C2
Saturated reservoir
contains both
oil and free gas
Note that :
-the reservoir temperature is constant
-reservoir pressure declines with production
Determination of the original gas in place(OGIP):
As with oil reservoirs, there are two main ways to estimate the original gas in
place : the volumetric method and Material balance method.
1-Volumetric method :
this approach is used early in the life of the reservoir ( for instance , before
5% of the reserves have been produced).
The standard cubic feet of gas in a reservoir which has a gas pore volume
of Vg ( cu.ft) is simply :
G = Vg  g
Where: g =gas formation volume factor ( Scf/ft 3) .
As (g) changes with pressure , the gas in place also changes as pressure
declines.
Also, the gas pore volume (Vg) may also be changing owing to water influx
into the reservoir.
The simplest form of the equation for the volumetric or pore volume method
that assumes a homogenous , isotropic reservoir is:
G = 43,560 A h  (1-Sw)
gi
where :
G = original gas in place ,scf
43,560 = conversion factor , sq.feet per acre.
A
= reservoir productive area , acres
h
= net thickness , feet

= porosity,%
Swi
= average water saturation ,% and
gi
= initial gas formation volume factor, cu.ft/scf.
(1)
The significance and derivation for gi is as follows:
(g) is used to signify gas formation volume factor which is equal to the
volume of gas at reservoir temperature and pressure divided by the
volume of the same amount of gas at standard conditions of temperature
and pressure. With this factor, we can relate gas reservoir volume to its
surface volume using the real gas law:
Z n R Tf
gi = Vreservoir =
Pi
= Zi Tf Psc
Vstandard
Zsc n R Tsc
Zsc Tsc P
Psc
gi = Psc . Tf Zi
cu.ft /scf
Pi Tsc
Normally , with field units , Tsc = 520 oR , Psc = 14.7 psia , Zsc =1.0
g = 0.0283 Z T
cu.ft/ scf
P
gi = 0.00504 Z T
bbl/scf
P
During the early life of the reservoir (such as zero , one or two wells drilled) ,
only one or two estimates for each of the parameters in equation (1) are
available . however, the geologist may have provide a structure map primarily
based on geophysical data from which a rough net- pay isopach can be
generated.
In this case , it would be suitable to use equation (1) together with estimates
of average values over the reservoir for all parameters except ( A) and (h) .
Net hydrocarbon (gas) , volume for this situation is determined by numerical
integration of the net pay isopach. This result in acre-ft is substituted into
the previous equation (1) in place of the product (A h) or VB.
The volumetric method makes use of subsurface and isopach maps based on
the data from electric logs , cores , and drill –stem and production tests .
- A subsurface contour map is a map showing lines connecting points of equal
elevations on the top of a marker bed , and it therefore , represents a map
showing geologic structure.
-A net isopach map is a map showing lines connecting points of equal net
formation thickness.
Two equations are used to estimate reservoir bulk volume (VB):
- Trapezoidal formula :
 VB = h ( An +
- Pyramidal formula :
2
 VB = h (( An +
3
An+1 )
An+1 ) + (An x
An+1)1/2
Calculation of unit recovery from volumetric gas reservoirs:
During the development period , VB is not known , so, it is better to place the
reservoir calculations on a unit basis (acre-ft).
Therefore ,
Connate water =
43,560  Sw
ft 3/ ac-ft
Reservoir gas volume = 43,560  (1- Sw)
Reservoir pore volume = 43,560 
ft 3/ ac-ft
ft 3/ ac-ft
Calculation of the initial gas in place (G) :
G= 43,560  (1- Sw) gi
Ga = 43,560  (1- Sw) ga
scf/ ac-ft
scf/ ac-ft
The unit recovery = the difference between the initial gas in-place and the
gas remaining at abandonment pressure.
Therefore , the unit recovery ,U.R = 43,560  (1- Sw) (gi - ga )
And the recovery factor , R.F.= ( G-Ga ) X100
G
scf/ ac-ft
=100 (gi - ga )
gi
%
Calculation of unit recovery from water drive gas reservoirs:
In many reservoirs under water drive , the pressure suffers an initial decline ,
after which water enters the reservoir at a rate equal the production and the
pressure stabilizes.
In this case , the stabilize pressure equals abandonment pressure.(Pb).
