A Pressure-Driven Recovery Factor Equation for Enhanced Oil Recovery Estimation in Depleted Reservoirs: A Practical Data-Driven Approach

Abstract

This study presents a new equation, the dynamic recovery factor (DRF), for evaluating the recovery factor (RF) in homogeneous and heterogeneous reservoirs. The DRF method’s outcomes are validated and compared using the decline curve analysis (DCA) method. Real measured field data from 15 wells in a homogenous sandstone reservoir and 10 wells in a heterogeneous carbonate reservoir are utilized for this study. The concept of the DRF approach is based on the material balance principle, which integrates several components (weighted average cumulative pressure drop (ΔP_cum), total compressibility (C_t), and oil saturation (S_o)) for predicting RF. The motivation for this study stems from the practical restrictions of conventional RF valuation techniques, which often involve extensive datasets and use simplifying assumptions that are not applicable in complex heterogeneous reservoirs. For the homogenous reservoir, the DRF approach predicts an RF of 8%, whereas the DCA method predicted 9.2%. In the heterogeneous reservoir, the DRF approach produces an RF of 6% compared with 5% for the DCA technique. Sensitivity analysis shows that RF is very sensitive to variations in C_t, ΔP_cum, and S_o, with values that vary from 6.00% to 10.71% for homogeneous reservoirs and 4.43% to 7.91% for heterogeneous reservoirs. Uncertainty calculation indicates that errors in C_t, ΔP_cum, and S_o propagate to RF, with weighting factor (W_i) uncertainties causing changes of ±3.7% and ±4.4% in RF for homogeneous and heterogeneous reservoirs, respectively. This study shows the new DRF approach’s ability to provide reliable RF estimations via pressure dynamics, while DCA is used as a validation and comparison baseline. The sensitivity analyses and uncertainty analyses provide a strong foundation for RF estimation that helps to select well-informed decisions in reservoir management with reliable RF values. The novelty of the new DRF equation lies in its capability to correctly estimate RFs using limited available historical data, making it appropriate for early-stage development and data-scarce situations. Hence, the new DRF equation is applied to various reservoir qualities, and the results show a strong alignment with those obtained from DCA, demonstrating high accuracy. This agreement validates the applicability of the DRF equation in estimating recovery factors through different reservoir qualities.

1. Introduction

In reservoir management, production forecasting, and economic decision making, recovery factor estimation is a critical reservoir engineering tool. Hydrocarbon recovery and EOR technology (enhanced oil recovery) are mainly determined by RF deduction; this is the most important parameter for all reservoir development projects. Recovery factors are defined as disturbances from oil reserves (OOIPs) that can be produced economically from certain reserves and mark key parameters for the evaluation of any potential reservoirs and their development. Declining curve analysis, combined material balance, and volumetric methods are traditional RF estimation methods that are widely used in the oil industry. Most of these methods, however, imply high data demand on production history, fluid properties, and reservoir geometry, which are not always available and/or reliable. Furthermore, in complex reservoir systems, such assumptions are not valid, leading to uncertainty in RF estimation [1,2]. RF estimation is of the utmost importance. It is directly related to the economic viability of oil and gas projects. For example, an overestimation of the recovery factor can lead to unrealistic production forecasts, which may cause companies to spend more money than necessary and lose profits [3,4].

1.1. Importance of Recovery Factor Estimation

This study presents a new equation, the DRF, for evaluating the recovery factor in homogeneous and heterogeneous reservoirs. The DRF method’s outcomes are validated and compared using the decline curve analysis method. Real measured field data from 15 wells in a homogenous sandstone reservoir and 10 wells in a heterogeneous carbonate reservoir were utilized for this study. The concept of the DRF approach is based on the material balance principle and integrates several components (weighted average cumulative pressure drop, total compressibility, and oil saturation) for predicting RF. The recovery factor is a significant parameter in a reservoir production plan, as it reveals the sum of hydrocarbons that can be efficiently recovered from a reservoir. It is influenced by some factors, such as initial reservoir pressure, reservoir rock and fluid properties, and production strategies. Accurate RF assessment is critical for selecting better recovery strategies and governing the commercial sustainability of reservoir development plans. Therefore, in waterflooding and gas injection development plans, RF is employed to determine incremental oil recovery and estimate the overall project feasibility [1,2].

Furthermore, RF is the key factor in reservoir simulation models. It is applied to accommodate the optimum reservoir management strategies and forecast potential future oil production. A detailed assessment of RF is crucial to mimic the underground reservoir behavior at different fluid flow conditions. This will help the reservoir engineers make the right decision on investment strategy plans and improve production efficiency [5,6].

1.2. Challenges in Recovery Factor Estimation

In heterogeneous reservoirs, it is very challenging to estimate reliable RF due to the uncertainty of the complex reservoir characteristics and inconsistent data. The current available methodologies to estimate the RF are not applicable to be applied in all oil and gas reservoirs, such as decline curve analysis, material balance, and volumetric methods. For example, the DCA method assumes that hydrocarbon production from complex geological formations declines exponentially over time [7,8]. Also, material balance needs very accurate reservoir rock and fluid data, along with historical production field data, which may not be accessible at the early field stage. For the volumetric method, the main concepts are based on the identifications of reservoir geometry, porosity, and fluid saturation, which can produce large errors in RF estimation [1,9]. Consequently, a unique RF methodology to handle all these challenges is required.

