# 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

Authors: Victor Torkiowei Biu; Shi-Yi Zheng
Journal: Journal of Petroleum Engineering
 (2015)

---

## 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 estimatek 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 k
ave 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 K estimation. This is within the k 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 k 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 d
p
' stabilisation points, the relative k can be generated.


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## Body

## 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(1)
∂
p
∂
ln
⁡
t
=
t
∂
p
∂
p
i
-
A
,
A
=
ln
⁡
t
i
/
t
i
-
k
Δ
p
i
+
j
ln
⁡
t
i
+
j
/
t
i
ln
⁡
t
i
+
j
/
t
i
-
k
+
B
B
=
ln
⁡
t
i
+
j
t
i
-
k
/
t
i
2
Δ
p
i
ln
⁡
t
i
+
j
/
t
i
ln
⁡
t
i
/
t
i
-
k
-
ln
⁡
t
i
+
j
/
t
i
Δ
p
i
+
k
ln
⁡
t
i
/
t
i
-
k
ln
⁡
t
i
+
j
/
t
i
-
k
.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:(2)
StatDev
i
=
SQRT
pdd
i
x
Δ
dev
i
x
Δ
2
P
i
.Model 2 (the exponential function).
Consider the following:(3)
StatExp
i
=
SQRT
EXP
SQRT
Δ
2
P
x
pdd
i
x
Δ
2
P
i
.Model 3.
Consider the following:(4)
StatdDev
i
=
SQRT
pdd
i
x
Δ
dev
i
x
Δ
2
P
i
x
Δ
2
P
i
.Model 4 (the time function).
Consider the following:(5)
StattDev
i
=
STDEV
Δ
t
t
i
,
Δ
t
t
i
+
1
,
StatDev
i
,
StatDev
i
+
1
.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 z-directions neglected as shown in Figure 1; the equations are given as follows:(6)
Mass  rate  in
-
Mass  rate  out
=
Mass  rate  storage
.Equation (6) represents the conservation of mass. Since the fluid is moving, the equation(7)
q
=
-
k
μ
A
∂
p
∂
ris applied. By conserving mass in an elemental control volume as shown in Figure 1 and applying transport rate equation, the following equation is obtained:(8)
-
2
π
r
h
k
μ
ρ
∂
p
∂
r
r
=
-
2
π
r
k
h
μ
ρ
∂
p
∂
r
r
+
Δ
r
+
2
π
r
Δ
r
h
∂
∂
t
ρ
ϕ
.Expand the equation using Taylor series:(9)
1
r
∂
∂
r
r
k
ρ
μ
∂
p
∂
r
=
∂
∂
t
ρ
ϕ
.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,(10)
ρ
=
ρ
o
e
C
p
-
p
o
.Substituting for pressure in the equation, the diffusivity equation in terms of density is given as(11)
∂
2
ρ
∂
r
2
+
1
r
∂
ρ
∂
r
=
ϕ
μ
c
+
c
r
k
∂
ρ
∂
t
,
(12)
∂
2
ρ
∂
r
2
+
1
r
∂
ρ
∂
r
=
ϕ
μ
c
t
k
∂
ρ
∂
t
.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(13)
∂
2
ρ
∂
r
+
1
r
∂
ρ
∂
r
=
1
r
∂
∂
r
r
∂
ρ
∂
r
=
0
.For inner boundary condition,(14)
r
∂
ρ
∂
r
r
w
=
q
μ
2
π
k
h
=
Constant
.For outer boundary condition,(15)
ρ
=
ρ
e
at
r
=
r
e
.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 coefficientC, in terms of density, (16)
C
=
1
ρ
∂
ρ
∂
p
.


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

Consider the following:(17)
-
C
∫
p
o
p
d
p
=
∫
ρ
0
ρ
∂
ρ
ρ
.Integrating(18)
e
C
P
0
-
P
=
ρ
ρ
0
,
C
P
0
-
P
=
ln
⁡
ρ
ρ
0
,
(19)
P
=
P
0
-
ln
⁡
ρ
/
ρ
0
C
,or applying the e
x expansion series,(20)
e
x
=
1
+
x
+
x
2
2
!
+
x
3
3
!
+
⋯
+
x
n
n
!
.Because the term C
P
0
-
P is very small, the e
x term can be approximated as(21)
e
x
=
1
+
x
.Therefore (14) can be rewritten as (22)
ρ
=
ρ
0
1
-
C
P
0
-
P
,
ρ
ρ
0
=
1
-
C
P
0
-
P
,
∴
P
=
P
0
-
ρ
/
ρ
0
+
1
C
.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:(23)
C
g
=
-
1
v
∂
v
∂
p
T
.For real gas equation of state,(24)
v
=
n
R
T
z
P
.Differentiating with respect to pressure at constant temperature,(25)
∂
v
∂
p
T
=
n
R
T
1
p
∂
z
∂
p
-
z
p
2
.Substituting (23) into (25),(26)
C
g
=
1
p
-
1
z
d
z
d
p
.In terms of density,(27)
C
g
=
1
p
-
1
ρ
∂
ρ
∂
p
.This equation is applicable for real gas condition.For compressible fluid,(28)
C
g
=
1
p
-
1
ρ
∂
ρ
∂
p
,
1
ρ
∂
ρ
∂
p
=
1
p
-
C
g
,
∫
ρ
0
ρ
∂
ρ
ρ
=
∫
p
0
p
∂
p
p
-
∂
p
C
g
,
ln
⁡
ρ
ρ
0
=
ln
⁡
p
p
0
-
p
-
p
0
C
g
.Applying the power series for l
n
p,(29)
ln
⁡
p
=
p
-
1
-
p
-
1
2
2
+
⋯
+
-
1
n
p
-
1
n
n
+
⋯
0
<
p
≤
2
.Limit l
n
x to the 1st term only:(30)
ρ
ρ
0
-
1
=
p
p
o
-
1
-
p
-
p
0
C
g
,
ρ
ρ
0
=
p
p
o
-
p
-
p
0
C
g
,
(31)
ρ
ρ
0
=
p
-
p
ρ
o
c
g
-
p
o
2
c
g
p
o
,
p
o
ρ
ρ
o
-
p
o
2
c
g
=
p
1
-
p
o
c
g
,
(32)
P
=
P
o
ρ
-
P
o
2
ρ
o
c
g
ρ
o
1
-
P
o
c
g
.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].


