Water Influx Impact on Oil Production in Hamzeh Oil Reservoir in Northeastern Jordan: Case Study

Abstract

This paper was conducted to delimit the water influx in the Hamzeh oil reservoir, located in northeastern Jordan approximately 150 km east of Amman. Petroleum reservoirs are frequently encompassed by water aquifers that back up the reservoir pressure through water inflow. When the pressure declines in a petroleum reservoir, the water aquifer responds by providing an influx of water. Gradually, the damage is reduced and then eliminated, and more oil is produced from the reservoir. The material balance equation (MBE) is used as the fundamental method for this study, predicting reservoir performance for a period of 11 years. The results for this study prove that the reservoir has a water drive mechanism and that the original oil in place (OOIP) was 24,958,290 m³. The projected oil recovery factor ranges from 10.9 to 25 percent for the Hummar and Shueib formations, respectively, depending on the areal efficiency assumed in the calculations. The water influx for the 11-year period was predicted by an MBE, an unsteady-state model, and the results of the performance reservoir.

1. Introduction

Petroleum reservoirs are often surrounded by water aquifers that support the reservoir pressure through water influx. In response to a pressure drop at a petroleum reservoir, a water aquifer reacts to offset or retard the pressure decline by providing a water influx or encroachment [1]. This work is important because it defines the relationships between the reservoir pressure, the quantities of oil and gas produced, the oil and gas content of the reservoir, and the characteristics of the reservoir fluids. This study uses an equation that, with sufficient data, makes it possible to calculate the oil content of the reservoir and to quantitatively predict the influence of the gas–oil ratio on the decline of reservoir pressure. Active oil and reservoir energy are the essential parameters for the determination of the rate of water encroachment into a field and its total amount [2,3]. This approach to water influx calculations offers a useful and flexible method for forecasting and analyzing the performance of water drive reservoirs [4].

The data, which provides the skin factors, indicates that the wells have suffered severe wellbore damage according to the assessment of oil well producing by matrix acidizing. The acid treatment was performed on the Hummar and Shueib formations, which improved the productivity of their wells [5].

The aims of this study are as follows:

• To limit the drive mechanisms and to promote a conformity within suitable water influx models and unknown parameter calculations;
• To calculate the water influx using an MBE; and
• To predict the oil production for the next 11 years.

This study was conducted at the Hamzeh oil reservoir to delimit its water influx, to predict its performance under active water drive, and to match its production and pressure decline history to the aquifer model. Water influx models simulate and predict aquifer performance. Most importantly, they predict the cumulative water influx history. When a water influx model is successfully integrated with a reservoir simulator, the net result is a model that effectively simulates the performance of water drive reservoirs [6]. Two geometries are considered: radial- and linear-flow systems. The radial-flow system assumes that the reservoir is a right cylinder and that the aquifer surrounds the reservoir.

An integrated analytical model was developed to simulate the combination of the water drive mechanism and the oil reservoir’s performance. It couples the general material balance equation with equations for the water influx, the water-invaded pore volume, and the gas-invaded pore volume [7]. The method that was chosen for optimizing and accelerating history matching is a material balance analysis. Based on simple measurements such as average pressure, daily production rate, and pressure, volume, and temperature (PVT) tests, a material balance model can be constructed for a three-phase system of a hydrocarbon reservoir [8]. A similar study was conducted to determine the water influx in the Volve oil field, which is situated in the South Viking Graben in the North Sea in block 15/9, approximately 200 km west of Stavanger at the southern end of the Norwegian sector [9].

Description of the Hamzeh Oil Reservoir

The Azraq Basin is located in northeastern Jordan and covers an area of approximately 200 km². Hydrocarbons were discovered in the Hamzeh field, and the total oil production from this field is approximately 1.49 × 10⁶ bbl = 230,405 m³. The main reservoir consists of the Hummar and Shueib dolomite and limestone formations as show in Figure 1. lithostratigraphic cross section of the Hamzeh field [10]. and contains six wells. Three of these wells produce oil (Well A, Well B, and Well C), and the other three wells need to be worked over.

The properties of the studied wells, such as the rock formation and fluid properties, were extracted and tested. The data are shown in

Table 1. Rock and fluid characteristics of reservoir.

Properties	Shueib	Hummar	Unit
Porosity	15	11	%
Permeability	290	132	md
Oil gravity	31.86°	32°	API°
Gas gravity	0.741	0.741	sp. gravity
Gas–oil ratio (produced)	4.994	5350	Nm3/m3
Check pressure	262.6	260.7	bar
Check temperature	90.56	90.55	°C
Water compressibility	2.12 × 10−2	2.12 × 10−2	Pa−1
Oil saturation	60	56	%
Water saturation	20	17	%
Oil density	0.8662	0.8662	gr/cm3
Oil viscosity	0.00217	0.00217	mPa.s
Oil formation volume factor	0.1677	0.1814	m3/m3
Bubble point pressure	68	51	Bar
Thickness	26.5	16.7	ft

.

2. Methodology

It has been confirmed that the main driving energy responsible for oil displacement in the Hamzeh reservoir is water influx or a water drive mechanism. Water influx occurs when the reservoir pressure drops due to a long interval of fluid production; therefore, the water influx rate in the general material balance equation must amount to the volumetric withdrawal rate. The OOIP is calculated using a volumetric method and a material balance equation. These methods depend on the PVT data and the analysis of the conventional core samples.

Amerada used other tools to measure the built-up pressure and quartz pressure gauges to predict future reservoir performance [5].

Next, the reserve estimates are performed. These estimates can be classified according to the pre- and post-production stages (static and dynamic methods). Static methods indicate the analogy and volumetric calculations that are used before the reservoir production process, while dynamic methods include the performance techniques that are applied after production has started and typically require production data and pressure analysis of the wells.

A.

Volumetric method

The volumetric method is used to estimate reserves in the early periods of production, before pressure is obtained and the production data is applied to the performance techniques. The hydrocarbon estimation is acquired from the reservoir rock volume and then the final hydrocarbon total can be predicted.

B.

Material Balance Equation (MBE)

The MBE has long been considered a basic method for predicting and analyzing reservoir performance. The MBE can be used to calculate hydrocarbon content (N) and water influx (We) to forecast future reservoir performance, to evaluate reservoir pressure, and to predict the ultimate hydrocarbon recovery under many kinds of drive mechanisms. The volumes of formation waters and crude oils are affected by temperature, pressure, and the quantity of gas in solution. Compressibility of the formation waters, pore volume compressibility, volume factors, predictions of oil displacement by water influx, and decline curve calculations should all be entered into the material balance calculations. These calculations were based on the oil produced by liquid expansion above the bubble point and on the oil produced by rock and fluid compressibility.

