The Output of a Particular Producing Oil Well Decreases at the Continuous Rate of 12

Oil Ratio

The GOR is defined as the volume ratio of gas and liquid phase obtained by taking petroleum from one equilibrium pressure and temperature, in the reservoir, to another, at the surface, via a precisely defined path.

From: Elements of Petroleum Geology (Third Edition) , 2015

Black Oils

Ahmed El-Banbi , ... Ahmed El-Maraghi , in PVT Property Correlations, 2018

Estimating Solution Gas–Oil Ratio

Total solution GOR is obtained from oil and gas sales data. In theory, the solution GOR is constant as long as the reservoir pressure is above the bubble-point pressure; in practice, however, some wells produce below the bubble-point pressure and therefore have higher producing GOR than solution GOR. It is usually advisable to make a plot of producing GOR versus time or cumulative oil production (the latter is preferred) to help determine a good value to use as initial solution GOR. An example is given in Fig. 7.8. The figure shows both the instantaneous (producing) GOR and the cumulative GOR. Cumulative GOR is defined according to the following equation. The same data is plotted against time in Fig. 7.9.

Figure 7.8. Example producing GOR and cumulative GOR versus cumulative oil production. GOR, gas–oil ratio.

Figure 7.9. Example producing GOR and cumulative GOR versus time. GOR, gas–oil ratio.

(7.4) $R_p=\frac{G_p}{N_p}$

Fig. 7.8 shows that the field producing GOR is a constant value (around 580 scf/STB) for some time until cumulative oil production has reached 0.5 MMSTB. GOR starts to increase and reaches 1000 scf/STB. The GOR data generally show fluctuations. In this example, considerable variation in GOR data occurred in 2005 and after, due to changes in wells completion and poor maintenance of gas meters.

Consideration of the entire field GOR rather than individual wells GOR is important for selecting an appropriate value for use in PVT correlations. In addition, taking a 3 or 4-month moving average for the data reduces the scattering in GOR data and obtains a more representative average initial GOR. The correct PVT correlations input is the total GOR (and not the separator GOR). In the majority of oil field operations, the gas produced from the stock-tank (and sometimes the low-pressure separator) is vented or sent to the flare. Therefore, the value of the stock-tank GOR is not usually available. Correlations are used to estimate the stock-tank GOR. These correlations are dependent on primary separator conditions in addition to the usual correlations input parameters.

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Predicting Oil Reservoir Performance

Tarek Ahmed , D. Nathan Meehan , in Advanced Reservoir Management and Engineering (Second Edition), 2012

5.1.1 Instantaneous GOR

The produced GOR at any particular time is the ratio of the standard cubic feet of total gas being produced at any time to the stock-tank barrels of oil being produced at the same instant—hence, the name instantaneous GOR. Eq. (1.53) describes the GOR mathematically by the following expression:

(5.1) $\text{GOR}=R_{\mathrm{s}}+\left(\frac{k_{\text{rg}}}{k_{\text{ro}}}\right)\left(\frac{{\mu }_{\mathrm{o}}{B}_{\mathrm{o}}}{{\mu }_{\mathrm{g}}{B}_{\mathrm{g}}}\right)$

where

GOR=instantaneous gas–oil ratio, scf/STB

R _s=gas solubility, scf/STB

k _rg=relative permeability to gas

k _ro=relative permeability to oil

B _o=oil formation volume factor, bbl/STB

B _g=gas formation volume factor, bbl/scf

µ _o=oil viscosity, cp

µ _g=gas viscosity, cp

The instantaneous GOR equation is of fundamental importance in reservoir analysis. The importance of Eq. (5.11) can appropriately be discussed in conjunction with Figures 5.1 and 5.2. Those illustrations show the history of the GOR of a hypothetical depletion drive reservoir that is typically characterized by the following points:

Figure 5.1. Characteristics of solution gas drive reservoirs.

Figure 5.2. History of GOR and R _s for a solution gas drive reservoir.

Point 1. When the reservoir pressure p is above the bubble point pressure p _b, there is no free gas in the formation, i.e., k _rg=0, and therefore:

(5.2) $\text{GOR}=R_{\text{si}}=R_{\text{sb}}$

The GOR remains constant at R _si until the pressure reaches the bubble point pressure at point 2.

Point 2. As the reservoir pressure declines below p _b, the gas begins to evolve from solution and its saturation increases. However, this free gas cannot flow until the gas saturation S _g reaches the critical gas saturation S _gc at point 3. From point 2 to point 3, the instantaneous GOR is described by a decreasing gas solubility, as:

(5.3) $\text{GOR}=R_{\mathrm{s}}$

Point 3. At this point, the free gas begins to flow with the oil and the values of GOR progressively increase with the declining reservoir pressure to point 4. During this pressure decline period, the GOR is described by Eq. (5.1), or:

$\text{GOR}=R_{\mathrm{s}}+\left(\frac{k_{\text{rg}}}{k_{\text{ro}}}\right)\left(\frac{{\mu }_{\mathrm{o}}{\beta }_{\mathrm{o}}}{{\mu }_{\mathrm{g}}{B}_{\mathrm{g}}}\right)$

Point 4. At this point, the maximum GOR is reached due to the fact that the supply of gas has reached a maximum and marks the beginning of the blow-down period to point 5.

Point 5. This point indicates that all the producible free gas has been produced and the GOR is essentially equal to the gas solubility and continues to decline following the R _s curve.

There are three types of GORs, all expressed in scf/STB, which must be clearly distinguished from each other. These are:

•

instantaneous GOR (defined by Eq. (5.1));

•

solution GOR, i.e., gas solubility R _s;

•

cumulative GOR R _p.

The solution GOR is a PVT property of the crude oil system. It is commonly referred to as "gas solubility" and denoted by R _s. It measures the tendency of the gas to dissolve in or evolve from the oil with changing pressures. It should be pointed out that as long as the evolved gas remains immobile, i.e., gas saturation S _g is less than the critical gas saturation, the instantaneous GOR is equal to the gas solubility. That is:

$\text{GOR}=R_{\mathrm{s}}$

The cumulative GOR R _p, as defined previously in the MBE, should be clearly distinguished from the producing (instantaneous) GOR. The cumulative GOR is defined as:

$R_{\mathrm{p}}=\frac{\text{Cumulative}\mathrm{(}\mathrm{total}\mathrm{)}\text{gas}\text{produced}}{\text{Cumulative}\text{oil}\text{produced}}$

or

(5.4) $R_{\mathrm{p}}=\frac{G_{\mathrm{p}}}{N_{\mathrm{p}}}$

where

R _p=cumulative GOR, scf/STB

G _p=cumulative gas produced, scf

N _p=cumulative oil produced, STB

The cumulative gas produced G _p is related to the instantaneous GOR and cumulative oil production by the expression:

(5.5) $G_{\mathrm{p}}={\int }_0^{N_{\mathrm{p}}}(\text{GOR})\mathrm{d}{N}_{\mathrm{p}}$

Eq. (5.5) simply indicates that the cumulative gas production at any time is essentially the area under the curve of the GOR vs. N _p relationship, as shown in Figure 5.3. The incremental cumulative gas produced, ΔG _p, between N _p1 and N _p2 is then given by:

Figure 5.3. Relationship between GOR and G _p.

(5.6) $\mathrm{\Delta }{G}_{\mathrm{p}}={\int }_{N_{\mathrm{p}1}}^{N_{\mathrm{p}2}}(\text{GOR})\mathrm{d}{N}_{\mathrm{p}}$

This integral can be approximated by using the trapezoidal rule, to give:

$\mathrm{\Delta }{G}_{\mathrm{p}}=\left[\frac{{(\text{GOR})}_1+{(\text{GOR})}_2}{2}\right](N_{\mathrm{p}2}-N_{\mathrm{p}1})$

or

$\mathrm{\Delta }{G}_{\mathrm{p}}={(\text{GOR})}_{\text{avg}}\mathrm{\Delta }{N}_{\mathrm{p}}$

Eq. (5.5) can then be approximated as:

(5.7) $G_{\mathrm{p}}={\displaystyle \sum }_0{(\text{GOR})}_{\text{avg}}\mathrm{\Delta }{N}_{\mathrm{p}}$

Example 5.1

The following production data is available on a depletion drive reservoir:

p (psi)	GOR (scf/STB)	N p (MMSTB)
	1340	0
2600	1340	1.380
2400	1340	2.260
	1340	3.445
1800	1936	7.240
1500	3584	12.029
1200	6230	15.321

The initial reservoir pressure is 2925   psia with a bubble point pressure of 2100   psia. Calculate cumulative gas produced G _p and cumulative GOR at each pressure.

Solution

Step 1.

Construct the following table by applying Eqs. (5.4) and (5.7):

$\begin{array}{l}{R}_p=\frac{G_p}{N_p}\hfill \\ \Delta G_p=\left[\frac{{(\text{GOR})}_1+{(\text{GOR})}_2}{2}\right](N_{p2}-N_{p1})={(\text{GOR})}_{avg}\Delta N_p\hfill \\ G_p={\displaystyle \sum _0{(GOR)}_{avg}\Delta N_p}\hfill \end{array}$

P (psi)	GOR (scf/STB)	(GOR)avg (scf/STB)	N p (MMSTB)	ΔN p (MMSTB)	ΔG p (MMscf)	G p (MMscf)	R p (scf/STB)
2925	1340	1340	0	0	0	0	–
2600	1340	1340	1.380	1.380	1849	1849	1340
2400	1340	1340	2.260	0.880	1179	3028	1340
2100	1340	1340	3.445	1.185	1588	4616	1340
1800	1936	1638	7.240	3.795	6216	10,832	1496
1500	3584	2760	12.029	4.789	13,618	24,450	2033
1200	6230	4907	15.321	3.292	16,154	40,604	2650

It should be pointed out that the crude oil PVT properties used in the MBE are appropriate for moderate–low volatility "black oil" systems, which, when produced at the surface, are separated into oil and solution gas. These properties, as defined mathematically below, are designed to relate surface volumes to reservoir volumes and vice versa.

