Condition2 : Build-up duration should be smaller than the duration of the last production period before shut-in.
Other than the above condition, it would be incorrect to use for Build-up interpretation without incorporating certain changes. The effect of short production time can be seen in a flattening out of the type curve, the build-up curve under the drawdown curve. Force match between the build-up data and a draw down curve would result in a type curve located too high on the set of curves and therefore in inaccurate results.
The most useful method of using drawdown type curves for build-up is Agrawal’s method.
It consists of plotting each measurement versus an equivalent time ∆te as defined below instead of ∆t.
p e
t t t t
+∆
= ∆
∆ 1
There is a condition to be satisfied before which this equivalent time can be used in Gringarten Type curve or any Type curve. The condition is; the semi-log straight line should have reached during the previous drawdown before build-up.
Advantages of Type Curves lie in the fact that it allows the interpreter to make a diagnosis about the type of reservoir and understand the flow regimes. It also allows the interpreter to use the flow concept regime in a conventional interpretation method with ease and confidence. However, assumption of constant well bore storage effect in a Type curve puts severe limitation to the interpretation.
Conventional Method of Well Test Interpretation
This particular section will study the response of flow/pressure behaviour at constant rate (drawdown) or when rate is zero (Build up).
Drawdown test
The solution of diffusivity equation in the transient pressure regime is given as;
Type curve from Type curve set
Type curve calculated for a shut-in well
− = + − + S
Hence, if pressure measured at the bottom of well bore is plotted against log of time, it would result in a straight line with slope, m
This slope m can be used for calculating flow capacity, kh of a reservoir and skin. P1hr is the pressure at 1hr from the start of drawdown test, read from the straight line equation.
Transient state is of short duration. If the test is extended and the compressible zone is allowed to travel and reach the boundary of the reservoir, the flow regime changes to pseudo steady state regime in absence of any support from the outer boundary. The solution of pressure response in a drawdown test in a pseudo-steady state is given as;
+
The above equation suggests that a plot of pressure against time in the pseudo-steady state region would result in a straight line whose slope is given as;
φ
hA is nothing but pore volume of the reservoir. If this is represented as Vp, thenthat RLT is valid for only for pseudo-steady state only and not for steady state. There are different ways to calculate the pore volume in a steady state condition.
Build-up Test: Horner’s Method
Most of the information from a well test comes from the interpretation of a pressure build-up. The reason is the fluctuation in the production rate which is inherent to the production. Fluctuation may cause large variation in bottom hole pressure during draw down test. This is not the case in a build-up test. In a build up test, the well is allowed to flow at sufficiently large time to allow the flowing pressure almost constant.
Subsequently, the well is closed and the continuous recording of bottom hole shut in pressure is done till the surface tubing shut-in pressure stabilizes.
The equation and analysis method was given by Horner.
The following expression for pressure is given ;
162.6 log( )
The value of pressure measured at the bottom is plotted versus the logarithm of t
on a graph, once the wellbore storage effect has ended a straight line with a slope of m can be observed
kh m=162.6qB
µ
This helps to know the flow capacity (kh) of the well. The thickness h is called the effective thickness and is obtained by subtracting the noncontributing length from gross thickness of the formation encountered in the well. Skin is determined from the following expression;
log of t
t Tp
∆
∆
+ is considered negligible while determining skin through Horner’s method
and p1Hr, must be calculated from the Horner straight line at ∆t 1= hr Extrapolated Pressure
If the slope of the Horner’s straight line is extrapolated at t
t Tp
∆
∆
+ =1 (i.e. when∆t ∞),
the value of the pressure read, is called initial reservoir pressure (P*) in most initial tests, where amount of fluid produced before shut-in is usually negligible compared with the amount in place. The idea is that if the build-up would have been continued for infinitely long time, the pressure would have stabilized to initial reservoir pressure. However, when substantial amount of oil has been produced, the value of P* is not the reservoir pressure existing at that point of time, rather this value is used to calculate the average reservoir pressure. There are conditions, when the value of P* is found to be less than average reservoir pressure −P−! So reservoir engineers should use this value with great caution and understanding.
Miller Dyes and Hutchinson Method of Build-Up Horner showed that build-up varies linearly with log(
t t Tp
∆
∆
+ ). When Tp >>> ∆t, the
term Tp +∆tcan be approximated as TP. Physically it means that the pressure drop due to previous production is neglected.