1.3. Advancements in Recovery Factor Estimation

In the last decade, there have been numerous research projects that have been conducted to estimate more reliable RF at different reservoir conditions. These methods include machine learning data science and simulation model methods. The simulation model requires a huge amount of data to initiate a reservoir model and simulate the actual geological and dynamic fluid conditions to predict hydrocarbon recovery [5,10].

However, data science and machine learning tools are dependent on the production database records. The results of these methods indicate promising potential for predicting more reliable RF, specifically for complex reservoirs. For instance, Al-Fattah and Startzman [11] presented a neural network model for assessing natural gas productivity, while Zhang et al. [6] proposed a machine learning method for calculating RF in unconventional reservoirs. These tools create new opportunities for RF estimation. However, they also need massive databases and more intricate algorithms, which cannot be available in all cases.

1.4. The Need for Simplified Methods

Despite the above-mentioned studies, there is a need for more attempts to produce a reliable RF estimation, specifically in cases where there is limited data accessible. The new DRF equation presented in this study meets the requirements by incorporating the pressure depletion history data, which is the key driver of reservoir recovery. The methodology concepts were based on reservoir pressure observations and cumulative production data, which are directly related to hydrocarbon recovery [3,12].

The approach will be validated using real measured reservoir and well production data and comparing the DRF results with the DCA method’s results. Both sensitivity analysis and uncertainty calculations will be performed to verify the effectiveness of the new approach. The results of this study have significant effects for reservoir engineers, permitting them to make optimum decisions about reservoir management and development production strategies [10,11].

The DRF-solving method proposed in this work achieves competitive accuracy with a much smaller amount of data input than in previous work. Garcia et al. [13] proposed an easier method based on pressure estimates, but it was not tested for homogeneous and heterogeneous reservoirs. In the meantime, Zhang et al. [6] used ML-based RF models, depending significantly on having extensive training data to yield RF estimates of comparable accuracy. Instead, the DRF equation reliably produced RF values of 8% for sandstone and 6% for carbonate reservoirs, aligning with DCA estimates. This result shows that the DRF is a reliable and flexible option, particularly in field circumstances where data are restricted or at the beginning of field production stages.

2. Literature Review

Various studies were developed to estimate recovery factors in the last decade. Each of these trials had its strengths and weaknesses. This section provides a comprehensive review of the most applied approaches for RF estimation, such as DCA, material balance, volumetric, simulation models, and machine learning.

2.1. Conventional Methods for RF Estimations

Decline curve analysis is one of the approaches that has been widely applied for recovery factor predictions in the oil and gas sector. This method was presented by Arps [14], where it was made based on the observations of the fluid production decline rate with time, and utilized mathematical principles to model the decline rate. The typical form of the DCA formula is as follows:

$${Q\left(t\right)=Q_{i}e}^{-Dt}$$

(1)

where Q(t) denotes the flow rate at time t, D is the decline rate, and Q_i denotes the initial flow rate.

The DCA approach has shown responsible results in many applications, mostly for reservoirs that have steady production rates. On the other hand, the method is predicted based on several assumptions that may not be applicable in all cases. For example, in heterogeneous reservoirs or unconventional resources, the hydrocarbon production decline may not have an exponential trend as assumed by the DCA approach. This could result in inaccurate RF estimation [2,15].

The material balance approach, first proposed by Schilthuis [16], is another common method that was applied for RF estimation. This approach assumes conservation of mass and includes adjusting the fluid volume extracted from the reservoir to account for the changes in reservoir pressure and fluid characteristics. The material balance equation in its generic form is as follows:

$$N_P\left[B_o+\left(R_p-R_s\right)B_g\right]+W_P{B}_w-W_i{B}_w-G_i{B}_g=N\left[E_o+E_{fw}+m{E}_g\right]+W_e$$

(2)

where N is the original oil in place; B_oi is the initial oil formation volume factor; B_w, B_o, and B_g are the water, oil, and gas formation volume factors at current pressure, respectively; and N_p, W_p, and R_p are the cumulative oil, water, and gas production, respectively. W_e is the aquifer inflow volume, W_iB_w is the water injection volume, G_iB_g is the gas injection volume, R_s is the solution gas, and E_fw is the rock and connate water expansion term.

The material balance approach is favorably applied to closed reservoirs, where the primary recovery process is pressure depletion [1,2]. However, this approach requires accurate input data on reservoir fluid properties, pressure, and fluid production history, which may not be accessible or dependable. Also, the approach presupposes that the reservoir is in equilibrium, which may not be the case in dynamic reservoirs with active aquifers or gas caps. These constraints indicated the necessity for different approaches that can estimate a reasonable RF [9,17].

The volumetric method is a straightforward method used to predict RF. The concepts of this method are based on the rock properties and reservoir geometry. The OOIP can be determined by the volumetric method by incorporating reservoir rock and fluid parameters. The recovered volume will be determined by using a predicted recovery factor. The following is the general form of the OOIP equation:

$$N={\displaystyle \frac{ 7758\cdot A\cdot h\cdot \varphi \cdot (1-Swi)}{B_{oi}}}$$

(3)

where N represents the OOIP, A denotes the reservoir area, h denotes the reservoir thickness, ϕ is the porosity, and S_wi represents the initial water saturation. This method is straightforward to apply, and requires little production data. The method is often used for the initial stages of reservoir appraisal [1,9].

The results of this method can produce a high uncertainty in RF estimation because it assumes that the reservoir is homogeneous and has constant recovery productivity, which is not the case with heterogeneous reservoirs. Therefore, it requires a robust practice that includes the uncertainties in recovery mechanisms and reservoir features [3,4].

