Journal of Petroleum Engineering
Volume 2015 (2015), Article ID 214084, 16 pages
http://dx.doi.org/10.1155/2015/214084

Research Article

A New Approach in Pressure Transient Analysis: Using Numerical Density Derivatives to Improve Diagnosis of Flow Regimes and Estimation of Reservoir Properties for Multiple Phase Flow

Victor Torkiowei Biu and Shi-Yi Zheng

London South Bank University, UK

Received 5 April 2015; Accepted 7 June 2015

Academic Editor: Alireza Bahadori

Copyright © 2015 Victor Torkiowei Biu and Shi-Yi Zheng. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

This paper presents the numerical density derivative approach (another phase of numerical welltesting) in which each fluid’s densities around the wellbore are measured and used to generate pressure equivalent for each phase using simplified pressure-density correlation, as well as new statistical derivative methods to determine each fluid phase’s permeabilities, and the average effective permeability for the system with a new empirical model. Also density related radial flow equations for each fluid phase are derived and semilog specialised plot of density versus Horner time is used to estimate relative to each phase. Results from 2 examples of oil and gas condensate reservoirs show that the derivatives of the fluid phase pressure-densities equivalent display the same wellbore and reservoir fingerprint as the conventional bottom-hole pressure BPR method. It also indicates that the average effective ranges between 43 and 57 mD for scenarios (a) to (d) in Example 1.0 and 404 mD for scenarios (a) to (b) in Example 2.0 using the new fluid phase empirical model for estimation. This is within the value used in the simulation model and likewise that estimated from the conventional BPR method. Results also discovered that in all six scenarios investigated, the heavier fluid such as water and the weighted average pressure-density equivalent of all fluid gives exact effective as the conventional BPR method. This approach provides an estimate of the possible fluid phase permeabilities and the % of each phase contribution to flow at a given point. Hence, at several stabilisation points, the relative can be generated.

1. Introduction

Several sets of well and reservoir models have been generated with pressure derivatives with different boundary conditions. Likewise, several type curves, which account for different combinations of wellbore, reservoir characteristics, and boundary effects with associated flow regimes for computation of well and reservoir parameters, have been used to simplify well test interpretation. This demonstrates that the log-log plot of the pressure derivative is a powerful tool for reservoir model identification in pressure transient analysis.

However, in practice, each current method of transient data analysis has its own strengths and limitations with no single pressure and production data analysis method capable of handling all types of data and reservoir types with clear reliable results [1]. The log derivative and derivative type curve, which have remained reference flow regime’s diagnostic tools for over four decades, are the only unified approach for welltest interpretation and are applicable in a wide range of situations.

The derivative method, which is the greatest breakthrough in welltest analysis, was first introduced by Tiab in 1976 [2, 3] and developed by French mathematician Dominique Bourdet in 1983 [4]. It has remained the reference solution for identifying flow regime, boundary response, and use for diagnosing complex reservoir features till date. This approach has helped to reduce the uncertainties surrounding the interpretation of welltest data because key regions of radial flow and boundary features have been adequately diagnosed. However, due to the nonunique solution of the mathematical fluid flow equation, mostly in heterogeneous reservoir, most engineers in the industry are compelled to use analytical model and type curve solutions to match complex model, which is oftentimes not realistic. Assumptions made are ignored while pursuing a perfect match and results obtained from this approach are often misleading [5].

This marked the beginning of numerical well testing in the industry by Zheng, 2006 [5], although the approach started from the early 1990s [6–10]. Zheng made more advances in 2006, providing more solutions to the nonunique problems mostly in heterogeneous reservoirs through numerical welltesting, thereby promoting its application. More papers have been published by researchers on the subject, thereby reflecting the advancement of numerical welltesting and its application in solving various reservoir engineering practical problems.

One of the main limitations of the pressure derivative is that the measured pressure data must be constructed into derivative data, by means of numerical differentiation. Oftentimes derivative data from real field are very noisy and difficult to interpret, resulting in various smoothing techniques developed by researchers on this subject. It is practically believed that smoothing of pressure derivative data often alters the characteristics of the data. Also, it is difficult to distinguish between fluid and reservoir fingerprints in critical saturated reservoirs.