Hence the Horner’s equation becomes 162.6 (ln ln )
p ws
i t T
kh p qB
p − =−
µ
∆ − This equation was proposed by Miller, Dyes and Hutchinson and the particular method of build up is called MDH method.Pressure Shapes and Interpretation Methods in Various Characteristic Boundaries When compressible zone created by the perturbation reaches reservoir boundary, it is perceived as a characteristic response in the pressure at the well. This nature of the response in the well bore pressure depends upon the characteristics of the boundary.
Few of the characteristic responses observed in different types of boundaries are explained below;
Linear Sealing Fault
The boundary condition corresponding to linear fault is the linear no-flow boundary.
Linear sealing fault and disappearing facies, unconformities are few of the examples of the characteristic boundaries. In such type of situation two different straight line segments are seen with slopes having approximate ratio of 2:1 in the semi-log straight line. The flow capacity and the skin should be calculated on the basis of first line.
However, P* should be calculated based on the second straight line in case of only one fault. Flow capacity in both the drawdown and the build-up should be calculated based on the following data;
m kh=162.6qB
µ
Skin in drawdown
time Tp
Pws
∆ t
∆ p
∆pMDH=1.151 − −log 2 +3.23
Gray suggested that the distance to the fault or barrier can be approximated using the following equation.
D = Distance to the barrier or fault, ft K = Formation permeability, mD φ = Porosity, fraction
= Fluid viscosity, cP ct =Total compressibility, psi-1
=
∆tt End of first straight line segment, hr
If the two barrier/faults are approximately the same distance, the characteristic doubling of slope will not be seen in the plot. In such case after the initial straight line is seen, the slope of the second line would increase to more than two times. In such case the second line suggests presence of more than one fault.
In a type curve the derivative of the slopes goes up from 0.5 to 1.
Pressure Build-up Data from a Well Producing from a Long Narrow Reservoir Such as Channel Sand
The pressure transient data collected from a well producing from a long narrow reservoir as shown below have characteristics that show combination of radial flow and linear flow.
During radial flow the pressure varies as logarithm of time. In a linear flow the pressure varies linearly with square root of time. The channel can be due to number cases such as;
1. Two parallel sealing faults.
2. a sedimentary deposit channels.
3. two parallel lateral variations in facies. etc.
w d
The channel is defined by its width w and by the distance, d, from the well to one of its edges.
During a well test inside a channel, following characteristics in the pressure patterns are observed;
• A semi-log straight line with stabilization of derivative at 0.5 is observed.
• As the compressible zone reaches the first edge of the channel, fault effect is seen. The boundary has exactly the same effect as sealing fault in an infinite reservoir. The slope of the line doubles. This is observed only when the well is very off centered in the channel.
• When the compressible zone reaches the two edges of the channel, it expands linearly parallel to the edges of the channel. The pressure varies linearly with square root time. Plot of pressure vs. t shows a straight line suggesting of a channel.
A plot of P vs. t+∆t − ∆t should be made in case of build-up. If the late time data becomes a straight line on this plot it along with doubling of slope in radial flow indicates channel reservoir. P* is determined from the linear plot by extrapolating t+∆t − ∆t to 0.
Linear flow is used to determine the width of the channel and the eccentricity of the well
The width of linear channel can be calculated by
ct
h qBm w m
φ
12
638 .
=0 ft :for a oil well
m1 = slope of ( pws vs.
∆
∆ +
t t tp
log ) psi/cycle
m2 = slope of (pws vs. t+∆t − ∆t ) q = oil flow rate, STB/D
B = oil formation volume factor, bbl/STB H = net pay thickness, ft
φ = porosity, fraction
Ct = total compressibility, psi-1
Pressure Build-up Data from a Hydraulically Fractured Well
Natural fractures are distributed homogeneously in the reservoir. Artificially fractures are, however, located in the vicinity of the well bore. They are created by the operations carried out on the well. They are an effective technique for increasing the productivity of damaged wells or wells producing from low-flow-capacity formation.
Fractures can be created both in vertical and horizontal direction. At depths of less than 1000m it is possible to create horizontal fractures. However, at great depths, the overburden weight makes the fractures develop only along vertical planes.
Flow around an Artificially Fractured Well
The presence of an artificial fracture modifies the flows near the well bore considerable.