Then , under abandonment conditions ,
water volume = 43,560  (1- Swi)
,and
Reservoir gas volume = 43,560  Sgr
Surface units of gas = 43,560  Sgr x ga
The Unit recovery ,U.R= 43,560  [ (1-Swi) gi - Sgr x ga] scf/ac-ft
The Recovery factor ,R.F = 100 [ (1-Swi) gi - Sgr x ga]
(1- Swi) gi
%
3
PETROLEUM
RESERVOIR ENGINEERING
Lectures
Selected-Compiled –Edited
By
Dr.Eng.Mostafa Mahmoud A. Kinawy
CHAPTER II
FLUIDS FLOW
IN
POROUS MEDIA
-Steady State Flow
-Pseudo- Steady State Flow
-Unsteady State Flow
Dimensionless Quantities:
Td = 0.0002637 K t
 Ct r w2
ct = total system compressibility ,psi-1
ct =cf + co so +cw sw +cf (for single phase oil flow)
tdA = (rw2)
, A = total drainage area
A
rD = r
rw
PD =
P
141.2 q 
Kh
with these dimensionless quantities , the diffusivity equation becomes :
2 PD
+ 1  PD
= PD
2
 rD
rD rD
tD
1-dimensionless quantities provide a convenient way to summarize the
increasing number of solutions being developed to depict well or reservoir
pressure behavior over a broad range of time , reservoir properties ,
boundary , geometry conditions.
2- an unfortunate consequence of the generalized dimensionless solution
approach is that the dimensionless parameters do not provide the engineer
with the physical feel available when normal dimensional parameters are
used.
3- Note that 141.2 q  =
0.8686 m , where m is the slope of the semi –
kh
log plot , =
162.6 q 
kh
Solutions to the diffusivity equation:
1-
for infinite reservoirs:
1-1-Constant Terminal Rate Solution:
initial and boundary conditions:
initial conditions :
p = pi at
t  0
every where
boundary conditions
q = constant for t  0 ( or r p ) = constant
r
p  pi as r   for all t
the solution is :
pD = fn (rD , tD)
note that :
= -1/2 Ei ( - rd2 )
4td
pD = pi – p
141.2 q 
kh
 1/2 [ ln ( td )
rD2
+ 0.80907]
for td  100
This is the "exponential integral "solution , also known as the line source or"
Theis" solution , where the exponential integral is defined as :

Ei (-x) = - x 
e –u du
u
and its values may be taken from tables , graphs or approximated:
Ei (-x)  ln (x) + 0.5772 ( for x  0.0025)
for rd  20 , , td  0.5
rD2
or for td  25 , the exponential integral solution
rD2
is sufficiently accurate. At the operating well , rD = 1.0 and so , td = td
rD2
1-2 Constant Terminal Pressure Solution:
The initial and boundary conditions are :
initial conditions :
p = pi
at t  0
every where
Boundary conditions :
p = constant at r = rw for t  0
p  pi as r   for all t
A solution was found by Van Everdingen & Hurst , to give the dimensionless
cumulative production ( water influx ) Q(t) as a function of dimensionless
time tD . A set of curves relating the dimensionless rate qD against
dimensionless time tD for a constant pressure test , where :
q D = 141.2 
k h (pi –pwf)
q
the following figures , after Craft& Hawkins , are useful to show the difference
between the constant terminal rate and constant terminal pressure cases.
t=0
time
t=1
t=10
t= 100
P=10
P =100
The steady state radial flow equation
Well bore damage and improvement effects:
Ways to quantify damage or improvement :
2- skin factor (S):
The pressure drop at a damaged or improved well differs
from that at undamaged well by the additive amount :
 Ps = 141.2
q . S
k h
From Darcy, s equation :
Q = 0.00708 k h (pe -pwf )
 o ln re/rw
pe = pwf + 141.2 q  o ln re/rw
kh
pat r = pwf + 141.2 q  o ln r / rw
kh
if there is a skin ( altered zone) around the well :
Pe
P
Ka
P1
K
Pwf
Pwf1
R1
rw
Ra
Re
Re
141.2 q  o ln re/rw
K.h
P1 – Pwf =
(1)
P1 - Pwf1 = 141.2 q  o ln re/rw
Ka . h
Subtract equation (2) from equation (1)
141.2 q  o ln ra/rw  1 h
Ka
Pwf – Pwf1 =
Pwf- Pwf1
(2)
1 
K
= pressure drop due to skin . Multiply by K and divide by K
Pwf – Pwf1 =
PS
PS
141.2 q  o
K h

K Ka
1  ln ra/rw
skin factor(s)
= 141.2 q  o
K h
.