3. Calculation to Determine Oil in Place with the Material Balance Equation during Water Influx

3.1. Drive Mechanism

At pressures above the saturation pressure, the material balance equation can be written as follows:

$$\mathrm{Np}\mathrm{Bo}-\mathrm{We}+\mathrm{Wp}\mathrm{Bw}=\mathrm{N}\mathrm{Bio}\mathrm{Coe}\Delta \mathrm{P}$$

(1)

where the effective compressibility of the oil, Coe, is defined as follows:

$$\mathrm{Coe}=\mathrm{Soi}\mathrm{Co}+\mathrm{Sw}+\mathrm{Cw}+\frac{\mathrm{Cf}}{\mathrm{Soii}}$$

(2)

and

$$\mathrm{Co}=\frac{\mathrm{dBo}}{\mathrm{dP}\mathrm{Boi}}$$

(3)

$$\mathrm{Cw}=\frac{\mathrm{dBw}}{\mathrm{dp}\mathrm{Bwi}}$$

(4)

With the tank oil in place, one method for estimating N OOIP is to apply Equation (1) at several different successive reservoir pressures. Each calculation set for oil production assumes that the water influx (We) is equal to zero. Afterward, the calculated values of the tank oil in place (Ncalc, oil production [Np]) are plotted.

If there has been water encroachment, the Ncalc vs. Np relationship would slope upward to the right and indicate that the driving mechanisms are the water drive and solution gas drive. If there has been no water encroachment, the relationship would be horizontal, indicating a volumetric-type reservoir with a solution gas drive.

The extrapolation of either curve to the Ncalc axis would give the true N as shown below.

$$\mathrm{N}=\frac{\mathrm{NP}\mathrm{BO}-\mathrm{We}+\mathrm{WP}\mathrm{BW}}{\mathrm{Boi}\mathrm{COe}\mathsf{\Delta }\mathrm{P}}$$

(5)

where Coe, the coefficient oil compressibility, is defined as

$$\mathrm{Coe}=\frac{\mathrm{Soi}\mathrm{Co}+\mathrm{Swi}\mathrm{Cw}+\mathrm{Cf}}{\mathrm{Soi}}$$

(6)

and

$$\mathrm{Soi}=\left(1-\mathrm{Swi}\right)$$

(7)

$$\mathrm{Co}=\frac{\mathrm{Bo}}{\mathrm{Boi}\mathsf{\Delta }\mathrm{P}}$$

(8)

where NP = cumulative oil produce (m³), N = OOIP = original of oil in place (m³), We = cumulative water influx (reservoir condition), Wp = cumulative water production (m³), BW = water formation volume factor (m³/m³), BoI = initial oil formation volume factor (m³/m³), BO = oil formation volume factor (m³), ΔP = cumulative pressure reduction from initial pressure (bar), Co, Cw, and Cf = compressibility of oil, water, and formation, respectively, Soi = initial oil saturation percentage, Swi = initial water saturation percentage, and Ncalc = calculated values of tank oil in place (m³ × 10⁶). The results are presented in tables for the Hummar formation using Hummar PVT data, the Hummar formation using Shueib PVT data, and the Shueib formation.

The relationship between the calculated values of tank oil in place (Ncalc) and the cumulative oil production (Np) for both the Hummar and Shueib formations is shown on

Table 2. Calculation of material balance for the Hummar formation using Hummar PVT data, where Swi = 11.8%, Cr = 0.01620 Pa⁻¹, and N = 6.12 × 10⁶ m^3..

P Bar	Δp Bar	Np m3	Coe Pa−1	Ncalc × 106 m3	Np × Bo	NBoi Coe Δp	We m3
271.29	-	-	-	-	-	-	-
267.82	3.46	4947.21	0.04680	14.297	522,415	1,045,260.70	2984.33
267.55	3.74	5789.18	0.04449	16.162	6113.37	1,214,199.59	3799.79
267.34	3.94	8955.64	0.05228	20.373	9448.20	1,894,717.82	6609.04
266.66	4.62	15,898.25	0.04999	32.259	16,788.55	3,363,827.89	13,589.00

and

Table 3. Calculation of material balance for the Shueib formation using Shueib PVT Data, where N = 3.25 × 10⁶ m³ Swi = 11.6%, and Cr = 0.01337 Pa⁻¹.

P Bar	Δp Bar	Np m3	Coe × 10−2 Pa−1	Ncalc × 106 m3	Np × Bo	NBoi Coe Δp	We m3
265.30	-	-	-	-	-	-	-
263.26	2.04	528.13	0.04449	2.72	602.59	106,399.29	-
261.22	4.08	1901.74	0.03945	5.54	2169.88	384,321.07	894.05
259.18	6,12	4329.88	0.03711	8.94	4940.39	875,098.95	3141.97
254.42	0.88	21,115.42	0.03568	25.51	24,092.69	4,268,167.65	21,030.31

and in Figure 2 and Figure 3. These figures explain that the calculated OOIP increases along with the cumulative oil production for both formations. These graphs slope upward, indicating that the drive mechanisms are water drive and solution gas drive.

3.2. Pore Volume Compressibility

The results show that the Hummar and Shueib formations are dual-porosity systems. The secondary porosity in these formations is believed to be attributable to fractures. The total pore volume compressibility of such systems can be expressed as follows:

$${\mathrm{C}}_{\mathrm{rt}}={\mathrm{F}}_{\mathrm{f}}\times {\mathrm{C}}_{\mathrm{f}}+{\mathrm{F}}_{\mathrm{fm}}\times {\mathrm{C}}_{\mathrm{m}}$$

(9)

where C_rt = total pore volume compressibility (Pa⁻¹), F_f = the fraction of total pore volume consisting of fractures, C_f = fracture compressibility (Pa⁻¹), F_fm = the fraction of total pore volume that consists of matrix, and C_m = matrix pore volume compressibility (Pa⁻¹).

The values of Cm were approximated by the correlation previously established by Teeuw [11], and the values of C_f were approximated with the following correlation established by Jones [12]. Cf is calculated as follows:

$${\mathrm{C}}_{\mathrm{f}}=\frac{-1}{\mathrm{NOB}\mathrm{l}\mathrm{n}\left(\frac{\mathrm{NOB}}{{\mathrm{P}}_{\mathrm{H}}}\right)}$$

(10)

where NOB = net overburden pressure and P_H = fracture healing pressure. The NOB was assumed to be 340 bars. This data came from the evaluation of the special core analyses. For P_H, a value of 2721 bars was assumed, as suggested by Jones [12]. The fractured volume (F_f) value is based on the well test interpretation.