2.2. Advanced Techniques for RF Estimation

In recent years, many new techniques for RF estimation have been increased, such as data-driven models, machine learning, and simulation models. To forecast production recovery, a simulation model is a completely static and dynamic reservoir model that needs to be built. This method requires a huge database, which might not be feasible for many applications, to generate more reliable results [5,10]. Also, the current studies have confirmed the use of systems analysis approaches to construct virtual models of oil fields, thus enhancing simulation capabilities. However, machine learning and data science algorithms require historical production data to estimate the RF and predict future production performance. These methods have proven their ability to estimate more reliable RF, specifically in heterogeneous reservoirs with reasonable sources of data. A machine learning method was presented by Zhang et al. [6] to estimate RF in unconventional reservoirs, while Al-Fattah and Startzman [11] proposed a neural network model for forecasting natural gas production performance. These techniques need big datasets, which may not always be available, to build intricate algorithms [12,18,19].

2.3. Addressing Research Gaps in RF Estimation

The applicability of existing methods to heterogeneous and unconventional reservoirs is one of the most demanding types of research required in RF estimation. These reservoirs often challenge the presumptions of available methods due to their poor reservoir quality, complex fracture network system, and non-linear flow behaviour. The DCA method is used for the complex dynamics of shale gas reservoirs, where hydraulic and natural fractures interact in irregular ways [20]. Advanced techniques such as simulation models and machine learning have shown promising results by counting these complexities; however, their high computational and data requirements sometimes make them unrealistic [21,22]. Another significant flaw in conventional RF estimation techniques is the lack of efficient uncertainty evaluation and sensitivity analysis. Most of the available techniques yield single-point estimates that do not consider the uncertainties associated with reservoir type, recovery techniques, or production data. This could result in imperfect reservoir management decisions and unreliable results [4,23]. The DRF equation facilitates reservoir engineers to analyze by considering the effect of these features on RF calculations and make the right decisions based on a detailed knowledge of uncertainty.

The literature on the combination of various data sources, including production history data, well logs, and seismic data, also has some significant gaps. The existing approaches normally depend on a single data source, which cannot provide a broad representation of the reservoir’s behavior. Cutting-edge methods like data science models and machine learning show promise in combining various data sources; however, their ability in many current cases is limited by the requirement for huge databanks and building complex algorithms [21,24]. The DRF approach fills this gap by offering a simpler methodology that only needs basic reservoir and well data, making it applicable in cases where there are few data available [24,25].

In general, by introducing a straightforward and understandable approach to RF estimates, the DRF equation overcomes all the above-mentioned difficulties and makes it accessible to any type of reservoir quality. Because of its reliability, efficiency, and ease of use, the DRF equation is a useful tool for improving reservoir management and development plans.

3. Methodology for Deriving the DRF Equation

The DRF equation, which is introduced to calculate RF based on pressure depletion and cumulative production data, is developed and validated using a process that is fully explained in this section. The material balance theory is the base of the DRF equation. It combines changes in a reservoir’s pressure with total production by considering changes in hydrocarbon volume inside the reservoir. The development of the new equation started with a fundamental form of the material balance equation for an undersaturated oil reservoir. The DRF approach is then validated, sensitivity analysis is carried out, and a comparison with the DCA method is conducted. Actual field data from both homogeneous and heterogeneous reservoirs are used to show the accuracy of the DRF method. Below, we provide a detailed and step-by-step derivation of the DRF equation.

3.1. Derivation of the DRF Equation from the Material Balance Principle

The DRF equation has been derived from the material balance principle, which links pressure changes in a reservoir to total production. The typical form of the material balance equation for an undersaturated oil reservoir is the starting point of the DRF equation.

Rearrange the Material Balance Equation

The general rearranged material balance equation is as follows:

$$NpBo=Ct.Vp\left({Pi-Pf}_i\right)$$

(4)

Divide both sides by B_o to solve for Np:

$$Np=Ct. Vp\left({Pi-Pf}_i\right)/Bo$$

(5)

$$C_t=C_o{S}_o+C_w{S}_w+C_g{S}_g+C_f$$

(6)

where C_t denotes total compressibility and C_o, C_w, C_g, and C_f denote oil, water, gas, and rock compressibility, respectively. Also, S_o, S_w, and S_g denote oil, water, and gas situations, respectively.

Express $Np$ in terms of recovery factors. The RF is identified as the fraction of original oil in place (N) that has been produced:

$$RF=Np/N$$

(7)

By substituting Np from the material balance equation into the RF equation:

$$RF={\displaystyle \frac{Ct.Vp\left({Pi-Pf}_i\right)}{ Bo.N}}$$

(8)

Now N can be expressed in terms of pore volume (Vp) and oil saturation (S_o):

$$N={\displaystyle \frac{Vp.So}{Bo}}$$

(9)

where S_o is the initial oil saturation (fraction). By substituting N into the RF equation:

$$RF={\displaystyle \frac{\frac{Ct.Vp.\left({Pi-Pf}_i\right)}{Bo}}{\frac{Vp.So}{Bo}}}={\displaystyle \frac{Ct.\left({Pi-Pf}_i\right)}{So}}$$

(10)

Consequently, the RF equation becomes:

$$RF={\displaystyle \frac{Ct. {(Pi-Pf}_i)}{S_o}}$$

(11)

To account for the cumulative pressure drops over time, replace (P_i − P_fi) with the weighted average cumulative pressure drop (ΔP_cum). Therefore, the final form of the DRF equation is:

$$SPDRF={\displaystyle \frac{Ct. \left(\mathsf{\Delta }Pcum\right)}{S_o}}$$

(12)

where DRF is the dynamic recovery factor (fraction), ΔP_cum (psi) is the weighted average cumulative pressure drop (psi), C_t (psi⁻¹) is the total compressibility, and S_o (fraction) is the oil saturation.

