3.2. Changing Wellbore Storage
3.2.3. Use of a numerical model
The principle is to use a wellbore model which, at any time, uses the flowing pressure to define the wellbore storage parameter. In order for the model to be stable, the wellbore storage has to be calculated implicitly at each time step. As the problem is not linear, this can only be done using Saphir Nonlinear .
This is by far the best way to simulate pressure related wellbore storage. However there are a couple of drawbacks:
•
The model is slower than an analytical model or a change of time variable•
It is inflexible: once you have entered the PVT and the wellbore volume there is no parameter to control the match. The model works, or it does not.Chapter 4
METHODS FOR THE ANALYSIS OF WELLBORE STORAGE DISSTORTED WELL TEST DATA
4.1. Russell Method (1966):
this method does not provide results which can be considered useful in the context of modern well test analysis and interpretation methods. Russell made the following assumptions in the derivation of his wellbore storage "correction" solution:
Completely penetrating well in an infinite reservoir
Slightly compressible liquid (constant compressibility)
Constant fluid viscosity
Single-phase liquid flow in the reservoir
Gravity and capillary pressure neglected
Constant permeability
Horizontal radial flow (no vertical flow)
Ideal gas (for the gas cushion in the well) Russell's [2] wellbore storage correction is given as:
Combining Eqs. 4.1 and 4.2 into a plotting function format, we obtain:
)
Russell treated the C2-term as an arbitrary constant to be optimized for analysis — in other words, the C2-term is the "correction" factor for the Russell method.
As prescribed by Russell, the C2-term is obtained using a trial-and-error sequence which yields a straight line when the left-hand-side term of Eq. 4.3 is plotted versus log(Δt).
Where the general form of the y-axis correction term prescribed by Eq. 4.3 is:
A schematic of the Russell method is shown in Fig. 4.1, where we note Russell's interpretation of the effect of the C2-term (i.e., where C2 is too large and C2 is too small).
Fig.4. 1. Schematic plot showing determination of the correct C2 value[2]
Once the C2-term is established, the kh-product is estimated using:
msl
kh qBµ
6 .
=162 (4.5)
And the skin factor can be estimated using:
In short, the Russell method has an elegant mathematical formulation, but ultimately, we believe that this formulation does not represent the wellbore storage condition, and hence, we do not recommend the Russell method under any circumstances.
4.2. Rate Normalization
4.2.1. Glatfelter Rate Normalization
Glatfelter, Tracy and Wilsey [18] introduced the "rate normalization" deconvolution approach which, in their words "permits direct measurement of the cause of low well productivity."
The objective of rate normalization is to remove/correct the effects of the variable rate from the observed pressure data. Rate normalization can also be defined as an approximation to convolution integral [18].
)
Where pu is the constant rate pressure response.
The afterflow rate-normalized pressure equation proposed by Glatfelter et al to analyze pressure buildup data dominated by afterflow was given as [19]
should be linear with slope equal to
m 162kh.6µ
′= (4.10)
The skin is determined from
Rate normalization has been employed for a number of applications in well test analysis.
For the specific application of "rate normalization" deconvolution, we must recognize that the approach is approximate — and while this method does provide some
"correction" capabilities, it is basically a technique that can be used for pressure data influenced by continuously varying flowrates.
4.2.2. Fetkovich Rate Normalization
Rate normalization techniques and procedures are best illustrated by first examining their application to drawdown data. Although the nature of the rate variation of drawdown data with time is different than that of afterflow rate variation, the end result is the same.
Also, drawdown rate variations generally last much longer than afterflow rate variations.[19]
Most notably, Fetkovich and Vienot [19], and Doublet et al.[18] ,have demonstrated the effectiveness of "rate normalization" deconvolution (albeit for specialized cases). In particular, for the wellbore storage domination and distortion regimes, rate normalization can provide a reasonable approximation of the no wellbore storage solution. For this infinite-acting radial flow case, rate normalization yields an erroneous estimate of the skin factor by introducing a shift on the semi log straight line (obviously, the sandface rate profile must be known). This last point, however, makes the application of rate normalization techniques very limited in our particular problem — we do not have measurements of sandface flowrate. Therefore, this method must be applied using an estimate of the downhole rate— which will definitely introduce errors in the deconvolution process. Such issues make rate normalization a "zero-order"
approximation that is, rate normalization results should be considered as a guide, but not relied upon as the best methodology.