Table 4. Calculated pore volume compressibility.

Formation	Porosity %	Ff %	Crt, V/v/Pa × 10−2
Shueib	12.25	0	0.0137
Shueib	12.25	1.9	0.0239
Hummar	10.45	0	0.0162
Hummar	10.75	1.9	0.0286

shows the calculated pore volume compressibility. For an average porosity of 12 percent, the rock compressibility is 0.0310365 Pa⁻¹. A compressibility of 0.02869 Pa⁻¹ means that the volume of one million barrels of reservoir fluid will increase by 0.02869 m³ for every one pascal decrease in pressure.

3.3. Calculation of the Natural Water Influx

The pore volume compressibility is predicted by a simple model and is used to predict the water influx into a reservoir. The compressibility is defined mathematically as follows:

$$\mathsf{\Delta }\mathrm{V}=\mathrm{C}\mathrm{V}\mathsf{\Delta }\mathrm{P}$$

(11)

When the production of oil is conducted, a drop in the pressure of the reservoir results in the water in the aquifer expanding and entering the inside of the reservoir.

$$\mathrm{Ni}=\frac{\left(\mathrm{ra}2-\mathrm{re}2\right)\mathrm{h}Ø}{5.615}$$

(12)

$$\mathrm{Ne}=\left(\mathrm{CW}+\mathrm{Cf}\right)\mathrm{Wi}\left(\mathrm{Pi}-\mathrm{P}\right)$$

(13)

where We = cumulative water influx (m³), CW = aquifer water compressibility (Pa⁻¹), Cf = aquifer rock compressibility (Pa⁻¹), Wi = initial volume of water in the aquifer (m³), Pi = initial reservoir pressure (bar), P = current reservoir pressure (bar), ra = radius of the aquifer (m), re = radius of the reservoir (m), h = thickness of the aquifer (m), and Ø = percentage of porosity.

Since water influx occurs when the reservoir pressure drops due to a long period of fluid production, the water influx rate in the general material balance equation must be equal to the volumetric withdrawal rate. This is caused by the aquifer water influx expansion and influx into the reservoir, so the water influx average, or dWe/dt, = [average of active oil volumetric voidage] + [average of free gas volumetric voidage] + [average of water volumetric voidage] [13].

When written in terms of single-phase oil volume factors [14], the formula is as follows:

$$\frac{\mathrm{dWe}}{\mathrm{dt}}=\mathrm{Bo}\frac{\mathrm{dNp}}{\mathrm{dt}}+\left(\mathrm{R}-\mathrm{Rs}\right)\frac{\mathrm{dNp}}{\mathrm{dt}}\mathrm{Bg}+\frac{\mathrm{dWp}}{\mathrm{dt}}\mathrm{Bw}$$

(14)

where dNp/dt = daily oil rate (m³/d), dWe/dt = rate of water influx (m³/d), dNp/dt = the daily free gas rate (m³/d), R = current gas–oil ratio (m³/d), Rs = solution gas–oil ratio subtracted from the net daily or current gas–oil ratio (m³/d), and Bg = gas volume factor of the gas voidage term.

In the case of the steady-state aquifer, the rate of water flow dWe/dt is directly proportional to the decline in reservoir pressure (Pi–P). This assumes that the pressure is maintained at the initial Pi value somewhere in the aquifer and that the flow to the reservoir is, according to Darcy’s law, proportional to the pressure differential assuming that the water viscosity, average permeability, and aquifer geometry remain constant. This is calculated as follows:

$$\frac{\mathrm{dWe}}{\mathrm{dt}}=\mathrm{K}\left(\mathrm{Pi}-\mathrm{P}\right)$$

(15)

$$\mathrm{We}=\mathrm{K}{{\displaystyle \int }}_0^1\left(\mathrm{Pi}-\mathrm{P}\right)\mathrm{dt}$$

(16)

using the technique of Havlena and Odeh [15]. The following material balance equation is for the Hamzeh undersaturated reservoir, which is always producing above the bubble point:

$$\mathrm{Np}\mathrm{Bo}=\mathrm{N}\left(\mathrm{B}0-\mathrm{Boi}\right)+\mathrm{We}$$

(17)

This will be used as the following equation of a straight line:

$$\mathrm{If}=\mathrm{NEo}+\mathrm{We},\mathrm{then}\mathrm{We}=\mathrm{F}-\mathrm{NEo}$$

(18)

or

$$\frac{\mathrm{F}}{\mathrm{E}0}=\mathrm{N}+\frac{\mathrm{We}}{\mathrm{E}0}$$

(19)

$$\mathrm{F}=\mathrm{Np}\mathrm{Bo}\left(\mathrm{Bo}-\mathrm{Boi}\right)$$

(20)

$$\mathrm{Eo}=\left(\mathrm{Bo}-\mathrm{Boi}\right)$$

(21)

All the parameters included in the equation are taken from the well test and the core and PVT analyses for the Hummar and Shueib formations. The results of the OOIP calculations are presented in

Table 5. Reserve estimation using water influx calculation for Hummar formation.

Time Days	Pavg Bar	$\mathbf{\Delta }\mathbf{P}$ Bar	Np m3	Bo m3/m3	F m3	Eo m3/m3	F/Eo m3	tD/t	TD	U	Qt	We m3	We/Eo
0	271.29	0	-	0.12576	-	-	-	-		121.3	-	-	-
100	267.68	3.6	4897.21	0.12579	5166.21	0.0000262	1.9 × 108	0.0497	4.970	121.3	4.5	4599.92	1.7 × 108
117	267.34	3.94	5805.63	0.12579	6124.70	0.0000286	2.1 × 108	0.0497	5.815	121.3	5	5593.23	1.9 × 108
404	266.87	4.42	15,873.67	0.12579	16746.47	0.0000321	5.2 × 108	0.0497	20.08	121.3	12.4	15,545.38	4.8 × 108
660	266.73	4.55	22,822.21	0.12579	24077.36	0.0000333	7.2 × 108	0.0497-	32.80	121.3	17.8	23,130.95	6.9 × 108

and

Table 6. Reserve estimations with water influx calculations using the Havlena–Odeh technique [13] (Shueib formation).

