Transient Flow

Transient flow is defined as a flow condition under which the radius of pressure wave
propagation from the wellbore has not reached the boundaries of the reservoir.

From: Well Productivity Handbook (Second Edition), 2019

Related terms:

Drawdown, Oil Production, Porosity, Equilibrium Flow, Compressibility, Unsteady

State, Gas Reservoir

View all Topics

Fundamentals of Drawdown Test Analy-

sis Methods
Amanat U. Chaudhry, in Gas Well Testing Handbook, 2003

Drawdown Test Analysis Using Pseudopressure Approach

Transient flow at constant rate from an infinite-acting reservoir in terms of pseudo-
pressure (pwf) is modeled by Eq. 5-39:

(5-39)

A plot of Δ [= (pi) − (pwf)] versus t on semilog coordinates should give a straight

line of slope m, from which

(5-40)

The apparent skin factor s can then be calculated using Eq. 5-41:

(5-41)

where Δ 1 is the value of Δ at t = 1. This value must be obtained from the

straight-line portion of the plot (extrapolated, if necessary). The pressure drop due
to skin effects may be obtained from

(5-42)
Similarly, the pressure drop due to IT flow effects may be obtained from

(5-42a)

The total pressure drop directly attributed to skin and IT flow effects may then be
obtained from

(5-43)

The well flow efficiency, FE, may be calculated from

(5-44)

Sometimes it is convenient to express the drawdown in dimensionless forms. This

is easily done as follows:

(5-45)

The analysis of a single-rate test in terms of pwf, , and (pwf) is illustrated by the
following Example 5-2.

> Read full chapter

Reservoir deliverability
Boyun Guo, in Well Productivity Handbook (Second Edition), 2019

3.2.1 Transient flow

Transient flow is defined as a flow condition under which the radius of pressure wave
propagation from the wellbore has not reached the boundaries of the reservoir. Dur-
ing transient flow the developing pressure funnel is small, relative to the reservoir
size. Therefore, the transient pressure behaves as if the reservoir were infinitely large.

Assuming single-phase oil flow in the reservoir, several analytical solutions have
been developed for describing transient flow behavior. These are available from
classic textbooks, such as Dake (1978). A constant-rate solution expressed by Eq. (3.1)
is frequently used in reservoir engineering:

(3.1)

where

pwf = flowing bottom-hole pressure (psia),

pi = initial reservoir pressure (psia),
q = oil production rate (stb/day),
μo = viscosity of oil (cp),
k = effective horizontal permeability to oil (md),
h = reservoir thickness (ft),
t = flow time (hours),
 = porosity (fractional),
ct = total compressibility (psi−1),
rw = wellbore radius to the sand face (ft),
S = skin factor, and
Log = 10-based logarithm (log10).

The fixed choke size used in typical production oil wells results in constant wellhead
pressure, which, in turn, results in constant bottom-hole pressure. A constant-bot-
tom-hole pressure solution is therefore more desirable for well inflow performance
analysis. Using an appropriate inner boundary condition arrangement, Earlougher
(1977) developed a constant-bottom-hole pressure solution, which is similar to Eq.
(3.1), expressed as

(3.2)

which is used for transient well performance analysis in reservoir engineering.

Eq. (3.2) indicates that at a constant-bottom-hole pressure, the oil production rate
decreases with flow time. This is because the radius of the pressure funnel (over
which the pressure drawdown (pi − pwf) acts) increases with time. In other words,
the overall pressure gradient in the drainage area decreases with time.

For gas wells, the transient solution is expressed as

(3.3)

where qg is the production rate in Mscf/d, k is the effective permeability to gas in

md, T is the temperature in °R, μg is the gas viscosity in cp, and m (p) is the real-gas
pseudo-pressure, defined as

(3.4)

where pb is the base pressure, usually taken as 14.7 psia. The real-gas pseudo-pres-
sure can be readily determined using spreadsheet program PseudoPressure.xls.

> Read full chapter

Reservoir Deliverability
Boyun Guo PhD, ... Xuehao Tan PhD, in Petroleum Production Engineering (Second
Edition), 2017

3.2.1 Transient Flow

“Transient flow” is defined as a flow regime where/when the radius of pressure wave
propagation from wellbore has not reached any boundaries of the reservoir. During
transient flow, the developing pressure funnel is small relative to the reservoir
size. Therefore, the reservoir acts like an infinitively large reservoir from transient
pressure analysis point of view.