4.3. Material Balance Deconvolution
Material balance deconvolution is an extension of the rate normalization method.
Johnston[19] defines a new x-axis plotting function (material balance time) which provides an approximate deconvolution of the variable-rate pressure transient problem. The general form of material balance deconvolution is provided for the pressure drawdown case in terms of the material balance time function and the rate-normalized pressure drop function. The material balance time function is given as:
q
tmb= Np (4.12)
The wellbore storage-based, material balance time function for the pressure buildup case is given as:
And the wellbore storage-based, rate-normalized pressure drop function for the pressure buildup case is given as:
ws
Plotting the rate-normalized pressure function versus the material balance time function (on log(tmb) scales) shows that the material balance time function does correct the erroneous shift in the semilog straight-line obtained by rate normalization. We believe that the material balance deconvolution technique is a practical approach (perhaps the most practical approach) for the explicit deconvolution of pressure transient test data which are distorted by wellbore storage and skin effects.
4.4. Power Deconvolution
This development assumes that variable (or constant) rate flow conditions exist in the reservoir. We will use the familiar convolution integral to develop our new deconvolution technique, The convolution integral is given as
∫
′ −The development of this method requires the Laplace transform and if we find that the inverse Laplace transform it becomes
D In order to apply Eq. 18 to field data, i.e., time, pressure, and flowrate, we merely compute qD= q(t)/q and substitute Δpw and Δps for pwD(tD) and pws(tD).
now we will describe how to use this deconvolution method, and apply it to field well test data .we should outline a general procedure to convert wellbore storage dominated data to the data that would have been obtained if storage effects were non-existent, and to recommend methods of calculating the sandface rate during afterflow.[17]
1. Obtain mwbs and Po:
2. Compute the pressure drop function, Δpw, and the time function, t:
a. Drawdown
4. Compute Δps where is the pressure drop for the deconvolved, constant rate data, by the expression:
4.5. β - Deconvolution
Van Everdingen[18] and Hurst[19] demonstrated empirically that the sandface rate profile can be modeled approximately using an exponential relation for the duration of wellbore storage distortion during a pressure transient test. The van Everdingen/Hurst exponential rate model is given in dimensionless form as:
tD
D
D t e
q ( )=1− −β (during wellbore storage distortion) (4.20) A similar approximation can be used for pressure buildup data:
tD
D
D t e
q ( )= −β (4.21) The convolution integral is given as
∫
′ −The β-deconvolution formula, which computes the undistorted pressure drop function directly from the wellbore storage affected data, is given as:
D
And in terms of field variable
dt
Chapter 5
INVESTIGATION OF WELLBORE STORAGE EFFECTS ON WELL TEST DATA USING TEST DESIGN
TECHNIQUE
All Saphir analytical and numerical models may be used to generate a virtual gauge on which a complete analysis may be simulated. Simulation options taking into account gauge resolution, accuracy and potential drift can be the basis for selecting the appropriate tools or to check if the test objectives can be achieved in practice but here we change model parameters to see the sensitivity of model to these parameters. Main parameter that we change is Wellbore storage coefficinet.we want to investigate the effect of wellbore storage coefficient on pressure response of models(including reservoir model and well model).in other words we determine the effect of wellbore storage coefficient on the wellbore storage disappearing of different models and investigate the extent of this effect on model recognition.
For test Design we have some basic parameter of these reservoir that is common in all cases listed below
Table5. 1.Reservoir properties
Wellbore radius
Net pay
thickness Porosity Permeability Formation compressibility
rw(ft) h(ft) φ K(md) Cf(psi-1)
0.25 50 0.15 20 4E-6
Initial
Pressure(psi) Temperature(˚F)
Pi(psi) T(˚F)
5000 210
Table5. 2.Reservoir initial condition
Fig5. 1.input reservoir characteristic Fig5. 2.input Temperature an Pressure
After we introduce the basic parameters, we should calculate the fluid properties using default correlations of software(Saphir) so we need the fluid properties at reservoir initial condition.