S
II) if the skin is viewed as a zone of finite thickness with permeability Ka:
s = 
III)
K - 1  ln ra/rw
Ka
An apparent well bore radius ,rwa may be defined so that the
correct pressure drop results:
rwa = rw e-s
notice that : ln re/rwa = ln relrw + s
since re/rwa =re/rw x rwlrwa
for damaged zone: ka<k
which means +ive skin factor
for a simulated zone , ka>k which means –ive skin factor
the value of skin factor ,s is an indication about how much is the severity
of damage or improvement , ra can be obtained from resistively logs.
s can vary from about -5 to + for too badly well, s is +ive for damage ,
and negative for improvement>
hydraulically fractured wells often show values of s from -3 to -5.
introducing the skin factor ,s:
pd =
pi-pwf
141.2 qo
kh
=
[
1/2 {ln td +0.80907} +s ]
Build –Up Tests
Pressure
Introduction:
Pressure build-up testing is probably the most familiar transient well testing
technique. the test requires shutting –in a producing well . it should be noticed
that:
(1) the well is produced at a constant rate ,q , either from the start-up , or
for a long enough period to establish a pressure distribution before
shutting-in ;
(2) the pressure is measured immediately before shut-in, and is recorded
as a function of time during shut-in period. the resulting pressure buildup curve is analyzed for reservoir properties , well bore conditions as
follows:
figure (i) schematically shows rate and pressure behavior for an ideal
pressure build-up test.
in that figure, tp is the production time and t is the running shut-in time.
flow rate
q
idealId case
0
actualIi case
( history case)
tp
t
shut –in time
Pressure
Pwf ( t =0)
0
tp
t
figure (i) represents the idealized pressure history for a pressure build-up test
as in transient well tests , knowledge of surface and subsurface mechanical
conditions is important in build-up test data interpretation . therefore , it is
recommended that tubing and casing size , well depth , packer locations , etc
, be determined before data interpretation starts .
stabilizing the well at a constant rate before testing is important for reliable
test. to determine the degree and adequacy of the stabilization , one way is to
check the length of the pre-shut-in constant rate period against the time
required for stabilization, as follows:
 ct A
ts = 380
k
where : ts =the minimum shut-in time ,hrs for a well in the center of a
symmetrical drainage area(A) , and
tR
= 946  ct A
k
where : tR
=
readjustment time , the time required for a short lived
transient to die out, hrs, for a single well in the center of a constant pressure
square.
Basic Analysis Method: (Horner ,s method )
it is applicable to an infinite , homogeneous , one –well
reservoir containing
a fluid of small and constant compressibility , and so, it applies well to newly
completed well in oil reservoirs above bubble point pressure(Pb).
derivation:
the exponential integtal solution given before equation 12 gives the pressure
drop for a well flowing for a time tp
dimensionless) form:
Pi-Pwf = -q 
4k h
 -q 
4k h
Ei ( -   cf rw2 )
4 k tp
ln (    ct rw2 )
4 k tp
where : = Euler,s constant = 1.78
as follows in dimensional
(not
if now , the well is shut-in for a time t , after having produced for a time t p ,
then pressure drop at t can be obtained by the principle of superposition as
follows:
Pi-Pws = (pressure drop caused by rate q for time (tp+t) ) + (pressure drop
caused by rate change -q for time t) or:
Pi - Pws = -q 
ln (    ct rw2 ) + q 
4k h
4 k (tp+t)
ln (    ct rw2)
4k h
4 k (t)
and
Pws
=
- q 
pi
ln ( tp + t)
t
4k h
Where : Pws = the well pressure after shut-in
Pi =
the well flowing pressure before shut-in
In the usual practical oilfield units , the last equation became:
Pws
=
Pi
- 162.6 q 
ln ( tp + t)
t
kh
Pws
=
Pi - m log
( tp + t)
t
This is Horner,s equation , applicable for ideal Build-Up test.
this equation indicates that a plot of observed shut-in bottom hole pressure ,
Pws Vs log [tp + t] should have a straight –line portion with slope = -m
t
that can be used to estimate reservoir permeability,
k
= 162.6 q 
m h
As indicated by the Horner , s equation , the straight line portion of the Horner
plot , see Figure(ii) , may be extrapolated to tp + t = 1.0 , log tp + t = 0
t
t
, the equivalent to infinite shut-in time , to obtain an estimate of (Pi ) . That is
an accurate estimate only for short production periods. It is also noticed that ,
as a result of using the principle of superposition , the skin factor does not
appear in Horner , s equation.
Pi
Pwf,
Psi
slope =slope =-m = slope
per cycle.
P1hr
(tp +t)/ t
However, the skin factor may be estimated from the build-up test data plus
the flowing pressure immediately before the build-up test (Pwf) , since the skin
does affect this flowing pressure.
S = 1.153 P1hr - Pwf t = zero - log (
K
) + 3.227 S
2
m
 ct rw
The first part of the build-up plot is usually non-linear resulting from the
combined effect of the skin factor and well bore storage. the latter is due to
the normal practice of closing in the well at surface rather than down hole ,
which results in the flow rate not being reduced to zero instantaneously .