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

Consider the following:(17)
-
C
∫
p
o
p
d
p
=
∫
ρ
0
ρ
∂
ρ
ρ
.Integrating(18)
e
C
P
0
-
P
=
ρ
ρ
0
,
C
P
0
-
P
=
ln
⁡
ρ
ρ
0
,
(19)
P
=
P
0
-
ln
⁡
ρ
/
ρ
0
C
,or applying the e
x expansion series,(20)
e
x
=
1
+
x
+
x
2
2
!
+
x
3
3
!
+
⋯
+
x
n
n
!
.Because the term C
P
0
-
P is very small, the e
x term can be approximated as(21)
e
x
=
1
+
x
.Therefore (14) can be rewritten as (22)
ρ
=
ρ
0
1
-
C
P
0
-
P
,
ρ
ρ
0
=
1
-
C
P
0
-
P
,
∴
P
=
P
0
-
ρ
/
ρ
0
+
1
C
.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:(23)
C
g
=
-
1
v
∂
v
∂
p
T
.For real gas equation of state,(24)
v
=
n
R
T
z
P
.Differentiating with respect to pressure at constant temperature,(25)
∂
v
∂
p
T
=
n
R
T
1
p
∂
z
∂
p
-
z
p
2
.Substituting (23) into (25),(26)
C
g
=
1
p
-
1
z
d
z
d
p
.In terms of density,(27)
C
g
=
1
p
-
1
ρ
∂
ρ
∂
p
.This equation is applicable for real gas condition.For compressible fluid,(28)
C
g
=
1
p
-
1
ρ
∂
ρ
∂
p
,
1
ρ
∂
ρ
∂
p
=
1
p
-
C
g
,
∫
ρ
0
ρ
∂
ρ
ρ
=
∫
p
0
p
∂
p
p
-
∂
p
C
g
,
ln
⁡
ρ
ρ
0
=
ln
⁡
p
p
0
-
p
-
p
0
C
g
.Applying the power series for l
n
p,(29)
ln
⁡
p
=
p
-
1
-
p
-
1
2
2
+
⋯
+
-
1
n
p
-
1
n
n
+
⋯
0
<
p
≤
2
.Limit l
n
x to the 1st term only:(30)
ρ
ρ
0
-
1
=
p
p
o
-
1
-
p
-
p
0
C
g
,
ρ
ρ
0
=
p
p
o
-
p
-
p
0
C
g
,
(31)
ρ
ρ
0
=
p
-
p
ρ
o
c
g
-
p
o
2
c
g
p
o
,
p
o
ρ
ρ
o
-
p
o
2
c
g
=
p
1
-
p
o
c
g
,
(32)
P
=
P
o
ρ
-
P
o
2
ρ
o
c
g
ρ
o
1
-
P
o
c
g
.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 (33)
P
i
=
ρ
g
p
g
+
ρ
o
p
o
+
ρ
w
p
w
ρ
g
+
ρ
o
+
ρ
w
.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 PhasesK

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(34)
k
ave
=
k
o
k
w
k
g
2
4
,where k
o = oil phase permeability, k
g = gas phase permeability, and k
w = 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 relativek 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 Figure2.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.
Table1 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 Example1.
Parameters
Design value
Eclipse model
Black oil
Model dimension
10 × 5 × 5
Length by width, ft by ft
500 × 400
Thicknessh, ft
250
PermeabilityK
x by K
y, mD
50.0  by 50.0
Porosity %
20
Well diameter, ft
0.60
Initial water saturationS
w
i,  %
22
Permeability,K, mD
50
Gas oil contact (GOC), ft
8820
Oil water contact (OWC), ft
9000.0
Initial pressure,P
i, psia
4000.0
Formation temperature,T, oF
200.0Assumption. (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 perforatedh
p = 30 ft between oil and water layer. Net sand thickness h = 250 ft.
(b)
Flowing + buildup sequence: well perforatedh
p = 30 ft inside the oil layer. Net sand thickness h = 250 ft.
(c)
Flowing + buildup sequence: well perforatedh
p = 30 ft between gas and oil layer. Net sand thickness h = 250 ft.
(d)
Falloff test; flowing + buildup sequence: well perforatedh
p = 30 ft inside the oil layer. Net sand thickness h = 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.Figure 3
Schematics of well perforation interval and sand thickness for oil + gas cap + water reservoir.
(a)
(b)
(c)
(d)Figure 4
Schematics of well perforation interval and sand thickness for gas condensate + water reservoir.
(a)
(b)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 permeabilitiesk using the specialised plot and k
a
v
e from the empirical model for three-phase conditions.The derivative in Figure6 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 d
p
' 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 andK estimation for scenario (a).A permeability value of 50.8 mD is estimated from the bottom-hole pressure BPR wherek
=
162.7
q
B
μ
/
m
h and m 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 k value while PDENGAS and PDENOIL differ, giving 15.8 and 403.7 mD, respectively. At h
=
50
f
t, the best k estimate is obtained depicting h
=
50
f
t as the thickness contributing to flow. At h
>
50 ft, k drops below the k imputed in the model.Using (34), k
a
v
e
=
47.3 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
K estimates for new approach versus conventional approach for scenario (a).
Parameters
Numerical density
k (mD)
Equivalent
h (ft)
BHP
50.8
PDENA
50.8
PDENG
15.8
50
PDENO
403.7
PDENW
50.8In scenario (b), the well is completed within the oil section to capture the pressure and fluid densities changes around the well. The derivative in Figure7 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 andK estimation for scenario (b).Using the estimated fluid phase permeabilities,k
a
v
e 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 k 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 h
=