Time Days	Pavg Bar	$\mathbf{\Delta }\mathbf{P}$ Bar	Np m3	Bo m3 /m3	F m3	Eo m3/m3	F/Eo m3	tD/t	TD	U m3/bar	Qt m3	We	We/Eo
0	265.30	0	-	0.1360	-	-	-	-	-	32.58	-	-	-
8	263.60	1.76	528.13	0.1360	602.69	0.0000119	5 × 107	0.0684	0.54	32.58	1.06	3236.8	3 × 107
33	261.90	3.40	1901.71	0.1360	2170.39	0.0000250	8 × 107	0.0684	2.25	32.58	2.5	15,268	7 × 107
76	261.22	4.08	4329.82	0.1360	4941.73	0.0000298	1.6 × 108	0.0684	5.20	32.58	4.6	33,712	1 × 108
377	260.27	5.03	21,115.08	0.1361	24,100.73	0.0000369	6 × 108	0.0684	25.7	32.58	15	135,582	4 × 108
581	259.79	5.51	29,996.66	0.1361	34,239.20	0.0000405	8 × 108	0.0684	39.7	32.58	20.7	204,802	6 × 108

for the Shueib and Hummar formations, respectively. The results from the material balance equation calculations match the water influx calculations extremely well.

The plot for F/Eo versus We/Eo should be linear with the intercept F/Eo = N (original oil in place) when We/Eo = 0 and the function has a unit slope. Table 4 and Table 5 show the reserve estimations of water influx Havlena and Odeh [15] for the Hummar and Shueib formations, respectively. Figure 4 and Figure 5 show that the reservoir mechanism is determined by plotting F/Eo versus water influx (We/Eo). From these figures, we can conclude that the reservoir mechanism is water drive because the shape of the curve is a straight line. The material balance equation calculations match the water influx calculations extremely well.

$$\mathrm{We}={{\displaystyle \int }}_0^{\mathrm{t}}\mathsf{\Delta }\mathrm{P}\mathrm{dt}=\mathrm{Cs}\mathsf{\Sigma }\mathsf{\Delta }\mathrm{P}\mathsf{\Delta }\mathrm{t}$$

(22)

where Cs is the water influx constant (m³/kg/m²) and ΔP is the pressure drop (bar).

The water influx is calculated using the unsteady-state water influx equation as illustrated by Van Everdingen, A.F. and W. Hurst [6].

$$\mathrm{We}=\mathrm{U}\mathrm{Dp}\mathrm{WD}\left(\mathrm{tD}\right)$$

(23)

where WD = the dimensionless water influx, tD = dimensionless time, and

U = 1.119 Ø h Ct rw2 × Θ/360

(24)

which is the aquifer constant for radial geometry (m³/kg/m²). Θ = the angle subtended by the reservoir circumferences, h = thickness of the pay zone (m), rw = well radius (m), and Ct = oil compressibility, volume per volume per pound per square inch (Pa⁻¹).

A comparative study of oil recovery factor determination for edge and bottom water drive mechanisms using water influx models shows that aquifer volume and aquifer permeability have a linear relationship with both bottom and edge water drives [16]. Bottom water drive is more efficient than edge water drive, and therefore, bottom water drive reservoirs tend to give higher oil recovery compared to edge water drive reservoirs [16]. The results of this study show that the mechanism of oil production is water, that the relationship is linear, and that this is consistent with other research.

Water influx modeling is highly needed to predict the future performance of the reservoir. Huge uncertainties exist when modeling water influx due to several reasons. First, we rarely drill wells into an aquifer to gain the necessary information about the aquifer reservoir properties. Second, properties are frequently inferred from what has been observed in the reservoir, and finally, geometry and areal continuity of the aquifer itself is a big question [17]. Natural influx of water in oil reservoirs surrounded by water-bearing rocks (aquifers) plays a very important role in increasing oil recovery. Such reservoirs are referred to as water drive reservoirs. The pressure drop in a reservoir caused by production of hydrocarbons prompts the influx of water from the surrounding aquifer to offset it. This pushes more hydrocarbons towards the surface, thus enhancing the oil recovery [18].

The dimensionless water influx (WD), a function of dimensionless time (tD) for a range of the aquifer-to-reservoir radius, is found from the charts done by Van Everdingen, A. and W. Hurst [6]. Successive dimensionless time (tD) is calculated from the following diffusivity equation:

tD = 0.00634 Kx Tx/Ø μCo rw²

(25)

where Kx = permeability (millidarcies), T = time (days), Ø = porosity fraction, μ = viscosity centipoise (mPa.s), Co = oil compressibility, volume per pound per square inch (Pa⁻¹), and rw² = reservoir radius (m).

The water influx is then found using the following equation:

We = UΣΔP × Q(t)

(26)

where the Q(t) values are found from the dimensionless time tables. The results are presented in

Table 7. Results of water influx match for Hummar formation.

Time Days	Pressure Bar	Material Balance Water Influx m3	Calculated Water Influx m3, Steady State	Calculated Water Influx m3, Unsteady State
134	267.89	523.04	324.95	379.00
305	265.91	1325.25	1395.20	1513.33
520	264.89	2062.76	3199.63	3245.25
540	264.42	2885.16	3388.81	3429.65
654	263.40	4538.08	4588.79	4551.91
712	262.78	6413.20	5372.25	5280.67

and

Table 8. Results of water influx match for Shueib formation.

Time Days	Pressure Bar	Material Balance Water Influx m3	Calculated Water Influx m3, Steady State	Calculated Water Influx m3, Unsteady State
25	263.12	161.52	171.22	213.65
46	261.42	582.18	574.32	687.11
97	259.52	1712.53	2121.90	2293.28
112	258.50	2558.14	2730.02	2907.58
160	257.14	4590.73	5002.53	5095.62
187	256.05	6954.97	6600.05	6599.41
237	255.10	9611.90	9505.25	9287.61

.

The prediction for the oil and water that is produced as a function of pressure is shown in Figure 6 and Figure 7. The reservoir pressure declines rapidly and continuously while the amounts of produced oil and water increase. This is because the amount of water that encroached into the reservoir is not enough to replace the produced oil and water.

3.4. The Mechanism of Oil Displacement by Water

After the prediction of the future reservoir performance under the active water drive mechanism, it is worthwhile to describe the mechanism of oil displacement by water. The fluid oil and water saturation distribution dip in the normal direction as the flood moves through the reservoir, so at each increment of oil production (Np) corresponding to a pressure step at different time intervals, the water and oil saturation will change at each point in the reservoir. The relative permeability to oil and water, which are themselves a function of the water saturation, will be distributed at the same point in the reservoir.

