Simulation of composition changes in reservoirs with large hydrocarbon columns and temperature gradient

This paper compares three methods for calculation of initial composition variation with depth in hydrocarbon reservoirs: considering thermal diffusion, considering temperature gradient without thermal diffusion effects; and by gravity forces only. Newton method-based numerical algorithm was implemented for solution of thermodynamic equations to evaluate pressure and hydrocarbon composition. Test calculations are performed for main gas-condensate reservoir of Vuktylskoye field with a gas column of 1350 m. The results obtained with the numerical algorithm indicate that gravity segregation impact is the strongest for all the cases considered. concentration decreases with depth for low molecular weight components and increases for high molecular weight components. The higher molecular weight of the component, the stronger variation of its concentration with depth. Initial reservoir pressure also changes accordingly. However, thermal diffusion also has a significant influence on variation of hydrocarbon composition with depth and initial reservoir pressure. For the test case considered, thermal diffusion magnifies the impact of gravity and results in strongly nonlinear dependencies of component concentrations on depth. When thermal gradient is taken into account without thermal diffusion effects, the results are only slightly different from those with the isothermal gravity segregation calculations. None of the calculation methods were successful in matching estimates of initial composition variation with depth obtained from well exploitation data. Physical mechanisms governing variation of composition within the main reservoir of the Vuktylskoye field require additional investigation. Despite the long history of the reservoir development, this problem was previously studied based only on field development data.


introduction
Hydrocarbon reservoirs with large oil or gas columns are characterized by significant changes in reservoir pressure and temperature with depth.As a result, the initial composition of the reservoir fluid also changes significantly with depth.
The distribution of components in depth in massive hydrocarbon reservoirs is greatly influenced by gravitational forces.Initial composition of the fluid in such reservoirs is formed in such a way that the concentration of light components decreases in the direction from the top to the bottom, while concentration of heavy components increases on the contrary.Accordingly, the content of condensate in the reservoir gas increases and the solutiongas content in oil decreases.
In 1954, A.Y. Namiot made calculations for mixtures modeling oils of different composition (Namiot, 1954).It was concluded that gravity forces have a significant impact on the composition of oils containing significant amounts of heavy hydrocarbons and dissolved gas.In case of reservoirs consisting mainly of light hydrocarbons, the composition varies slightly with depth.Subsequently, the mathematical apparatus for calculating changes in multicomponent mixtures composition under the influence of gravity has been widely developed and applied (Whitson, Belery, 1994;Brusilovskii, 2002).
The method of calculating gravitational distribution of pressure and composition by reservoir depth is based on the assumption of the system thermodynamic equilibrium in the gravity field.This state of the system can be achieved at a constant system temperature throughout the entire volume.
For most formations, the temperature increases substantially with depth.Usually, the natural vertical temperature gradient (geothermal gradient) is 0.02-0.03°C/m.In such conditions, thermodynamic equilibrium is not achieved, so there should be vertical heat and mass transfer within the hydrocarbon-saturated part of reservoir.
Many authors have shown a significant influence of the thermal diffusion effect on the distribution of component composition (Pedersen, Hjermstad, 2006;Belery, Da Silva, 1990;Whitson, Belery, 1994;etc.).However, unlike the effect of gravity, there is no generally accepted opinion about the nature of this effect as well as there is no correct mechanism for its consideration for hydrocarbon reservoirs, and therefore a number of conclusions in the above-mentioned works are contradictory.Therefore, there are no established approaches for taking into account the temperature factor influence on the change in the initial composition with depth when calculating reserves and designing the development of specific reservoirs.In addition, the methodology of the corresponding options in the PVTmodeling packages (Schlumberger PVTi, Roxar PVTx, etc.) are poorly documented.
One interesting example in terms of assessing the temperature factor influence is the Vuktylskoye oiland gas-condensate field.For the initial conditions in the main gas-condensate reservoir, the difference in pressures within the gas column was 4.3 MPa and in temperatures -25.65°C.The depth of the reservoir varies in the range from 2000 m to 3350 m.However, despite the 50-year history of the field development, until now the change in the initial composition with depth has been studied mainly on the basis of the actual well data analysis.As for mathematical modeling, there remains the problem of setting thermodynamically consistent input data for performing calculations with a reservoir simulation model.
In this paper, using the example of the Vuktylskoye field, the authors estimate thermal diffusion effect on the component composition distribution in a gascondensate reservoir with a large gas column.The results of mathematical modeling are compared with actual field data.