After defining of reservoir and well parameters next we can choose different model for test design and derive pressure response according to this model(including reservoir model and well model).
5.1.Oil well Data
As mentioned , we need the fluid properties at initial condition of reservoir . these parameters are listed below
At Reservoir Condition :(T=210˚F , P=5000psi)
Table5. 3.Fluid properties
Formation volume factor
Oil compressibility
Bo(bbl/STB) Co(psi-1)
1.25 5E-5
Also well production data is required :
Table5. 4.Production data
Time Flow Rate
t (hr) qo(STBD)
200 500
300 0
Fig5. 4.test design screen(Saphir software) Fig5. 3.schematic of model chosen in test design(Saphir software)
5.1.1. Constant wellbore storage, Homogenous reservoir, Vertical well, Infinite acting
Fig. 5.4 with various constant wellbore storage constants is illustrated below. Pure wellbore storage is characterized by the merge of both Pressure and Bourdet Derivative curves on the same unit slope.
At a point in time, and in the absence of any other interfering behaviors, the Derivative will leave the unit slope and transit into a hump which will stabilize into the horizontal line corresponding to Infinite Acting Radial Flow.
The horizontal position of the curve is only controlled by the wellbore storage coefficient C.
Taking a larger C will move the unit slope to the right, hence increase the time at which wellbore storage will fade. More exactly, multiplying C by 10 will translate the curve to one log cycle to the right.
The figure below presents the response with wellbore storage values, C of 0.0001, 0.001, 0.01,0.1 and 1 (bbl/psi).
Fig5. 5.Sensitivity to C for homogenous reservoir
The value of C has a major effect, which is actually exaggerated by the logarithmic time scale. When the influence of wellbore storage is over both the pressure change and the derivative merge together. Wellbore storage tends to masks infinite acting radial flow on a time that is proportional to the value of C.
According to derivative curve of above figure the radial flow for C = 0.0001bbl/psi starts at t = 0.01hr and wellbore storage almost not be seen and for other value of C we have
C=0.01 C=0.1 C=1
C=1E-4 C=0.001
For C=0.001 bbl/psi Radial flow Starts At t=1hr For C=0.01 bbl/psi Radial flow Starts At t=10hr For C=0.1 bbl/psi Radial flow Starts At t=100hr For C=1 bbl/psi Radial flow not be seen
And we see that characterization of reservoir behavior for C=1 bbl/psi is impossible and wellbore storage effect distorted pressure response of reservoir.
5.1.2. Constant wellbore storage, Homogenous reservoir, H.C fracture, Infinite acting
For high conductivity the linear flow should be seen in early time region .the characteristics of this flow regime is +1/2 slope and the distance of log2 between the pressure and pressure derivative curve.
Fig5. 6.Sensitivity to C for high conductivity fracture
we see in the Fig 5.6 for C=0.001 according to pressure derivative curve radial flow starts at t = 0.1hr and the two curve have slope of +1/2 and the distance of log2 but if we go to larger value of C the slope tend to 1 and the distance tend to zero between two curves. For example the slope of Curve for C=0.01 bbl/psi is about +3/4 and the distance is lower than log2.
For wellbore storage coefficient C ,0.1 bbl/psi, 1 bbl/psi we can not see the linear flow.
Start of radial flow for different value of C is listed below : For C=0.001 bbl/psi Radial flow Starts At t=0.1hr
C=0.001 C=0.01
C=0.1
C=1 1E-3 bbl/ psi (current)
0.01 bbl/ psi 0.1 bbl/ psi 1 bbl/ psi
For C=0.01 bbl/psi Radial flow Starts At t=1hr For C=0.1 bbl/psi Radial flow Starts At t=10hr For C=1 bbl/psi Radial flow Starts At t=100hr
And we see that characterization of reservoir behavior for C= 1bbl/psi is almost impossible.
5.1.3. Constant wellbore storage, Homogenous reservoir, L.C fracture, Infinite acting
For Low conductivity the bilinear flow should be seen in early time region .the characteristics of this flow regime is +1/4 slope (both curves) and the distance of log4 between the pressure and pressure derivative curves.