By depicting this distribution, we construct the mathematical model that describes the movement of water toward the highest point of the reservoir and its influence on the mechanism of oil and water production during the operating life of the reservoir. The parameters for this enable the depiction of the process of oil displacement by water.

In addition to water saturation distribution, the following parameter is calculated: the fractional water flow or the water cut at each time step. The fractional flow equation can qualitatively show the ratio of the flux rate of water to the total flux rate of fluids at definite periods and spaces in a linear system of water injection. The equation demonstrates the relation between the volume of water flux in every zone of the reservoir and the amount of water saturation in a homogenous two-phase isothermal system.

Initially, only oil was produced from the wells in the part of the reservoir where only the connate (immobile) water saturation prevailed. However, in the later stage of production, when the water saturation (Sw) becomes higher than the connate water saturation (Swc), oil and water are produced from the same well. Therefore, in the Hamzeh reservoir, the wells will produce oil only at 11.8% and 11.6% in the Hummar and Shueib formations, respectively. They will then produce oil with water cut percentages corresponding to a higher water saturation (Sw) until the residual oil saturation (Sor) is achieved, and only water will be produced. Thus, the fraction of water flow or water cut corresponding to any water saturation is calculated as follows:

Fw = Qw/Qo + Qw = 1/1+ Ko/Kwx µw/µo

(27)

where Ko and Kw are the permeabilities to oil and water at different saturations—as measured in the laboratory—of the core, µo and µw are oil and water viscosities, and Qo and Qw are the oil and water volume factors, which are obtained from the PVT analysis. At each cumulative phase of oil production (Np) and at every corresponding pressure step, the oil saturation that prevails in this part of the reservoir becomes as follows (the water saturation at this step increases as much as the oil saturation decreases):

$$\mathrm{So}=\frac{\left(\mathrm{N}-\mathrm{Np}\right)\mathrm{Bo}\left(1-\mathrm{Swi}\right)}{\mathrm{NBoi}}$$

(28)

Thus, the water saturation is calculated as follows:

$$\mathrm{Sw}=1-\mathrm{So}$$

(29)

Ossai, P.G. et al. [19] show the relationship between the relative permeability ratio and water saturations. This was input as data for the prediction of the water flooding scheme performance.

If the water saturation distribution at any point and the relevant water fraction flow or water cut (fw) are known, the water–oil ratio (WOR) can be calculated as follows:

$$\mathrm{WOR}=\frac{\mathrm{fw}}{1+\mathrm{fw}}$$

(30)

A water cut (wco) is the ratio of water production to the production of hydrocarbon fluids. A water cut in exploration wells can be calculated using the following equations:

$$\mathrm{wco}=\frac{\mathrm{WOR}}{1+\mathrm{WOR}}$$

(31)

where wco = water cut for oil well and WOR = water–oil ratio, and KRwo = permeability ratio of water to oil fraction.

$$\mathrm{KRwo}=\frac{\mathrm{Krw}}{\mathrm{Kro}}$$

(32)

where Kro = relative permeability for oil and Krw = relative permeability for water.

$$\mathrm{WOR}=\frac{\mathrm{fw}}{1-\mathrm{fw}}$$

(33)

$$\mathrm{wc}=\frac{\mathrm{WOR}}{\mathrm{WOR}+1}=\frac{\mathrm{fw}}{\mathrm{fw}+1-\mathrm{fw}}=\mathrm{fw}$$

(34)

The calculated values for both of the Hummar and Shueib formations are given in

Table 9. Resulting relative permeabilities, fractional flow data of oil. sw versus fw for Shueib formation, μo = 2.54 cp, μw = 0.3 cp.

sw%	fw	wc	kro	Krw
11.6	0.000	0.000	1.00	0.00
20.0	0.321	0.346	0.483	0.027
25.0	0.631	0.659	0.307	0.062
30.0	0.804	0.824	0.1833	0.089
35.0	0,913	0.923	0.100	0.125
40.0	0.968	0.972	0.43	0.1567
45.0	0.992	0,993	0.012	0.193
50.0	0.997	0.998	0.004	0.230
56.0	1.000.	0.9992	0.000	0.272

and

Table 10. Resulting relative permeabilities, fractional flow and water cut values of oil. Sw versus fw for Hummar formation, μo = 2.8 cp, μw = 0.3 cp.

sw	fw	wc	Kro	Krw
11.8	0.000	0.469	1.000	0.000
20.0	0.4573	0.674	0.587	0.053
25.0	0.6629	0.815	0.413	0.087
30.0	0.8073	0.898	0.274	0.123
35.0	0.8932	0.960	0.173	0.155
40.0	0.9589	0.993	0.080	0.200
45	0.9928	0.998	0.017	0..253
52	1.0000	0.9994	0.000	0.320

. “Sw” and “fw” are the water saturation and fractional flow at the producing end of the system, and “Pv” is the water invasion at the portion fraction of pore volume.

Table 11. Summary calculation of the data of the breakthrough of water saturation.

Formation	Swbt	fw	dfw/dsw	Wi
Hummar	0.21	o.573	5.435	0.184
Shueib	0.26	0.674	4.681	0.214

shows the calculations of fractional water flow, fractional flow slope, and water influx.

A.

Water breakthrough time

Water breaks through to the producing wells when a certain value of water saturation is achieved. This is called breakthrough water saturation, and it is obtained using the technique used by S.E. Buckley and M.C. Leverette [20] as well as Welge, H.J [21]. This technique can estimate when the water breakthrough can occur. The time of water breakthrough is a function of the water influx (Wi), itself a function of pore volume during the water flood. The water influx is defined as follows:

$$\mathrm{Wi}=\left(1/\mathrm{dfw}/\mathrm{dsw}\right)\mathrm{swbt}$$

(35)

$$\mathrm{t}=\frac{\left(\mathrm{Wi}\times \mathrm{cum}.\mathrm{rock}\mathrm{vol}.\times \mathrm{porosity}\right)}{\left(\mathrm{q}\times \mathrm{Bo}\times 5.615\times 365\right)}$$

(36)

where t is the time in years, Wi is the water influx, and Bo is the formation volume factor.

The time of breakthrough of water saturation (swbt) depends on the flow rate q/d and the withdrawal rate from the reservoir, so the time of breakthrough is calculated in the range of 5.50 to 16.50 years and 5.40 to 11 years for the Hummar and Shueib formations, respectively.

B.