Another limitation of these derivatives is diagnosing flow regimes in complex reservoir structures such as complex faulted systems and high permeability streak with interbedded shales, which is common in deep water turbidite systems, channel-levee, lobe, and channelized deposits. Also, in situations of multiphase flow around the wellbore, the derivative data are always noisy and difficult to interpret, resulting in the application of deconvolution and various smoothing techniques to obtain a perceived representative model which often might not be. Additionally, the analytical solution for transient pressure analysis is limited to single phase flow, which in real case is never the situation. Presently there are few literatures or research on multiphase transient pressure analysis. However, the combination of the new statistical approach [11] and the density derivative approach serves as a support tool for better interpretation and estimation of reservoir properties in these conditions.

The diagnosis of flow, which appears as distinctive patterns in the pressure derivative curve, is a vital point in welltest interpretations since each flow regime reflects the geometry of the flow streamlines in the tested formation. Hence, for each flow regime identified, a set of well and/or reservoir parameters can be estimated using the region of the transient data that exhibits the characteristic pattern behaviour [11]. In the study, the pressure derivative formulation from Horne (1995) [12] and the new statistical approach by Biu and Zheng (2015) [11] would be used throughout the analysis.

The mathematical formulation for pressure derivative by Horne (1995) [12] is given as

Also the mathematical formulation for the new statistical derivative approach by Biu and Zheng (2015) [11] is given as follows.

Model 1. Consider the following:

Model 2 (the exponential function). Consider the following:

Model 3. Consider the following:

Model 4 (the time function). Consider the following:

Equations (1) to (5) are the derivative and statistical models used for flow regime diagnosis, behaviours, and estimation of wellbore and reservoir properties using the log-log derivative plot.

2. Theoretical Concept of the Density Derivatives

The basic concepts involved in the derivation of fluid flow equation include

(i)conservation of mass equation,(ii)transport rate equation (e.g., Darcy’s law),(iii)equation of state.

Consider flow in a cylindrical coordinate with flow but with flow in angular and -directions neglected as shown in Figure 1; the equations are given as follows:

Equation (6) represents the conservation of mass. Since the fluid is moving, the equation

is applied. By conserving mass in an elemental control volume as shown in Figure 1 and applying transport rate equation, the following equation is obtained:

Expand the equation using Taylor series:

Equations (6) to (9) apply to both liquid and gas. Equation (9) is known as the general diffusivity equation and for each fluid; the density or pressure term in (9) can be replaced by the correct expression in terms of density or pressure.

Figure 1: Schematics of basic fluid flows concept.

For small or constant compressibility liquid,

Substituting for pressure in the equation, the diffusivity equation in terms of density is given as

Equation (12) is known as the density diffusivity equation, which can also be rewritten in the form of pressure.

Over the decades, the transient test analysis has applied the general diffusivity equation in pressure term to generate several nonunique solutions using several pressure-rate data.

Invariably, as in pressure term, the density term also implored

For inner boundary condition,

For outer boundary condition,

Presently there are permanent downhole gauges (PDG) with density measurement tool along with pressure and temperature installed during flowing and shut-in testing conditions but the data are not interpreted or used for reservoir monitoring. For simplification and application of the density derivative in existing welltest software, the density-pressure equivalent equation is formulated.

3. Software Suitability (Pressure Equivalent)

To apply the numerical density approach in existing software, the pressure equivalent of the fluid density changes at the wellbore is generated from the relationship below.

Using the isothermal compressibility coefficient , in terms of density,

3.1. For Small or Constant Compressibility Fluid Such as Oil and Water

Consider the following:

Integrating

or applying the expansion series,

Because the term is very small, the term can be approximated as

Therefore (14) can be rewritten as

For small compressibility fluid such as oil and water, either (19) or (22) is used to generate the pressure equivalent from well fluid density obtained from reservoir simulation or PDG tool. This pressure is then analyzed in any available well test softwares.

3.2. For Compressible Fluid in Isothermal Conditions

Consider the following:

For real gas equation of state,

Differentiating with respect to pressure at constant temperature,

Substituting (23) into (25),

In terms of density,

This equation is applicable for real gas condition.