150 ft (60% of sand thickness and 83% of oil thickness), the estimated k is still within range (k
=
48.9 mD). This indicates h
=
50 ft or 83% of oil thickness contributing to flow.Table 3
K estimates for new approach versus conventional approach for scenario (b).
Parameters
Numerical density
k (mD)
Equivalent
h (ft)
BHP
50.0
PDENA
50.0
PDENG
12.1
50
PDENO
484.2
PDENW
50.0To 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 Figure8. 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 K
a
v
e value of 57 and 52 mD from (34) is obtained for scenarios (c) and (d), respectively. This is slightly higher than k
=
50 mD imputed in the simulation model.Figure 8
Derivative andK estimation for scenario (c).For each of the fluid phase permeabilities estimates,k value of 50 mD is obtained using the conventional BPR for scenario (c), which is in line with the uniform k in the simulation model. This is the same for the PDENDWA and PDENWAT. Also, at h
=
150 ft (60% of sand thickness and 83% of oil thickness), the estimated k is still within range (k
=
41.2 mD). This also indicates that h
=
150 ft or 83% of oil thickness is contributing to flow. The k values for PDENGAS and PDENOIL are 20.9 and 484.2 mD, respectively. Summary of the result is shown in Table 4.Table 4
K estimates for new approach versus conventional approach for scenario (d).
Parameters
Numerical Densityk (mD)
Equivalenth (ft)
Scenario (d)
BHP
50.0
100
PDENA
50.0
PDENG
17.5
50
PDENO
490
250
PDENW
50.0
100Table 5
Comparison ofK  estimates between conventional and numerical density parameters.
Scenario
Numerical density method
Calculated
K (mD)
Conventional
k (mD)
Simulationmodel K (mD)
RelativeK
%  phasecontribution to flow
Phases
K (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
9.0However, 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 ifh
=
100 ft (40% of sand thickness and 56% of oil thickness). PDENGAS and PDENOIL differ, giving 35.0 mD at h
=
50 ft and 196.0 mD at h
=
250 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 andK estimation for scenario (d).In summary, for scenarios (a) to (d), thek 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 k 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 k
a
v
e 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 k 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. Table6 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
Thicknessh, ft
150
PermeabilityK
x by K
y, mD
400.0  by 300.0
Porosity, %
30
Well diameter, ft
1.15
Initial water saturationS
w
i, %
60
Permeability,K, mD
400
Gas oil contact (GOC), ft
6890
Oil water contact (OWC), ft
6890
Initial pressure,P
i, psi
4495
Formation temperature,T, °C 120
200.0Assumption. (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 perforatedh
p
=
30 ft between gas condensate and water layer. Net sand thickness h
=
150 ft.
(b)
Flowing + buildup sequence: well perforatedh
p
=
30 ft inside gas condensate layer. Net sand thickness h
=
150 ft.Figure 10 shows the production and shut-in sequence for 2 scenarios in Example 2.Figure 10
Production test sequence (buildup) for Example2.As in Example1, 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 andK 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 differentd
p
' stabilisation.Permeability value of 370.7 mD was obtained from the conventional method BPR wherek
=
1637
q
T
/
m
h is the same for the PDENDWA and PDENWAT, and m is obtained from the specialised plot. This is in line with the uniform k (k
x
=
400 mD and k
y
=
300 mD) imputed in the simulation model. However PDENGAS and PDENOIL differ, giving 35.0 mD and 3574.7 mD, respectively, if h
=
148 ft.Also the derivative in Figure12 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 d
p
' stabilisation.Figure 12
Derivative andK estimation for scenario (b).For each of the fluid phase permeabilities,k value of 340.9 mD at h
=
148 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 h
=
148 ft, respectively.For scenarios (a) and (b), a goodk
a
v
e 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 k 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
k (mD)
Equivalent
h (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
341This approach estimates the fluid phase permeabilities and the % of each phase contribution to flow at a given point; hence at several points the relativek can be generated as shown in Table 8.Table 8
Comparison ofK  estimates between conventional and numerical density parameters.
Scenario
Numerical density method
Calculated
K (mD)
Conventional
k (mD)
Simulationmodel K (mD)
RelativeK
%  phasecontribution to flow
Phases
K (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