Waterfront distance from the water–oil connection or aquifer

Since the oil is dislodged by the water, the water–oil saturation starts to increase from the point of contact in the reservoir. The water saturation at any point in the reservoir is proportional with the net rate of water gain at that point.

At the initial condition before production starts, the water influx (We) equals zero and the time (t) equals zero. Therefore, at any given time after the beginning of production, as the reservoir pressure declines, water flows from the aquifer and moves toward the reservoir through the water–oil contact in the linear system. The rate of the water movement is directly proportional to the cross section of the water breakthrough area and the porosity of the reservoir; therefore, the position of the constant water saturation plane or the distance of the waterfront from the aquifer can be estimated for the Hummar and Shueib reservoirs.

The following equation uses the Buckley–Leverett dfw/dsw:

$$\mathrm{X}=\frac{5.615\times \mathrm{Qt}}{\text{Ø}\mathrm{A}}\times \frac{\mathrm{dfw}}{\mathrm{dsw}}\times \mathrm{sw}$$

(37)

This is the Welge equation that is applicable up to the flood front (swf):

$$\mathrm{X}=\frac{5.615\times \mathrm{Qt}}{\text{Ø}\mathrm{A}}\times \frac{\mathrm{dfw}}{\mathrm{dsw}}\times \mathrm{swf}$$

(38)

These calculations are for different flow rates at different time steps. We tested the oil in place, the driving mechanism, the water influx, reservoir performance, and oil displacement.

The Hummar and Shueib formations are not homogenous, and in order to predict the flow rates of wells and their effects on pressure distribution, we must consider the structural contour map. The map uses an aerial grid system for both reservoirs and is extremely important for the description of the average petrophysical properties. The hydrocarbons of the both the Hummar and Shueib reservoirs are simulated using an aerial grid system with 200 grids, with an area representing 62,500 sq. ft. or 15.444 acres for each.

3.5. Prediction Pressure and Flow Rate Distribution

At any point in the reservoir, the distribution of pressure with distances from the bore hole in the formation is independent of formation permeability, but the velocity of this distribution is directly dependent on the formation permeability between wells. If one of the wells is producing at rate Q1, it will induce a pressure drop at the other wells located in the same reservoir or drainage area.

Using the Darcy’s law rate equation, a pressure drop in the second well can be calculated with the following:

$$\mathrm{Pe}-\mathrm{Pd}=\frac{\mathrm{Q}1\times \mathrm{Uo}\times \mathrm{Bo}\times \frac{\mathrm{Ln}\mathrm{re}}{\mathrm{rd}}}{7.08\times \mathrm{k}\times \mathrm{h}}$$

(39)

4. Results

The results obtained during the analysis of the data from Table 1, Table 2 and Table 3 are presented in this text to make a suitable model of reservoir water influx. To understand the effects of water influx on oil production, the following steps must be implemented: delimit the drive mechanism, make a model of the water influx, and predict oil production.

A.

Delimitation of reservoir drive mechanism

Table 2 and Table 3 show the reserve estimations for water influx for the Hummar and Shueib formations, respectively. One method for estimating N = original oil in place is to apply Equation (1) at a number of different successive reservoir pressures. In each calculation, We = 0 and the calculated values of the tank oil in place were plotted, and Ncalc = calculated values of tank oil in place versus Np = cumulative oil produced. If there has been water encroachment, the Ncalc vs. Np relationship would slope upward to the right, indicating water drive and solution gas drive.

Figure 2 and Figure 3 show that the calculated oil in place increases with cumulative production. Since these reservoirs are undersaturated, this indicates water encroachment and that the drive mechanisms for both reservoirs, Shueib and Hummar, are a combination of water drive and expansion drive.

B.

Model, parameter determination, and water influx modeling as shown in Table 4 and Table 5 show the reserve estimation of water influx, Havlen and D.S. Odeh model [15] for Hummar and Shueib formations, respectively. During the interpretation of the data, it appears that the appropriate water influx model for this case is the Pot aquifer model. Therefore, a plot of the term (F/Eo) against (We/Eo) is shown in Figure 4 and Figure 5. This plot produced a straight line. The value of the true N may be substituted in Equation (1), and the equation is solved for the water influx “(We)” at identical reservoir pressures. This would make water influx a function of Np, pressure, or time. Figure 4 and Figure 5 are graphs of the plots for F/Eo versus We/Eo, which determines what the reservoir mechanism is. From these figures, we can tell that the reservoir mechanism is water drive. This is due to the shape of the curve, which is a straight line. The material balance equation calculations match the water influx calculations extremely well. The reservoir energy is from fluid expansion and also accounts for the presence of water influx as source of energy.

C.

Prediction of reservoir behavior

The water influx is calculated using Equation (22), which is for the unsteady-state water influx. The calculation points of F/Eo vs. We/Eo increase as demonstrated. This indicates that the reservoir has been motivated by water influx. The water influx that was calculated with the material balance equation was first calculated utilizing both models of the Schilthuis method ((22) and (23)) and the unsteady-state equation. Results of the water influx are presented in Table 6 and Table 7. Performance predictions were made using the unsteady-state water influx model (23). Figure 8 and Figure 9 show these results, and the points almost make a straight line. Therefore, the water influx that results from each method (MBE and the unsteady-state model) corresponds very well with the reserve estimation that uses water influx calculations along with the D. Havlena and A.S Odeh techniques [15]. for the Hummar and Shueib formation oil reservoir.

D.

Predict oil production

The time of breakthrough of water saturation or the time of achieving such (swbt) depends on the flow rate q/d and the withdrawal rate from the reservoir, so the time of breakthrough is calculated in the range from 5.50 to 16.50 years and 5.40 to 11 years for the Hummar and Shueib formations, respectively.

5. Discussion

Table 1 show the characteristic reservoir; the reservoir is naturally fractured, and log–log plots and type-curve matching indicate the double porosity behavior of the reservoir (transient interporosity flow). Fractures are mostly vertical as obtained by core samples analysis. Initial reservoir pressures in the area are estimated to be 268.36 bar at 9629.73 ft drilled depth in Hummar formation, bottom hole pressure of 264.41 bar 9491.93 ft drilled depth, formation thickness of 16.4 ft, bubble point pressure of 51 bar, oil viscosity 3 cp, formation volume factor 1.075 vol/vol, permeability 132 mD, water saturation 17%, and average flow rate 34.57 m³/day. In the Shueib formation, the fractured permeability around the wellbore is very high compared with Hummar formation. The fractures are horizontal, fractured permeability Kf = 2083 mD, average Permeability Kav = 290 mD, average permeability of all reservoirs about 285 mD, and formation volume factor is 1.141 vol/vol. Initial reservoir pressures in the area are estimated to be 262.58 bar at 9485.37 ft drilled depth in Shueib formation, and bottom hole pressure 244.89 bar at 9508.33 ft drilled depth. Formation thickness is 26.25 ft, bubble point pressure is 68 bar, and production rate 51.62 m³ /day. The gas oil ratio (GOR) in the Shueib formation is relatively higher, 2.148 sm³/m³ compared to 0.402 sm³/m³ in the Hummar formation. There is an indication of water oil contact observed by electrical logging; in addition to that, in most Hamzeh wells the Shueib formation produced water. This study confirmed the existence of water and improved the existing understanding of drive mechanisms and fluid expansion in Hamzeh oil field.