For compressible fluid,

Applying the power series for ,

Limit to the 1st term only:

For compressible fluids such as gas, (32) is used to generate the pressure equivalent from the fluid density obtained from reservoir simulation or PDG tool at the well and then the pressure is analyzed in any available well test softwares.

Also, outside the available welltest software, the pressure derivative can be generated from the pressure equivalent obtained from the oil, gas, and water densities at the well by applying (1) formulated by Horne (1995) [12] or (2) to (4) by Biu and Zheng (2015) [11].

4. Density Weighted Average (DWA)

While (19) or (22) and (32) give the pressure equivalent for independent fluid phases such as gas, oil, and water, the weighted average method is used to obtain the density equivalent for a two- or three-phase combination. The equivalent pressure derived from the density for all three fluid components such as gas, oil, and water is given as

Equation (33) comprises all the fluid phases in the system and as such will be comparable to the conventional bottom-hole pressure measurement during the derivative analysis.

4.1. Empirical Model Correlation between Fluid Phases

An empirical model integrating the fluid phase’s permeabilities for a given system is formulated to determine the average reservoir permeability. The mathematical model is given as

where = oil phase permeability, = gas phase permeability, and  = water phase permeability.

With the estimation of the phase’s permeabilities, it is therefore possible to estimate the possible relative permeability to each phase and the percentage contribution to flow by each phase at one point analysis; hence, at several points, the relative can be generated.

To illustrate applicability of this approach, 6 scenarios in conventional oil reservoir and gas condensate reservoir are investigated using numerical model built with a commercial simulator. Local grid refinement (LGR) is imposed around the well to capture sharp change in fluid densities as shown in Figure 2.

Figure 2: Simulation model for gas cap + oil + water reservoir and gas condensate + water showing local grid refinement around the well.

The simulation software keywords LBPR, LDENO, LDENW, LDENG AND WBHP were outputs to obtain the density and pressure change around the well and as far as the perturbation could extend.

Example 1. Table 1 presents a summary of the well and reservoir synthetic data used for the buildup and drawdown simulated scenarios with additional information given below. It is required to generate the pressure equivalent and derivative for each phase, compare their diagnostic signatures, and also determine the phases permeabilities and average reservoir permeability.

Table 1: Reservoir and fluid data for Example 1.

Parameters
Design value

Eclipse model
Black oil

Model dimension
10 × 5 × 5

Length by width, ft by ft
500 × 400

Thickness , ft
250

Permeability by , mD
50.0 by 50.0

Porosity %
20

Well diameter, ft
0.60

Initial water saturation , %
22

Permeability, , mD
50

Gas oil contact (GOC), ft
8820

Oil water contact (OWC), ft
9000.0

Initial pressure, , psia
4000.0

Formation temperature, , oF
200.0

Assumption. (i) Oil reservoir + gas cap is completed with one well.

(ii) LGR is imposed around the well and far across to account for pressure and density changes.

Gas, oil, and water densities around the local grid refinement (wellbore) and WBHP were output using the simulator keywords. The following scenarios were evaluated.

(a)Flowing + buildup sequence: well perforated = 30 ft between oil and water layer. Net sand thickness = 250 ft.(b)Flowing + buildup sequence: well perforated = 30 ft inside the oil layer. Net sand thickness = 250 ft.(c)Flowing + buildup sequence: well perforated = 30 ft between gas and oil layer. Net sand thickness = 250 ft.(d)Falloff test; flowing + buildup sequence: well perforated = 30 ft inside the oil layer. Net sand thickness = 250 ft.

The schematic of the 4 scenarios is depicted in Figures 3 and 4. Figure 5 shows the production, shut-in and injection sequence for the 4 scenarios in Example 1.

(a)

(b)

(c)

(d)

Figure 3: Schematics of well perforation interval and sand thickness for oil + gas cap + water reservoir.

(a)

(b)

Figure 4: Schematics of well perforation interval and sand thickness for gas condensate + water reservoir.

Figure 5: Production and injection test sequence for scenarios (a) to (d) (buildup and falloff).

In scenario (a), the well is completed between the oil and water layer to mimic multiphase conditions at the wellbore and some distance away from the well and also see the effect on pressure distribution, fluid densities changes around the wellbore, and estimate fluid phase permeabilities using the specialised plot and from the empirical model for three-phase conditions.