$\begin{array}{l}{R}_s=\frac{\begin{array}{l}\text{Volume of solution gas dissolved}\hfill \\ \text{in the oil at reservoir condition}\hfill \end{array}}{\text{Volume of the oil at stock}-\text{tank conditions}}\\ B_o=\frac{\text{Volume of oil at reservoir condition}}{\text{Volume of the oil at stock}-\text{tank conditions}}\\ B_g=\frac{\text{Volume of the free gas at reservoir condition}}{\text{Volume of free gas at stock}-\text{tank conditions}}\end{array}$

Whitson and Brule (2000) point out that the above three properties constitute the classical (black oil) PVT data required for various type of applications of the MBE. However, in formulating the material balance equation, the following assumptions were made when using the black oil PVT data:

(1)

Reservoir gas does not yield liquid when brought to the surface.

(2)

Reservoir oil consists of two surface "components"; stock-tank oil and total surface separator gas.

(3)

Properties of stock-tank oil in terms of its API gravity and surface gas do not change with depletion pressure.

(4)

Surface gas released from the reservoir oil has the same properties as the reservoir gas.

This situation is more complex when dealing with volatile oils. This type of crude oil system is characterized by significant hydrocarbon liquid recovery from their produced reservoir gases. As the reservoir pressure drops below the bubble point pressure, the evolved solution gas liberated in the reservoir contains enough heavy components to yield appreciable condensate dropout at the separators that is combined with the stock-tank oil. This is in contrast to black oils for which little error is introduced by the assumption that there is negligible hydrocarbon liquid recovery from produced gas. Also, volatile oils evolve gas and develop free-gas saturation in the reservoir more rapidly than normal black oils as pressure declines below the bubble point. This causes relatively high GORs at the wellhead. Thus, performance predictions differ from those discussed for black oils mainly because of the need to account for liquid recovery from the produced gas. Conventional material balances with standard laboratory PVT (black oil) data underestimate oil recovery. The error increases for increasing oil volatility.

Consequently, depletion performance of volatile oil reservoirs below bubble point is strongly influenced by the rapid shrinkage of oil and by the large amounts of gas evolved. This results in relatively high gas saturation, high producing GORs, and low-to-moderate production of reservoir oil. The produced gas can yield a substantial volume of hydrocarbon liquids in the processing equipment. This liquid recovery at the surface can equal or exceed the volume of stock-tank oil produced from the reservoir liquid phase. Depletion-drive recoveries are often between 15% and 30% of the original oil-in-place.

For volatile oil reservoir primary-performance prediction methods, the key requirements are correct handling of the oil shrinkage, gas evolution, gas and oil flow in the reservoir, and liquids recovery at the surface. If

Q _o=black oil flow rate, STB/day

$Q_{\mathrm{o}}^{\backslash }$ =total flow rate including condensate, STB/day

R _s=gas solubility, scf/STB

GOR=total measured gas–oil ratio, scf/STB

r _s=condensate yield, STB/scf

then

$Q_{\mathrm{o}}=Q_{\mathrm{o}}^{\backslash }-(Q_{\mathrm{o}}^{\backslash }\text{GOR}-Q_{\mathrm{o}}{R}_{\mathrm{s}})r_{\mathrm{s}}$

Solving for Q _o gives:

$Q_{\mathrm{o}}=Q_{\mathrm{o}}^{\backslash }\left[\frac{1-(r_{\mathrm{s}}\text{GOR})}{1-(r_{\mathrm{s}}{R}_{\mathrm{s}})}\right]$

The above expression can be used to adjust the cumulative "black oil" production, N _p, to account for the condensate production. The black oil cumulative production is then calculated from:

$N_{\mathrm{p}}={\int }_0^t{Q}_{\mathrm{o}}\mathrm{d}t\approx {\displaystyle \sum }_0^t(\mathrm{\Delta }{Q}_{\mathrm{o}}\mathrm{\Delta }t)$

The cumulative total gas production "G _p" and the adjusted cumulative black oil production "N _p" is used in Eq. (5.4) to calculate the cumulative gas–oil ratio, i.e.:

$R_{\mathrm{p}}=\frac{G_{\mathrm{p}}}{N_{\mathrm{p}}}$

See Whitson and Brule (2000).

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The Re-refining Process Experimental Results

Firas Awaja , Dumitru Pavel , in Design Aspects of Used Lubricating Oil Re-Refining, 2006

4.2.1 The Optimum Solvent to Oil Ratio

The solvent to oil ratio investigation is conducted at a solvent composition of 25% 2-propanol, 50% 1-butanol and 25% butanone as reported by Whisman et al. (1978). The results for mass balance for the optimum solvent to oil ratio experiments are tabulated in Table 4.2. While the tests for these experiments are tabulated in Table 4.3. The properties of produced solvent treated oil, i.e., oil recovery, ash reduction and sulfur reduction in relation to solvent to oil ratio are shown in Fig. 4.1.

Table 4.2. Measurements of mass balance for optimum solvent to oil ratio experiments

Solvent to oil ratio	Oil feed (g)	Solvent (g)	Extract (g)	Raffinate (g)	Extract Oil (g)	Solvent (g)	Loss (g)
2:1	45.11	79.29	108.95	15.45	39.35	69.50	0.10
3:1	45.10	117.26	150.14	12.22	40.18	109.70	0.27
4:1	45.00	158.12	192.80	10.32	42.77	149.85	0.18
5:1	45.00	197.53	237.50	5.04	42.72	194.61	0.17
6:1	45.00	239.20	276.90	7.30	43.27	233.82	0.00

Table 4.3. Test analysis of the optimum solvents to oil ratio experiments

Solvent to oil ratio	Oil recovery (wt%)	Solvent recovery (wt%)	Ash content (wt%)	Ash reduction (wt%)	Sulfur content (wt%)	Sulfur reduction (wt%)
2:1	87.22	87.65	0.272	22.73	1.389	14.73
3:1	89.10	93.55	0.207	41.93	1.422	12.71
4:1	95.05	94.77	0.163	53.69	1.483	8.96
5:1	94.94	98.52	0.194	44.89	1.238	24.00
6:1	96.15	97.75	0.279	20.80	1.213	25.54

Fig. 4.1. Solvent to oil ratio vs. weight percent product.

The results of the investigation, Table 4.3 and Fig. 4.1 indicate that the maximum ash reduction is achieved for solvent to oil ratio of 4:1. The oil recovery and ash reduction for the same ratio are better than that obtained for solvent to oil ratio of 3:1 and 2:1. This indicates that by increasing the solvent amount, the solvency power is improved. The percentage of oil recovery for the solvent to oil ratio of 6:1 is further improved, but this solvent to oil ratio produces an ash reduction lower than that obtained for the solvent to oil ratio of 4:1 and 5:1 as shown in Fig. 4.1. That means that solvent to oil ratio larger than 4:1 will lead to dissolution of some contaminants in the solvent phase especially the ash forming material, which was considered to be undesirable. As a result of the above mentioned facts, the solvent to oil ratio of 4:1 was considered to be the better solvent to oil ratio used for treatment of used lubricating oil. Nevertheless, ratios above 3:1 were not considered economically feasible by industry. Thus a solvent to oil ratio of 3 to 1 is considered the optimum ratio in this study because it gives good ash reduction, good oil recovery and low cost.

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Petroleum reservoir properties

Boyun Guo , in Well Productivity Handbook (Second Edition), 2019

2.2.1.1 Solution gas–oil ratio

The solution GOR is the fundamental parameter used to characterize oil. It is defined as the volume of gas, normalized to standard temperature and pressure (STP), which will dissolve in a unit volume of oil at the prevailing pressure and temperature of the actual reservoir. That is,

(2.1) $R_s=\frac{V_{gas}}{V_{oil}}$

where

R _s   =   solution GOR (scf/stb),

V _gas   =   gas volume at STP (scf), and

V _oil   =   oil volume at STP (stb).

In most states in the United States, STP is defined as 14.7   psia and 60°F. At a given temperature, the solution GOR of a particular oil remains constant at pressures greater than bubble-point pressure. In the pressure range less than the bubble-point pressure, the solution GOR decreases as the pressure decreases.

PVT laboratories can provide actual solution GORs from direct measurement, or empirical correlations can be made based on PVT laboratory data. One of the correlations is expressed as

(2.2) $R_s={\gamma }_g{\left[\frac{p}{18}\frac{{10}^{0.0125(°\mathrm{API})}}{{10}^{0.00091t}}\right]}^{1.2048}$

where γ _g and °API are gas-specific gravity and oil-API gravity (defined in later sections of this chapter), and p and t are pressure and temperature in psia and °F, respectively.

Solution GORs are often used for volumetric oil and gas calculations in reservoir engineering, and as a base parameter for estimating other fluid properties such as oil density.

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Properties of Petroleum Fluids

Boyun Guo PhD , ... Xuehao Tan PhD , in Petroleum Production Engineering (Second Edition), 2017

2.2.1 Solution Gas–Oil Ratio

"Solution GOR" is defined as the amount of gas (in standard condition) that will dissolve in unit volume of oil when both are taken down to the reservoir at the prevailing pressure and temperature; that is,

(2.1) $R_s=\frac{V_{gas}}{V_{oil}},$

where R _s =solution GOR (in scf/stb); V _gas =gas volume in standard condition (scf); V _oil =oil volume in stock tank condition (stb)

The "standard condition" is defined as 14.7 psia and 60 °F in most states in the United States. At a given reservoir temperature, solution GOR remains constant at pressures above bubble-point pressure. It drops as pressure decreases in the pressure range below the bubble-point pressure.