calculation of changes in component composition with depth in isothermal conditions
In isothermal equilibrium conditions, fugacity values of i-th component of the mixture at depths h 1 and h 2 are interrelated by the following equality (Brusilovskii, 2002): (1) where f i and M i are the fugacity and molecular weight of the i-th component, respectively, p is the reservoir pressure, T is the reservoir temperature, is the vector of molar fractions (concentrations) for components in the gas-condensate mixture, g is the gravity acceleration, R is the universal gas constant, N is the number of components of the mixture.
Let's denote: (2) The relation (1) corresponds to the balance between the changes in chemical and gravitational potentials for each component of the mixture between two depth under conditions of thermodynamic equilibrium in the field of gravity.For molar concentrations of the components, the normalization condition is also satisfied: (3) If the mixture's component composition y 1 (h 1 ),..., y N (h 1 ) and pressure p(h 1 ) at the level h 1 are known, the composition and pressure at the level h 2 are determined by solving the following set of N+1 non-linear algebraic equations: (4) where p(h 2 ),y 1 (h 2 ),…,y N (h 2 ) are the pressure and component composition (mole fractions of the components) of the mixture at h 2 , respectively.
The Newton method is used to effectively solve the system (4).This method has a high convergence rate in the presence of a good initial approximation, which is achieved by adjusting the depth step in the calculation (Brusilovskii, 2002).

Solution by the newton's method
From the last equation of system (4) one can express: (5) and reduce its order by one.The base (iterative) unknowns are p 1 , y 2 ,...y N .
After transformations, the system of equations for calculating the component composition and pressure of the mixture at a depth of h 2 is reduced to solving the following system of N transcendental equations: (6) Successive approximation of the variables is performed by solving at each step the system of equations JS=F obtained by linearizing the equations of system (6).The matrix J and the vectors S, F have the following form: ; ; where m = 0,1,2,... is the iteration number.
The components of the matrix J are calculated as follows: ; . ( 9) The apparatus of cubic equations of state is used for calculation of fugacities and their derivatives (Brusilovskii, 2002).
Within the approximate approach, it is possible to take into account temperature changes with depth in the system (4).For this, hydrocarbon column is divided into a number of small intervals and the temperature is considered constant within each interval.The difference in temperatures between the intervals should be taken into account when calculating fugacities; however, thermodynamic equilibrium conditions (1) are considered approximately valid.However, such a calculation neglects the heat transfer and the accompanying mass transfer of components caused by thermal diffusion.

equation of state
The following Peng-Robinson cubic equation of state was applied to calculate the fugacities in this work: .(10) For a given equation of state, the fugacities are calculated by the following formula: , (11) where , ( 12) , ( 16) , ( 17) , ( 18) where Z is the compressibility factor; p is the reservoir pressure; T is the reservoir temperature; R is the universal gas constant; T r is the reduced temperature; Tc i is the critical temperature of the component; pc i is critical component pressure; w i is acentric factor; y i is the mole concentration of the component in the mixture.