The derivative in Figure 6 shows a good radial flow but drop in derivative at late time due to constant pressure support (likely aquifer support). A continuous drop is seen from 10 hrs in the model parameters. The derivatives for all output parameters display the same well and reservoir signatures (3 flow periods, early to late time response) but with different stabilisation. The well bottom-hole pressure (BPR → BHP) response shows good overlay with pressure equivalent of density weighted average (PDENDWA → PDENA) and pressure equivalent of water density (PDENWAT → PDENW) while pressure equivalent of gas density (PDENGAS → PDENG) and pressure equivalent of oil density (PDENOIL → PDENO) differ completely. The PDENDWA gives a better fingerprint that is less noisy.

Figure 6: Derivative and estimation for scenario (a).

A permeability value of 50.8 mD is estimated from the bottom-hole pressure BPR where and is obtained from the specialised plot. This is an approximate of the input value in the simulation model. Also the simulator outputs PDENDWA and PDENWAT in the simulation give the same value while PDENGAS and PDENOIL differ, giving 15.8 and 403.7 mD, respectively. At , the best estimate is obtained depicting as the thickness contributing to flow. At  ft, drops below the imputed in the model.

Using (34),  mD is obtained, which is approximately close to that of BPR, hence a good estimate of the fluid phase permeabilities. A summary of the result is shown in Tables 2 and 5.

Table 2: estimates for new approach versus conventional approach for scenario (a).

Parameters
Numerical density
(mD)
Equivalent
(ft)

BHP
50.8

PDENA
50.8

PDENG
15.8
50

PDENO
403.7

PDENW
50.8

In scenario (b), the well is completed within the oil section to capture the pressure and fluid densities changes around the well. The derivative in Figure 7 shows a good radial flow but with noisy numerical artefact. It is likely that the boundary response is masked by numerical artefact. Also a continuous drop is seen after 10 hrs in the derivative.

Figure 7: Derivative and estimation for scenario (b).

Using the estimated fluid phase permeabilities, from (34) is 47.2 mD as shown in Table 3, which is the same as scenario (a). This is in line with the uniform used in the simulation model. Also the BPR gives a permeability value of 50.0 mD, which is the same for PDENDWA and PDENWAT but differs with PDENGAS and PDENOIL that give 12.1 and 484.4 mD, respectively. At  ft (60% of sand thickness and 83% of oil thickness), the estimated is still within range ( mD). This indicates  ft or 83% of oil thickness contributing to flow.

Table 3: estimates for new approach versus conventional approach for scenario (b).

Parameters
Numerical density
(mD)
Equivalent
(ft)

BHP
50.0

PDENA
50.0

PDENG
12.1
50

PDENO
484.2

PDENW
50.0

To test this approach in the gas column, the well was completed in between the gas and oil layer, which is considered as scenario (c) and scenario (d) with the well completed within the oil layer but with water injection after flowing and shut-in sequence. In both scenarios, the multiphase fluid distribution is triggered at the wellbore in order to capture the density changes for each phase and calculate fluid phase permeabilities. First, the well fluid densities equivalent pressures and pressures at bottom-hole for flowing and buildup test are generated.

The derivative for both scenarios declines after 3 hours probably due to fluid redistribution but is noisy (numerical artefact) in scenario (c) as shown in Figure 8. A good radial stabilisation for both the conventional method BPR and the density outputs PDENDWA, PDENWAT, PDENGAS, and PDENOIL is observed. Likewise, as obtained in scenarios (a) and (b), a good value of 57 and 52 mD from (34) is obtained for scenarios (c) and (d), respectively. This is slightly higher than  mD imputed in the simulation model.

Figure 8: Derivative and estimation for scenario (c).

For each of the fluid phase permeabilities estimates, value of 50 mD is obtained using the conventional BPR for scenario (c), which is in line with the uniform in the simulation model. This is the same for the PDENDWA and PDENWAT. Also, at  ft (60% of sand thickness and 83% of oil thickness), the estimated is still within range ( mD). This also indicates that  ft or 83% of oil thickness is contributing to flow. The values for PDENGAS and PDENOIL are 20.9 and 484.2 mD, respectively. Summary of the result is shown in Table 4.

Table 4: estimates for new approach versus conventional approach for scenario (d).