The reservoir power was supplied mainly by rock and fluid expansion assisted by a certain volume of water influx. However, the water influx greatly influences reservoir pressure, thus affecting predictions for production. Oloro and Ukrakpor [22] worked on the determination of water influx in the Niger Delta, and they proposed that, to determine the total water influx into a reservoir at any given time, it is necessary to determine the water influx as a result of each successive pressure drop that has been imposed on the reservoir and aquifer. In calculating accumulative water influx, the total water influx must be calculated from the start. This is required for the different times during which the various pressure declines were effective, which is in line with the approach used in this study.

The water influx was predicted using the material balance equation and the Van Everdingen–Hurst unsteady-state equation, and the results for the cumulative oil production and water influx prediction were greater for the model than for the material balance equation, which does not bear all points obtained from the results in the study [1]. At the same time, a prediction for water influx was made by using the material balance equation and the Van Everdingen–Hurst unsteady-state equation. The results for the accumulative oil production and water influx prediction for the MBE and the unsteady-state model have an average absolute error of 1.214%. The material balance model is 9611.90 m³ and 9287.61 m³ with the unsteady-state model. The MB equation has greater priority than the equation model but not at all points with the results obtained in this study.

A Brownfield development plan was investigated by mining the data pack from the Equinor Volve field repository, and models and simulations were used to establish the quantity of hydrocarbon that is still recoverable therefrom [23]. In this study, models and simulations must be used to establish the quantity of hydrocarbon in the Hamzeh oil field.

A.

The mechanism of oil displacement by water

Utilizing the Buckley–Leverette frontal advance theory, we evaluated the water saturation distribution and residual oil saturation during the immiscible displacement process as well as the oil displacement efficiency. The displacement sweeps at each WOR were calculated from the relative permeability data shown in Table 9 and Figure 6 and Figure 7 for the Shueib and Hummar reservoirs, respectively.

B.

Fractional water flow or the water cut at each time step

By using Equations (32)–(34), the water cut value can be calculated. The resulting fractional flow values from these calculations are summarized in Table 9. In Figure 10, a tangent to the fractional flow curve from irreducible water saturation (Swir) is shown, and the point of tangency represents the water saturation at the front (Swf) and the value of the fractional flow for producing the water cut.

Table 9 shows the resulting relative permeabilities and the fractional flow data of oil. Figure 10 presents the variation in relative permeabilities and the portion of water flux that has the change in water saturation. The fractional water flow (fw) was calculated with Equation (26). The relative permeability diagram has curves of oil and water present in its usual curves, and the endpoint rate of relative water permeability was a fraction of 0.272. At this point, only water is injected into the reservoir.

These results indicate that the fractional flow amount curve was a 0.54 fraction of the oil manifests. The uniform method shows that a major quantity of movable oil in the reservoir is displaced by water injection. In addition, in Figure 10, the Sw versus fw relationship shows a water saturation cutoff. Table 10 presents the resulting relative permeabilities, the fractional flow, and the water cut values of oil.

The diagram of fractional water flux or the water cut versus water saturation was then plotted. Hence, the cutoff value for the water saturation for the whole oil field was realized. The diagram for fractional flow versus water saturation is shown in the following figure. In Figure 11, the Sw versus WC relationship also shows a water saturation cutoff of 40%. Utilizing fractional flow and water cut information together, we were able to notice that there is only small difference between the two values. This evidence shows that fractional flow is equivalent to the water cut; Fw = 0.751 and WC = 0.735. The resulting fractional flow value from the calculation above is summarized in Table 9.

The oil and water relative permeability curves under different displacement pressures were compared, which is shown in Figure 12 and Figure 13. Its ability is visible when the experiment’s displacement pressure is incremental, the range of the two-phase zone raises, and the residual oil saturation lowers. The relative permeabilities of oil and water rise, and the curve shifts to the upper right region. As the displacement pressure increases, the range of the pore sizes that permit the fluid to flow increases, so the remaining oil saturation will decrease and the oil recovery factor will be enhanced.

Next, the diagram of fractional flow or water cut versus water saturation was plotted. Hence, the cutoff value of the water saturation for the field and the fractional flow value equivalent to the water cut value can be obtained, as previously explained. Utilizing both fractional flow and water cut information, it can be observed that there is only a small difference between the two values. This shows that the fractional flow is equal to the water cut.

According to both the cumulative and average of production oil and water, daily oil and water production is calculated for both the Hummar and Shueib formations. Using the different WOR values of 0.79 m³, 3.97 m³, and 15.89 m³, water saturation (Sw) is obtained from the fractional flow curves for each corresponding fractional water flow (fw), or in other words, for each WOR.

Table 9 and Figure 13 show the changes in relative permeabilities and fractional water flow along with the change in water saturation. The fractional water flow (fw) was calculated with Equation (26). Relative permeability curves of oil and water present ordinary curves, and the endpoint rate of relative water permeability was the 0.320 fraction.

At this point, only water was pumped into the reservoir. The results indicate that the fractional flow value curve was 0.673 fraction for oil, manifesting a regular shape, and the application of this curve to the water flooding method shows that a large amount of movable oil in the reservoir is displaced by water injection. In addition, in Figure 14, the Sw versus fw relationship shows a water saturation cutoff of 40%.

In Figure 15, the Sw versus WC relationship also shows a water saturation cutoff of 40%. Utilizing fractional flow and water cut information together, it is possible to notice that there is only one small variance between the two values. This indicates that fractional flow is equal to the water cut.

Figure 10 presents the evidence for the fractional flow curve in the displacement of oil by water: the curve has an elongated fluid (Swi < Sw < 1-Sor). The stage of saturation at the tangent point of a straight line drawn from the irreducible saturation on the fractional flow curve is utilized to define the saturation rate at the waterfront, which conforms to the Buckley–Leverett theory.