The use of the weighted average cumulative pressure drop in the DRF equation replaces the instant pressure drop (P_i − P_fi) to capture the reservoir’s pressure depletion record. As oil production increases, reservoir pressure gradually decreases over time, and ΔP_cum integrates these variations, indicating the cumulative effect of pressure depletion rather than a single record. This method is more accurate since reservoir pressure decline is regularly non-linear, affected by factors such as fluid flow, drive mechanisms, and heterogeneity, where ΔP_cum is considered by weighting pressure drops along the production period.

By using ΔP_cum, the DRF equation aligns more closely with the material balance principle, which needs an accounting system for the total volume of fluids produced due to pressure changes. C_t tells how much the reservoir expands in response to pressure depletion, and ΔP_cum validates that the cumulative expansion is accurately represented. Dividing C_t ΔP_cum by S_o ensures that RF is proportional to the accessible OOIP, giving the computation process a physical meaning consistent with reservoir behavior.

The new DRF equation demonstrates a reliable result for RF estimation. It shows acceptable accuracy in non-linear pressure decline reservoir conditions. By addressing total pressure depletion, the DRF method provides a simpler and more effective recovery factor calculation by accurately capturing the reservoir’s pressure history.

This approach intrinsically accounts for variations between homogeneous and heterogeneous reservoirs by integrating total compressibility and oil saturation, which are influenced by the mineralogical composition and petrophysical characteristics of the reservoir rock. These factors allow the DRF equation to adjust to different geological settings by capturing the impacts of lithology, fluid content, and pore structure.

3.2. Background of the Field Data

The DRF approach has been validated using real data from two different oil reservoirs: a homogeneous sandstone reservoir and a heterogeneous carbonate reservoir. The homogeneous reservoir is suitable for testing the DRF approach because of its consistent permeability and porosity. However, the heterogeneous reservoir represents a challenging condition for this new DRF approach due to significant variations in porosity and permeability.

Table 1. Reservoir and well data.

Reservoir Properties	Homogeneous Reservoir	Heterogeneous Reservoir
h	50 ft	80 ft
Swi	0.25	0.30
So	0.75	0.70
Boi	1.25 RB/STB	1.30 RB/STB
Bw	1.02 RB/STB	1.03 RB/STB
Bg	0.005 RB/SCF	0.006 RB/SCF
API gravity	38° (light oil)	32° (medium oil)
μo	1.2 cP	2.5 cP
k	500 mD (uniform)	50–500 mD (variable)
ϕ	22% (uniform)	15–25% (variable)
Number of wells	15	10
Pi	3000 psi	2000 psi
Ct	7.95 × 10−5 psi−1	9.03 × 10−5 psi−1
N	108,535,000 STB	75,600,000 STB

shows the homogeneous and heterogeneous reservoir and well data.

The sandstone reservoir used in this study is well-organized and comprises quartz, features minimal organic content, and lacks natural cracks that enable consistent fluid flow. On the contrary, the carbonate reservoir varies in its mineral character, comprising various natural fractures, a moderate to high level of organic material, calcite, dolomite, and a small amount of clay. These differences in rock types and the existence of fractures lead to variations in permeability and porosity, which significantly affect how the reservoir operates and complicate recovery predictions. Rather than depend on geometric consistency to tackle these challenges, the DRF equation uses how compressibility acts along with pressure reduction.

3.3. Comparison Analysis

The DCA method is commonly used to estimate RF in reservoir engineering. This method was used to validate its results with the DRF results. The exponential DCA approach will be used for this study, with a presumption that the decline rate will stay constant over time. A more dynamic approach to recovery factor calculation is offered by the DRF technique, which considers well-specific production contributions and pressure depletion data. The same field data will be used for both approaches to make a fair comparison between the heterogeneous and homogeneous reservoirs. In heterogeneous reservoirs, conventional approaches like DCA may not be effective.

3.4. Sensitivity Analysis

A sensitivity analysis will be performed to assess the influence of DRF parameters on the new equation outputs: C_t, ΔP_cum, and S_o. Every parameter will be varied within realistic ranges to evaluate its effect on the recovery factor. For C_t, values will range from 7.62 × 10⁻⁵ to 1.02 × 10⁻⁴ psi⁻¹, reflecting standard compressibility values for oil reservoirs. The ΔP_cum will be changed from 500 psi to 700 psi for the homogeneous reservoir and from 400 psi to 600 psi for the heterogeneous reservoir, indicating different levels of reservoir depletion. S_o will be varied from 0.6 to 0.8, covering the normal range for oil reservoirs. The sensitivity analysis will show how variations in these parameters affect the recovery factor, providing insights into the strength of the DRF approach, and find which parameters have the highest effect on RF calculations. This assessment will also assist in establishing practical guides for parameter estimation and uncertainty management.