Use of Horner,s method for analysis of pressure build-up tests
during the infinite acting period:
During the infinite acting period of time, after :
-
Well bore storage effects have diminished , and
-
tD > 100 ( which occurs after few minutes for most unfractured
systems
Pws = Pi - m
log (tp +t)/ t
slope = -m and intercept = Pi
which is a straight line semi-log plot ,
From this plot , we make the following estimates:
1- K = 162.6 q
m.d (undamaged zone permeability)
m h
(tp +t)/ t = 1 ( this is accurate only for
2- Pi : by extrapolation to
short production periods , otherwise, extrapolated values give P*= false
pressure.
3-
S = 1.153 P1hr - Pwf t = zero - log (
K
) + 3.227 S
2
m
 ct rw
This equation is good as long as tp >> 1 hr , otherwise , for example in
case of drill stem tests (DST).
S = 1.153 P1hr - Pwf t = zero + log (tp+1 ) - log (
K
) + 3.227 S
2
m
tp
 ct rw
The value of tp may be approximated from :
tp = 24 x
Vp
q
Where : Vp = cumulative volume produced since the last pressure
equalization, or the Np
4- The average permeability , Kavg = q  ln (re/rw)
7.08 h (Pe=Pwf)
Here , the
Pwf is the stabilized well pressure corresponding to the
stabilized flow rate (q).
5-Productivity ratio (PR) = Kaverage
=
0.868 m
Kundamaged
ln (re/rw)
Pe -Pwf
= 2 m log re/rw
Pe -Pwf
6- Damage factor = 1
- PR
Also , S =
Ka
Kund
-
ln ( ra/rw)
Ka
and PR
= ln re/rw
ln (re/rw) +S
If the productivity ratio ,PR  1
this means stimulation , sinc3e Kavg >
Kun.
If PR  1
, This means damage , also ,
Damage factor = 1-
PR
,
Positive +
Damage
: may be
Negative - Stimulation
Also ,
S may be
Positive +
Kun > Kavg
Negative -
Kun < Kavg
Damage
Stimulation
Remarks:
In all pressure build-up test analysis , the log-log data plot ( log (Pws- Pwf)
Vs log t ) should be made . when well bore storage dominates , the plot
will have a unit slop straight line.
This helps choosing the data for the straight line semi-log plot ; use the 1-1.5
cycles in time rule of thumb or the following equation to estimate the line for
the beginning of the straight line semi-log plot:
t
>
170000 C
e0.145
(Kh / )
Actual build up tests , definition of the infinite –acting region of the test ,
(middle times region , or infinite –acting period or semi-log straight linen
region)
Pws
E
Early
Time Region
ETR
True formation
Well interference
MiddleTime Region
MT R
Skin or well storage
Log to
(tp +t)/ t
Late y Time Region
LTR
We can divide a build-up curve into three regions , as shown in the above
figure.
1- An early time region : during which a pressure transient is moving
through the formation nearest the well bore
2- A middle –time region ,: during which , the pressure transient has
moved away from the well bore into the bulk formation and
3- A late –time region : in which the radius of investigation has reached
the well,s drainage boundaries.
well storage:
-well bore storage affects short-time transient pressure behavior , also
called after f low ,after production , after injection , and well bore unloading
or loading.
if it is not considered in transient test design , analysis , wrong conclusions
may be made.
Well –bore storage constant(coefficient or factor) "C":
C= V
or  V = C  P
P
where :
 V =
change in volume of fluid in the well bore , bbls at well bore
conditions.
P
= change in bottom hole pressure , psi
C
= well –bore storage coefficient , bbl/psi
for a well –bore with changing liquid level:
iC =
Vu
(
=
g)
144 gc
Vu
( if g =gc)

144
where :
= the well –bore volume per unit length , bbl/ft
Vu

= density , g = acceleration of gravity , gc= conversion factor
ii-
for a well bore completely full of a single phase fluid:
C =
Vw .
c
where :
Vw = total well –bore volume , bbls
c = compressibility of the well bore fluid at well bore conditions psi-1
Dimensionless well bore storage coefficient (CD):
By definition:
CD
=
5.6146
C
2   ct h rw2
where :  , ct , h
are relevant to the reservoir.
Surface flow rate (q) and sand face flow rate ( qsf ) :
Well bore storage causes qsf to change more slowly than q . The following
figure shows ( qsf/q ) as a function of time when the well is opened and
surface rate changed from 0
to q
at time = zero.
Note that :
When
C
=
1.0
zero , qsf /q
= 1.0 at time = 0.0
C1
C2
C3
qsf/q
0
0
td
Effect of well bore storage on sand –face flow rate C3 > C2 > C1
The sand face flow rate may be calculated from:
Qsf
= q +
24 C
.