Assuming single-phase oil flow in the reservoir, several analytical solutions have
been developed for describing the transient flow behavior. They are available from
classic textbooks such as that of Dake (1978). A constant rate solution expressed by
Eq. (3.1) is frequently used in production engineering:

(3.1)

where pwf=flowing bottom-hole pressure, psia; Pi=initial reservoir pressure, psia;

q=oil production rate, stb/day; μo=viscosity of oil, cp; k=effective horizontal perme-
ability to oil, md; h=reservoir thickness, ft; t=flow time, hour; =porosity, fraction;
ct=total compressibility, psi−1; rw=wellbore radius to the sand face, ft; S=skin factor;
Log=10-based logarithm log10

Because oil production wells are normally operated at constant bottom-hole pres-
sure because of constant wellhead pressure imposed by constant choke size, a con-
stant bottom-hole pressure solution is more desirable for well-inflow performance
analysis. With an appropriate inner boundary condition arrangement, Earlougher
(1977) developed a constant bottom-hole pressure solution, which is similar to Eq.
(3.1):

(3.2)

which is used for transient well performance analysis in production engineering.

Eq. (3.2) indicates that oil rate decreases with flow time. This is because the radius of
the pressure funnel, over which the pressure drawdown (pi–pwf) acts, increases with
time, that is, the overall pressure gradient in the reservoir drops with time.

For gas wells, the transient solution is

(3.3)

where qg is production rate in Mscf/d, T is temperature in °R, and m(p) is real gas
pseudopressure defined as
(3.4)

The real gas pseudo-pressure can be readily determined with the spreadsheet
program PseudoPressure.xls.

> Read full chapter

Analysis of Decline and Type Curves

Tarek Ahmed, in Reservoir Engineering Handbook (Fifth Edition), 2019

Solution
During transient flow, Equation 6-78 is designed to describe the pressure at any
radius r and any time t, as given by

or

Values of “pi – p(r,t)” are presented as a function of time and radius (i.e., at r = 10
feet and 100 feet) in the following table and graphically in Figure 16-14.

Figure 16-14. Pressure profile at 10 feet and 100 feet as a function of time.

Assumed t, r = 10 feet r = 100 feet

hours
t/r2 Ei[–0.0001418pr2i-p(r,t)
/t] t/r2 Ei[–0.0001418pri2-p(r,t)
/t]
0.1 0.001 –1.51 157 0.00001 0.00 0
0.5 0.005 –3.02 314 0.00005 –0.19 2
1.0 0.010 –3.69 384 0.00010 –0.12 12
2.0 0.020 –4.38 455 0.00020 –0.37 38
5.0 0.050 –5.29 550 0.00050 –0.95 99
10.0 0.100 –5.98 622 0.00100 –1.51 157
20.0 0.200 –6.67 694 0.00200 –2.14 223
50.0 0.500 –7.60 790 0.00500 –3.02 314
100.0 1.000 –8.29 862 0.00100 –3.69 386

Figure 16-14 shows different curves for the two radii. Obviously, the same calcula-
tions can be repeated for any number of radii and, consequently, the same number
of curves will be generated. However, the solution can be greatly simplified by
examining Figure 16-15. This plot shows that when the pressure difference pi –
p(r,t) is plotted versus t/r2, the data for both radii form a common curve. In fact,
the pressure difference for any reservoir radius will plot on this exact same curve.

Figure 16-15. Pressure profile at 10 feet and 100 feet as a function of t/r2.

For example, in the same reservoir, to calculate the pressure p at 150 feet after 200
hours of transient flow:

From Figure 16-15:

Thus,

Several investigators have employed the dimensionless-variables approach to deter-

mine reserves and to describe the recovery performance of hydrocarbon systems
with time, notably the following:

○ Fetkovich (1980)

○ Carter (1985)
○ Palacio and Blasingame (1993)

○ Mattar and Anderson’s Flowing Material Balance (2003)

○ Anash et al. (2000)

○ Decline-Curve Analysis for Fractured

All the methods are based on defining a set of decline-curve dimensionless variables
that includes:

○ Decline-curve dimensionless rate, qDd

○ Decline-curve dimensionless cumulative production, QDd

○ Decline-curve dimensionless time, tDd

The aforementioned methods were developed with the objective of providing the
engineer with an additional convenient tool for estimating reserves and determining
other reservoir properties for oil and gas wells using the available performance data.
A review of these methods and their practical applications is given next.

> Read full chapter

Analysis of Decline and Type Curves

Tarek Ahmed, in Reservoir Engineering Handbook (Fourth Edition), 2010

Solution
During transient flow, Equation 6-78 is designed to describe the pressure at any
radius r and any time t, as given by

or

Valuesof “pi − p(r,t)” are presented as a function of time and radius (i.e., at r = 10 feet
and 100 feet) in the following table and graphically in Figure 16-14.
Figure 16-14. Pressure profile at 10 feet and 100 feet as a function of time.