## 4.1. Empirical Model Correlation between Fluid PhasesK

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(34)
k
ave
=
k
o
k
w
k
g
2
4
,where k
o = oil phase permeability, k
g = gas phase permeability, and k
w = 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 relativek 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 Figure2.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.
Table1 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 Example1.
Parameters
Design value
Eclipse model
Black oil
Model dimension
10 × 5 × 5
Length by width, ft by ft
500 × 400
Thicknessh, ft
250
PermeabilityK
x by K
y, mD
50.0  by 50.0
Porosity %
20
Well diameter, ft
0.60
Initial water saturationS
w
i,  %
22
Permeability,K, mD
50
Gas oil contact (GOC), ft
8820
Oil water contact (OWC), ft
9000.0
Initial pressure,P
i, psia
4000.0
Formation temperature,T, oF
200.0Assumption. (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 perforatedh
p = 30 ft between oil and water layer. Net sand thickness h = 250 ft.
(b)
Flowing + buildup sequence: well perforatedh
p = 30 ft inside the oil layer. Net sand thickness h = 250 ft.
(c)
Flowing + buildup sequence: well perforatedh
p = 30 ft between gas and oil layer. Net sand thickness h = 250 ft.
(d)
Falloff test; flowing + buildup sequence: well perforatedh
p = 30 ft inside the oil layer. Net sand thickness h = 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.Figure 3
Schematics of well perforation interval and sand thickness for oil + gas cap + water reservoir.
(a)
(b)
(c)
(d)Figure 4
Schematics of well perforation interval and sand thickness for gas condensate + water reservoir.
(a)
(b)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 permeabilitiesk using the specialised plot and k
a
v
e from the empirical model for three-phase conditions.The derivative in Figure6 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 d
p
' 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 andK estimation for scenario (a).A permeability value of 50.8 mD is estimated from the bottom-hole pressure BPR wherek
=
162.7
q
B
μ
/
m
h and m 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 k value while PDENGAS and PDENOIL differ, giving 15.8 and 403.7 mD, respectively. At h
=
50
f
t, the best k estimate is obtained depicting h
=
50
f
t as the thickness contributing to flow. At h
>
50 ft, k drops below the k imputed in the model.Using (34), k
a
v
e
=
47.3 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
K estimates for new approach versus conventional approach for scenario (a).
Parameters
Numerical density
k (mD)
Equivalent
h (ft)
BHP
50.8
PDENA
50.8
PDENG
15.8
50
PDENO
403.7
PDENW
50.8In scenario (b), the well is completed within the oil section to capture the pressure and fluid densities changes around the well. The derivative in Figure7 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 andK estimation for scenario (b).Using the estimated fluid phase permeabilities,k
a
v
e 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 k 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 h
=
150 ft (60% of sand thickness and 83% of oil thickness), the estimated k is still within range (k
=
48.9 mD). This indicates h
=
50 ft or 83% of oil thickness contributing to flow.Table 3
K estimates for new approach versus conventional approach for scenario (b).
Parameters
Numerical density
k (mD)
Equivalent
h (ft)
BHP
50.0
PDENA
50.0
PDENG
12.1
50
PDENO
484.2
PDENW
50.0To 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 Figure8. 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 K
a
v
e value of 57 and 52 mD from (34) is obtained for scenarios (c) and (d), respectively. This is slightly higher than k
=
50 mD imputed in the simulation model.Figure 8
Derivative andK estimation for scenario (c).For each of the fluid phase permeabilities estimates,k value of 50 mD is obtained using the conventional BPR for scenario (c), which is in line with the uniform k in the simulation model. This is the same for the PDENDWA and PDENWAT. Also, at h
=
150 ft (60% of sand thickness and 83% of oil thickness), the estimated k is still within range (k
=
41.2 mD). This also indicates that h
=
150 ft or 83% of oil thickness is contributing to flow. The k values for PDENGAS and PDENOIL are 20.9 and 484.2 mD, respectively. Summary of the result is shown in Table 4.Table 4
K estimates for new approach versus conventional approach for scenario (d).
Parameters
Numerical Densityk (mD)
Equivalenth (ft)
Scenario (d)
BHP
50.0
100
PDENA
50.0
PDENG
17.5
50
PDENO
490
250
PDENW
50.0
100Table 5
Comparison ofK  estimates between conventional and numerical density parameters.
Scenario
Numerical density method
Calculated
K (mD)
Conventional
k (mD)
Simulationmodel K (mD)
RelativeK
%  phasecontribution to flow
Phases
K (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
9.0However, 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 ifh
=
100 ft (40% of sand thickness and 56% of oil thickness). PDENGAS and PDENOIL differ, giving 35.0 mD at h
=
50 ft and 196.0 mD at h
=
250 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 andK estimation for scenario (d).In summary, for scenarios (a) to (d), thek 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 k 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 k
a
v
e 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 k 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. Table6 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
Thicknessh, ft
150
PermeabilityK
x by K
y, mD
400.0  by 300.0
Porosity, %
30
Well diameter, ft
1.15
Initial water saturationS
w
i, %
60
Permeability,K, mD
400
Gas oil contact (GOC), ft
6890
Oil water contact (OWC), ft
6890
Initial pressure,P
i, psi
4495
Formation temperature,T, °C 120
200.0Assumption. (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 perforatedh
p
=
30 ft between gas condensate and water layer. Net sand thickness h
=
150 ft.
(b)
Flowing + buildup sequence: well perforatedh
p
=
30 ft inside gas condensate layer. Net sand thickness h
=
150 ft.Figure 10 shows the production and shut-in sequence for 2 scenarios in Example 2.Figure 10
Production test sequence (buildup) for Example2.As in Example1, 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 andK 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 differentd
p
' stabilisation.Permeability value of 370.7 mD was obtained from the conventional method BPR wherek
=
1637
q
T
/
m
h is the same for the PDENDWA and PDENWAT, and m is obtained from the specialised plot. This is in line with the uniform k (k
x
=
400 mD and k
y
=
300 mD) imputed in the simulation model. However PDENGAS and PDENOIL differ, giving 35.0 mD and 3574.7 mD, respectively, if h
=
148 ft.Also the derivative in Figure12 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 d
p
' stabilisation.Figure 12
Derivative andK estimation for scenario (b).For each of the fluid phase permeabilities,k value of 340.9 mD at h
=
148 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 h
=
148 ft, respectively.For scenarios (a) and (b), a goodk
a
v
e 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 k 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
k (mD)
Equivalent
h (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
341This approach estimates the fluid phase permeabilities and the % of each phase contribution to flow at a given point; hence at several points the relativek can be generated as shown in Table 8.Table 8
Comparison ofK  estimates between conventional and numerical density parameters.
Scenario
Numerical density method
Calculated
K (mD)
Conventional
k (mD)
Simulationmodel K (mD)
RelativeK
%  phasecontribution to flow
Phases
K (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(35)
1
r
∂
ρ
∂
r
r
∂
ρ
∂
r
=
ϕ
μ
c
k
∂
ρ
∂
t
.For oil,(36a)
1
r
∂
ρ
∂
r
r
∂
ρ
o
∂
r
=
ϕ
μ
o
c
k
o
∂
ρ
o
∂
t
.For water phase, (36b)
1
r
∂
ρ
∂
r
r
∂
ρ
w
∂
r
=
ϕ
μ
w
c
k
w
∂
ρ
w
∂
t
.For gas phase,(36c)
1
r
∂
ρ
∂
r
r
∂
ρ
g
∂
r
=
ϕ
μ
g
c
k
g
∂
ρ
g
∂
t
.