=
q
[
dP
dt
1-CD d
d td
(
td , CD ……)]
where:
PD is a special function which accounts for well bore storage , and is
shown in the given figure:
Notice that , by definition:
C .  P
=  V or
C
dP
dt
, the additional flow rate ,
The slope of PD
Vs
=
dV
dt
t , in hours and
q
in bbls per day.
tD on log –log graph is 1.0 , during well bore
storage domination .
The dimensional values  P and  t are proportional to PD
and tD
The location of the log-log unit slope line does not give any information about
the reservoir , but can be used to estimate the apparent well bore storage
coefficient from :
C
=
q
24
where :  t

 t
 P
,  P are values read for a point on the log-log unit slope line .
In the above figure ,
C D = tD
at PD = 1.0
The log –log plot of data is valuable , when early pressure data is available
to recognize well bore storage effect.
Make this plot is a part of the transient test analysis. It helps delineate
important periods of the analysis.
1- slope = 1.0
------------- well bore storage dominant
2- after the end of that period by 1 to 1.5 cycles in time , the standard
semi-log plot starts. Also , time for the beginning of standard semi-log plot
technique may be estimated from:
Td
>
(60+3.5 S) CD
or
t > (200,000+12000S) C
( k h/ )
approximately for drawdown and injection tests , and from:
tD > 50 CD e 0.14s
or
t > 170000 C
e
0.14S
( k h / )
for pressure build-up and fall off tests.
3- In between , type –curve matching techniques apply.
approximately
The Pressure Build-Up Procedure:
1-Estimate the probable shut –in time for accurate data . Minimum shut-in
time , ts  190  co  re2
ko
where : ts
=
ko
required shut-in time , hrs
= permeability to oil ,md
o = oil viscosity ,cp
 = porosity ,%
co = oil compressibility, psi-1
2-put the well on a particular production rate and hold until the well is
stabilized with a pressure bomb in place at the mid point of the perforation .
3-shut the well in and measure the build-up in pressure vs time
4-plot pressure versus log 10 { tp + t } , tp
is the production time of
t
the well prior to shut-in , and  t is the time since the well was shut-in.
5-determine the slope per cycle (m) from the straight line portion of the curve.
6-determine the flow capacity , k h = 162.6 q 
m
7-determine the static reservoir pressure , the skin effect , flow efficiency , the
productivity index , and damage ratio.
the flow efficiency , FE : also called " the condition ratio , productivity ratio
,PR , or completion factor.
FE = Jactual
=
P - Pwf -  Ps
J ideal
Where :
J = productivity index.
P - Pwf
=
K actual
k
(avg)
Draw Down Testing
Often the first significant event at a production well is the initial production
period that results in a pressure drawdown at the formation face.
Thus, it seems logical to investigate what can be learned about the well and
reservoir from pressure drawdown data.
Here , we will consider the drawdown test analysis for infinite acting and
pseudo-steady state periods, and we will deal only with constant rate
drawdown testing.
Drawdown testing may provide information about:
1-formation permeability
2-skin factor and
3-the reservoir volume communicating with the well.
The attached figure schematically illustrates the production and pressurehistory during a drawdown test.
Producing
Rate,q
Shut-in
0
0
time , t
Bottom
hole flowing
pressure,
Pws
Pwf
Pws
0
time , t
Ideally , the well is shut-in until it reaches static pressure of the reservoir
before the test. That requirement is met in new reservoirs , it is less met in
old reservoirs. The drawdown test is run by producing the well at a constant
flow rate wile continuously recording bottom –hole pressure.
While most reservoir information obtained from a drawdown test also can be
obtained from a pressure build-up test , there is an economic advantage to
drawdown test , that the well is produced during the test.
Advantages of D/D Test:
1-the well is producing during the test
2-the possibility for estimating reservoir volume
Disadvantages of D/D Test:
1. the difficulty of maintaining a constant production rate.
Pressure drawdown analysis in infinite –acting reservoirs:
The pressure at a well producing at a constant rate in an infinite acting
reservoir is given by:
pi - pwf = 141.2 q
[ Pd (td …….)+ S]………..(1)
k h
If the reservoir is at Pi
initially , the dimensionless pressure at the well
( rD=1.0) is:
PD = 1/2 [ ln (td) +0.80907] ……………………(2)
When
(td/rD2) > 100 and after well bore storage effects have diminished:
td =
t , hours
0.002637 k t
……….(3)
 Cf rw2
By combining equation (1) and (3) and rearranged , we get the familiar form
of the pressure drawdown equation:
Pwf = Pi – 162.6 q
[
log t + log (
k
) - 3.2275 +0.86858 S]
 Cf rw2
k h
Equation (4) describes a straight line relationship between ( P wf ) and log(t).