3.5. Uncertainty Assessment and Error Propagation

Uncertainty analysis is calculated by investigating the propagation of errors in each used parameter. The uncertainty in C_t will be calculated based on laboratory measurements and historical data records. Also, the uncertainty in ΔP_cum will be calculated by accurate pressure measurements and W_i applied in its computations. Similarly, the uncertainty in S_o will be calculated based on core data analysis and well log data. The weighting factor (Wi) physically characterizes the relative contribution of each well to the total production, counting the effect of individual wells on the overall recovery factor. It is calculated as the ratio of the cumulative production (or pressure drop) from every well to the total cumulative production (or pressure drop) of all wells, permitting the model to account for variations in well performance across the reservoir.

A statistical approach, such as Monte Carlo simulations, will be applied to measure the uncertainty in the recovery factor computations. This valuation will give a range of realistic RF values rather than a single-point valuation, providing a more satisfactory recovery potential. By addressing these uncertainties, the DRF approach can enhance its dependability in real reservoir management conditions. Furthermore, the findings of the uncertainty evaluation will help reservoir engineers to generate the optimum recovery strategies and supervise the risks more effectively.

4. Analysis and Discussion

This section discusses the RF calculations for both homogeneous and heterogeneous reservoirs by using the new DRF equation. The analysis obtained is based on real production measured data from 2 different oil fields, using 15 oil wells in the homogeneous reservoir and 10 wells in the heterogeneous reservoir. Then the results are compared with DCA for each reservoir type. The discussion focuses on the clarification of the new methodology for reservoir management and production optimization.

4.1. Application of the DRF Method

The new equation is applied using real production measured data from two types of reservoirs: homogeneous and heterogeneous. The process incorporates the RF estimation based on the weighted average cumulative pressure drop, initial reservoir pressure, and the weighting factor.

Table 2. The homogenous reservoir production data.

Well No.	Production Rate (stb/Day)	Cumulative Production (stb)	Pfi, (psi)
1	1200	516,240	2700
2	1350	580,770	2800
3	1400	602,280	2600
4	1250	537,750	2650
5	1300	559,260	2500
6	1450	623,790	2550
7	1500	645,300	2400
8	1550	666,810	2450
9	1600	688,320	2300
10	1650	709,830	2350
11	1700	731,340	2200
12	1750	752,850	2250
13	1800	774,360	2100
14	1850	795,870	2150
15	1900	817,380	2000
Cum		10,002,150

and

Table 3. The heterogenous reservoir production data.

Well No.	Initial Production Rate (stb/Day)	Cumulative Production (stb)	Pfi, (psi)
1	800	288,000	1950
2	850	306,000	1850
3	900	324,000	1820
4	950	342,000	1710
5	1000	360,000	1620
6	1050	378,000	1530
7	1100	396,000	1440
8	1150	414,000	1350
9	1200	432,000	1260
10	1250	450,000	1170
Cum		3,690,000

display the final pressures, well production rate, and cumulative production data for a homogenous reservoir.

DRF Calculation Steps in Homogeneous vs. Heterogeneous Reservoir

The following section will discuss the process for calculating RF using the new DRF equation. Below are the steps to calculate the ΔP_i, W_i, and ΔP_cum for each oil well. These steps are important for calculating RF, which reflects the reservoir’s recovery productivity for both homogeneous and heterogeneous reservoirs.

Step 1: Calculate the cumulative pressure drop.

The cumulative pressure drop, ΔP_i, for every oil well is defined as:

$${\mathsf{\Delta }P}_i=P_i-{Pf}_i$$

(13)

where P_i is the initial reservoir pressure and P_fi is the final pressure at each well (psi).

Step 2: Calculate the weighting factor.

The weighting factor for each well is formulated as:

$$W_i={\displaystyle \frac{{NP}_i}{\sum {NP}_i}}$$

(14)

where NP_i is denoted as cumulative oil production for each well (stb) and $\sum NPi$ is denoted as total cumulative production from all the wells.

Step 3: Calculate the weighted average cumulative pressure drop.

The weighted average cumulative pressure drop (

Table 4. The DRF process calculations for the homogenous reservoir.

Well No.	Cumulative Production (stb)	Pfi, (psi)	ΔPi (psi)	Wi	Δpcum (psi)
1	516,240	2700	300	0.0516	15
2	580,770	2800	200	0.0581	12
3	602,280	2600	400	0.0602	24
4	537,750	2650	350	0.0538	19
5	559,260	2500	500	0.0559	28
6	623,790	2550	450	0.0624	28
7	645,300	2400	600	0.0645	39
8	666,810	2450	550	0.0667	37
9	688,320	2300	700	0.0688	48
10	709,830	2350	650	0.0710	46
11	731,340	2200	800	0.0731	58
12	752,850	2250	750	0.0753	56
13	774,360	2100	900	0.0774	70
14	795,870	2150	850	0.0796	68
15	817,380	2000	1000	0.0817	82
10,002,150				630

and

Table 5. The DRF process calculations for the heterogenous reservoir.