Parameters
Numerical Density (mD)
Equivalent (ft)

Scenario (d)
BHP
50.0
100

PDENA
50.0

PDENG
17.5
50

PDENO
490
250

PDENW
50.0
100

Table 5: Comparison of estimates between conventional and numerical density parameters.

Scenario
Numerical density method
Calculated
(mD)
Conventional
(mD)
Simulation
model (mD)
Relative
% phase
contribution to flow

Phases
(mD)

(a)
Gas
15.8

0.03
3.0

Oil
403.7
47.3
50.8
50.0
0.86
86.0

Water
50.8

0.11
11.0

(b)
Gas
12

0.03
2.0

Oil
484.4
43.2
50.0
50.0
1.13
89.0

Water
50

0.12
9.0

(c)
Gas
20.9

0.04
4.0

Oil
484.2
57.0
50.0
50.0
0.86
87

Water
50

0.09
9.0

(d)
Gas
17.5

0.03
3.0

Oil
490
52.3
50.0
50.0
0.95
88.0

Water
50

0.10
9.0

However, in scenario (d), the permeability value of 50.0 mD using the conventional BPR, which is the same for PDENDWA and PDENWAT, is only achieved if  ft (40% of sand thickness and 56% of oil thickness). PDENGAS and PDENOIL differ, giving 35.0 mD at  ft and 196.0 mD at  ft, respectively. This indicates the impact of water injection on densities and pressures changes around the well and consequently its impact on sand thickness contributing to flow. The drop in derivative curve in Figure 9 depicts the impact of the injected water.

Figure 9: Derivative and estimation for scenario (d).

In summary, for scenarios (a) to (d), the value of 50.0 mD is achieved if the thickness contributing to flow ranges from 50 to 150 ft. Generally, results indicate that, in all 4 scenarios investigated, the heavier fluid such as water and the weighted average pressure-density equivalent of all fluid give exact effective as the BPR, likewise the density outputs PDENDWA and PDENWAT. Also results from Table 5 show that the empirical model from (34) with all fluid phase permeabilities gives an average effective of 47–57 mD, which is within that used in the simulation model and also the same as the estimated conventional approach. This approach provides an estimate of the possible fluid phase permeabilities and the % of each phase contribution to flow; hence, at several points, the relative can be generated as shown in Table 5.

Example 2. To capture the influence of highly compressible fluid on estimated fluid phase permeabilities, an example on gas condensate reservoir (volatile system) was tested. Table 6 presents a summary of the well and reservoir synthetic data used for the buildup and drawdown simulated scenarios with additional information given below. It is required to generate the pressure equivalent and derivative for each fluid phase, compare their diagnostic signatures, and also determine the phases permeabilities and average reservoir permeability.

Table 6: Summary of reservoir simulations data.

Parameters
Design value

Eclipse model
Black oil

Model dimension
9 × 3 × 3

Length by width, ft by ft
1312 × 984

Thickness , ft
150

Permeability by , mD
400.0 by 300.0

Porosity, %
30

Well diameter, ft
1.15

Initial water saturation , %
60

Permeability, , mD
400

Gas oil contact (GOC), ft
6890

Oil water contact (OWC), ft
6890

Initial pressure, , psi
4495

Formation temperature, , °C 120
200.0

Assumption. (i) Condensate reservoir is completed with one well.

(ii) LGR is imposed around the well and far across to account for pressure and density changes.

Gas, condensate, and water densities around the local grid refinement (wellbore) and WBHP were output using the simulator's keywords. The following scenarios were evaluated.

(a)Flowing + buildup sequence: well perforated  ft between gas condensate and water layer. Net sand thickness  ft.(b)Flowing + buildup sequence: well perforated  ft inside gas condensate layer. Net sand thickness  ft.

Figure 10 shows the production and shut-in sequence for 2 scenarios in Example 2.

Figure 10: Production test sequence (buildup) for Example 2.

As in Example 1, the derivative in scenario (a) of Example 2 also shows good radial flow with no late time effect as seen in Figure 11. A negative half-slope fingerprint is seen at 0.3 to 3.0 hrs of buildup and a good stabilisation from 4.0 hrs in the model parameters.

Figure 11: Derivative and estimation for scenario (a).