A fractional flow curve (Fw versus Sw) is utilized to characterize the immiscible fluid displacement operation. Developing an illustrative fractional flow curve for a specific reservoir perfectly may be challenging after a liquid and special core information analysis has been performed. Then, the technique that measures fractional flow curve from the historic production information must be used.

The curve becomes the primary one for a similar model that permits the estimations of oil rate production and reserves for present or proposed wells. The relative permeability curves can be formed mainly due to the resultant fractional flow curve. A comparison of relative permeability curves can be made from the special core analysis information, which may raise confidence.

By extrapolating this tangent on the fractional flow curve to 1, Figure 10 and Figure 11 will yield the value of the average water saturation at breakthrough. The values that were obtained are as follows: fw = 0.56 for the Hummar formation, fw = 0.674 for the Shueib formation, and SWf = 0.21 and 0.26 for Hummar and Shueib formations, respectively.

Figure 16 shows the extrapolation of the WOR versus Np plot and changes in its decline of the ability to indicate increased oil recovery. Therefore, an inspection of the WOR versus Np plot is helpful in defining the increased recovery due to infill drilling or operational changes, as shown in Figure 16.

The variable slope of the curve indicates increased reserves after infill drilling in additional successful recovery efforts (including completions or treatments to restrain water as well as infill drilling).

As shown in Figure 17, the reservoir pressure declines rapidly and continuously with the amount of production time. Due to this, the water that enters the reservoir is not enough to replace the produced oil.

A.

Water breakthrough time

Table 12. The results of calculation for breakthrough time.

Formation	Sw	Cum.rock vol. m3 × 106	Prod.Rate m3/d	Water Breakthrough Time Years for Hummar Formation	Prod.Rate m3/d	Water Breakthrough Time Years for Shueib Formation
Hummar	0.118	28.12	15.89	16.50	14.80	11
			23.84	11.00	21.40	8.50
Shueib	0.116	21.23	31.79	8.300	29.30	8.20
			39.74	6.600	35.70	5.65
			47.69	5.500	43.65	5.40
			55.64	4.700	52.60	4.70

and Figure 18 and Figure 19 show that the oil production rate was varied, beginning from a principle state of 15.89 to 23 m³/day. It moved to 84 m³/day, and then to a range of 31.79 m³/day to 55.64 m³/day. This resulted in six simulation runs for the Hummar and Shueib formations, respectively.

This relation is visible in the diagram that shows performance of and change in oil production rate relative to breakthrough time. This increase in oil rate is inversely proportional to the breakthrough time. This is indicative of the actuality that increasing the production rate will result in a single breakthrough time and coning taking place.

However, the cumulative oil recovery increases with the influx rate, and the volume of oil produced before the synchronous production of the reservoir fluids is affected by the oil column height below and above perforations at breakthrough time. Thus, as the production rate rises, a smaller amount of “water-free” oil and “gas-free” oil will be produced before the ultimate concurrent coning into the well occurs in spite of the oil recovery increase along with the influx rate. This further backs up the actuality that rising the oil production rate will raise the rate of concurrent coning into the well.

The impact of oil influx rate on the slope of the WOR curve is minimal. This indicates that the differences in influx rates influence only the times at which water coning starts, or when water breaks into the well, and changes in the oil influx rate have a minimal effect on gas–oil ratio (GOR) and WOR. The time of breakthrough or the time of achieving the swbt depends on the flow rate q and the withdrawal rate from the reservoir, so the time of breakthrough is calculated in a range from 5.50 to 16.50 years and 5.40 to 11 years for the Hummar and Shueib formations, respectively.

Where time is expressed in years, “We” is water influx, and the average water influx range is between 18.4% and 21.4%. The breakthrough water saturation was 21% to 26% for the Hummar and Shueib formations, respectively. The oil displacement efficiency (ED) for the Hummar formation was 0.85 to 0.94% and was 0.90 to 0.99% for the Shueib formation. The time of water breakthrough, assuming cumulative production rate, with q = 160 to 240 m³/d, was calculated in a range between 5.65 and 16.50 years for the Hummar formation and from 5.40 to 11 years for the Shueib formation. The prediction of oil recovery performance under the energy of water influx is the main portion of the actual reservoir concerning oil displacement by water and recovery factors.

6. Conclusions

The problem that arises is how to know whether the quantity of water influx in the reservoir influenced future performance. With the master method of the material balance equation being applied to a group of available data (pressure, volume, and temperature data, production and reservoir data). Drive mechanisms are determined by the analysis of historical production data, primarily reservoir pressure data and fluid production ratios [24]. A solution to this problem was found using a methodology that was defined according to the main objectives, which consisted of the following:

• Identify the drive mechanism to discover the presence and size of the aquifer so as to know which energy mainly drives the fluids.
• Elect a suitable water influx model, water influx modeling method, and oil production prediction. The results showed that the drive mechanism (fluid and rock expansion) for Hamzeh oil reservoir has been confirmed. This study confirms that the reservoir energy comes from fluid expansion and also accounts for the presence of water influx as a source of energy. Similarly, the results of the oil production prediction seem to be confirmed by the total increase in production. The oil production rate was varied, beginning from a principle state of 15.89 to 23 m³/day. It moved to 84 m³/day and then to a range of 31.79 m³/day to 55.64 m³/day, resulting in six simulation runs for the Hummar and Shueib formations, respectively, which increased the recovery factor accompanied by a slight decrease in pressure.
• The water breakthrough time was calculated to be in a range between 5.65 and 16.50 years for the Hummar formation and 5.40 to 11 years for the Shueib formation.
• The fractional flow curve (fw) will yield the value of average water saturation at breakthrough. These values obtained were fw = 0.56 for the Hummar formation, fw = 0.674 for the Shueib formation, and SWf = 0.21 and 0.26 for the Hummar and Shueib formations, respectively. A fractional flow curve (Fw versus Sw) is utilized to characterize the immiscible fluid displacement operation.
• The recovery factor will be increased to 25% in Shueib formation, relatively higher than the 23% in Hummar formation. The results in this study were confirmed with other researchers in the same field. That natural influx of water in oil reservoirs surrounded by water-bearing rocks (aquifers) play a very important role in increasing oil recovery. This study examines the Alwyn North Field and presents a comparative analysis for producing crude oil using water drive, gas drive, and natural drive or depletion drive; profitability index for water injection was 1.73 and was 1.36 for gas injection. These economic tools help us to arrive at the conclusion that the best drive mechanism for the production of the field was water drive [25].