However because of the short distance extension of the fracture, these fractures have finite conductivities, unlike natural fractures which have infinite conductivities.
In an artificially fractured well, initially, there is a fracture linear flow. This period is quite short and is normally dominated by wellbore storage. Flow from the reservoir causes the matrix to contribute to the flow of fluid to the fracture. This period is featured by linear flows in both fractures and the formation and the fracture tip still has not affected the flow behaviour of the well. These bilinear flow regimes are experienced only by fractures of finite conductivity. Bilinear flow is followed by linear flow. During the start of this flow period, the flow behaviour starts getting affected by the fracture tip. There is a linear flow from matrix to the fracture. This flow is very often seen during testing of artificially fractured wells. Finally, at, long times the pseudo-radial flow is reached by all fractured systems regardless of the fracture conductivity or damage. The system developed for
the radial homogeneous system is equally applicable for interpreting data of this flow period, albeit with minor modification.
Flow Model for Each Flow Pattern a. Linear flow in the Fracture
The flow exists theoretically at the very beginning of the test. During this flow most of the fluids produced at the well come from expansion in the fracture. The flow is linear. The pressure varies linearly with t
The variation can be expressed as
Dxf
r
D t
P =2C
π η
or
f t f wf
i k C
t wh
p qB
p ( )
128 . 8
φ
=
µ
−
Fracture
Xf
where
η is the ratio of diffusivity inside the fracture and diffusivity in the reservoir.
And Cr is the relative conductivity and is expressed as flow regime is seen on the plot. A fracture with relative conductivity of over 100 behaves as if it had infinite conductivity. At low fracture conductivity, linear flow regime is not seen.
The concept of relative conductivity explains why the smaller the formation permeability, the more effective the hydraulic fracturing is.
b. Bilinear Flow
It is called bilinear because it corresponds to two simultaneous linear flows;
• an incompressible linear flow in the fracture
• a compressible linear flow in the formation
Bilinear flow lasts as long as the ends of the fracture do not affect the flows. This flow period occurs only in case of finite fracture conductivity cases and where there is no well bore storage distortion. In this flow regime, the pressure behaviour is featured by the linear relationship when data are plotted by using the pw and t1/4 coordinates
The equation suggests that slope of the bilinear plot would lead to the estimation of kfw and fracture half length. However, it should also be noted that determination of fracture characteristics by this method requires knowledge of reservoir properties.
c. Linear Flow in the Formation
This flow is very often visible during testing of artificially fractured well. It is an integral part of the conventional analysis methods of these tests. The flow regime occurs in the
fracture itself and in the formation proper. This type flow is exhibited by only highly conductive fractures (Cr > 100). This flow period if exists, should be used for calculation of fracture properties. It is characterized by a linear variation of the pressure versus t The flow is characterized by following expression
pD =
π
tDxf ork x
t c h
p qB p
t f wf
i − = 4.064
φ µ
d. Pseudo Radial Flow
At long time and end of bilinear and linear flows, pseudo-radial flow regime starts. The reason why it is called pseudo radial flow is that flow period is not fully radial (Russel and Truitt). Nevertheless, all curves approach a common value of maximum slope which is dependent on the length of the fracture penetration. Raghvan et. al. constructed a graph of correction factors fc , which must be used to obtain the correct permeability factor.
i.e. k=kH fc
Russel-Truitt Method of Permeability Determination from Pseudo Radial Flow Russel Truitt method for the determination of true permeability of the reservoir is given below;
The graph is a plot between R=(Measured slope of build-up data/theoretical slope of build up data) versus ( Lf/D=fracture length/spacing between wells). A prototype of slope is shown below and is not to the scale;
Procedure
R
Lf/D
Russel-Truitt Plot for slope Correction
For an oil well, the equation relating fracture length with reservoir test parameter is given
log plot psi/cycle for drawdown
M2 = slope of the pwf vs ∆t for drawdown
slope of the pws vs t+∆t − ∆t for build up φ = porosity, fraction
R = Correction factor from Russel-Truitt plot
Steps to solve:
Step 1. : Assume an Lf value Step 2. : Calculate Lf/D value
Step 3. : From the graph of Russel-Truitt calculate R Step 4. : Calculate Lf from the equation
Step 5 : The assumed value and the calculated value if found equal gives the correct value of fracture length. Otherwise, repeat the iterative process.
Step 6: Put the value of R in following equation to know correct value of permeability of