Conductivity Ratio

𝐺16= 𝑁𝐹=

𝜙𝐾ℎ𝑓

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Rock Thermal Conductivity to Effective Thermal

Conductivity Ratio 𝐺17= 𝑁𝑇= (1 − 𝜙)𝐾ℎ𝑟 𝐾ℎ𝑒 Dimensionless Time _𝐺 18= 𝑡𝐷 = 𝑈𝑡𝑡 𝐿𝜙(1 − 𝑠𝑤𝑖− 𝑠𝑜𝑟)

Different rock and fluid properties are employed to find out the value of each dimensionless groups which is reported in table 5.2 and 5.3. Four different cases are studied where the reservoir has four different dimensions and different rock and fluid properties depending on temperature and pressure conditions (Table 5.3) Only ten dimensionless numbers give the same value for each case. These ten dimensionless numbers are evaluated, and their relative effect on oil recovery are estimated. The common parameters which are used in dimensionless number evaluation for all four cases are noted in table 5.2.

Table 5.2: Common parameters required in scaling groups

Parameters Values Parameters values Parameters Values

𝑘_𝑧 (mD) 800 𝐾_𝑤 =(Kj/h-m-k) 3.7758 𝐶_𝑝𝑤(Kj/kg-k) 4.1868 𝑘_𝑥 (mD) 800 𝐾_𝑔 =(Kj/h-m-k) 0.0143 fR 0.2 𝑘_𝑟𝑔 0.150 𝐾_ℎ𝑓 =(Kj/h-m-k) 1.7288 ms (ft2/s) 17.20 𝑘𝑟𝑜 0.085 𝐾ℎ𝑒 =(Kj/h-m-k) 6.7936 Lv (Btu/lb) 837.3 𝑘𝑟𝑤 0.400 ℎ𝐿 (Kj/h-m2k) 280.87 M1 (Btu/ft3 ℉) 3.69 𝑔 (cm/s2_{) 980.7 𝑝} 𝑖 (pa) 1.7×107 𝜂𝑅 0.35 𝜎𝑜𝑤 (dyne/cm) 49.0 𝐶𝑝𝑔 (Kj/kg-k) 29.7263 α 0.2 𝐾𝑟=(Kj/h-m-k) 9.346 𝐶𝑝𝑟(Kj/kg-k) 0.8792 fs 0.8 𝐾_𝑜=(Kj/h-m-k) 1.3962 𝐶_𝑝𝑜(Kj/kg-k) 2.0934 𝐶_𝑝𝑤(Kj/kg-k) 4.1868

Four different dimensions of reservoir is considered with their corresponding rock and fluid properties are reported in table 5.3. Depending on temperature and pressure conditions the properties should be different from each other. Reservoir length, width, total fluid velocity, longitudinal and transvers dispersion, porosity, viscosity and density of each phases are noted in table 5.3.

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Table 5.3: Parameters required in scaling groups for four different cases

Parameters Case 1 Case 2 Case 3 Case 4

L, cm (ft) 7.65 1500 (49.2) 1800 (59.0) 3735 (122.5) H, cm (ft) 2.56 502 (16.5) 602.4 (19.8) 1250 (41.0) UT, ft/s 3.36×10-4 1.71×10-5 1.43×10-5 2.50×10-6 𝐷𝐿𝑜1𝑅, ft2/s 1.12×10-4 1.12×10-3 9.67×10-4 3.38×10-4 𝐷_{𝑇𝑜1𝑅}, ft2/s 3.53×10-6 3.53×10-5 3.05×10-5 1.067×10-5 𝜙 0.332 0.332 0.385 0.400 µ_𝑤 at 2500 psi, cP 0.99 0.99 0.99 0.99 µ_𝑜 at 2500 psi, cP 1.03 1.03 1.03 1.03 µ_𝑔 at 2500 psi, cP 0.125 0.125 0.125 0.125 𝜌𝑤 at 2500 psi, g/cm3 1.020 1.005 1.005 1.005 𝜌_𝑜at 2500 psi, g/cm3 0.683 0.988 0.991 0.998 𝜌_𝑔at 2500 psi, g/cm3 0.942 1.001 1.002 1.003 𝜌_𝑟at 2500 psi, g/cm3 2.680 2.681 2.682 2.683

5.4. Scaling group for reservoir heterogeneity

Reservoir heterogeneity is a key factor in determining the oil recovery from petroleum reservoirs. Most of the Canadian heavy oil reservoirs sands comprises a considerable amount of heterogeneity (Akram, 2012) which must be counted when proposing parameters for field development using steam flooding process. The Dykstra-Parsons coefficient is comprised as an additional dimensionless number to account for reservoir heterogeneity. It can be defined as:

𝐹_𝐷𝑃 = 𝑘0.5−𝑘0.84

𝑘0.5 (13)

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𝑘_0.84= One standard deviation from the median permeability

The value of 𝐹𝐷𝑃 should be lies on 0 to 1. Here 0 representing homogeneous reservoir and 1

represents heterogeneous reservoir. The minimum and maximum value of 𝐹_𝐷𝑃 lies between 0.1 to 0.8 for this study.