Assumed t, r = 10 feet r = 100 feet

hours
t/r2 Ei[− pi − p(r, t) t/r2 Ei[− pi− p(r, t)
0.0001418 0.0001418
r2/t] r2/t]
0.1 0.001 − 1.51 157 0.00001 0.00 0
0.5 0.005 − 3.02 314 0.00005 − 0.19 2
1.0 0.010 − 3.69 384 0.00010 − 0.12 12
2.0 0.020 − 4.38 455 0.00020 − 0.37 38
5.0 0.050 − 5.29 550 0.00050 − 0.95 99
10.0 0.100 − 5.98 622 0.00100 − 1.51 157
20.0 0.200 − 6.67 694 0.00200 − 2.14 223
50.0 0.500 − 7.60 790 0.00500 − 3.02 314
100.0 1.000 − 8.29 862 0.00100 − 3.69 386

Figure 16-14 shows different curves for the two radii. Obviously, the same calcula-
tions can be repeated for any number of radii and, consequently, the same number
of curves will be generated. However, the solution can be greatly simplified by
examining Figure 16-15. This plot shows that when the pressure difference pi −
p(r,t) is plotted versus t/r2, the data for both radii form a common curve. In fact,
the pressure difference for any reservoir radius will plot on this exact same curve.
Figure 16-15. Pressure profile at 10 feet and 100 feet as a function of t/r2.

For example, in the same reservoir, to calculate the pressure p at 150 feet after 200
hours of transient flow:

t/r2 = 200/1502 = 0.0089

From Figure 16-15:

pi − p(r,t) = 370 psi

Thus,

p(r,t) = pi − 370 = 5000 − 370 = 4630 psi

Several investigators have employed the dimensionless-variables approach to deter-

mine reserves and to describe the recovery performance of hydrocarbon systems
with time, notably the following:

• Fetkovich (1980)

• Carter (1985)

• Palacio and Blasingame (1993)

• Mattar and Anderson's Flowing Material Balance (2003)

• Anash et al. (2000)

• Decline-curve analysis for fractured reservoirs

All the methods are based on defining a set of decline-curve dimensionless variables
that includes:

• Decline-curve dimensionless rate, qDd

• Decline-curve dimensionless cumulative production, QDd

• Decline-curve dimensionless time, tDd

The aforementioned methods were developed with the objective of providing the
engineer with an additional convenient tool for estimating reserves and determining
other reservoir properties for oil and gas wells using the available performance data.
A review of these methods and their practical applications is given next.

> Read full chapter

Forecast of Well Production

Boyun Guo PhD, ... Xuehao Tan PhD, in Petroleum Production Engineering (Second
Edition), 2017

7.3.1 Oil Production During Single-Phase Flow Period

Following a transient flow period and a transition time, oil reservoirs continue to
deliver oil through single-phase flow under a pseudo-steady-state flow condition.
The IPR changes with time because of the decline in reservoir pressure, while the
TPR may be considered constant because fluid properties do not significantly
vary above the bubble-point pressure. The TPR model can be chosen from simple
ones such as Poettmann–Carpenter and sophisticated ones such as the modified
Hagedorn–Brown. The IPR model for vertical wells is given by Eq. (3.9), in Chapter
3, Reservoir Deliverability that is,

(7.2)

The driving mechanism above the bubble-point pressure is essentially the oil expan-
sion because oil is slightly compressible. The isothermal compressibility is defined
as

(7.3)

where V is the volume of reservoir fluid and p is pressure. The isothermal compress-
ibility c is small and essentially constant for a given oil reservoir. The value of c can
be measured experimentally. By separating variables, integration of Eq. (7.3) from
the initial reservoir pressure pi to the current average-reservoir pressure results in

(7.4)
where Vi is the reservoir volume occupied by the reservoir fluid. The fluid volume V
at lower pressure includes the volume of fluid that remains in the reservoir (still Vi)
and the volume of fluid that has been produced, that is,

(7.5)

Substituting Eq. (7.5) into Eq. (7.4) and rearranging the latter gives

(7.6)

where r is the recovery ratio. If the original oil in place (OOIP) N is known, the
cumulative recovery (cumulative production) is simply expressed as Np=rN.

For the case of an undersaturated oil reservoir, formation water and rock also expand
as reservoir pressure drops. Therefore, the compressibility c should be the total
compressibility ct, that is,

(7.7)

where co, cw, and cf are the compressibilities of oil, water, and rock, respectively, and
So and Sw are oil and water saturations, respectively.