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

From (22),(37)
P
=
P
o
-
ρ
/
ρ
o
+
1
c
o
.Differentiating with respect to ρ,(38)
∂
P
∂
ρ
=
-
1
c
o
ρ
o
,
∂
P
=
-
∂
ρ
c
o
ρ
o
.From Darcy flow equation,(39)
q
=
-
2
π
k
h
μ
r
∂
ρ
∂
r
.Substitute for ∂
P:(40)
q
=
2
π
k
h
μ
r
c
o
ρ
o
∂
ρ
∂
r
.To derive the analytical density transient equation for each phase, the following assumption/conditions are applicable.For oil phase, from (35),(41)
1
r
∂
ρ
∂
r
r
∂
ρ
∂
r
=
ϕ
μ
c
k
∂
ρ
∂
t
.Initial condition is as follows:(42a)
ρ
r
,
t
=
0
=
ρ
i
.BC at the wellbore is as follows:(42b)
lim
r
→
0
⁡
2
π
k
h
μ
r
c
o
ρ
o
∂
ρ
∂
r
=
Q
.BC at infirmity is as follows:(42c)
lim
r
→
∞
⁡
ρ
r
,
t
=
ρ
i
.Using the Boltzmann transformation, consider the following.Assumingη
=
ϕ
μ
c
r
2
/
k
t,(43)
∂
ρ
∂
r
=
∂
ρ
∂
η
∂
η
∂
r
=
2
ϕ
μ
c
r
k
t
∂
ρ
∂
η
=
ϕ
μ
c
r
2
k
t
2
r
∂
ρ
∂
η
=
2
η
r
∂
ρ
∂
η
.Therefore(44)
1
r
∂
∂
r
r
∂
ρ
∂
r
=
2
η
r
2
∂
∂
η
2
η
∂
ρ
∂
η
=
4
η
r
2
∂
∂
η
η
∂
ρ
∂
η
.Differentiating with respect to r is equal to differentiating with respect to η; multiply by 2
η
/
r.From (35), the L.H.S is resolved as follows:(45)
∂
ρ
∂
t
=
∂
ρ
∂
η
∂
η
∂
t
=
ϕ
μ
c
r
k
t
2
∂
ρ
∂
η
=
-
η
t
∂
ρ
∂
η
,
(46)
ϕ
μ
c
k
∂
ρ
∂
t
=
-
ϕ
μ
c
k
η
t
∂
ρ
∂
t
=
-
ϕ
μ
c
r
2
k
t
η
r
2
∂
ρ
∂
η
=
-
η
2
r
2
∂
ρ
∂
η
.Equating (44) and (46), we have(47)
∂
∂
η
η
∂
ρ
∂
η
=
-
η
4
∂
ρ
∂
η
.This is the simplified ordinary differential equation of ρ as a function of η.Apply boundary conditions as follows.Initial condition is as follows:(48a)
lim
r
→
∞
⁡
ρ
η
=
ρ
i
.Substituting (43) into (40),(48b)
lim
r
→
0
⁡
2
π
k
h
μ
r
c
o
ρ
o
∂
ρ
∂
r
=
Q
,
(49)
lim
r
→
0
⁡
η
∂
ρ
∂
η
=
ϕ
μ
c
o
ρ
o
4
π
k
h
.Assume that m
=
η
∂
ρ
/
∂
η.From (47),(50)
∂
m
∂
η
=
-
m
4
.Integrating from η
=
0 to η,(51)
⟹
∫
m
0
m
η
∂
m
∂
η
=
-
∫
0
η
∂
η
4
⟹
ln
⁡
m
η
m
0
=
-
η
4
⟹
m
η
=
m
0
e
-
η
/
4
.Recall boundary condition(52)
lim
r
→
0
⁡
η
∂
ρ
∂
η
=
Q
μ
c
o
ρ
o
4
π
k
h
,
m
0
=
Q
μ
c
o
ρ
o
4
π
k
h
,
m
η
=
Q
μ
c
o
ρ
o
4
π
k
h
e
-
η
/
4
.Recall that m
=
η
∂
ρ
/
∂
η.Therefore,(53)
∂
ρ
η
∂
η
=
μ
Q
c
o
ρ
o
4
π
k
h
e
-
η
/
4
η
.Integrating from η
=
∞ where ρ
=
ρ
i,(54)
∫
ρ
i
ρ
η
∂
ρ
=
-
∫
∞
η
μ
Q
c
o
ρ
o
4
π
k
h
e
-
η
/
4
η
∂
η
,
ρ
η
=
ρ
i
-
μ
Q
c
o
ρ
o
4
π
k
h
-
∫
η
∞
e
-
η
/
4
η
∂
η
,where(55)
η
=
ϕ
μ
c
r
2
k
t
.Assuming u
=
η
/
4, then ∂
η
/
η
=
∂
u
/
u and u
=
ϕ
μ
c
r
2
/
4
k
t.Hence,(56)
ρ
=
ρ
i
-
μ
Q
c
o
ρ
o
4
π
k
h
-
∫
ϕ
μ
c
r
2
/
4
k
t
∞
e
-
η
/
4
η
∂
η
,where(57)
-
E
i
-
x
=
∫
x
∞
e
-
u
u
∂
u
.Known as the exponential integral function defined by [13],(58)
ρ
r
,
t
=
ρ
i
+
μ
Q
c
o
ρ
o
4
π
k
h
E
i
-
x
,where x
=
ϕ
μ
c
r
2
/
4
k
t and (59)
-
E
i
-
x
=
∫
x
∞
e
-
u
u
∂
u
.At large times, x will be small applying Taylor’s series and integrating(60)
-
E
i
-
x
=
ln
⁡
γ
,
ln
⁡
γ
=
ln
⁡
1.781
=
0.5772
,where γ is known as Euler’s number.For1
/
4
x
>
25, consider the following.Then(61)
ρ
r
,
t
=
ρ
i
+
μ
Q
c
o
ρ
o
4
π
k
h
ln
⁡
x
+
ln
⁡
γ
,
ρ
r
,
t
=
ρ
i
+
μ
Q
c
o
ρ
o
4
π
k
h
ln
⁡
ϕ
μ
c
r
2
γ
4
k
t
,
ρ
r
,
t
=
ρ
i
-
μ
Q
c
o
ρ
o
4
π
k
h
ln
⁡
2.246
k
t
ϕ
μ
c
r
2
,
ρ
r
,
t
=
ρ
i
-
μ
Q
c
o
ρ
o
4
π
k
h
ln
⁡
2.246
k
t
ϕ
μ
c
r
2
+
0.80907
.Plotting ρ
w
t versus l
n
t will yield a straight line at longer time and the slope of the line is given as(62)
∂
ρ
w
∂
ln
⁡
t
=
m
oil
=
μ
Q
c
o
ρ
o
4
π
k
h
.Therefore,(63)
k
h
=
μ
Q
c
o
ρ
o
4
π
m
oil
.