The derivatives for all output parameters display the same well and reservoir fingerprint (2 flow periods, early to middle time response) but with different stabilisation.

Permeability value of 370.7 mD was obtained from the conventional method BPR where is the same for the PDENDWA and PDENWAT, and is obtained from the specialised plot. This is in line with the uniform ( mD and  mD) imputed in the simulation model. However PDENGAS and PDENOIL differ, giving 35.0 mD and 3574.7 mD, respectively, if  ft.

Also the derivative in Figure 12 shows good radial flow and a negative half-slope fingerprint is seen at 0.5 to 2.0 hrs of buildup and a good stabilisation from 3.0 hrs in the model parameters. After 3 hrs of shut-in, some noisy data (numerical artefact) is seen in all the fluid phase derivatives but still good enough to identify stabilisation.

Figure 12: Derivative and estimation for scenario (b).

For each of the fluid phase permeabilities, value of 340.9 mD at  ft is obtained using BPR, which is the same for the PDENDWA and PDENWAT. PDENGAS and PDENOIL differ, giving 149.0 and 3514.1 mD at  ft, respectively.

For scenarios (a) and (b), a good value of 405.0 and 403.0 mD is obtained, respectively, from the empirical model (34) integrating all fluid phase permeabilities, which is within that used in the simulation model and the estimated conventional approach. From the 2 scenarios investigated, it has been demonstrated that the heavier fluid such as water and the weighted average pressure-density equivalent of all fluid give exact effective as the conventional method BPR. A summary of the result is shown in Table 7.

Table 7: Summary of reservoir modelling properties imputed in eclipse model for case study 2.

Parameters
Numerical density
(mD)
Equivalent
(ft)

Scenario (a)
BHP
371

PDENA
371

PDENG
142
148

PDENO
3575

PDENW
371

Scenario (b)
BHP
341

PDENA
341

PDENG
149
148

PDENO
3514

PDENW
341

This approach estimates the fluid phase permeabilities and the % of each phase contribution to flow at a given point; hence at several points the relative can be generated as shown in Table 8.

Table 8: Comparison of estimates between conventional and numerical density parameters.

Scenario
Numerical density method
Calculated
(mD)
Conventional
(mD)
Simulation
model (mD)
Relative
% phase
contribution to flow

Phases
(mD)

(a)
Gas
142.2

0.04
3.0

Oil
3574.7
404.6
370.7
400
0.89
87.0

Water
370.7

0.09
9.0

(b)
Gas
149

0.04
4.0

Oil
3514.1
403.8
340.9
400
0.89
88.0

Water
340.9

0.08
9.0

5. Density Related Radial Flow Equation Derivation for Each Fluid Phase

Radial diffusivity equation is given as

For oil,

For water phase,

For gas phase,

5.1. From Equation for Slight and Small Compressibility Such as Water and Oil

From (22),

Differentiating with respect to ,

From Darcy flow equation,

Substitute for :

To derive the analytical density transient equation for each phase, the following assumption/conditions are applicable.

For oil phase, from (35),

Initial condition is as follows:

BC at the wellbore is as follows:

BC at infirmity is as follows:

Using the Boltzmann transformation, consider the following.

Assuming ,

Therefore

Differentiating with respect to is equal to differentiating with respect to ; multiply by .

From (35), the L.H.S is resolved as follows:

Equating (44) and (46), we have

This is the simplified ordinary differential equation of as a function of .

Apply boundary conditions as follows.

Initial condition is as follows:

Substituting (43) into (40),

Assume that .

From (47),

Integrating from to ,

Recall boundary condition

Recall that .

Therefore,

Integrating from where ,

where

Assuming , then and .

Hence,

where

Known as the exponential integral function defined by [13],

where and

At large times, will be small applying Taylor’s series and integrating

where is known as Euler’s number.

For , consider the following.

Then

Plotting versus will yield a straight line at longer time and the slope of the line is given as

Therefore,

5.2. For Water Phase

The radial density equation is given as

Plotting versus will yield a straight line at longer time and the slope of the line is given as

where

5.3. For Gas Phase

From equation for compressible fluid such as gas, consider the following.

