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

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 pressuredensity 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 k 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 kave ranges between 43 and 57mD for scenarios (a) to (d) in Example 1.0 and 404mD for scenarios (a) to (b) in Example 2.0 using the new fluid phase empiricalmodel forK estimation.This is within the k value used in the simulationmodel and likewise that estimated from the conventional BPRmethod. 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 BPRmethod.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 dp stabilisation points, the relative k can be generated.


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][7][8][9][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: 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.

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: Mass rate in − Mass rate out = Mass rate storage.(6) 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.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.

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 [ 0 − ] 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 ln, Limit ln 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].

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.

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   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.
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.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.
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  ave from the empirical model for threephase 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 (PDEN-WAT → 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.
A permeability value of 50.8 mD is estimated from the bottom-hole pressure BPR where  = 162.7/ℎand  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 ℎ = 50 ft, the best  estimate is obtained depicting ℎ = 50 ft as the thickness contributing to flow.At ℎ > 50 ft,  drops below the  imputed in the model.
Using (34),  ave = 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.
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.
Using the estimated fluid phase permeabilities,  ave 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   ( = 48.9mD).This indicates ℎ = 50 ft or 83% of oil thickness contributing to flow.
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  ave value of 57 and 52 mD from (34) is obtained for scenarios (c) and (d), respectively.This is slightly higher than  = 50 mD imputed in the simulation model.
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 ℎ = 150 ft (60% of sand thickness and 83% of oil thickness), the estimated  is still within range ( = 41.2 mD).This also indicates that ℎ = 150 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.
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 ℎ = 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.
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  ave 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.

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 ℎ  = 30 ft between gas condensate and water layer.Net sand thickness ℎ = 150 ft. Figure 10 shows the production and shut-in sequence for 2 scenarios in 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.
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  = 1637/ℎ is the same for the PDENDWA and PDENWAT, and  is obtained from the specialised plot.This is in line with the uniform  ( = 400 mD and  = 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 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.
For each of the fluid phase permeabilities,  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 good  ave 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.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.

Density Related Radial Flow Equation Derivation for Each Fluid Phase
Radial diffusivity equation is given as For water phase, For gas phase,

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.
Differentiating with respect to  is equal to differentiating with respect to ; multiply by 2/.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 .
Plotting   () versus ln() will yield a straight line at longer time and the slope of the line is given as where Assume that  = (/4) (/) = / and  =  2 /4.Hence where −(−) = ∫ ∞  ( − /).Known as the exponential integral function defined by [13], where  =  The calculated slope from the semilog plot shown in Figure 13  (90) Also, for the gas phase, consider the following.
The calculated slope from the semilog plot shown in Figure 15  (93)

Conclusion
The following inferences were drawn from the six scenarios reviewed: (i) Results from the three-phase fluid empirical model developed for average effective  ave 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)),  = 50.0mD 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 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 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, PDE-NOIL, PDENGAS, PDENWAT, and PDENDW) depict different fingerprint, the density derivative will serve as support to distinguish between reservoir and nonreservoir response.

Figure 1 :
Figure 1: Schematics of basic fluid flows concept.

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

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 ℎ, 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 9000.0Initial pressure,   , psia 4000.0Formation temperature, , oF 200.0 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.
production test Net sand thickness h = 250 ft water layer Well perforated h p = 30 ft between oil and Flowing + buildup sequence (a) Gas Oil Water Case 2: production test Flowing + buildup sequence Net sand thickness h = 250 ft Well perforated h p = 30 ft inside the oil layer.
production test Flowing + buildup sequence Net sand thickness h = 250 ft oil layer Well perforated h p = 30 ft between gas and (c) Gas Oil Water Case 4: falloff test Flowing + buildup sequence Net sand thickness h = 250 ft water layer Well perforated h p = 30 ft between oil and

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

Water Case 1 Figure 4 :Figure 5 :
Figure 4: Schematics of well perforation interval and sand thickness for gas condensate + water reservoir.

Table 2 :
estimates for new approach versus conventional approach for scenario (a).
484.4 mD, respectively.At ℎ = 150 ft (60% of sand thickness and 83% of oil thickness), the estimated  is still within range

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

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

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

Table 6 :
Summary of reservoir simulations data.

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

Table 8 :
Comparison of  estimates between conventional and numerical density parameters.
sc , and  sc are known as temperature, pressure, and gas constant at standard condition.Therefore, integrating from  = ∞ where  =   , used for  calculation is the average rate for all flowing periods.
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  oil = 162.7   ℎ oil = 435.2mD,  water = 162.7   ℎ water = 49 mD.