### 5.2. For Water Phase

The radial density equation is given as(64)
ρ
r
,
t
=
ρ
i
-
μ
Q
c
o
ρ
o
4
π
k
h
ln
⁡
2.246
k
t
ϕ
μ
c
r
2
+
0.80907
.Plotting ρ
w
t versus l
n
t will yield a straight line at longer time and the slope of the line is given as(65)
∂
ρ
w
∂
ln
⁡
t
=
m
water
=
μ
Q
c
o
ρ
o
4
π
k
h
,where(66)
k
h
=
μ
Q
c
o
ρ
o
4
π
m
water
.


### 5.3. For Gas Phase

From equation for compressible fluid such as gas, consider the following.From (32),(67)
P
=
P
o
ρ
-
P
o
2
ρ
o
c
g
ρ
o
1
-
P
o
c
g
,
∂
P
∂
ρ
=
P
o
ρ
o
1
-
P
o
c
g
,
∂
P
=
P
o
ρ
o
1
-
P
o
c
g
∂
ρ
.From the fundamental,(68)
n
=
P
V
z
R
T
.At standard condition,(69)
P
V
z
T
=
P
sc
V
sc
T
sc
,
P
5.615
q
z
T
=
P
sc
Q
sc
T
sc
,
q
=
P
sc
T
sc
z
T
P
Q
sc
5.615
,
q
=
P
sc
T
sc
Q
sc
5.615
z
T
P
=
2
π
k
h
μ
r
∂
p
∂
r
,
q
=
P
sc
T
sc
Q
sc
5.615
z
T
=
2
π
k
h
μ
r
P
∂
p
∂
r
,where(70)
∂
P
=
P
o
ρ
o
1
-
P
o
c
g
∂
ρ
,
α
=
P
o
ρ
o
1
-
P
o
c
g
,
∂
P
=
α
∂
ρ
,
P
=
ρ
R
z
T
.Substitute into the equation, where(71)
m
ρ
i
-
m
ρ
w
f
=
2
μ
∫
ρ
w
f
ρ
i
ρ
∂
ρ
,
m
ρ
i
-
m
ρ
w
f
=
ρ
i
2
-
ρ
w
f
2
μ
,
ρ
¯
=
ρ
i
2
+
ρ
w
f
2
2
.From the diffusivity equation for gas,(72)
1
r
∂
ρ
∂
r
r
∂
ρ
g
∂
r
=
ϕ
μ
g
c
k
g
∂
ρ
g
∂
t
.Initial condition is as follows:(73)
ρ
r
,
t
=
0
=
ρ
i
.BC at the wellbore is as follows:(74)
lim
r
→
0
⁡
4
π
k
h
α
R
μ
5.615
T
sc
P
sc
r
ρ
∂
ρ
∂
r
=
Q
.BC at infirmity is as follows:(75)
lim
r
→
∞
⁡
ρ
r
,
t
=
ρ
i
,where R, P
s
c, and T
s
c are known as temperature, pressure, and gas constant at standard condition.Therefore, integrating fromη
=
∞ where ρ
=
ρ
i,(76)
∫
ρ
i
ρ
η
ρ
∂
ρ
=
-
∫
∞
η
μ
Q
4
π
k
h
α
R
P
sc
5.615
T
sc
e
-
η
/
4
η
∂
η
,
ρ
2
η
=
ρ
i
2
-
μ
Q
4
π
k
h
α
R
P
sc
5.615
T
sc
-
∫
η
∞
e
-
η
/
4
η
∂
η
,where(77)
η
=
ϕ
μ
c
r
2
k
t
.Assume that u
=
η
/
4  ∂
η
/
η
=
∂
u
/
u and u
=
ϕ
μ
c
r
2
/
4
k
t.Hence(78)
ρ
2
=
ρ
i
2
-
μ
Q
4
π
k
h
α
R
P
sc
5.615
T
sc
-
∫
ϕ
μ
c
r
2
/
4
k
t
∞
e
-
η
/
4
η
,where -
E
i
-
x
=
∫
x
∞
e
-
u
/
u
∂
u.Known as the exponential integral function defined by [13],(79)
ρ
2
=
ρ
i
2
-
μ
Q
4
π
k
h
α
R
P
sc
5.615
T
sc
-
E
i
-
x
,where x
=
ϕ
μ
c
r
2
/
4
k
t and -
E
i
-
x
=
∫
x
∞
e
-
u
/
u
∂
u.At large times,x will be small applying Taylor’s series and integrating(80)
-
E
i
-
x
=
ln
⁡
γ
,
ln
⁡
γ
=
ln
⁡
1.781
=
0.5772
,where γ is known as Euler’s number.For1
/
4
x
>
25, consider the following.Then,(81)
ρ
2
r
,
t
=
ρ
i
2
+
μ
Q
4
π
k
h
α
R
P
sc
5.615
T
sc
ln
⁡
x
+
ln
⁡
γ
,
(82)
ρ
2
r
,
t
=
ρ
i
2
+
μ
Q
4
π
k
h
α
R
P
sc
5.615
T
sc
ln
⁡
ϕ
μ
c
r
2
γ
4
k
t
,
ρ
2
r
,
t
=
ρ
i
2
-
μ
Q
4
π
k
h
α
R
P
sc
5.615
T
sc
ln
⁡
2.246
k
t
ϕ
μ
c
r
2
,
m
ρ
i
-
m
ρ
w
f
=
2
μ
∫
ρ
w
f
ρ
i
ρ
∂
ρ
,
m
ρ
w
f
=
m
ρ
i
-
μ
Q
4
π
k
h
α
R
P
sc
5.615
T
sc
·
ln
⁡
2.246
k
t
ϕ
μ
c
r
2
+
0.80907
,
∂
m
ρ
w
∂
ln
⁡
t
=
m
gas
=
μ
Q
4
π
k
h
α
R
P
sc
5.615
T
sc
.Plotting ρ
w
2
t or m
ρ
w versus l
n
t will yield a straight line at longer time and the slope of the line is given as follows:(83)
k
h
=
μ
Q
4
π
k
h
α
R
P
sc
5.615
T
sc
1
m
gas
.Example 3.
Data from Example1(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.
Q
∗
∗ used for k calculation is the average rate for all flowing periods.From Figures13 and 14, the calculated slopes of the radial flow for oil and water phases are given as(84)
∂
ρ
w
∂
ln
⁡
t
=
m
oil
=
162.7
μ
Q
c
o
ρ
o
k
h
=
-
0.009
,
∂
ρ
w
∂
ln
⁡
t
=
m
water
=
162.7
μ
Q
c
o
ρ
o
k
h
=
-
0.0003
.The estimated phase permeabilities are(85)
k
oil
=
162.7
μ
Q
c
o
ρ
o
h
m
oil
=
445
mD
,
k
water
=
162.7
μ
Q
c
o
ρ
o
h
m
water
=
50
mD
.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 Figure13 is given as(86)
∂
m
ρ
w
∂
ln
⁡
t
=
m
gas
=
1637
μ
Q
k
h
α
R
=
-
1.2195
.And estimated gas permeability is(87)
k
gas
=
1637
μ
Q
o
h
α
R
m
gas
=
17.0
mD
.Using the empirical model for all three phases, the average estimated permeability is as follows:(88)
k
ave
=
50
×
445
×
17
.
0
2
4
,
k
ave
=
50
mD
.Example 4.
Data from Example1(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 Figures15 and 16, the calculated slopes of the radial flow for oil and water phases are given as(89)
∂
ρ
w
∂
ln
⁡
t
=
m
oil
=
162.7
μ
Q
c
o
ρ
o
k
h
=
-
0.009
,
∂
ρ
w
∂
ln
⁡
t
=
m
water
=
162.7
μ
Q
c
o
ρ
o
k
h
=
-
0.0003
.The estimated phase permeabilities are(90)
k
oil
=
162.7
μ
Q
c
o
ρ
o
h
m
oil
=
435.2
mD
,
k
water
=
162.7
μ
Q
c
o
ρ
o
h
m
water
=
49
mD
.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 Figure15 is given as(91)
∂
m
ρ
w
∂
ln
⁡
t
=
m
gas
=
1637
μ
Q
k
h
α
R
=
-
1.2502
.And estimated gas permeability is(92)
k
gas
=
1637
μ
Q
o
h
α
R
m
gas
=
17.9
mD
.Using the empirical model for all three phases, the average estimated permeability is as follows:(93)
k
ave
=
49
×
435.2
×
17
.
9
2
4
,
k
ave
=
51
mD
.