Well No.	Cumulative Production (stb)	Pfi, (psi)	ΔPi (psi)	Wi	Δpcum (psi)
1	288,000	1950	50	0.078	3.902
2	306,000	1850	150	0.083	12.439
3	324,000	1820	180	0.088	15.805
4	342,000	1710	290	0.093	26.878
5	360,000	1620	380	0.098	37.073
6	378,000	1530	470	0.102	48.146
7	396,000	1440	560	0.107	60.098
8	414,000	1350	650	0.112	72.927
9	432,000	1260	740	0.117	86.634
10	450,000	1170	830	0.122	101.220
Cum	3,690,000				465.122

) is considered as:

$${\mathsf{\Delta }P}_{cum}=\sum (W_i.{\mathsf{\Delta }P}_i)$$

(15)

Step 4: Calculate RF using the DRF equation.

$$DRF={\displaystyle \frac{{Ct.\mathsf{\Delta }P}_{cum}}{S_o}}$$

(16)

where for a homogenous reservoir, the total compressibility, C_t, = 7.95 × 10⁻⁵ Psi⁻¹ and $So$ = 0.62.

$$\mathrm{D}\mathrm{R}\mathrm{F}={\displaystyle \frac{7.95\times {10}^{-5}\ast 630}{0.62}}=0.08074 or 8\%$$

For the heterogeneous reservoir, C_t = 9.03 × 10 ⁻⁵ Psi⁻¹, and $So$ = 0.7.

$$DRF={\displaystyle \frac{9.03\times {10}^{-5}\ast 465.122}{0.7}}=0.06 or 6\%$$

The DRF method produces an RF of 8% for the homogeneous reservoir. This is reliable with expectations, as homogeneous reservoirs naturally have higher recovery factors as a result of their uniform properties, which simplify effective fluid flow and pressure distribution. Also, the DRF approach generates an RF of 6% for the heterogeneous reservoir because of the intricate and non-uniform properties of heterogeneous reservoirs, which delay effective fluid flow and pressure maintenance. The inclusion of the DRF calculation confirms that the influence of each well on the total pressure drop is proportional to its cumulative oil production. This offers a more accurate representation of reservoir performance and recovery potential. The results show the significance of considering well-specific production data and pressure changes in RF formulation. Accurate valuation is crucial for dependable RF estimations and effective reservoir management.

4.2. Validation of DRF Equation

The DRF method’s outcomes are validated by comparing them with the conventional method, the DCA method. The validation focuses on three important parts: the accuracy of the approach in predicting recovery factor, the simplicity of application, the data supplies for the DCA method, and the ability of the DCA method to apply to various reservoir natures.

Estimation RF Homogeneous and Heterogenous Reservoirs Using DCA

The exponential DCA method predicts RF based on the decline rate of the production rate with time. The general formula for the exponential decline is:

$$Q \left(t\right)=Q_i{ . e}^{-D.t}$$

(17)

where Q(t) denotes the production rate at time t (stb/day), Q_i denotes the initial production rate (stb/day), D denotes decline rate (1/month), and t is time (months).

The cumulative production equation for the exponential decline is:

$$NP(t)={\displaystyle \frac{Q_i-Q\left(t\right)}{D}}$$

(18)

Now, calculate the decline rate, D, for each well using the production data at t = 430.2 days or 1.18 years. Rearrange the exponential decline equation to solve for D:

$$D=-{\displaystyle \frac{1}{t}} \mathit{ln}\left({\displaystyle \frac{Q\left(t\right)}{Q_i}}\right)$$

(19)

Since t = 1 year, this shortens to:

$$D=-\mathit{ln}\left({\displaystyle \frac{Q\left(t\right)}{Q_i}}\right)$$

(20)

The production rates at t = 430.2 days or 1.18 years are equal to the initial production rates q(t) = qi. This indicates no decline (D = 0) over the first year. This proposes that the oil wells are producing at a constant rate with no decline throughout the first year. The predicted RF based on the cumulative production at t = 430.2 days or 1.18 years is:

$$RF={\displaystyle \frac{\sum NP}{N}} {\displaystyle \frac{\mathrm{10,002,150}}{\mathrm{108,530,000}}}=0.0921 or (9.2\%)$$

For a heterogeneous reservoir, the production rates at t = 1 year are the same, q(t) = qi. Subsequently, D = 0 over the first year, where the oil wells are producing with no decline during the first year. Therefore, RF is:

$$RF={\displaystyle \frac{\sum NP}{N}} {\displaystyle \frac{\mathrm{3,690,000}}{\mathrm{75,600,000}}}=0.048 or (5\%)$$

The DCA technique predicts an RF of 9.2% for the homogeneous reservoir. This is closer to the DRF method’s RF (8.1%). Also, the DCA method results in an RF of 5% for the heterogeneous reservoir, which is almost close to the DRF method’s RF (6%). The results indicate that the RF result of DCA is in agreement with the estimated recovery factor obtained with the DRF equation.

4.3. Sensitivity Analysis

The sensitivity analysis for the new DRF approach assesses exactly how changes in the main equation parameters affect the RF result. The key parameters are P_i, ΔP_cum, C_t, and S_o.

Each parameter will be verified separately while maintaining the other parameters constant and monitoring the effect on the RF value. Additionally, a depth analysis of errors and uncertainties linked with each parameter and their propagation via the DRF equation will be carried out. Pi has been verified from 2000 psi to 3500 psi in increments of 100 psi for homogenous and heterogenous reservoirs. Figure 1 illustrates the estimated RF for every Pi using the DRF equation at S_o = 0.62, C_t = 7.95 × 10⁻⁵ psia⁻¹, and ΔP_cum = 630.1 psi, and S_o = 0.7, C_t = 9.03 × 10⁻⁵ psia⁻¹, and ΔP_cum = 465 psi for heterogeneous reservoir.

The initial reservoir pressure is normally measured by downhole pressure gauges, which can have an error margin of ±1% to ±2%. This indicates that the RF of 10.30% could have an uncertainty of ±0.060.