Fig5. 7.Sensitivity to C for low conductivity fracture
we see in Fig 5.6 for C=0.001bbl/psi according to pressure derivative curve radial flow starts at t = 1hr and the two curve have slope of +1/4 and the distance of log2 but if we go to larger value of C the slope tend to 1 and the distance tend to zero between two curves. for example the slope of Curve for C=0.01 bbl/psi is about +1/2,maybe confused with high conductivity fracture, and the distance is lower than log4.for C ,0.1 bbl/psi, 1bbl/psi the linear flow disappear and wellbore storage overcome this flow regime.
Start of radial flow for different value of C is listed below : For C=0.001 bbl/psi Radial flow Starts At t=1hr For C=0.01 bbl/psi Radial flow Starts At t=10hr For C=0.1 bbl/psi Radial flow Starts At t=100hr
C=0.001 C=0.01
C=0.1
C=1 1E-3 bbl/ psi (current)
0.01 bbl/ psi 0.1 bbl/ psi 1 bbl/ psi
For C=1 bbl/psi Radial flow is not seen
And we see that characterization of reservoir behavior(Middle time Region) for C= 1bbl/psi is impossible.
5.1.4. Constant wellbore storage, Homogenous reservoir, Limited entry well, Infinite acting
Fig5. 8.Sensitivity to C for limited entry well
For the wellbore storage coefficient C= 0.0001bbl/psi wellbore storage is not seen . For the Partial penetration ,Wellbore storage will quickly mask the spherical flow regime. If we look at the curve of C=0.0001bbl/psi ,we can see the 1st stabilization occurs at t=0.04hr(In practice this flow regime is more often than not masked by wellbore storage.) then spherical flow is seen (slope = -1/2). next to spherical flow ,second stabilization is seen at t=100hr.as wellbore storage coefficient increases ,the spherical flow seen at later time until at C=1 bbl/psi spherical flow disappears . the similarity of all curves is 2nd stabilization at which all of them reach together about t= 100hr .start of 1st Stabilization, Spherical Flow and 2nd Stabilization for different value of wellbore storage coefficient are listed below :
Table5. 5.Start of flow regimes for limited entry well
Time(hr) C(bbl/psi)
1st Stabilization Spherical Flow 2nd Stabilization
C=0.0001bbl/psi 0.04 2 100
C=0.001bbl/psi 0.7 4.5 100
C=0.01bbl/psi 2.8 6 100
C=0.1bbl/psi Not Be Seen Not Be Seen 100
C=1 bbl/psi Not Be Seen Not Be Seen Not Be Seen
C=0.0001bbl/psi
C=0.001bbl/psi C=0.01bbl/psi C=0.1bbl/psi C=1bbl/psi
1E-4 bbl/ psi 1E-3 bbl/ psi 0.01 bbl/ psi (current) 0.1 bbl/ psi 1 bbl/ psi
5.1.5. Constant wellbore storage, Double porosity reservoir, Vertical well, Infinite acting
Wellbore storage will invariably mask the fissure response in the double porosity reservoir. The transition can thereby easily be misdiagnosed and the whole interpretation effort can be jeopardized. we can see for C=0.0001bbl/psi there is almost no wellbore storage and two stabilization on 0.5 and valley shaped transition are seen .1st stabilization starts at t=0.01hr if we change wellbore storage coefficient to C=0.001bbl/psi ,start of 1st stabilization shift to t= 0.1 hr.for C=0.01bbl/psi the 1st stabilization disappear immediately and we will face to transition valley .finally for C=0.1bbl/psi we can not see 1st stabilization.
Fig5. 9.Sensitivity to C for Double porosity reservoir
At higher wellbore storage coefficients even the whole transition period may be lost such C=0.1bbl/psi 1st that stabilization and transition valley disappear.the similarity of curves with different value of wellbore storage coefficient is 2nd stabilizatiion at which all of them meet eachother at 0.5 .