Solution GOR is measured in PTV laboratories. Empirical correlations are also available based on data from PVT labs. One of the correlations is,

(2.2) $R_s={\gamma }_g{\left[\frac{p}{18}\frac{{10}^{0.0125(°API)}}{{10}^{0.00091t}}\right]}^{1.2048}$

where γ _g and °API are defined in the latter sections, and p and t are pressure and temperature in psia and °F, respectively.

Solution GOR factor is often used for volumetric oil and gas calculations in reservoir engineering. It is also used as a base parameter for estimating other fluid properties such as density of oil.

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Productivity of Intelligent Well Systems

In Well Productivity Handbook, 2008

8.3.5.4 Applications of Multiphase Choke Flow

Whether using the Sachedeva's and Perkins' models, or that of Sun et al., the major drawback of these multiphase choke flow models is that they require free gas quality (the mass faction of gas within the mixture) as an input parameter to determine the flow regime and flow rates. This parameter is usually unknown before production data such as GOR and WOR is available. In addition, as the down-hole flow control valve is normally working at high down-hole pressure and temperature conditions, the hydrocarbon phase behavior properties must be taken into account. Even with good production data, this feature makes estimation of the in-situ free gas quality more difficult.

Depending upon available information, two different approaches can be used to estimate the in-situ free gas quality, oil quality and water quality upstream of the down-hole valve. One method is to apply surface production data, while another is to use data from hydrocarbon P-V-T relations.

Figure 8-15 shows the calculation flow charts proposed by Sun et al. (2006) for the in-situ x_g, x_o , and x_w , when surface production data, that is, the gas-oil ratio (GOR), water-oil ratio (WOR), oil API and gas-specific gravity, are available. The oil API and gas-specific gravity (γ _g ) can be used to estimate the in-situ oil solution gas ratio (R_s ), oil formation volume factor (B_o) and oil density (ρ _o1) at the valve upstream by applying the black oil correlations of Standing (1947) or Vasquez-Beggs (1980). The water in-situ properties (water viscosity, volume factor B_w , and water density) can be estimated using the correlations of Gould (1974) and Van Wingen (1950), or to simplify calculation by using a constant value. Knowing the gas components or gas-specific gravity at surface (γ _g ), the pseudoreduced pressure (P_pr ) and pseudoreduced temperature (T_pr ) can be calculated per Guo-Ghalambor (2005). The gas compression factor can be calculated from the Brill-Beggs z-factor correlation (1974) or the Hall-Yarborough z-factor correlation (1973). Then, the in-situ free gas, oil, and water qualities can be estimated using the equations below:

Figure 8-15. Procedure for calculating free gas quality.

(8.42) $x_g=\frac{\xi }{\xi +5.615{\rho }_{o1}{B}_o+350.52{\gamma }_{w}WOR}$

(8.43) $x_o=\frac{5.615{\rho }_{o1}{B}_o}{\xi +5.615{\rho }_{o1}{B}_o+350.52{\gamma }_{w}WOR}$

(8.44) $x_w=1-x_o-x_g$

(8.45) $\xi =0.0765{\gamma }_g(GOR\times {10}^6-R_s)$

where

GOR = gas-oil ratio (MMscf/stb)

WOR = water oil ratio (stb/stb)

R_s = solution gas-oil ratio (scf/stb)

B_w = water in-situ volume factor (bbl/stb)

B_o = oil in-situ volume factor (bbl/stb)

γ _w = water specific gravity (pure water = 1)

γ _g = gas-specific gravity (air = 1)

ρ _o1 = oil in-situ density at valve upstream (lbm/ft³)

ξ = parameters group (lbm/STB)

8-4

SAMPLE PROBLEM

A 5-1/2-in shrouded down-hole control valve is installed to control hydrocarbon production from one pay zone. Oil, water and gas are measured at the surface, where the GOR and the WOR from this pay zone are 0.00286 MMSCF/STB and 0.1 STB/STB. The oil, water and gas properties are listed in Table 8-1. The valve has a full open flow area equal to 16.34 in², and is opened remotely to 4%. The upstream hydraulic flow area of the valve is calculated as 22.3 in². After compensating for the hydrostatic/frictional pressure drop, the in-situ upstream/downstream pressures measured by the down-hole gauges are 2920 psi and 2810 psi, respectively. The measured fluid temperature through the valve is 185°F. If the valve discharge coefficient is given as 0.843, estimate the mixture production rate and flowing velocity through the valve. The oil properties are assumed to be represented by Standing's correlation.

Table 8-1. Fluid Property Data

Oil API Gravity	30 API 0.848 (pure water = 1)
Oil Bubble point, pb	3500 psi
Water Specific Gravity	1.020 (pure water = 1)
Gas Specific Gravity	0.872 (air = 1)
Cvo (heat capacity of oil)	0.430 BTU/(lbm-R)
Cvw (heat capacity of water)	1.000 BTU/(lbm-R)
Cvg (heat capacity of gas)	0.583 BTU(lbm-R)
k (heat specific gravity of gas)	1.254

SOLUTION

1)

Gas quality, water quality and oil quality calculations

The pressure of interest (upstream choke pressure) is less than the oil bubble pressure, and solution gas is also released at the point of interest. Applying Standing's correlation to estimate R_s and formation volume factor B_o at the upstream choke gives

$\begin{array}{c}{R}_s={\gamma }_g{\left(\frac{p}{18\times {10}^{(0.00091T-0.0125{\gamma }_{APL})}}\right)}^{1.204}\\ =0.872\times {\left(\frac{2920}{18\times {10}^{(0.0091\times 185-0.0125\times 30)}}\right)}^{1.204}\\ =708\text{S}\text{C}\text{F}/\text{S}\text{T}\text{B}\end{array}$

$\begin{array}{c}{B}_o=0.972+0.000147{\left(R_s{\left(\frac{{\gamma }_g}{{\gamma }_o}\right)}^{0.5}+1.25T\right)}^{1.175}\\ =0.972+0.000147\times {\left(708\times {\left(\frac{0.872}{0.848}\right)}^{0.5}+1.25\times 185\right)}^{1.175}\\ \mathrm{=}1.429\text{r}\text{b}/\text{S}\text{T}\text{B}\end{array}$

Gould (1974) correlation is used for water density and water formation volume factor:

$\begin{array}{c}{B}_w=1+0.00012\times (185-60)+0.000001\times (185-60){}^2-0.00000333\times 2920\\ =1.021\text{r}\text{b}/\text{S}\text{T}\text{B}\end{array}$

Water density (lbm/ft³) at the valve is

${\rho }_w=1.02\times \frac{62.4259}{1.021}=62.371\text{b}\text{m}/\text{f}{\text{t}}^3$

Brill and Beggs (1974) correlation is used to estimate z-factor at 2920 psi and 185°F:

$z_i=0.792$

Gas density at the valve is estimated by

$\begin{array}{c}{\rho }_{G1}=\frac{2.7{\gamma }_g{p}_1}{z_1{T}_1}\\ =\frac{2.7\times 0.872\times 2920}{0.792\times (185+459.67)}\\ =13.4651{\text{bm/ft}}^{\text{3}}\end{array}$

Oil density at the valve is estimated by

$\begin{array}{c}{\rho }_{o1}=\frac{350{\gamma }_o+0.0764{\gamma }_g{R}_s}{5.615B_o}\\ =\frac{350\times 0.848+0.0764\times 0.872\times 708}{5.615\times 1.429}\\ =44.11{\text{\hspace{0.5em}1bm/ft}}^3\end{array}$

Apply Equations (8.42), (8.43), and (8.44) to give

$\begin{array}{c}{x}_g=\frac{0.0765\times 0.872\times (0.00286\times {10}^6-708)}{0.0765\times 0.872\times (0.00286\times {10}^6-708)+5.615\times 44.11\times 1.429+350.376\times 62.37\times 1.021\times 0.1}\\ =0.269\\ x_o=\frac{5.615\times 44.11\times 1.429}{0.0765\times 0.872\times (0.00286\times {10}^6-708)+5.615\times 44.11\times 1.429+350.376\times 62.37\times 1.021\times 0.1}\\ =0.664\end{array}$

$\begin{array}{c}{x}_w=1-x_g-x_o\\ =1-0.269-0.664\\ =0.067\end{array}$

2)

Gas polytropic exponent calculations

Apply Equation (8.32) to calculate the gas polytropic exponent (n):

$\begin{array}{c}n=\left(\frac{x_{g}k{c}_{vg}+x_o{c}_{vo}+x_w{c}_{vw}}{x_g{c}_{vg}+x_o{c}_{vo}+x_w{c}_{vw}}\right)=\left(\frac{0.269\times 1.254\times 0.538+0.664\times 0.43+0.067\times 1}{0.269\times 0.538+0.664\times 0.43+0.076\times 1}\right)\\ =1.074\end{array}$

3)

Using Equations (8.39) and (8.40) to determine the critical-subcritical ratio

The upstream-downstream fluid velocity ratio (β) is

$\begin{array}{c}\beta =\frac{A_2}{A_1}\left(\frac{V_{G1}{x}_g+\frac{x_o}{{\rho }_o}+\frac{x_w}{{\rho }_w}}{V_{g1}{y}^{-\frac{1}{n}}{x}_g+\frac{x_o}{{\rho }_o}+\frac{x_w}{{\rho }_w}}\right)\\ =\frac{16.34\times 0.04}{22.3}\left(\frac{\frac{1}{13.465}\times 0.269+\frac{0.664}{44.11}+\frac{0.067}{62.37}}{\frac{1}{13.465}\times y^{\left(-\frac{1}{1.074}\right)}\times 0269+\frac{0.664}{44.11}+\frac{0.067}{62.37}}\right)\end{array}$

The critical-subcritical ratio is involved in

$\begin{array}{l}\frac{1}{13.465}{y}_c{}^{-\frac{1}{1.074}}0.269\left(\frac{1.074}{1.074-1}\right)\left[1-{\left(\frac{1}{y_c}\right)}^{\frac{1.074-1}{1.074}}\right]+\left(\frac{0.664}{44.11}+\frac{0.067}{62.37}\right)\left(1-\frac{1}{y_c}\right)\\ -\frac{1.074({\beta }^2-1){\left[0.269\left(\frac{1}{13.465}{y}_c{}^{-\frac{1}{1.074}}-\left[\frac{0.664\times 62.37+0.067\times 44.11}{62.37\times 44.11\times (0.664+0.067)}\right]\right)+\left(\frac{0.664\times 62.37+0.067\times 44.11}{62.37\times 44.11\times (0.664+0.067)}\right)\right]}^2}{2\times 0.269\times \frac{1}{13.465}{y}_c{}_{\frac{1}{1.074}}}\\ =0\end{array}$

The critical-subcritical ratio (y_c ) can be solved numerically to yield y_c = 0.509 and β = 0.020.