By grouping the intercept and slope terms , together , it may be written as:
Pwf =
m log t +
P1hr
 m
=
Pwf -P1hr
log t
Theoretically , a plot of Pwf
vs
log t , on a semi-log paper should be a
straight line with a slope , m and intercept , P 1hr as shown in the following
figure:
pwf , psi
Deviation from straight line caused by damage and
well bore storage effects
slope = m = 162.6 q
kh
P1hr
0.1
1
10
time , t ,hours
Figure (2)
Pwf,
psi
ET R
0
ETR = Early
M T R
Time
, t , hours
time region , affected by :
1- damage zone
2- well storage
LTR = Late time region , affected by:
1- No flow boundary
2- well interference
L T R
A plot of Pwf
Vs
log
t
yields a straight line portion appears after well
bore damage and storage effects have diminished.
The slope of the straight line in figure (2)
m = -162.6 q 
k h
The intercept at log t = 0
, which occurs at t = 1 hr is also determined :
p 1 hr = pi + m [ log (
k
) - 3.2275 + 0.86859 S]
 Cf rw2
Formation permeability ; k
= 162.6 q 
k h
S =1.1513 [ P1hr –Pi - log (
The skin factor,
k
) - 3.2275 ]
 Cf rw2
m
Example:
Estimate the oil permeability and skin factor from the drawdown data:
Known reservoir data are:
h =130 ft ,  = 20% , rw =
0.25 ft , Pi = 1154 psi , qo = 348 stb/D ,
m = -22 psi/cycle, o = 1.14 bbl/stb , o= 3.93 cp , P1hr = 954 psi ,
Cf = 8.74 x 10 -6 psi-1
Solution
ko = 162.6 x 348 x 3.93 x 1.14
= 89 md
(-22) x( 130)
S= 1.1513 [ (954-1154)
- log (
89
)
+ 3.2275]
0.2 X 3.93X8.74X10 -6 X(0.25)2
-22
S = 4.6
Two graphs of pressure drawdown data are required:
1. log-log data plot [ log (Pi-Pwf) Vs log (t) ] to estimate when well bore
storage effects are no longer important.
Well bore storage
C = q 
p
24
t
Log  P
slope /cycle
log t ,hrs
We can estimate when the semi-log straight line begin:
t >
(200,000 + 12000 S)C
( k h / )
Reservoir limit test:
A drawdown test run specifically to determine the reservoir volume
communicating with the well is called a reservoir limit test. The dimensionless
pressure during pseudo-steady state
flow is a linear function of
dimensionless time. The following equation is obtained:
Pwf =
m* t
+ Pint…………………..(5)
where :
m*
=
-0.23395 q 
……………..(6)
 Cf h A
Flowing
m
Pressure
Pwf,psi
P1hr
2060
2040
2020
Flow time , hours
2000 0.1
1
10
102
103
Pint = Pi - 70.6 q  [ ln( A ) +ln ( 2.2458 ) + 2S] …..(7)
and
rw2
k h
Equation (5) Pwf =
m* t
CA
+ Pint
of bottom hole flowing pressure (Pwf)
indicates
that a Cartesian plot
Vs time (t) should be a straight line
during pseudo-steady state flow , with slope m* given by equation (6).
m* = -0.23395 q 
, and intercept Pint given by equation (7).
 Ct h A
Pwf
m*
Pint
0
5
10
15
20
Flowing
25
30
35
40
time ,hours
Note that :
cf = dv . 1
Vp
dp
q = dv /dt
cf =
q
dt o
Vp . dp
Vp
=
q o
cf (dp/dt)
The slope may be used to estimate the connected reservoir drainage volume:
 h A = - 0.23395 q 
where the volume in cubic feet.
ct x m*
If ( h) is known , the drainage area may be estimated. Another technique has
been proposed for analyzing pseudo-steady state data , but this one appears to
be the simplest . If pressure data are available during both the infinite acting
period
, and PSS period , it is possible to estimate the drainage shape for
the test well. The semi-log plot is used to estimate (m)
and (P 1hr) ; the
Cartesian plot is used to get (m*) and (Pint).
The system shape factor is estimated from:
CA = 5.456 m
exp [ 2.303 (P1hr-Pint) ]
m*
m
By knowing the shape factor , use table (C-1) , to determine the reservoir
configuration with the shape factor closest to that calculated. This process may
by refined by computing ( tda)pss = 0.1833 m*
tpss
m
and using the exact value for (tDA ) , from the table (C-1) , the time tpss is when
the Cartesian straight line starts.
Drill Stem Testing (DST)
Different methods are used for evaluating formation productivity includes
core analysis , well logging and drill stem testing. Despite the tremendous
value of core analysis and logging , some shadow of doubt always remains
concerning the potential productivity of an exploratory well , and this doubt is
not dispelled until a sizable sample of oil has been delivered to the surface.