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

From (22),(37)
P
=
P
o
-
ρ
/
ρ
o
+
1
c
o
.Differentiating with respect to ρ,(38)
∂
P
∂
ρ
=
-
1
c
o
ρ
o
,
∂
P
=
-
∂
ρ
c
o
ρ
o
.From Darcy flow equation,(39)
q
=
-
2
π
k
h
μ
r
∂
ρ
∂
r
.Substitute for ∂
P:(40)
q
=
2
π
k
h
μ
r
c
o
ρ
o
∂
ρ
∂
r
.To derive the analytical density transient equation for each phase, the following assumption/conditions are applicable.For oil phase, from (35),(41)
1
r
∂
ρ
∂
r
r
∂
ρ
∂
r
=
ϕ
μ
c
k
∂
ρ
∂
t
.Initial condition is as follows:(42a)
ρ
r
,
t
=
0
=
ρ
i
.BC at the wellbore is as follows:(42b)
lim
r
→
0
⁡
2
π
k
h
μ
r
c
o
ρ
o
∂
ρ
∂
r
=
Q
.BC at infirmity is as follows:(42c)
lim
r
→
∞
⁡
ρ
r
,
t
=
ρ
i
.Using the Boltzmann transformation, consider the following.Assumingη
=
ϕ
μ
c
r
2
/
k
t,(43)
∂
ρ
∂
r
=
∂
ρ
∂
η
∂
η
∂
r
=
2
ϕ
μ
c
r
k
t
∂
ρ
∂
η
=
ϕ
μ
c
r
2
k
t
2
r
∂
ρ
∂
η
=
2
η
r
∂
ρ
∂
η
.Therefore(44)
1
r
∂
∂
r
r
∂
ρ
∂
r
=
2
η
r
2
∂
∂
η
2
η
∂
ρ
∂
η
=
4
η
r
2
∂
∂
η
η
∂
ρ
∂
η
.Differentiating with respect to r is equal to differentiating with respect to η; multiply by 2
η
/
r.From (35), the L.H.S is resolved as follows:(45)
∂
ρ
∂
t
=
∂
ρ
∂
η
∂
η
∂
t
=
ϕ
μ
c
r
k
t
2
∂
ρ
∂
η
=
-
η
t
∂
ρ
∂
η
,
(46)
ϕ
μ
c
k
∂
ρ
∂
t
=
-
ϕ
μ
c
k
η
t
∂
ρ
∂
t
=
-
ϕ
μ
c
r
2
k
t
η
r
2
∂
ρ
∂
η
=
-
η
2
r
2
∂
ρ
∂
η
.Equating (44) and (46), we have(47)
∂
∂
η
η
∂
ρ
∂
η
=
-
η
4
∂
ρ
∂
η
.This is the simplified ordinary differential equation of ρ as a function of η.Apply boundary conditions as follows.Initial condition is as follows:(48a)
lim
r
→
∞
⁡
ρ
η
=
ρ
i
.Substituting (43) into (40),(48b)
lim
r
→
0
⁡
2
π
k
h
μ
r
c
o
ρ
o
∂
ρ
∂
r
=
Q
,
(49)
lim
r
→
0
⁡
η
∂
ρ
∂
η
=
ϕ
μ
c
o
ρ
o
4
π
k
h
.Assume that m
=
η
∂
ρ
/
∂
η.From (47),(50)
∂
m
∂
η
=
-
m
4
.Integrating from η
=
0 to η,(51)
⟹
∫
m
0
m
η
∂
m
∂
η
=
-
∫
0
η
∂
η
4
⟹
ln
⁡
m
η
m
0
=
-
η
4
⟹
m
η
=
m
0
e
-
η
/
4
.Recall boundary condition(52)
lim
r
→
0
⁡
η
∂
ρ
∂
η
=
Q
μ
c
o
ρ
o
4
π
k
h
,
m
0
=
Q
μ
c
o
ρ
o
4
π
k
h
,
m
η
=
Q
μ
c
o
ρ
o
4
π
k
h
e
-
η
/
4
.Recall that m
=
η
∂
ρ
/
∂
η.Therefore,(53)
∂
ρ
η
∂
η
=
μ
Q
c
o
ρ
o
4
π
k
h
e
-
η
/
4
η
.Integrating from η
=
∞ where ρ
=
ρ
i,(54)
∫
ρ
i
ρ
η
∂
ρ
=
-
∫
∞
η
μ
Q
c
o
ρ
o
4
π
k
h
e
-
η
/
4
η
∂
η
,
ρ
η
=
ρ
i
-
μ
Q
c
o
ρ
o
4
π
k
h
-
∫
η
∞
e
-
η
/
4
η
∂
η
,where(55)
η
=
ϕ
μ
c
r
2
k
t
.Assuming u
=
η
/
4, then ∂
η
/
η
=
∂
u
/
u and u
=
ϕ
μ
c
r
2
/
4
k
t.Hence,(56)
ρ
=
ρ
i
-
μ
Q
c
o
ρ
o
4
π
k
h
-
∫
ϕ
μ
c
r
2
/
4
k
t
∞
e
-
η
/
4
η
∂
η
,where(57)
-
E
i
-
x
=
∫
x
∞
e
-
u
u
∂
u
.Known as the exponential integral function defined by [13],(58)
ρ
r
,
t
=
ρ
i
+
μ
Q
c
o
ρ
o
4
π
k
h
E
i
-
x
,where x
=
ϕ
μ
c
r
2
/
4
k
t and (59)
-
E
i
-
x
=
∫
x
∞
e
-
u
u
∂
u
.At large times, x will be small applying Taylor’s series and integrating(60)
-
E
i
-
x
=
ln
⁡
γ
,
ln
⁡
γ
=
ln
⁡
1.781
=
0.5772
,where γ is known as Euler’s number.For1
/
4
x
>
25, consider the following.Then(61)
ρ
r
,
t
=
ρ
i
+
μ
Q
c
o
ρ
o
4
π
k
h
ln
⁡
x
+
ln
⁡
γ
,
ρ
r
,
t
=
ρ
i
+
μ
Q
c
o
ρ
o
4
π
k
h
ln
⁡
ϕ
μ
c
r
2
γ
4
k
t
,
ρ
r
,
t
=
ρ
i
-
μ
Q
c
o
ρ
o
4
π
k
h
ln
⁡
2.246
k
t
ϕ
μ
c
r
2
,
ρ
r
,
t
=
ρ
i
-
μ
Q
c
o
ρ
o
4
π
k
h
ln
⁡
2.246
k
t
ϕ
μ
c
r
2
+
0.80907
.Plotting ρ
w
t versus l
n
t will yield a straight line at longer time and the slope of the line is given as(62)
∂
ρ
w
∂
ln
⁡
t
=
m
oil
=
μ
Q
c
o
ρ
o
4
π
k
h
.Therefore,(63)
k
h
=
μ
Q
c
o
ρ
o
4
π
m
oil
.


## 5.2. For Water Phase

The radial density equation is given as(64)
ρ
r
,
t
=
ρ
i
-
μ
Q
c
o
ρ
o
4
π
k
h
ln
⁡
2.246
k
t
ϕ
μ
c
r
2
+
0.80907
.Plotting ρ
w
t versus l
n
t will yield a straight line at longer time and the slope of the line is given as(65)
∂
ρ
w
∂
ln
⁡
t
=
m
water
=
μ
Q
c
o
ρ
o
4
π
k
h
,where(66)
k
h
=
μ
Q
c
o
ρ
o
4
π
m
water
.