4)

Calculating the liquid and gas rates

Because $\left(\frac{2810}{2920}\right)=0.962>0.506$ , subcritical flow exists and

Equation (8.41) is used to calculate mass rate:

$\begin{array}{l}{M}_2=\left(\frac{16.34\times 0.04}{144}\right)\times (0.843)\times \\ {\left\{\frac{288\times (32.17)\times 2920\left[\frac{1}{13.465}{(0.962)}^{\left(-\frac{1}{1.074}\right)}0.269\left(\frac{1.074}{1.074-1}\right)\left(0.962-{0.962}^{\left(\frac{1}{1.074}\right)}\right)+\left(\frac{0.664\times 62.37+0.067\times 44.11}{62.37\times 44.11\times (0.664+0.067)}\right)(1-0.269)(0.962-1)\right]}{({0.020}^2-1){\left[0.269\left(\frac{1}{13.465}\times {0.962}^{-\frac{1}{1074}}-\left(\frac{0.664\times 62.37+0.067\times 44.11}{62.37\times 44.11\times (0.664+0.067)}\right)\right)+\left(\frac{0.664\times 62.37+0.067\times 44.11}{62.37\times 44.11\times (0.664+0.067)}\right)\right]}^2}\right\}}^{\frac{1}{2}}\end{array}$

which gives M ₂ = 18.34 lbm/s. Thus,

$\begin{array}{c}{M}_g=M_2\times x_g=18.34\times 0.269=4.93\text{lbm}/\text{s}\\ M_o=M_2\times x_g=18.34\times 0.664=12.18\text{lbm}/\text{s}\\ M_w=M_2\times x_w=18.34\times 0.067=1.23\text{lbm}/\text{s}\end{array}$

In field units, the free gas, oil, and water flow rates through the valve are

$\begin{array}{c}{Q}_g=\frac{M_g\times 24\times 60\times 60}{{\rho }_{G1}}=\frac{4.93\times 24\times 60\times 60}{13.465\times 1,000,000}\approx 0.032\text{MMcf}/\text{D}\\ Q_o=\frac{M_o\times 24\times 60\times 60}{{\rho }_{o1}\times 5.615\times B_o}=\frac{12.18\times 24\times 60\times 60}{44.11\times 5.615\times 1.429}\approx 2973\text{STBD}\\ Q_w=\frac{M_w\times 24\times 60\times 60}{{\rho }_{w1}\times 5.615\times B_w}=\frac{1.23\times 24\times 60\times 60}{62.37\times 5.615\times 1.021}\approx 297\text{STBD}\end{array}$

5)

Calculating fluid velocities

Equation (8.37) can be used to calculate mixture fluid density:

$\begin{array}{c}{\rho }_{m2}=\frac{1}{x_g{V}_{G2}+\left(\frac{x_o}{{\rho }_o}\right)+\left(\frac{x_w}{{\rho }_w}\right)}\\ =\frac{1}{0.269\times \left(\frac{1}{13.465}\right){\left(0.962\right)}^{\left(\frac{-1}{1.074}\right)}+\left(\frac{0.664}{44.11}\right)+\left(\frac{0.067}{62.37}\right)}\\ =27.15\text{lbm}/{\text{ft}}^3\end{array}$

Then the mixture fluid velocity through the valve is estimated:

$\begin{array}{c}{u}_2=\frac{M_2}{A_2{\rho }_{m2}}\\ =\frac{18.34}{\frac{16.34\times 0.04}{144}\times 27.15}\\ \approx 149\text{ft}/\sec \end{array}$

If reservoir fluid P-V-T data is available, the in-situ free gas quality can be estimated by calculating the released gas volume and assuming that the gas-liquid phase is moving at the same velocity in the wellbore.

Note

The introduced method assumed that the gas and liquid phases are moving at the same velocity along the production string, which in many situations is not true. Therefore, the above method can be used as a rough estimate for the liquid-gas flow rate through the down-hole valve. To make the estimation more accurate, liquid holdup in the tubular string must be taken into account.

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Predicting Oil Reservoir Performance

Tarek Ahmed , in Reservoir Engineering Handbook (Fifth Edition), 2019

Instantaneous Gas-Oil Ratio

The produced gas-oil ratio (GOR) at any particular time is the ratio of the standard cubic feet of total gas being produced at any time to the stock-tank barrels of oil being produced at that same instant. Hence, the name instantaneous gas-oil ratio. Equation 6-54 in Chapter 6 describes the GOR mathematically by the following expression:

(12-1) $\mathrm{GOR}={\mathrm{R}}_{\mathrm{s}}+\left(\frac{{\mathrm{k}}_{\mathrm{rg}}}{{\mathrm{k}}_{\mathrm{ro}}}\right)\left(\frac{{\mathrm{\mu }}_{\mathrm{o}}{\mathrm{B}}_{\mathrm{o}}}{{\mathrm{\mu }}_{\mathrm{g}}{\mathrm{B}}_{\mathrm{g}}}\right)$

Where:

GOR = instantaneous gas-oil ratio, scf/STB

R_s = gas solubility, scf/STB

k_rg = relative permeability to gas

k_ro = relative permeability to oil

B_o = oil formation volume factor, bbl/STB

B_g = gas formation volume factor, bbl/scf

μ_o = oil viscosity, cp

μ_g = gas viscosity, cp

The instantaneous GOR equation is of fundamental importance in reservoir analysis. The importance of Equation 12-1 can appropriately be discussed in conjunction with Figures 12-1 and 12-2.

Figure 12-1. Characteristics of solution-gas-drive reservoirs.

Figure 12-2. Schematic illustration of the interrelationship between GOR and R_s in solution-gas drive reservoirs.

These illustrations show the history of the gas-oil ratio of a hypothetical depletion-drive reservoir that is typically characterized by the following points:

Point 1. When the reservoir pressure p is above the bubble-point pressure p_b, there is no free gas in the formation, i.e., k_rg = 0, and therefore:

(12-2) $\mathrm{GOR}={\mathrm{R}}_{\mathrm{si}}={\mathrm{R}}_{\mathrm{sb}}$

The gas-oil ratio remains constant at R_si until the pressure reaches the bubble-point pressure at Point 2.

Point 2. As the reservoir pressure declines below p_b, the gas begins to evolve from solution and its saturation increases. This free gas, however, cannot flow until the gas saturation S_g reaches the critical gas saturation S_gc at Point 3. From Point 2 to Point 3, the instantaneous GOR is described by a decreasing gas solubility as:

(12-3) $\mathrm{GOR}={\mathrm{R}}_{\mathrm{s}}$

Point 3. At Point 3, the free gas begins to flow with the oil and the values of GOR are progressively increasing with the declining reservoir pressure to Point 4. During this pressure decline period, the GOR is described by Equation 12-1, or:

$\mathrm{GOR}={\mathrm{R}}_{\mathrm{s}}+\left(\frac{{\mathrm{k}}_{\mathrm{rg}}}{{\mathrm{k}}_{\mathrm{ro}}}\right)\left(\frac{{\mathrm{\mu }}_{\mathrm{o}}{\mathrm{B}}_{\mathrm{o}}}{{\mathrm{\mu }}_{\mathrm{g}}{\mathrm{B}}_{\mathrm{g}}}\right)$

Point 4. At Point 4, the maximum GOR is reached due to the fact that the supply of gas has reached a maximum and marks the beginning of the blow-down period to Point 5.

Point 5. This point indicates that all the producible free gas has been produced and the GOR is essentially equal to the gas solubility and continues to Point 6.

There are three types of gas-oil ratios, all expressed in scf/STB, which must be clearly distinguished from each other. These are:

○

Instantaneous GOR (defined by Equation 12-1)

○

Solution GOR

○

Cumulative GOR

The solution gas-oil ratio is a PVT property of the crude oil system. It is commonly referred to as gas solubility and denoted by R_s. It measures the tendency of the gas to dissolve in or evolve from the oil with changing pressures. It should be pointed out that as long as the evolved gas remains immobile, i.e., gas saturation S_g is less than the critical gas saturation, the instantaneous GOR is equal to the gas solubility, i.e.:

$\mathrm{GOR}={\mathrm{R}}_{\mathrm{s}}$

The cumulative gas-oil ratio R_p, as defined previously in the material balance equation, should be clearly distinguished from the producing (instantaneous) gas-oil ratio (GOR). The cumulative gas-oil ratio is defined as:

${\mathrm{R}}_{\mathrm{p}}=\frac{\text{cumulative}\left(\text{TOTAL}\right)\mathrm{gas}\text{produced}}{\text{cumulative}\text{oil}\text{produced}}$

Where:

R_p = cumulative gas-oil ratio, scf/STB

G_p = cumulative gas produced, scf

N_p = cumulative oil produced, STB

The cumulative gas produced G_p is related to the instantaneous GOR and cumulative oil production by the expression:

(12-5) ${\mathrm{G}}_{\mathrm{p}}=\underset{\mathrm{o}}{\overset{{\mathrm{N}}_{\mathrm{p}}}{\int }}\left(\mathrm{GOR}\right)\mathrm{d}{\mathrm{N}}_{\mathrm{p}}$

Equation 12-5 simply indicates that the cumulative gas production at any time is essentially the area under the curve of the GOR versus N_p relationship, as shown in Figure 12-3.