This drawback is not inherent in drill stem testing. The decision to run a drill
stem test on a zone is often based on shows of oil in the cuttings which , in
the opinion of the geologist or engineer in charge , deserve detailed
investigation .
This may happen many times in the course of drilling a wildcat , with as many
as 20 or 30 tests being conducted on a single well. Although the cost of such
detailed testing is quite high , it is much better to test and be sure , rather than
miss a productive zone.
General procedure:
A drill stem test is a temporary completion whereby the desired section of the
open hole is isolated , relieved of the mud column pressure , and allowed to
produce through the drill pipe (drill stem).
The basic test tool assembly consists of:
1- a rubber packing element or packer which can be expanded against
the hole to segregate the annular sections above and below the
element.
2- A tester valve to control flow into the drill pipe , that is to exclude mud
during entry into the hole and to allow formation fluids to enter during
the test , and
3- By –pass valve to allow pressure equalization across the packer after
completion of the flow test.
4- Anchor : This is merely the extension below the tool which supports
the weight applied to set the packer.
5- Pressure recorders: These furnish a complete record of all events
which may occur during a particular test. This record is in the form of a
graph of pressure versus time. Two pressure recorders are usually
desirable and should be located so that one will measure the pressure
inside, and the other the pressure outside the anchor. These two
measurements allow accurate determination of weather or not the
perforations have become plugged during the test.
6- Safety Joints: These merely afford a means of unscrewing the drill
string at a point convenient for fishing operations , should the packers
become stuck.
Figure (1) illustrates the procedure for testing the bottom section of a hole.
While going in the hole, the
packer collapsed , allowing the displace mud
to rise as shown by the arrows. After the pipe reaches bottom and the
necessary surface preparations have made , the packer is set
(compressed and expanded) ; this isolates the lower zone from the rest of
the open hole. The compressive load is furnished by a slacking off the
desired amount of drill string weight which is transferred to the anchor pipe
below the packer. The tester valve is then opened and thus the isolated
section is exposed to the low pressure inside the empty or nearly empty
drill pipe .
Formation fluids can then inter the pipe as shown in the
second picture. At the end of the test , the tester valve is closed , trapping
any fluid above it and the by-pass valve is opened to equalize the pressure
across the packer.
Finally the setting weight is taken off and the packer is pulled free. The
pipe is then pulled from the hole until the fluid containing section reaches
surface. As each successive stand is then broken (unscrewed) , its fluid
content may be examined. Frequently such stand –by-stand sampling is
neither necessary nor desirable , and the test is reversed (circulated
opposite to the normal direction ) as shown. This reversal is performed by
closing the blow out preventers and pumping mud down the annulus, the
mud then enters the drill pipe through the reversing ports, thereby
displacing any formations fluid in the pipe. The recovered fluid samples as
they are discharged at the surface.
Although the above is a very common type of test , there are many other
variations of procedure as indicated in Figure (2). These are:
1-Straddle Packer Test:
is necessary when isolation from formations both above and below the test
zone is necessary. Such a situation commonly arises when evidence
(electric logs, radioactivity logs, detailed sample analysis etc.) indicates
that a zone previously passed by has productive possibilities. Straddle
testing is less desirable than conventional testing , from both a cost and an
operational hazard stand point. Two packers are more apt to become
stuck than one , since any material which sloughs or caves from the test
zone may accumulate between the packers.
2-The Cone Packer or Rat- Hole Method:
Is used when the test section is smaller in diameter than the hole above.
This situation commonly occurs as a consequence of coring operations in
which the corehead used was smaller than the regular bits. The coneshaped packer is compressed against the shoulder , forming the necessary
seal.
3-Wall Over Cone Packer Test:
if the formation opposite the shoulder is soft, an additional conventional
wall packer placed above the cone packer may be necessary to provide the
desired seal.
7- Testing Through Perforations in the Casing:
Zones behind casing may be tested through perforations by the same
basic procedures except that the packer used has slips which engage or
grab the casing wall. These are commonly called hook-wall packers. Since
the slips support the compressive load used to expand the packer element
, no anchor pipe is required, and the packer may be reset several times at
different depths if necessary. Testing inside casing is widely used in soft
rock areas where open hole testing is particularly hazardous.
Factors affecting drill stem testing:
1-Condition of the hole:
the close tolerance between the hole and the tool assembly requires a full
gage, clean , well bore if the tool is to reach bottom in an undamaged ,
unplugged condition. The cake and cavings shoved ahead of the packer
may plug the perforations and/or choke when the valve is opened. It is
common practice to circulate for some time prior to testing, so that all
cuttings are removed from the hole. The drilling mud should be
conditioned to the desired density and viscosity before the test is started.