The following procedure is taken to perform the production forecast during the
single-phase flow period:

1. Assume a series of average-reservoir pressure values between the initial

reservoir pressure pi and oil bubble-point pressure pb. Perform Nodal analyses
to estimate production rate q at each average-reservoir pressure and obtain
the average production rate over the pressure interval.
2. Calculate recovery ratio r, cumulative production Np at each average-reservoir
pressure, and the incremental cumulative production ΔNp within each aver-
age-reservoir pressure interval.
3. Calculate production time Δt for each average-reservoir pressure interval by
and the cumulative production time by t=Σ Δt.

Example Problem 7.2 Suppose the reservoir described in Example Problem 7.1
begins to produce oil under pseudo-steady-state flow conditions immediately after
the 6-month transient flow. If the bubble-point pressure is 4500 psia, predict the
oil production rate and cumulative oil production over the time interval before the
reservoir pressure declines to bubble-point pressure.

Solution Based on the transient flow IPR, Eq. (7.1), the productivity index will
drop to 0.2195 stb/d-psi and production rate will drop to 583 stb/day at the end
of the 6 months. If a pseudo-steady-state flow condition assumes immediately
after the 6-month transient flow, the same production rate should be given by the
pseudo-steady-state flow IPR, Eq. (7.2). The average pressure within the production
funnel is given by (Dake, 1978), or

Substituting this and other values of parameters into Eq. (7.2) gives a drainage area
of 1458 acres. Assuming an initial water saturation of 0.35, the OOIP in the drainage
area is estimated to be 87,656,581 stb.

Using these additional data, Nodal analyses were performed with spreadsheet pro-
gram Pseudo-Steady-1Phase ProductionForecast.xls at 10 average-reservoir pressures
from 5426 to bubble-point pressure of 4500 psia. Operating points are shown in
Fig. 7.3. The production forecast result is shown in Table 7.2. The production rate
and cumulative production data in Table 7.2 are plotted in Fig. 7.4.

Figure 7.3. Nodal analysis plot for Example Problem 7.2.

Table 7.2. Production Forecast for Example Problem 7.2

Reservoir Production Recovery Cumulative Incremen- Incremen- Pseudo–Steady-State Production Time (d

Pressure Rate Ratio Production tal tal
(psia) (stb/day) (stb) Production Production
(stb) Time (days)
5426 583 0.0010 84,366 0
5300 563 0.0026 228,204 143,837 251 251
5200 543 0.0039 342,528 114,325 207 458
5100 523 0.0052 457,001 114,473 215 673
5000 503 0.0065 571,624 114,622 223 896
4900 483 0.0078 686,395 114,771 233 1,129
4800 463 0.0091 801,315 114,921 243 1,372
4700 443 0.0105 916,385 115,070 254 1,626
4600 423 0.0118 1,031,605 115,220 266 1,892
4500 403 0.0131 1,146,975 115,370 279 2,171
Figure 7.4. Production forecast for Example Problem 7.2.

> Read full chapter

Application of Decline Curve Analysis

Methods
Amanat U. Chaudhry, in Gas Well Testing Handbook, 2003

Characteristics of Exponent b during Transient Flow

If rules during transient flow are used to compute the value of exponent b, then such
measurements may suggest that the value of b is greater than unity and is given by

In most cases the exponent b would be greater than unity, if transient responses
were used to predict performance. Behavior in rate will follow the b = 0 curve only
for the case ¯t /c¯t = constant as long as c¯t/ ¯t is a linear function of time.

> Read full chapter

Well Testing Terminology in Multilay-

ered Reservoir Systems
Amanat U. Chaudhry, in Gas Well Testing Handbook, 2003

Constant Producing Rate

Pressure performance for transient flow period is given by
 (13-19)

(13-20)

For larger times (semisteady state), the pressure behavior is described by

(13-21)

(13-22)

where (kh)t = k1h1 +k2h2; ( h)t = ( 1h1) + ( 2h2); ht = h1 + h2; t = time in hours; and gi

= 0.00504zi Tpsc/pi Tsc inrb/mmscfd.

The time at which the semisteady state starts is given by

(13-23)

For semisteady-state flow, the slope of the plot of flowing bottom-hole pressure
versus time is given by

(13-24)

Figure 13-11 shows an idealized constant-rate flowing bottom-hole pressure perfor-

mance curve, and reservoirs of the type should possess the properties shown on this
plot.

Figure 13-11. Idealized constant-rate pressure performance in two-layer reservoir

with crossflow.3(after Russell and Prats)

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