Figure 12-3. Relationship between GOR and G_p.

The incremental cumulative gas produced ΔG_p between N_p1, and N_p2 is then given by:

(12-6) $\mathrm{\Delta }{\mathrm{G}}_{\mathrm{p}}=\underset{{\mathrm{N}}_{\mathrm{p}1}}{\overset{{\mathrm{N}}_{\mathrm{p}2}}{\int }}\left(\mathrm{GOR}\right)\mathrm{d}{\mathrm{N}}_{\mathrm{p}}$

The above integral can be approximated by using the trapezoidal rule, to give:

$\mathrm{\Delta }{\mathrm{G}}_{\mathrm{p}}=\left[\frac{{\left(\mathrm{GOR}\right)}_1+{\left(\mathrm{GOR}\right)}_2}{2}\right]\left({\mathrm{N}}_{\mathrm{p}2}-{\mathrm{N}}_{\mathrm{p}1}\right)$

Equation 12-5 can then be approximated as:

(12-7) ${\mathrm{G}}_{\mathrm{p}}=\sum _{\mathrm{o}}{\left(\mathrm{GOR}\right)}_{\mathrm{avg}}\mathrm{\Delta }{\mathrm{N}}_{\mathrm{p}}$

Example 12-1

The following production data are available on a depletion-drive reservoir:

ppsi	GORscf/STB	NpMMSTB
2925(pi)	1340	0
2600	1340	1.380
2400	1340	2.260
2100(pi)	1340	3.445
1800	1936	7.240
1500	3584	12.029
1200	6230	15.321

Calculate cumulative gas produced G_p and cumulative gas-oil ratio at each pressure.

Solution

Step 1.

Construct the following table:

ppsi	GOR scf/STB	(GOR)avg scf/STB	Np MSTB	ΔNp MMSTB	ΔGp MMscf	Gp MMscf	Rp scf/STB
2925	1340	1340	0	0	0	0	―
2600	1340	1340	1.380	1.380	1849	1849	1340
2400	1340	1340	2.260	0.880	1179	3028	1340
2100	1340	1340	3.445	1.185	1588	4616	1340
1800	1936	1638	7.240	3.795	6216	10,832	1496
1500	3584	2760	12.029	4.789	13,618	24,450	2033
1200	6230	4907	15.321	3.292	16,154	40,604	2650

The Reservoir Saturation Equations

The saturation of a fluid (gas, oil, or water) in the reservoir is defined as the volume of the fluid divided by the pore volume, or:

(12-8) ${\mathrm{S}}_{\mathrm{o}}=\frac{\text{oil}\text{volume}}{\text{pore}\text{volume}}$

(12-9) ${\mathrm{S}}_{\mathrm{w}}=\frac{\text{water}\text{volume}}{\text{pore}\text{volume}}$

(12-10) ${\mathrm{S}}_{\mathrm{g}}=\frac{\mathrm{gas}\text{volume}}{\text{pore}\text{volume}}$

(12-11) ${\mathrm{S}}_{\mathrm{o}}+{\mathrm{S}}_{\mathrm{w}}+{\mathrm{S}}_{\mathrm{g}}+1.0$

Consider a volumetric oil reservoir with no gas cap that contains N stock-tank barrels of oil at the initial reservoir pressure p_i. Assuming no water influx gives:

${\mathrm{S}}_{\mathrm{oi}}=1-{\mathrm{S}}_{\mathrm{wi}}$

where the subscript i indicates initial reservoir condition. From the definition of oil saturation:

$1-{\mathrm{S}}_{\mathrm{wi}}=\frac{\mathrm{N}{\mathrm{B}}_{\mathrm{oi}}}{\text{pore volume}}$

or

(12-12) $\text{pore volume}=\frac{\mathrm{N}{\mathrm{B}}_{\mathrm{oi}}}{1-{\mathrm{S}}_{\mathrm{wi}}}$

If the reservoir has produced N_p stock-tank barrels of oil, the remaining oil volume is given by:

(12-13) $\text{remaining oil volume}=\left(\mathrm{N}-{\mathrm{N}}_{\mathrm{p}}\right){\mathrm{B}}_{\mathrm{o}}$

Substituting Equation 12-13 and 12-12 into Equation 12-8 gives:

(12-14) ${\mathrm{S}}_{\mathrm{o}}=\frac{\left(\mathrm{N}-{\mathrm{N}}_{\mathrm{p}}\right){\mathrm{B}}_{\mathrm{o}}}{\left(\frac{\mathrm{N}{\mathrm{B}}_{\mathrm{oi}}}{1-{\mathrm{S}}_{\mathrm{wi}}}\right)}$

or

(12-15) ${\mathrm{S}}_{\mathrm{o}}=\left(1-{\mathrm{S}}_{\mathrm{wi}}\right)\left(1-\frac{{\mathrm{N}}_{\mathrm{p}}}{\mathrm{N}}\right)\frac{{\mathrm{B}}_{\mathrm{o}}}{{\mathrm{B}}_{\mathrm{oi}}}$

(12-16) ${\mathrm{S}}_{\mathrm{g}}=1-{\mathrm{S}}_{\mathrm{o}}-{\mathrm{S}}_{\mathrm{wi}}$

Example 12-2

A volumetric solution-gas-drive reservoir has an initial water saturation of 20%. The initial oil formation volume factor is reported at 1.5   bbl/STB. When 10% of the initial oil was produced, the value of B_o decreased to 1.38. Calculate the oil saturation and gas saturation.

Solution

From Equation 12-5

$\begin{array}{l}{\mathrm{S}}_{\mathrm{o}}=\left(1-0.2\right)\left(1-0.1\right)\left(\frac{1.38}{1.50}\right)=0.662\\ {\mathrm{S}}_{\mathrm{g}}=1-0.662-0.20=0.138\end{array}$

It should be pointed out that the values of the relative permeability ratio k_rg/k_ro as a function of oil saturation can be generated by using the actual field production as expressed in terms of N_p, GOR, and PVT data. The proposed methodology involves the following steps:

Step 1.

Given the actual field cumulative oil production N_p and the PVT data as a function of pressure, calculate the oil and gas saturations from Equations 12-15 and 12-16, i.e.:

$\begin{array}{l}{\mathrm{S}}_{\mathrm{o}}=\left(1-{\mathrm{S}}_{\mathrm{wi}}\right)\left(1-\frac{{\mathrm{N}}_{\mathrm{p}}}{\mathrm{N}}\right)\frac{{\mathrm{B}}_{\mathrm{o}}}{{\mathrm{B}}_{\mathrm{oi}}}\\ {\mathrm{S}}_{\mathrm{g}}=1-{\mathrm{S}}_{\mathrm{o}}-{\mathrm{S}}_{\mathrm{wi}}\end{array}$

Step 2.

Using the actual field instantaneous GORs, solve Equation 12-1 for the relative permeability ratio as:

(12-17) $\frac{{\mathrm{k}}_{\mathrm{rg}}}{{\mathrm{k}}_{\mathrm{ro}}}=\left(\mathrm{GOR}-{\mathrm{R}}_{\mathrm{s}}\right)\left(\frac{{\mathrm{\mu }}_{\mathrm{g}}{\mathrm{B}}_{\mathrm{g}}}{{\mathrm{\mu }}_{\mathrm{o}}{\mathrm{B}}_{\mathrm{o}}}\right)$

Step 3.

Plot (k_rg/k_ro) versus S_o on a semilog paper.

Equation 12-15 suggests that all the remaining oil saturation be distributed uniformly throughout the reservoir. If water influx, gas-cap expansion, or gas-cap shrinking has occurred, the oil saturation equation, i.e., Equation 12-15, must be adjusted to account for oil trapped in the invaded regions.

Oil saturation adjustment for water influx

The proposed oil saturation adjustment methodology is illustrated in Figure 12-4 and described by the following steps:

Figure 12-4. Oil saturation adjustment for water influx.

Step 1.

Calculate the pore volume in the water-invaded region, as:

${\mathrm{W}}_{\mathrm{e}}-{\mathrm{W}}_{\mathrm{p}}{\mathrm{B}}_{\mathrm{w}}={\left(\mathrm{P}.\mathrm{V}\right)}_{\text{water}}\left(1-{\mathrm{S}}_{\mathrm{wi}}-{\mathrm{S}}_{\mathrm{orw}}\right)$

Solving for the pore volume of water-invaded zone (P.V)_water gives:

(12-18) ${\left(\mathrm{P}.\mathrm{V}\right)}_{\text{water}}=\frac{{\mathrm{W}}_{\mathrm{e}}-{\mathrm{W}}_{\mathrm{p}}{\mathrm{B}}_{\mathrm{w}}}{1-{\mathrm{S}}_{\mathrm{wi}}-{\mathrm{S}}_{\mathrm{orw}}}$

Where:

(P.V)_water = pore volume in water-invaded zone, bbl

S_orw = residual oil saturated in the imbibition water-oil system.

Step 2.

Calculate oil volume in the water-invaded zone, or:

(12-19) $\mathrm{volume}\mathrm{of}\mathrm{oil}=\left(\mathrm{P}.\mathrm{V}\right)\mathrm{water}{\mathrm{S}}_{\mathrm{orw}}$

Step 3.