5.5. Steam-oil viscosity ratio

Steam-oil viscosity ratio is an important dimensionless group which can characterize the steam flooding process. Steam is injected into the reservoir through the injection well which will reduce the viscosity of the oil and increase the flow efficiency and hence increase the oil recovery. It also largely depends on the contact time. Steam-oil viscosity ratio can be expressed as:

𝑅𝑆𝑇 = µ𝑔

µ𝑜 (14)

Where µ𝑔 and µ𝑜 is the viscosity of steam and oil respectively. 5.6. Proposed new group

The estimation of oil recovery is obtained through reservoir simulation and scaled model studies of steam flooding process. The new dimensionless number is developed that can capture important process controlling parameters. Steam-oil viscosity ratio, gravity number, capillary number, mobility ratio is used to develop this number. Oil recovery is proportional to gravity number and steam-oil viscosity ratio and inversely proportional to mobility ratios and capillary number. The main objective of proposing this number is it can capture physical process more effectively as more parameters are involved in this number than any other dimensionless numbers.

The recovery of oil obtained through the studies of scaling numbers were investigated using gravity and capillary numbers to improve a relationship that captures influential process controlling parameters. It is found that the oil recovery is directly proportional to gravity numbers and inversely proportional to the capillary number. The obtained results suggested that, though the gravity number can provide accurate and closely matched relationship, but few other variables should be studied for the estimation of oil recovery. The pore trapping of oil behind the steam flood front is caused by capillary retention which diminishes the oil recovery performance for steam flooding process. Thus, capillary force effects must be counted for assessing steam flooding process performance. Moreover, the results show that the mobility ratio and the oil viscosity changes have a profound effect on oil recovery. Figure 11 suggests

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that oil recovery is increased with increasing viscosity ratio (viscosity of steam to oil). Therefore, the mobility ratios and viscosity ratio should be considered for proposing new dimensionless number. A novel relationship is obtained in this study from the above findings to characterize and evaluate the performance of steam flooding EOR process. The capillary number, gravity number, mobility ratio and the viscosity ratio are considered to develop this correlation. The proposed new dimensionless number is presented as:

𝑁𝐴=

(µ𝑆𝑡𝑒𝑎𝑚

µ𝑜 )(𝑁𝐺𝑤𝑜+𝑁𝐺𝑜𝑔)

𝑎

𝑀𝑔𝑜𝑀𝑤𝑜𝑁𝐶 (15)

where a = 0.2 is a scaling factor which shows the effect of gravity number is less than any other numbers in the proposed new dimensionless group.

Table 5.4 describes different dimensionless numbers with maximum, minimum and mean value. It can also show the recovery factor coefficients along with standard error of each dimensionless group.

Table 5.4: Values of each group corresponding to minimum and maximum level

Groups Minimum Maximum Mean Regression Coefficient 𝑁𝑃𝐿𝑜1 50 700 375 1.09 × 10-4 𝑁𝑃𝑇𝑜1 200 2500 1332.32 -4.09 × 10-5 𝑀𝑜𝑤 0.98 1.09 1.04 -0.09 𝑀𝑔𝑜 8.67 14.48 11.32 -0.11 𝑄𝐿 0.349 0.863 0.546 0.0007 𝑁𝐺𝑤𝑜 2.00 × 108 3.49 × 108 2.69 × 108 2.92× 10-10 𝑁𝐺𝑜𝑔 3.5 × 107 5.5 × 107 4.52 × 107 1.22× 10-9 𝑅𝐿 2.00 5.84 3.97 1.02 × 10-2 𝑁𝐶 4.5 × 10-6 8.4 × 10-6 6.27 × 10-6 -0.03711 𝑉𝐷𝑝 0.1 0.8 0.45 0.00457 𝐶𝐴𝐷 0.1 0.5 0.282 0.133 𝑅𝑆𝑇 0.0965 0.1622 0.1356 0.150 𝐶𝐴 7.28 × 104 12.62 × 104 9.82 × 104 0.0345

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5.7. Sensitivity analysis of dimensionless numbers

It should be unrealistic to assess the effect of individually all scaling groups on oil recovery because of the extreme workload and large demand for resources. Only thirteen groups are evaluated to analyze their relative effect on oil recovery. A standardized effect is estimated by dividing every regression coefficient with its standard error. Figure 1 shows the value of standardized effect of each scaling group on oil recovery in a normal plot. The positive value indicates a proportional relationship and a negative value indicate the inversely proportional relationship. It also shows the relative importance and magnitude of each group on oil recovery. From figure 1 it should be noted that the most dominant groups are new proposed dimensionless group, steam-oil viscosity ratio, capillary number, mobility ratios and additive concentration groups that can largely affect a steam flooding process. On the other hand, there are some groups which have a minor or insignificant effect on oil recovery that can be termed as secondary scaling groups for steam flooding process.

Figure 5.1: Absolute value of standardized effect

5.8. Dominant scaling groups

Dominant dimensionless groups are summarized below: 5.8.1. New proposed number

Newly proposed number has the significant effect on enhanced oil recovery by steam flooding. It is the combination of five dimensionless numbers which is employed to predict the performance of a steam flooding process. The capillary number, gravity number, steam-oil viscosity ratio, mobility ratios are used to develop a relationship that can capture vital steam

-100 -50 0 50 100 1 Perc en t Standardized Effect