Concerning the weighted average cumulative pressure drop, ΔP_cum has been verified from 500 psi to 700 psi in increments of 20 psi for the homogenous reservoir. RF is computed for every ΔP_cum and C_t varies from 7.62 × 10⁻⁵ to 1.02 × 10⁻⁴ psi and So changes from 0.6 to 0.8, at Pi = 3000 psi. The weighted average, W_i, is 0.0702 ± 0.0026. The uncertainty in W_i leads to a ±3.7% uncertainty in RF (

Table 6. The homogeneous reservoir sensitivity.

Ct (psi−1)	So = 0.6	So = 0.65	So = 0.7	So = 0.75	So = 0.8
7.62 × 10−5	8.00%	7.38%	6.86%	6.40%	6.00%
8.25 × 10−5	8.66%	8.00%	7.43%	6.93%	6.50%
8.89 × 10−5	9.33%	8.62%	8.00%	7.47%	7.00%
9.52 × 10−5	10.00%	9.23%	8.57%	8.00%	7.50%
1.02 × 10−4	10.71%	9.88%	9.18%	8.57%	8.04%

).

For the heterogeneous reservoir, Ct varies from 7.62 × 10⁻⁵ to 1.02 × 10⁻⁴ psi⁻¹, and S_o changes from 0.6 to 0.8, at P_i = 2000 psi, the W_i is 0.102 ± 0.0045, and W_i leads to a ±4.4% uncertainty in the recovery factor. Figure 2 reveals the change of RF at different W_i.

For the homogeneous reservoir, RF has a constant uncertainty of ±0.00167 at any value of ΔP_cum, meaning RF can differ by this amount because of variations in cumulative pressure drop (Figure 3). For the heterogeneous reservoir, the uncertainty in RF is a little higher at ±0.00175, also constant at all ΔP_cum values. The uncertainty range for C_t is from 7.62 × 10⁻⁵ to 1.02 × 10⁻⁴ psi⁻¹. For S_o, the uncertainty changes from 0.6 to 0.8. Therefore, the uncertainty results in RF for both homogeneous and heterogeneous reservoirs vary from 6.00% to 10.71% and from 4.43% to 7.91%, respectively. These ranges (Table 6 and

Table 7. Heterogeneous reservoir sensitivity.

Ct (psi−1)	So = 0.6	So = 0.65	So = 0.7	So = 0.75	So = 0.8
7.62 × 10−5	5.91%	5.45%	5.06%	4.72%	4.43%
8.25 × 10−5	6.40%	5.91%	5.49%	5.12%	4.80%
8.89 × 10−5	6.89%	6.36%	5.91%	5.52%	5.18%
9.52 × 10−5	7.38%	6.81%	6.33%	5.91%	5.55%
1.02 × 10−4	7.91%	7.30%	6.78%	6.33%	5.94%

) reflect the sensitivity of RF to variations in C_t and S_o.

The observations for the homogeneous reservoir demonstrate that as C_t increases, RF increases at any value of S_o. And as S_o increases, RF decreases at any value of C_t. Consequently, RF varies from 6.0% to 10.71% based on C_t and S_o variation. Also, the observations for the heterogeneous reservoir demonstrate that, as C_t increases, RF increases at any value of S_o. And as S_o increases, RF decreases at any value of C_t. Consequently, RF varies from 4.43% to 7.91% depending on C_t and S_o. Hence, the higher C_t primes to higher RF because the reservoir can produce more liquids for a given pressure drop. Also, the higher S_o leads to lower RF because a bigger fraction of the pore volume is filled by oil, decreasing the relative effect of pressure depletion. Heterogeneous reservoirs have lower RF values compared with homogeneous reservoirs for equal C_t and S_o values, reflecting the effect of spatial alterations in reservoir properties.

The sensitivity analysis emphasizes the significance of observing reservoir pressure over time. A higher ΔP_cum shows larger reservoir depletion, which increases the RF. This highlights the necessity for effective pressure maintenance strategies to improve recovery.

In general, the sensitivity analysis, combined with a detailed error and uncertainty analysis, gives a broad understanding of how the DRF equation parameters (ΔP_cum, C_t, and S_o) change the recovery factor estimations. The results emphasize the significance of accurate parameter measurement to confirm reliable RF calculations. By considering the uncertainties associated with all parameters, reservoir engineers can make optimum decisions and boost recovery strategies.

5. Discussion

The DRF equation’s resilience is more supported by the fact that it can be applied to reservoirs of heterogeneous carbonate and homogeneous sandstone. The DRF equation adapts its calculations according to the particulars of each reservoir by including Ct and So, two factors that naturally characterize differences in rock type and fluid behavior.

The new approach provides a reliable basis for predicting RF in both homogeneous and heterogeneous reservoirs, indicating its dependability by validation with real oil field measured data and alignment with DCA predictions. For homogeneous reservoirs, the new approach predicts an RF of 8%, consistent with expectations because of uniform reservoir properties that simplify effective fluid flow. The RF for heterogeneous reservoirs is 6%, showing challenges resulting from differences in porosity and permeability throughout various areas. By including W_i in the equation, the impact of each well on pressure dynamics is pointed out. However, RF differences of ±3.7% and ±4.4% are caused by errors in W_i for both homogeneous and heterogeneous reservoirs.