Adjust Equation 12-14 to account for the trapped oil by using Equations 12-18 and 12-19:

(12-20) ${\mathrm{S}}_{\mathrm{o}}=\frac{\left(\mathrm{N}-{\mathrm{N}}_{\mathrm{p}}\right){\mathrm{B}}_{\mathrm{o}}-\left[\frac{{\mathrm{W}}_{\mathrm{e}}-{\mathrm{W}}_{\mathrm{p}}{\mathrm{B}}_{\mathrm{w}}}{1-{\mathrm{S}}_{\mathrm{wi}}-{\mathrm{S}}_{\mathrm{orw}}}\right]{\mathrm{S}}_{\mathrm{orw}}}{\left(\frac{\mathrm{N}{\mathrm{B}}_{\mathrm{oi}}}{1-{\mathrm{S}}_{\mathrm{wi}}}\right)-\left[\frac{{\mathrm{W}}_{\mathrm{e}}-{\mathrm{W}}_{\mathrm{p}}{\mathrm{B}}_{\mathrm{w}}}{1-{\mathrm{S}}_{\mathrm{wi}}-{\mathrm{S}}_{\mathrm{orw}}}\right]}$

Oil saturation adjustment for gas-cap expansion

The oil saturation adjustment procedure is illustrated in Figure 12-5 and summarized below:

Figure 12-5. Oil saturation adjustment for gas-cap expansion.

Step 1.

Assuming no gas is produced from the gas cap, calculate the net expansion of the gas cap, from:

(12-21) $\text{Expansion}\text{of}\text{the}\mathrm{gas}\mathrm{cap}=\mathrm{m}\mathrm{N}{\mathrm{B}}_{\mathrm{oi}}\left(\frac{{\mathrm{B}}_{\mathrm{g}}}{{\mathrm{B}}_{\mathrm{gi}}}-1\right)$

Step 2.

Calculate the pore volume of the gas-invaded zone, (P.V)gas, by solving the following simple material balance:

$\mathrm{m}\mathrm{N}{\mathrm{B}}_{\mathrm{oi}}\left(\frac{{\mathrm{B}}_{\mathrm{g}}}{{\mathrm{B}}_{\mathrm{gi}}}-1\right)={\left(\mathrm{P}.\mathrm{V}\right)}_{\mathrm{gas}}\left(1-{\mathrm{S}}_{\mathrm{wi}}-{\mathrm{S}}_{\mathrm{org}}\right)$

or

(12-22) ${\left(\mathrm{P}.\mathrm{V}\right)}_{\mathrm{gas}}=\frac{\mathrm{mN}{\mathrm{B}}_{\mathrm{oi}}\left(\frac{{\mathrm{B}}_{\mathrm{g}}}{{\mathrm{B}}_{\mathrm{gi}}}-1\right)}{1-{\mathrm{S}}_{\mathrm{wi}}-{\mathrm{S}}_{\mathrm{org}}}$

Where:

(P.V)_gas = pore volume of the gas-invaded zone

S_org = residual oil saturation in gas-oil system

Step 3.

Calculate the volume of oil in the gas-invaded zone.

(12-23) $\mathrm{oil}\mathrm{volume}={\left(\mathrm{P}.\mathrm{V}\right)}_{\mathrm{gas}}{\mathrm{S}}_{\mathrm{org}}$

Step 4.

Adjust Equation 12-14 to account for the trapped oil in the gas expansion zone by using Equations 12-22 and 12-23, to give:

(12-24) ${\mathrm{S}}_{\mathrm{o}}=\frac{\left(\mathrm{N}-{\mathrm{N}}_{\mathrm{p}}\right){\mathrm{B}}_{\mathrm{o}}-\left[\frac{\mathrm{mN}{\mathrm{B}}_{\mathrm{oi}}\left(\frac{{\mathrm{B}}_{\mathrm{g}}}{{\mathrm{B}}_{\mathrm{gi}}}-1\right)}{1-{\mathrm{S}}_{\mathrm{wi}}-{\mathrm{S}}_{\mathrm{org}}}\right]{\mathrm{S}}_{\mathrm{org}}}{\left(\frac{\mathrm{N}{\mathrm{B}}_{\mathrm{oi}}}{1-{\mathrm{S}}_{\mathrm{wi}}}\right)-\left[\frac{\mathrm{mN}{\mathrm{B}}_{\mathrm{oi}}}{1-{\mathrm{S}}_{\mathrm{wi}}-{\mathrm{S}}_{\mathrm{org}}}\right]\left(\frac{{\mathrm{B}}_{\mathrm{g}}}{{\mathrm{B}}_{\mathrm{gi}}}-1\right)}$

Oil saturation adjustment for combination drive

For a combination-drive reservoir, i.e., water influx and gas cap, the oil-saturation equation as given by Equation 12-14 can be adjusted to account for both driving mechanisms, as:

(12-25) ${\mathrm{S}}_{\mathrm{o}}\mathrm{=}\frac{\left(\mathrm{N}\mathrm{-}{\mathrm{N}}_{\mathrm{p}}\right){\mathrm{B}}_{\mathrm{o}}\mathrm{-}\left[\frac{\mathrm{mN}{\mathrm{B}}_{\mathrm{oi}}\left(\frac{{\mathrm{B}}_{\mathrm{g}}}{{\mathrm{B}}_{\mathrm{gi}}}\mathrm{-}1\right){\mathrm{S}}_{\mathrm{org}}}{1\mathrm{-}{\mathrm{S}}_{\mathrm{wi}}\mathrm{-}{\mathrm{S}}_{\mathrm{org}}}\mathrm{+}\frac{\left({\mathrm{W}}_{\mathrm{e}}\mathrm{-}{\mathrm{W}}_{\mathrm{p}}\right){\mathrm{S}}_{\mathrm{orw}}}{1\mathrm{-}{\mathrm{S}}_{\mathrm{wi}}\mathrm{-}{\mathrm{S}}_{\mathrm{orw}}}\right]}{\frac{\mathrm{N}{\mathrm{B}}_{\mathrm{oi}}}{1\mathrm{-}{\mathrm{S}}_{\mathrm{wi}}}\mathrm{-}\left[\frac{\mathrm{mN}{\mathrm{B}}_{\mathrm{oi}}\left(\frac{{\mathrm{B}}_{\mathrm{g}}}{{\mathrm{B}}_{\mathrm{gi}}}\mathrm{-}1\right)}{1\mathrm{-}{\mathrm{S}}_{\mathrm{wi}}\mathrm{-}{\mathrm{S}}_{\mathrm{org}}}\mathrm{+}\frac{{\mathrm{W}}_{\mathrm{e}}\mathrm{-}{\mathrm{W}}_{\mathrm{p}}{\mathrm{B}}_{\mathrm{w}}}{1\mathrm{-}{\mathrm{S}}_{\mathrm{wi}}\mathrm{-}{\mathrm{S}}_{\mathrm{orw}}}\right]}$

Oil saturation adjustment for shrinking gas cap

Cole (1969) points out that the control of the gas cap size is very often a reliable guide to the efficiency of reservoir operations. A shrinking gas cap will cause the loss of substantial amount of oil, which might otherwise be recovered. Normally, there is little or no oil saturation in the gas cap, and if the oil migrates into the original gas zone, there will necessarily be some residual oil saturation remaining in this portion of the gas cap at abandonment. The magnitude of this loss may be quite large, depending upon the:

○

Area of the gas-oil contact

○

Rate of gas-cap shrinkage

○

Relative permeability characteristics

○

Vertical permeability

A shrinking gas cap can be controlled by either shutting in wells that are producing large quantities of gas-cap gas or by returning some of the produced gas back the gas cap portion of the reservoir. In many cases, the shrinkage cannot be completely eliminated by shutting in wells, as there is a practical limit to the number of wells that can be shut in. The amount of oil lost by the shrinking gas cap can be very well the engineer's most important economic justification for the installation of gas return facilities.

The difference between the original volume of the gas cap and the volume occupied by the gas cap at any subsequent time is a measure of the volume of oil that has migrated into the gas cap. If the size of the original gas cap is m N B_oi, then the expansion of the original free gas resulting from reducing the pressure from p_i to p is:

$\text{Expansion}\text{of}\text{the}\text{original}\mathrm{gas}\mathrm{cap}=\mathrm{m}\mathrm{N}{\mathrm{B}}_{\mathrm{oi}}\left[\left({\mathrm{B}}_{\mathrm{g}}/{\mathrm{B}}_{\mathrm{gi}}\right)–1\right]$

Where:

m N B_oi = original gas-cap volume, bbl

B_g = gas FVF, bbl/scf

If the gas cap is shrinking, then the volume of the produced gas must be larger than the gas-cap expansion. All of the oil that moves into the gas cap will not be lost, as this oil will also be subject to the various driving mechanisms. Assuming no original oil saturation in the gas zone, the oil that will be lost is essentially the residual oil saturation remaining at abandonment. If the cumulative gas production from -the gas cap is G_pc scf, the volume of the gas-cap shrinkage as expressed in barrels is equal to:

$\mathrm{Gas}-\mathrm{cap}\text{shrinkage}={\mathrm{G}}_{\mathrm{pc}}{\mathrm{B}}_{\mathrm{g}}-\mathrm{mN}{\mathrm{B}}_{\mathrm{oi}}\left[\left({\mathrm{B}}_{\mathrm{g}}/{\mathrm{B}}_{\mathrm{gi}}\right)-1\right]$

From the volumetric equation:

${\mathrm{G}}_{\mathrm{pc}}{\mathrm{B}}_{\mathrm{g}}-\mathrm{mN}{\mathrm{B}}_{\mathrm{oi}}\left[\left({\mathrm{B}}_{\mathrm{g}}/{\mathrm{B}}_{\mathrm{gi}}\right)-1\right]=7758\mathrm{Ah\varphi }\left(1-{\mathrm{S}}_{\mathrm{wi}}-{\mathrm{S}}_{\mathrm{gr}}\right)$

Where:

A = average cross-sectional area of the gas-oil contact, acres

h = average change in depth of the gas-oil contact, feet

S_gr = residual gas saturation in the shrinking zone

The volume of oil lost as a result of oil migration to the gas cap can also be calculated from the volumetric equation as follows:

$\text{Oil lost}=7758\mathrm{Ah\varphi }{\mathrm{S}}_{\mathrm{org}}/{\mathrm{B}}_{\mathrm{oa}}$

Where:

S_org = residual oil saturation in the gas-cap shrinking zone

B_oa = oil FVF at abandonment

Combining the above relationships and eliminating the term 7,758 A h ϕ, give the following expression for estimating the volume of oil in barrels lost in the gas cap:

$\text{Oil}\text{lost}=\frac{\left[{\mathrm{G}}_{\mathrm{pc}}{\mathrm{B}}_{\mathrm{g}}-\mathrm{mN}{\mathrm{B}}_{\mathrm{oi}}\left(\frac{{\mathrm{B}}_{\mathrm{g}}}{{\mathrm{B}}_{\mathrm{gi}}}-1\right)\right]{\mathrm{S}}_{\mathrm{org}}}{\left(1-{\mathrm{S}}_{\mathrm{wi}}-{\mathrm{S}}_{\mathrm{gr}}\right){\mathrm{B}}_{\mathrm{oa}}}$

Where:

G_pc = cumulative gas production for the gas cap, scf

B_g = gas FVF, bbl/scf

All the methodologies that have been developed to predict the future reservoir performance are essentially based on employing and combining the above relationships that include the:

○

MBE

○

Saturation equations

○

Instantaneous GOR

○

Equation relating the cumulative gas-oil ratio to the instantaneous GOR

Using the above information, it is possible to predict the field primary recovery performance with declining reservoir pressure. There are three methodologies that are widely used in the petroleum industry to perform a reservoir study. These are:

○

Tracy's method

○

Muskat's method

○

Tarner's method

These methods yield essentially the same results when small intervals of pressure or time are used. The methods can be used to predict the performance of a reservoir under any driving mechanism, including:

○

Solution-gas drive

○

Gas-cap drive

○

Water drive

○

Combination drive

The practical use of all the techniques is illustrated in predicting the primary recovery performance of a volumetric solution-gas-drive reservoir. Using the appropriate saturation equation, e.g., Equation 12-20 for a water-drive reservoir, any of the available reservoir prediction techniques could be applied to other reservoirs operating under different driving mechanisms.

The following two cases of the solution-gas-drive reservoir are considered:

○

Undersaturated-oil reservoirs

○

Saturated-oil reservoirs

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PVT Properties of Crude Oils

Tarek Ahmed Ph.D., P.E. , in Equations of State and PVT Analysis, 2007

Correcting the Separator Tests Data

Stock-Tank Gas/Oil Ratio and Gravity

No corrections are needed for the stock-tank gas/oil ratio and the stock-tank API gravity.

Separator Gas/Oil Ratio

The total gas-oil ratio, R _sfb, is changed in the same proportion as the differential ratio was changed:

(4-126) $R_{\text{sfb}}^{\text{new}}=R_{\text{sfb}}^{\text{old}}\left(\frac{R_{\text{sfb}}^{\text{new}}}{R_{\text{sfb}}^{\text{old}}}\right)$

The separator gas/oil ratio then is the difference between the new (corrected) gas solubility $R_{\text{sfb}}^{\text{new}}$ and the unchanged stock-tank gas/oil ratio.

Formation Volume Factor

The separator oil formation volume factor, B _ofb, is adjusted in the same proportion as the differential liberation values:

(4-127) $B_{\text{ofb}}^{\text{new}}=B_{\text{ofb}}^{\text{old}}\left(\frac{B_{\text{odb}}^{\text{new}}}{B_{\text{odb}}^{\text{old}}}\right)$

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Properties of Oil and Natural Gas

Boyun Guo Ph.D. , ... Ali Ghalambor Ph.D. , in Petroleum Production Engineering, 2007

2.2 Properties of Oil

Oil properties include solution gas–oil ratio (GOR), density, formation volume factor, viscosity, and compressibility. The latter four properties are interrelated through solution GOR.

2.2.1 Solution Gas–Oil Ratio

"Solution GOR" is defined as the amount of gas (in standard condition) that will dissolve in unit volume of oil when both are taken down to the reservoir at the prevailing pressure and temperature; that is,

(2.1) $R_s=\frac{V_{gas}}{V_{oil}},$

where

R_s = solution GOR (in scf/stb)

V_gas = gas volume in standard condition (scf)

V_oil = oil volume in stock tank condition (stb)

The "standard condition" is defined as 14.7 psia and 60 °F in most states in the United States. At a given reservoir temperature, solution GOR remains constant at pressures above bubble-point pressure. It drops as pressure decreases in the pressure range below the bubble-point pressure.

Solution GOR is measured in PTV laboratories. Empirical correlations are also available based on data from PVT labs. One of the correlations is,

(2.2) $R_s={\gamma }_g{\left[\frac{p}{18}\frac{{10}^{0.0125(°API)}}{{10}^{0.00091t}}\right]}^{1.2048}$

where γ_g and °API are defined in the latter sections, and p and t are pressure and temperature in psia and °F, respectively.

Solution GOR factor is often used for volumetric oil and gas calculations in reservoir engineering. It is also used as a base parameter for estimating other fluid properties such as density of oil.

2.2.2 Density of Oil

"Density of oil" is defined as the mass of oil per unit volume, or lb_m/ft³ in U.S. Field unit. It is widely used in hydraulics calculations (e.g., wellbore and pipeline performance calculations [see Chapters 4 and 11]).

Because of gas content, density of oil is pressure dependent. The density of oil at standard condition (stock tank oil) is evaluated by API gravity. The relationship between the density of stock tank oil and API gravity is given through the following relations:

(2.3) $°API=\frac{141.5}{{\gamma }_o}-131.5$

and

(2.4) ${\gamma }_o=\frac{{\rho }_o{}_{,st}}{{\rho }_w},$

where

°API = API gravity of stock tank oil

γ_o = specific gravity of stock tank oil, 1 for freshwater

ρ_o,st = density of stock tank oil, lb_m/ft³

ρ_w = density of freshwater, 62.4 lb_m/ft³

The density of oil at elevated pressures and temperatures can be estimated on empirical correlations developed by a number of investigators. Ahmed (1989) gives a summary of correlations. Engineers should select and validate the correlations carefully with measurements before adopting any correlations.

Standing (1981) proposed a correlation for estimating the oil formation volume factor as a function of solution GOR, specific gravity of stock tank oil, specific gravity of solution gas, and temperature. By coupling the mathematical definition of the oil formation volume factor with Standing's correlation, Ahmed (1989) presented the following expression for the density of oil:

(2.5) ${\rho }_o=\frac{62.4{\gamma }_o+0.0136R_s{\gamma }_g}{0.972+0.000147{\left[R_s\sqrt{\frac{{\gamma }_g}{{\gamma }_o}+1.25t}\right]}^{1.175}},$

where

t = temperature, °F

γ_g = specific gravity of gas, 1 for air.

2.2.3 Formation Volume Factor of Oil

"Formation volume factor of oil" is defined as the volume occupied in the reservoir at the prevailing pressure and temperature by volume of oil in stock tank, plus its dissolved gas; that is,

(2.6) $B_o=\frac{V_{res}}{V_{st}},$

where

B_o = formation volume factor of oil (rb/stb)

V_res = oil volume in reservoir condition (rb)

V_st = oil volume in stock tank condition (stb)

Formation volume factor of oil is always greater than unity because oil dissolves more gas in reservoir condition than in stock tank condition. At a given reservoir temperature, oil formation volume factor remains nearly constant at pressures above bubble-point pressure. It drops as pressure decreases in the pressure range below the bubble-point pressure.

Formation volume factor of oil is measured in PTV labs. Numerous empirical correlations are available based on data from PVT labs. One of the correlations is

(2.7) $B_o=0.9759+0.00012\left[R_s\sqrt{\frac{{\gamma }_g}{{\gamma }_o}}+1.25t\right]\mathrm{1.2.}$

Formation volume factor of oil is often used for oil volumetric calculations and well-inflow calculations. It is also used as a base parameter for estimating other fluid properties.

2.2.4 Viscosity of Oil

"Viscosity" is an empirical parameter used for describing the resistance to flow of fluid. The viscosity of oil is of interest in well-inflow and hydraulics calculations in oil production engineering. While the viscosity of oil can be measured in PVT labs, it is often estimated using empirical correlations developed by a number of investigators including Beal (1946), Beggs and Robinson (1975), Standing (1981), Glaso (1985), Khan (1987), and Ahmed (1989). A summary of these correlations is given by Ahmed (1989). Engineers should select and validate a correlation with measurements before it is used. Standing's (1981) correlation for dead oil is expressed as

(2.8) ${\mu }_{od}=\left(0.32+\frac{1.8\times {10}^7}{AP{I}^{4.53}}\right){\left(\frac{360}{t+200}\right)}^A,$

where

(2.9) $A={10}^{(0.43+\frac{8.33}{API})}$

and

μ_od = viscosity of dead oil (cp).

Standing's (1981) correlation for saturated crude oil is expressed as

(2.10) ${\mu }_{ob}={10}^a{\mu }_{od}^b,$

where μ_ob = viscosity of saturated crude oil in cp and

(2.11) $a=R_s(2.2\times {10}^{-7}{R}_s-7.4\times {10}^{-4}),$

(2.12) $b=\frac{0.68}{{10}^c}+\frac{0.25}{{10}^d}+\frac{0.062}{{10}^e},$

(2.13) $c=8.62\times {10}^{-5}{R}_s,$

(2.14) $d=1.10\times {10}^{-3}{R}_s,$

and

(2.15) $e=3.74\times {10}^{-3}{R}_s,$

Standing's (1981) correlation for unsaturated crude oil is expressed as

(2.16) ${\mu }_o={\mu }_{ob}+0.001(p-p_b)(0.024{\mu }_{ob}^{1.6}+0.38{\mu }_{ob}^{0.56}).$

2.2.5 Oil Compressibility

"Oil compressibility" is defined as

(2.17) $c_o=-\frac{1}{V}{\left(\frac{\partial V}{\partial p}\right)}_T,$

where T and V are temperature and volume, respectively. Oil compressibility is measured from PVT labs. It is often used in modeling well-inflow performance and reservoir simulation.