## 5.3. For Gas Phase

From equation for compressible fluid such as gas, consider the following.From (32),(67)
P
=
P
o
ρ
-
P
o
2
ρ
o
c
g
ρ
o
1
-
P
o
c
g
,
∂
P
∂
ρ
=
P
o
ρ
o
1
-
P
o
c
g
,
∂
P
=
P
o
ρ
o
1
-
P
o
c
g
∂
ρ
.From the fundamental,(68)
n
=
P
V
z
R
T
.At standard condition,(69)
P
V
z
T
=
P
sc
V
sc
T
sc
,
P
5.615
q
z
T
=
P
sc
Q
sc
T
sc
,
q
=
P
sc
T
sc
z
T
P
Q
sc
5.615
,
q
=
P
sc
T
sc
Q
sc
5.615
z
T
P
=
2
π
k
h
μ
r
∂
p
∂
r
,
q
=
P
sc
T
sc
Q
sc
5.615
z
T
=
2
π
k
h
μ
r
P
∂
p
∂
r
,where(70)
∂
P
=
P
o
ρ
o
1
-
P
o
c
g
∂
ρ
,
α
=
P
o
ρ
o
1
-
P
o
c
g
,
∂
P
=
α
∂
ρ
,
P
=
ρ
R
z
T
.Substitute into the equation, where(71)
m
ρ
i
-
m
ρ
w
f
=
2
μ
∫
ρ
w
f
ρ
i
ρ
∂
ρ
,
m
ρ
i
-
m
ρ
w
f
=
ρ
i
2
-
ρ
w
f
2
μ
,
ρ
¯
=
ρ
i
2
+
ρ
w
f
2
2
.From the diffusivity equation for gas,(72)
1
r
∂
ρ
∂
r
r
∂
ρ
g
∂
r
=
ϕ
μ
g
c
k
g
∂
ρ
g
∂
t
.Initial condition is as follows:(73)
ρ
r
,
t
=
0
=
ρ
i
.BC at the wellbore is as follows:(74)
lim
r
→
0
⁡
4
π
k
h
α
R
μ
5.615
T
sc
P
sc
r
ρ
∂
ρ
∂
r
=
Q
.BC at infirmity is as follows:(75)
lim
r
→
∞
⁡
ρ
r
,
t
=
ρ
i
,where R, P
s
c, and T
s
c are known as temperature, pressure, and gas constant at standard condition.Therefore, integrating fromη
=
∞ where ρ
=
ρ
i,(76)
∫
ρ
i
ρ
η
ρ
∂
ρ
=
-
∫
∞
η
μ
Q
4
π
k
h
α
R
P
sc
5.615
T
sc
e
-
η
/
4
η
∂
η
,
ρ
2
η
=
ρ
i
2
-
μ
Q
4
π
k
h
α
R
P
sc
5.615
T
sc
-
∫
η
∞
e
-
η
/
4
η
∂
η
,where(77)
η
=
ϕ
μ
c
r
2
k
t
.Assume that u
=
η
/
4  ∂
η
/
η
=
∂
u
/
u and u
=
ϕ
μ
c
r
2
/
4
k
t.Hence(78)
ρ
2
=
ρ
i
2
-
μ
Q
4
π
k
h
α
R
P
sc
5.615
T
sc
-
∫
ϕ
μ
c
r
2
/
4
k
t
∞
e
-
η
/
4
η
,where -
E
i
-
x
=
∫
x
∞
e
-
u
/
u
∂
u.Known as the exponential integral function defined by [13],(79)
ρ
2
=
ρ
i
2
-
μ
Q
4
π
k
h
α
R
P
sc
5.615
T
sc
-
E
i
-
x
,where x
=
ϕ
μ
c
r
2
/
4
k
t and -
E
i
-
x
=
∫
x
∞
e
-
u
/
u
∂
u.At large times,x will be small applying Taylor’s series and integrating(80)
-
E
i
-
x
=
ln
⁡
γ
,
ln
⁡
γ
=
ln
⁡
1.781
=
0.5772
,where γ is known as Euler’s number.For1
/
4
x
>
25, consider the following.Then,(81)
ρ
2
r
,
t
=
ρ
i
2
+
μ
Q
4
π
k
h
α
R
P
sc
5.615
T
sc
ln
⁡
x
+
ln
⁡
γ
,
(82)
ρ
2
r
,
t
=
ρ
i
2
+
μ
Q
4
π
k
h
α
R
P
sc
5.615
T
sc
ln
⁡
ϕ
μ
c
r
2
γ
4
k
t
,
ρ
2
r
,
t
=
ρ
i
2
-
μ
Q
4
π
k
h
α
R
P
sc
5.615
T
sc
ln
⁡
2.246
k
t
ϕ
μ
c
r
2
,
m
ρ
i
-
m
ρ
w
f
=
2
μ
∫
ρ
w
f
ρ
i
ρ
∂
ρ
,
m
ρ
w
f
=
m
ρ
i
-
μ
Q
4
π
k
h
α
R
P
sc
5.615
T
sc
·
ln
⁡
2.246
k
t
ϕ
μ
c
r
2
+
0.80907
,
∂
m
ρ
w
∂
ln
⁡
t
=
m
gas
=
μ
Q
4
π
k
h
α
R
P
sc
5.615
T
sc
.Plotting ρ
w
2
t or m
ρ
w versus l
n
t will yield a straight line at longer time and the slope of the line is given as follows:(83)
k
h
=
μ
Q
4
π
k
h
α
R
P
sc
5.615
T
sc
1
m
gas
.Example 3.
Data from Example1(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.
Q
∗
∗ used for k calculation is the average rate for all flowing periods.From Figures13 and 14, the calculated slopes of the radial flow for oil and water phases are given as(84)
∂
ρ
w
∂
ln
⁡
t
=
m
oil
=
162.7
μ
Q
c
o
ρ
o
k
h
=
-
0.009
,
∂
ρ
w
∂
ln
⁡
t
=
m
water
=
162.7
μ
Q
c
o
ρ
o
k
h
=
-
0.0003
.The estimated phase permeabilities are(85)
k
oil
=
162.7
μ
Q
c
o
ρ
o
h
m
oil
=
445
mD
,
k
water
=
162.7
μ
Q
c
o
ρ
o
h
m
water
=
50
mD
.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 Figure13 is given as(86)
∂
m
ρ
w
∂
ln
⁡
t
=
m
gas
=
1637
μ
Q
k
h
α
R
=
-
1.2195
.And estimated gas permeability is(87)
k
gas
=
1637
μ
Q
o
h
α
R
m
gas
=
17.0
mD
.Using the empirical model for all three phases, the average estimated permeability is as follows:(88)
k
ave
=
50
×
445
×
17
.
0
2
4
,
k
ave
=
50
mD
.Example 4.
Data from Example1(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 Figures15 and 16, the calculated slopes of the radial flow for oil and water phases are given as(89)
∂
ρ
w
∂
ln
⁡
t
=
m
oil
=
162.7
μ
Q
c
o
ρ
o
k
h
=
-
0.009
,
∂
ρ
w
∂
ln
⁡
t
=
m
water
=
162.7
μ
Q
c
o
ρ
o
k
h
=
-
0.0003
.The estimated phase permeabilities are(90)
k
oil
=
162.7
μ
Q
c
o
ρ
o
h
m
oil
=
435.2
mD
,
k
water
=
162.7
μ
Q
c
o
ρ
o
h
m
water
=
49
mD
.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 Figure15 is given as(91)
∂
m
ρ
w
∂
ln
⁡
t
=
m
gas
=
1637
μ
Q
k
h
α
R
=
-
1.2502
.And estimated gas permeability is(92)
k
gas
=
1637
μ
Q
o
h
α
R
m
gas
=
17.9
mD
.Using the empirical model for all three phases, the average estimated permeability is as follows:(93)
k
ave
=
49
×
435.2
×
17
.
9
2
4
,
k
ave
=
51
mD
.


## 6. Conclusion

The following inferences were drawn from the six scenarios reviewed:(i)
Results from the three-phase fluid empirical model developed for average effectivek
a
v
e 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 samek 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 effectivek as the conventional method.
(iv)
For oil reservoir system (scenarios (a) to (d)),k
=
50.0 mD is achieved if the thickness contributing to flow ranges from 50 to 150 ft while, for gas condensate reservoir (scenarios (a) to (b)), k ranges between 340 and 371 mD which is achieved if the net sand thickness h
=
148 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 severald
p
' stabilisation points, the relative k 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.



---

*Source: 214084-2015-07-12.xml*