In spite of its practical advantages, the DRF equation has inherent limitations that should be known. The method depends on correct measurements of ΔP_cum and S_o; errors in these parameters can meaningfully affect the RF estimation. Moreover, the model assumes undersaturated reservoir conditions, meaning it might not be applied in saturated reservoirs. In extremely fractured reservoirs where flow dynamics are intricate and the reservoir heterogeneity is extreme, DRF may not be a proper method to represent real-time changes of reservoir recovery performance. Hence, while DRF is appropriate for various applications, it should be used with care in such reservoir characterization and complemented by other approaches when needed.

Sensitivity analysis shows that C_t clearly affects RF, whereas greater S_o values lower RF due to an increase in the amount of oil saturated in the pore volume. RF ranges between 4.43% and 7.91% in heterogeneous reservoirs and between 6.00% and 10.71% in homogeneous reservoirs. Valid measurements of P_i, ΔP_cum, C_t, and S_o are required as their uncertainties have major effects on RF estimation. This study shows the importance of adequate pressure maintenance approaches for heterogeneous reservoirs where recovery effectiveness is reduced by complex flow paths. The new approach enables engineers to further develop recovery strategies, improve reservoir management, and enhance fluid production by assessing parameter sensitivities and uncertainties. This provides effective thoughts for more accurate decisions in the oil and gas sector.

An acceptable range of deviation between the recovery factor estimated by the new method and available methods, such as decline curve analysis, is approximately ±1.5–2%. This range reflects typical uncertainties in reservoir data and modeling assumptions and is considered acceptable within industry standards. Small differences within this range do not cause a high effect on decision making and prove that the new approach provides reasonable estimates under different reservoir conditions.

6. Limitations and Drawbacks of the Proposed Approach

Despite the new method’s promising results, some limitations should be considered. For estimation accuracy, the new method depends on key input data such as pressure, initial oil saturation, and cumulative pressure differential. Uncertain input parameter data can cause deviations in the estimation recovery factor, mainly in low reservoir permeability or data-scarce reservoirs. In homogeneous reservoirs, the new approach might perform well; however, it might not capture the sort of heterogeneities in the reservoir that lead to the underestimation of the recovery factor. In reservoirs with high heterogeneity, mainly with variable mineralogy or natural fractures, the method’s predictive ability reduces as it does not completely account for complex flow paths in the porous media, permeability differences, or dynamic variations in reservoir properties through production. Furthermore, the method assumes that main reservoir properties stay constant over time and does not reflect variations such as wettability alteration, permeability decrease due to fine movements, or fracture progress through production. Additionally, the new method requires detailed reservoir rock and fluid data, which might not always be accessible, especially in the early phases of reservoir development.

7. Conclusions

The material balance principle is the fundamental basis for the new RF equation, incorporating Δ_Pcum, C_t, and S_o to provide reliable RF values. This study applies real-time measured data from 10 oil wells in a heterogeneous reservoir and 15 oil wells in a homogeneous reservoir.

This study verifies the new RF equation by calculating the recovery factors for homogeneous and heterogeneous reservoirs. The homogeneous reservoirs have uniform characteristics that enable better fluid flow and pressure distribution, to produce an RF of 8%. By assuming no production decline, the DCA method, on the other hand, calculates an RF of 9.2%. As the heterogenous reservoir reflects the more complex reservoir attributes, the RF calculations for the DCA method show 5% compared with 6% calculated by the DRF equation for the heterogeneous reservoir.

Sensitivity analysis confirms that RF is extremely sensitive to changes in C_t, ΔP_cum, and S_o. For homogeneous reservoirs, RF varies from 6.00% to 10.71% as the total compressibility changes from 7.62 × 10⁻⁵ to 1.02 × 10⁻⁴ psi⁻¹, and oil saturation changes from 0.6 to 0.8. For heterogeneous reservoirs, RF varies from 4.43% to 7.91% under the same parameter varieties. Higher C_t values increase RF as the reservoir produces more fluids for a given pressure drop, while higher S_o values decrease RF because of a more oil-occupied pore volume. Based on the uncertainty evaluation, variation in RF for homogeneous and heterogeneous reservoirs is ±3.7% and ±4.4%, respectively, as a result of errors in C_t, ΔP_cum, and S_o propagating to RF and W_i uncertainties. The importance of careful parameter measurement and prediction for reliable RF estimates is emphasized by these findings.

The new DRF method’s capability to include well-specific production data and pressure variations makes it particularly valuable for complicated heterogeneous reservoirs, where conventional approaches such as DCA may be deficient. The results show the significance of tailored recovery strategies for different reservoir types, with homogeneous reservoirs benefiting from effective pressure maintenance and unchanging fluid flow; however, heterogeneous reservoirs need more reliable methods to consider all variations and optimize recovery. This study also highlights the necessity for continuous pressure monitoring and advanced statistical approaches to decrease uncertainties in RF calculations. Recent advancements in recovery factor valuation have combined data-driven and artificial intelligence approaches, such as neural network models, that will deliver reliable results but are typically data-intensive and computationally challenging. The developed DRF technique, however, generates robust RF evaluations with few input data. Due to the simplicity of the DRF method, it can be used in situations with limited input data, and its potential for integration with artificial intelligence tools will increase its applications even further.

In conclusion, the new DRF equation, combined with sensitivity and uncertainty analyses, presents a strong basis for RF prediction. The findings emphasize the value of precise reservoir description, well-specific data incorporation, and advanced simulation practices in improving recovery approaches. Future research could discover the incorporation of machine learning and artificial intelligence to improve RF calculation and reservoir optimization practices, confirming more effective and sustainable resource recovery.