2-Pressure surges:
the drill stem test conditions represent a severe case of pressure surge,
because the lower end of the pipe is closed, necessitating displacement of
the total drill pipe volume. Special consideration should be givn to pipe
running and pulling speeds to avoid undue bottom hole pressure
variations.
3-operating conditions:
(a) The length and location of test section govern the amount of
tail pipe required and the choice of a conventional or straddle test. The
testing of short sections is more conclusive , and is generally preferable.
(b) The packer seat location , while of no particular importance for
tests run inside casing, is critical for a successful open hole test. The seat
should be placed in a true gage section of the hole opposite as dense and
consolidated formation as possible. As a general rule, electric or caliper
logs are not taken prior to open hole testing and the main guides to packer
seat location are the drilling time and sample logs.
(c)Size and number of packers: the pressure differential which a
packer can stand depends on the amount it must expand to furnish the
desired seal. The service companies which supply these tools have a
standard range of sizes for various hole diameters. The ratio of hole
diameter to unexpanded packer diameter is kept as low as possible, and
commonly ranges from 1.1 to 1.2. In deep , open hole tests , two closely
spaced packers are often run as a precautionary measure.
This often
eliminates test failure due to by-passing in fractured zones; also, if one
packer fails , the other may carry the load. The cost of the second packing
element is quite nominal compared to the cost of misrun.
(d)Choke sizes: the size of the bottom hole and surface orifices
selected depends on the anticipated test conditions. The bottom choke is
of primary importance and is used to govern the flow rate. The top choke
is used primarily as a safety measure and should be considerably larger
than the bottom choke in order to minimize surface pressure in case a
flowing test is obtained.
(e) Use of cushions: this refers to the practice of placing a certain
length or head of liquid inside the drill pipe , rather than running it dry. This
is commonly done for two reasons:
1-to reduce the collapse (external )pressure on the drill pipe in deep holes
and/or
2- to reduce the pressure drop on the formation and across the packer (s)
when the tool is first opened.
(f) length of test: this is difficult if not impossible to predict until after the
test has commenced and some observations are available.
While this is a safe and necessary practice in many cases, it should not
be carried to the extreme that back pressure becomes large enough to
prevent formation fluid flow.
The Material Balance Equation as a Straight Line
(The Havlena and Odeh Method )
Normally , when using the material balance equation, an engineer
considers each pressure and the corresponding production data as
being separate points from other pressure values. From each separate
point, a calculation for a dependent variable is made . the results of
the calculations are sometimes averaged. The Havlena –Odeh method
uses all the data points , with the further requirement that these points
must yield solutions to the material balance equation that behave
linearly to obtain values of the independent variable.
The straight-line method begins with the material balance equation
written as:
Np [Bt+ (Rp-Rsi)Bg ] + Wp Bw – Wi – GiBgi =
N
(Bt – Bti ) + (1+m) Bti { cw Sw + cf } P+ mBti (Bg-Bgi)
1-Swi
+ We
Bgi
Havlena –Odeh defined the following terms and rewrote the equation
as follows:
F = Np [Bt+ (Rp-Rsi)Bg ] + Wp Bw – Wi – Gi Bgi
Eo = Bt - Bti
Ef,w =
{ cw Swi + cf }
P
1-Swi
Eg = Bg –Bgi
And finally :
F
=
N Eo +N (1+m) Bti Ef,w + { N m Bti }
Eg + W e
-----(1)
Bgi
In the last equation , F represents the net production from the reservoir.
Eo , Ef,w , and Eg represent the expansion of oil, formation and water
and gas respectively.
Havlena and Odeh examined several cases of varying reservoir types
with this equation and found that the equation can be rearranged into
the form of a straight line. For instance, consider the case of no original
gas cap , no water influx , and negligible formation and water
compressibilities. The equation can be reduced to
F=N Eo
----------------(2)
This would suggest that a plot of F as the
the X
Y coordinate and Eo as
coordinate would yield a straight line with slope
N and
intercept equal to zero.
For the case of a saturated reservoir with an initial gas cap and
neglecting the compressibility term, Ef,w, equation (1) becomes
F
=
NEo
+ mNBti
Eg
+
We
Bgi
If N is factored out of the first two terms on the right –hand side and
both sides of the equation are divided by the expression remaining
after factoring, we get :
F
Eo+
m Bti
=
Eg
N
+
We
Eo + m Bti Eg
Bgi
Bgi
F/[Eo+(mBti/Bgi) Eg]
MM STB
Slope =1.0
N
We/[Eo+(mBti/Bgi) Eg] , MMSTB