Example Problem 2.1

The solution GOR of a crude oil is 600 scf/stb at 4,475 psia and 140 °F. Given the following PVT data, estimate density and viscosity of the crude oil at the pressure and temperature:

Bubble-point pressure:	2,745 psia
Oil gravity:	35 °ApI
Gas-specific gravity:	0.77 air = 1

Solution Example Problem 2.1 can be quickly solved using the spreadsheet program OilProperties.xls where Standing's correlation for oil viscosity was coded. The input and output of the program is shown in Table 2.1.

Table 2.1. Result Given by the Spreadsheet Program OilProperties.xls

OilProperties.xls
Description: This spreadsheet calculates density and viscosity of a crude oil.
Instruction: (1) Click a unit-box to choose a unit system; (2) update parameter values in the Input data section; (3) view result in the Solution section and charts.
Input data	U.S. Field units	SI units
Pressure (p):	4,475 psia
Temperature (t):	140 °F
Bubble point pressure (pb ):	2,745 psia
Stock tank oil gravity:	35 °API
Solution gas oil ratio (Rs ):	600 scf/stb
Gas specific gravity (γg ):	0.77 air = 1
Solution
${\gamma }_o=\frac{141.5}{{}^{ˆ}AOI+131.5}$	= 0.8498 H2O = 1
${\rho }_o=\frac{62.4{\gamma }_o+0.0136R_s{\gamma }_g}{0.972+0.000147\left[R_s\sqrt{\frac{{\gamma }_g}{{\gamma }_o}+1.25t}\right]1.175}$	= 44.90 lbm/ft3
A = 10(°.43+8.33/API)	= 4.6559
${\mu }_{od}=\left(0.32+\frac{1.8\times {10}^7}{AP{I}^{4.53}}\right){\left(\frac{360}{t+200}\right)}^A$	= 2.7956 cp
a = Rs (2.2 × 10−7 Rs – 7.4 × 10−4)	= −0.3648
c = 8.62 × 10−5 Rs	= 0.0517
d = 1.10 × 10−3 Rs	= 0.6600
e = 3.74 × 10−3 Rs	= 2.2440
$b=\frac{0.68}{{10}^c}+\frac{0.25}{{10}^d}+\frac{0.062}{{10}^e}$	= 0.6587
${\mu }_{ob}={10}^a{\mu }_{od}^b$	= 0.8498 cp	0.0008 Pa-s
${\mu }_o={\mu }_{ob}+0.001(p-p_b)(0.024{\mu }_{ob}^{1.6}+0.38{\mu }_{ob}^{0.56})$	= 1.4819 cp	0.0015 Pa-s

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Predicting Oil Reservoir Performance

Tarek Ahmed , in Reservoir Engineering Handbook (Fourth Edition), 2010

Instantaneous Gas-Oil Ratio

The produced gas-oil ratio (GOR) at any particular time is the ratio of the standard cubic feet of total gas being produced at any time to the stock-tank barrels of oil being produced at that same instant. Hence, the name instantaneous gas-oil ratio. Equation 6-54 in Chapter 6 describes the GOR mathematically by the following expression:

(12-1) $\text{GOR}={\text{R}}_{\text{s}}+\left(\frac{{\text{k}}_{\text{rg}}}{{\text{k}}_{\text{ro}}}\right)\left(\frac{{\mu }_{\text{o}}{\text{B}}_{\text{o}}}{{\mu }_{\text{g}}{\text{B}}_{\text{g}}}\right)$

where GOR = instantaneous gas-oil ratio, scf/STB

R_s = gas solubility, scf/STB

k_rg = relative permeability to gas

k_ro = relative permeability to oil

B_o = oil formation volume factor, bbl/STB

B_g = gas formation volume factor, bbl/scf

μ_o= oil viscosity, cp

μ_g = gas viscosity, cp

The instantaneous GOR equation is of fundamental importance in reservoir analysis. The importance of Equation 12-1 can appropriately be discussed in conjunction with Figures 12-1 and 12-2.

Figure 12-1. Characteristics of solution-gas-drive reservoirs

Figure 12-2. History of GOR and R_s for a solution-gas-drive reservoir.

These illustrations show the history of the gas-oil ratio of a hypothetical depletion-drive reservoir that is typically characterized by the following points:

Point 1.

When the reservoir pressure p is above the bubble-point pressure p_b, there is no free gas in the formation, i.e., k_rg = 0, and therefore:

(12-2) $\text{GOR}={\text{R}}_{\text{si}}={\text{R}}_{\text{sb}}$

The gas-oil ratio remains constant at R_si until the pressure reaches the bubble-point pressure at Point 2.

Point 2.

As the reservoir pressure declines below p_b, the gas begins to evolve from solution and its saturation increases. This free gas, however, cannot flow until the gas saturation S_g reaches the critical gas saturation S_gc at Point 3. From Point 2 to Point 3, the instantaneous GOR is described by a decreasing gas solubility as:

(12-3) $\text{GOR}={\text{R}}_{\text{s}}$

Point 3.

At Point 3, the free gas begins to flow with the oil and the values of GOR are progressively increasing with the declining reservoir pressure to Point 4. During this pressure decline period, the GOR is described by Equation 12-1, or:

$\text{GOR}={\text{R}}_{\text{s}}+\left(\frac{{\text{k}}_{\text{rg}}}{{\text{k}}_{\text{ro}}}\right)\left(\frac{{\mu }_{\text{o}}{\text{B}}_{\text{o}}}{{\mu }_{\text{g}}{\text{B}}_{\text{g}}}\right)$

Point 4.

At Point 4, the maximum GOR is reached due to the fact that the supply of gas has reached a maximum and marks the beginning of the blow-down period to Point 5.

Point 5.

This point indicates that all the producible free gas has been produced and the GOR is essentially equal to the gas solubility and continues to Point 6.

There are three types of gas-oil ratios, all expressed in scf/STB, which must be clearly distinguished from each other. These are:

•

Instantaneous GOR(defined by Equation 12-1)

•

Solution GOR

•

Cumulative GOR

The solution gas-oil ratio is a PVT property of the crude oil system. Itis commonly referred to as gas solubility and denoted by R_s. It measures the tendency of the gas to dissolve in or evolve from the oil with changing pressures. It should be pointed out that as long as the evolved gas remains immobile, i.e., gas saturation S_g is less than the critical gas saturation, the instantaneous GOR is equal to the gas solubility, i.e.:

The cumulative gas-oil ratio R_p, as defined previously in the material balance equation, should be clearly distinguished from the producing (instantaneous) gas-oil ratio (GOR). The cumulative gas-oil ratio is defined as:

(12-4) $\begin{array}{c}{\text{R}}_{\text{p}}=\frac{\text{cumulative}\left(\text{TOTAL}\right)\text{gas}\text{produced}}{\text{cumulative}\text{oil}\text{produced}}\\ \text{or}\\ {\text{R}}_{\text{p}}=\frac{{\text{G}}_{\text{p}}}{{\text{N}}_{\text{p}}}\end{array}$

where R_p = cumulative gas-oil ratio, scf/STB

G_p = cumulative gas produced, scf

N_p = cumulative oil produced, STB

The cumulative gas produced G_p is related to the instantaneous GOR and cumulative oil production by the expression:

Equation 12-5 simply indicates that the cumulative gas production at any time is essentially the area under the curve of the GOR versus N_p relationship, as shown in Figure 12-3.

Figure 12-3. Relationship between GOR and G_p.

The incremental cumulative gas produced ΔG_p between N_p1, and N_p2 is then given by:

(12-6) $\Delta {\text{G}}_{\text{p}}=\underset{\text{Np}1}{\overset{\text{Np}2}{\int }}\left(\text{GOR}\right)\text{d}{\text{N}}_{\text{p}}$

The above integral can be approximated by using the trapezoidal rule, to give:

$\begin{array}{c}\Delta {\text{G}}_{\text{p}}=\left[\frac{{\left(\text{GOR}\right)}_1+{\left(\text{GOR}\right)}_2}{2}\right]\left({\text{N}}_{\text{p}2}-{\text{N}}_{\text{p}1}\right)\\ \text{or}\\ \Delta {\text{G}}_{\text{p}}={\left(\text{GOR}\right)}_{\text{avg}}\Delta {\text{N}}_{\text{p}}\end{array}$

Equation 12-5 can then be approximated as:

Example 12-1

The following production data are available on a depletion-drive reservoir:

p psi	GOR scf/STB	Np MMSTB
2925 (Pi)	1340	0
2600	1340	1.380
2400	1340	2.260
2100 (Pi)	1340	3.445
1800	1936	7.240
1500	3584	12.029
1200	6230	15.321

Calculate cumulative gas produced G_p and cumulative gas-oil ratio at each pressure.

Solution

Step 1.

Construct the following table:

p psi	GOR scf/STB	(GOR)avg scf/STB	Np MMSTB	ΔNp MMSTB	ΔGp MMscf	Gp MMscf	Rp scf/STB
2925	1340	1340	0	0	0	0	—
2600	1340	1340	1.380	1.380	1849	1849	1340
2400	1340	1340	2.260	0.880	1179	3028	1340
2100	1340	1340	3.445	1.185	1588	4616	1340
1800	1936	1638	7.240	3.795	6216	10,832	1496
1500	3584	2760	12.029	4.789	13,618	24,450	2033
1200	6230	4907	15.321	3.292	16,154	40,604	2650

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