From (32),

From the fundamental,

At standard condition,

where

Substitute into the equation, where

From the diffusivity equation for gas,

Initial condition is as follows:

BC at the wellbore is as follows:

BC at infirmity is as follows:

where , , and are known as temperature, pressure, and gas constant at standard condition.

Therefore, integrating from where ,

where

Assume that    and .

Hence

where .

Known as the exponential integral function defined by [13],

where and .

At large times, will be small applying Taylor’s series and integrating

where is known as Euler’s number.

For , consider the following.

Then,

Plotting or versus will yield a straight line at longer time and the slope of the line is given as follows:

Example 3. Data from Example 1(a) was used for this case and the semilog plot of density for each phase (oil, water, and gas) is plotted against log of Horner time.
used for calculation is the average rate for all flowing periods.

From Figures 13 and 14, the calculated slopes of the radial flow for oil and water phases are given as

The estimated phase permeabilities are

Also, for the gas phase, consider the following.

Figure 13: Oil and gas phase density versus log Horner time.

Figure 14: Water phase density versus log Horner time.

The calculated slope from the semilog plot shown in Figure 13 is given as

And estimated gas permeability is

Using the empirical model for all three phases, the average estimated permeability is as follows:

Example 4. Data from Example 1(c) was used for this case and the semilog plot of density for each phase (oil, water, and gas) is plotted against log of Horner time.
From Figures 15 and 16, the calculated slopes of the radial flow for oil and water phases are given as

The estimated phase permeabilities are

Also, for the gas phase, consider the following.

Figure 15: Oil and gas phase density versus log Horner time.

Figure 16: Water phase density versus log Horner time.

The calculated slope from the semilog plot shown in Figure 15 is given as

And estimated gas permeability is

Using the empirical model for all three phases, the average estimated permeability is as follows:

6. Conclusion

The following inferences were drawn from the six scenarios reviewed:

(i)Results from the three-phase fluid empirical model developed for average effective estimation range between 47 and 57 mD for Example 1 and give 404 mD for Example 2, which is within that used in the simulation model and also estimated from BPR.(ii)The BPR, PDENDWA, and PDENWAT give the same stabilisation and the same estimates making it suitable for interpretation of pressure transient analysis.(iii)It has been demonstrated in all 6 scenarios investigated that the heavier fluid such as water and the weighted average pressure-density equivalent of all fluid give exact effective as the conventional method.(iv)For oil reservoir system (scenarios (a) to (d)),  mD is achieved if the thickness contributing to flow ranges from 50 to 150 ft while, for gas condensate reservoir (scenarios (a) to (b)), ranges between 340 and 371 mD which is achieved if the net sand thickness  ft is contributing to flow.(v)This approach also provides an estimate of the possible fluid phase permeabilities and the % of each phase contribution to flow at a given point; hence, at several stabilisation points, the relative can be generated.(vi)The derivatives for all output parameters display the same wellbore and reservoir fingerprint as the BPR method.(vii)Generally, where output parameters (BPR, PDENOIL, PDENGAS, PDENWAT, and PDENDW) depict different fingerprint, the density derivative will serve as support to distinguish between reservoir and nonreservoir response.

Nomenclature

: | Pressure, psi
: | Temperature, oF
: | Radius, ft
: | Permeability, mD
: | Porosity fraction
: | Viscosity, cp
: | Time, hrs
: | Production rate, bbl/day
: | Formation volume factor, rb/Stb
: | Total compressibility, psi⁻¹
: | Wellbore radius, ft
: | Change in pressure, psia
: | Formation thickness, ft
: | Drainage area acres
: | Bottom-hole flowing pressure, psi
: | Initial pressure, psi
: | Difference between two point/time series
: | Subscript of an observed variable
: | Subscript of a calculated variable
: | SSE of data point
: | Number of data points
: | Covariance of data point
: | Standard deviation
: | Cumulative production time
: | Wellbore storage constant
: | Mean of data point.

Abbreviations

: | Local grid bottom-hole pressure
: | Local grid oil density
: | Local grid water density
: | Local grid gas density
: | Well bottom-hole pressure
: | Well bottom-hole pressure
: | Pressure equivalent of LDENO
: | Pressure equivalent of LDENG
: | Pressure equivalent of LDENW
: | Pressure equivalent of density weighted average (LDENO, LDENG, and LDENW).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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