thermal diffusion
The temperature gradient makes a significant contribution to the change in the composition of the hydrocarbon mixture.Taking into account the vertical temperature gradient leads, as a rule, to a stronger dependence of the initial composition on the depth than can be explained only by gravitational forces.
The effect of thermal diffusion on the distribution of the reservoir components is described using nonequilibrium thermodynamical models.Different approaches to considering the effect of thermal diffusion have been proposed by such authors as Belery and Da Silva (Belery, Da Silva, 1990), Haase (Haase, 1990), Kempers (Kempers, 1989), Whitson (Whitson, Belery, 1994), etc.
Pedersen and Lindeloff (Pedersen, Lindeloff, 2003) proposed the following relationships to calculate the compositional change with depth in reservoirs under the influence of gravity and thermal diffusion: where is the absolute partial molar enthalpy of i-th component, H is the absolute molar enthalpy of the mixture, M is the average molecular weight of the mixture, ∆T is the temperature difference between the depths h 2 and h 1 , T 1 is the temperature at the depth h 1 , T 2 is the temperature at the depth h 2 .Relation (24) corresponds to the nonequilibrium stationary state of the system in a gravitational and geothermal field.
The authors (Pedersen, Lindeloff, 2003) suggested that the change in component composition under stationary conditions and influence of the temperature gradient is determined by the specific enthalpy of each component.components with a higher enthalpy than the average for the mixture tend to a warmer zone.Under typical reservoir conditions, high molecular weight components will have a higher specific enthalpy than low molecular weight ones.This corresponds to actual observations showing that the change in component composition with depth in reservoirs with a positive vertical temperature gradient is higher than can be explained only by the gravitational effect.
The partial molar enthalpy of i-th component in a mixture at temperature T can be represented by the following expression: where is the partial molar enthalpy of i-th component in the ideal-gas state at temperatures T and 273.15 K, respectively, -is the partial residual molar enthalpy: (26) where φ is the fugacity coefficient of the i-th component.
Enthalpy of the i-th component in an ideal-gas state at a temperature of 273.15K can be determined by the following formula: (27) and at temperature T, it can be determined from the following thermodynamic relation: (28) the heat capacity of an gas is determined by the correlation with (29) where the coefficients C 1,i -c 4,i are tabulated for c 1 -c 5 components of the mixture (Reid, Prausnitz, Poling, 1987).For C 6+ components, the coefficients are calculated using Kesler-Lee empirical formulas (Kesler, Lee, 1976).

Vuktylskoye field
Based on the described model, calculation were performed for changes in the initial composition with depth for the main gas-condensate reservoir of the Vuktylskoye field with a large gas column (Tables 1-3).Fig. 1 shows the changes in the initial component composition of the gas-condensate mixture with the reservoir depth according to the data of (Dolgushin, 2007).The plots shown were obtained from actual data analysis of wells completed in the reservoir at various depths.Fig. 2 shows the change in the initial reservoir pressure and temperature over the depth of the reservoir.It is interesting to compare the dependences of the initial composition and pressure on the depth according to Fig. 1-2 and obtained by mathematical modeling.The latter are necessary as initial conditions for forecast simulations on a multicomponent 3D reservoir flow model.The model of the gas-condensate mixture was adopted according to the latest development design document (Supplement to the field development plan ..., 2014) and is represented by hydrocarbon components (methane, ethane, propane, butane, pentane and six fractions), as well as nitrogen and carbon dioxide (Table 4).At a depth of 3000 m, the initial composition of the mixture is known, the temperature of the reservoir is 334 K, the reservoir pressure is 36.1 MPa.Geothermal gradient equals to 0.019°С/m.Physical properties of hydrocarbon fractions are presented in Table 5.The goal is to recalculate the pressure and composition from the initial depth of 3000 m to a depth of 2000 m (upper level of the reservoir), 2500 m and 3350 m (gas-liquid contact).The calculations were carried out using the following three methods: 1) Gravity method -the equation ( 1), 2) Taking into account the interval temperature change, but without thermal diffusion, 3) Taking into account thermal diffusion -the equation ( 24).

calculation results
Fig. 3-10 show some calculation results.The depth changes of pressure and initial concentrations of gascondensate mixture individual components are given for the main reservoir of the Vuktylskoye field.Fig. 9 corresponds to the group of c 5+ hydrocarbons, i.e. the sum of pentanes and all c 6+ fractions.
The concentration of light components (nitrogen and methane) decreases with depth (Fig. 3-4), while the concentration of heavier hydrocarbons starting with ethane increases with depth (Fig. 5-9).The higher the molecular weight of the component, the more its content increases with depth.Initial reservoir pressure changes consistently (Fig. 10).These features are associated with the influence of the gravitational field and are predominant in all variants of calculations.Supplement to the field development plan of the Vuktylskoye oil-, gasand condensate field (2014).Report.Ukhta: Gazprom VNIIGAZ (In Russ.)

Fig. 4 .
Fig. 4. Change in methane content with depth

Fig. 10 .
Fig. 9. Change in content of group of components C 5+ with depth

Table 1 .
Calculated component composition (molar fractions of the components in the mixture) at a depth of 2000 m

Constant temperature Interval temperatures Thermal diffusionTable 4 .
Model of gas condensate mixture and input data at a depth of 3000 m (Supplement to the field developmentplan ..., 2014)

Table 3 .
Calculated component composition (molar fractions of the components in the mixture) at a depth of 3350 m