Table5. 6.Start of flow regimes for double porosity reservoir
Time(hr) C(bbl/psi)
1st Stabilization Transition Valley 2nd Stabilization
C=0.0001bbl/psi 0.01 0.25 100
C=0.001bbl/psi 0.1 0.45 100
C=0.01bbl/psi 0.5 1 100
C=0.1bbl/psi Not be seen 4 100
C=1 bbl/psi Not be seen Not be seen 100
C=0.0001 C=0.001 C=0.01 C=0.1
1E-4 bbl/ psi 1E-3 bbl/ psi 0.01 bbl/ psi (current) 0.1 bbl/ psi
5.1.6. Constant wellbore storage, Double permeability reservoir, Vertical well, Infinit acting
For the double permeability, First, the behavior corresponds to two layers without cross flow .At intermediate times, when the fluid transfer between the layers starts, the response follows a transition regime. Later, the pressure equalizes in the two layers and the behavior describes the equivalent homogeneous total system. The derivative stabilizes at 0.5.if we look at Fig 5.10,for C =0.0001bbl/psi we have no wellbore storage and the reservoir behavior is obviously seen.
Fig5. 10.Sensitivity to C for Double permeability reservoir
for C=0.001bbl/psi we cannot see pure wellbore storage but wellbore storage is seen and the 1st stabilization starts at t=0.1hr.if we change the wellbore storage coefficient to C=0.01bbl/psi ,start of 1st stabilization shift to t=0.5hr .with increasing of value of wellbore storage coefficient to C= 0.1bbl/psi 1st stabilization disappear. For C=1bbl/psi transition disappear and we can see only 2nd stabilization.
Table5. 7.Start of flow regimes for double permeability reservoir
Time(hr) C(bbl/psi)
1st Stabilization Transition Valley
2nd Stabilization
C=0.0001bbl/psi 0.01 0.3 100
C=0.001bbl/psi 0.1 0.7 100
C=0.01bbl/psi Not be seen 1 100
C=0.1bbl/psi Not be seen 4.2 100
C=1 bbl/psi Not be seen Not be seen 100
C=0.0001
C=0.001 C=0.01 C=0.1 C=1
1E-4 bbl/ psi 1E-3 bbl/ psi 0.01 bbl/ psi (current) 0.1 bbl/ psi 1 bbl/ psi
Chapter 6
CONCLUSION AND RECOMMENDATION
6.1.Conclusions
As we have seen, the wellbore storage effect distort pressure data and make it difficult to interpret well test data. In Chapter 5 ,we have analyzed wellbore storage for different type of reservoir models and well models to see difference of wellbore storage effects between them. For homogenous model, curve would shift to right. For high conductivity and low conductivity in early time, wellbore storage have not be seen .With increasing the value of C linear and bilinear flow disappear which make it difficult to detect hydraulic fracturing . In some cases wellbore storage will prevent detection the type of fracture. In limited entry well for small value of C there is no wellbore storage but with increasing value of C the time of pure wellbore storage increases and more data is distorted. For double porosity and permeability reservoirs with increasing the value of C, there is common trend in wellbore storage effect .
6.2.Recommendation
1. For performing well test in any type of reservoir, first we should estimate the time for production or build up to see all reservoir behavior .for this we can obtain rock and fluid properties from laboratory and running a test design for estimating this time.
2. For preventing distortion of pressure data, we can set a flow rate estimator that records flow rate and pressure simultaneously .
3. In a well where there is no designed partial penetration the interpreter can easily miss the effect and as the limited entry can result in a high geometrical and thus a high total skin this can often be misdiagnosed as damage alone when coupled with the wellbore storage effect and lead to ineffective acidizing.
REFRENCE
1. Bourdet, D. : "Well Test Analysis: The Use of Advanced Interpretation Models"
,ELSEVIER 2002
2. Russell, D.G.: "Extensions of Pressure Build-Up Analysis Methods," paper SPE 1513 presented at the 1966 SPE Annual Meeting, Dallas, Texas, 2–5 October.
2. Russell, D.G.: "Extensions of Pressure Build-Up Analysis Methods," paper SPE 1513 presented at the 1966 SPE Annual Meeting, Dallas, Texas, 2–5 October.