Numerical Analysis on the Influence of Thermal Effects on Oil Flow Characteristic in High-Pressure Air Injection (HPAI) Process

In previous laboratory study, we have shown the thermal behavior of Keke Ya light crude oil Tarim oilfield, branch of CNPC for high-pressure air injection HPAI application potential study. To clarify the influences of thermal effects on oil production, in this paper, we derived a mathematical model for calculating oil flow rate, which is based on the heat conduction property in porous media from the combustion tube experiment. Based on remarkably limited knowledge consisting of very global balance arguments and disregarding all the details of the mechanisms in the reaction zone, the local governing equations are formulated in a dimensionless form. We use finite difference method to solve this model and address the study by way of qualitative analysis. The time-space dimensionless oil flow rate qD profiles are established for comprehensive studies on the oil flow rate characteristic affected by thermal effects. It also discusses how these findings will impact HPAI project performances, and several guidelines are suggested.


Introduction
Air injection in light oil reservoirs HPAI, referred to as high-pressure air injection has gained greater attention during the last decade 1-3 .HPAI projects have been steadily increasing in recent years, especially in light oil carbonate reservoirs in the USA 4 .However, some controversies still exist on recovery mechanisms, especially for thermal effects.Over the years, HPAI has been considered a simple fuel-gas flood, giving little credit to the thermal drive as a production mechanism 5 , where the flue gases provide the main driving mechanism for oil production and the thermal effects have very little impact on the oil recovery mechanism.For instance, Sakthikumar et al. 6 simulated the HPAI process as an immiscible nitrogen flood, and Glandt et al. 7 modeled the process as an isothermal flue conditions and to assess the validity of the assumptions made in classical macroscopic descriptions 15 .This is of great practical interest in the prospect of tackling large-scale problems.
In this study, a simplified model is developed for predicting the influences of thermal effects on oil flow rate and based on the heat conduction model in combustion tube experiments 17 .Many of the existing literatures report the combustion tube simulations all are based on this classical model.For instance, in consideration of fuel concentration in oxygen-enriched in situ combustion, Rodriguez and Mamora 18,19 developed a new analytical model of the combustion zone in combustion tube experiments.Jabbari et al. 20 modeled the toe-to-heel air injection process and the temperature distribution was analytically formulated and compared to the experimental data.But these models always are using analytical solutions.
The goals of this paper are to study the influences of thermal effects on oil flow characteristic by way of qualitative analysis.The rest of this paper is organized as follows.Section 2 describes the preliminary analysis and model derivation.Section 3 introduces the finite difference method and ascertains the main parameters for numerical calculation.Section 4 makes comprehensive theoretical analysis on how the thermal effects will impact oil production during HPAI process.Finally, what we get in our paper is summarized in Section 5, and future directions are proposed.

Theoretical Model Descriptions
Based on the feasibility of autoignition for Keke Ya light oil, the following model assumptions will be well founded.The physical model is based on the combustion tube model originally proposed in 17 and we briefly summarize this model first.Penberthy and Ramey developed an analytical heat model of movement of a burning front axially along a cylinder with heat loss through an annular insulation.A heat balance may be made for a differential element ahead of the burning front.It is well known that a stationary regime and a typical combustion front speed could exist in solid/gas combustion 15, 21, 22 .Hence, a moving coordinate system such that the burning front is always at the x-coordinate zero is used; correspondingly, the x-coordinate represents distance from the burning front.The model to describe the temperature profile ahead and behind the combustion front for combustion tube experiments is based on heat balance conduction and convection for a differential element in the combustion tube.The classical model is written by the following expression: The appropriate initial and boundary conditions are where k b is the thermal conductivity, kJ/m • • C; C f is the specific heat of the matrix, kJ/kg • • C; C g is the specific heat of air, kJ/kg • • C; ρ ma is the density of the matrix, g/cm 3 ; u is the superficial air velocity at burning front, m 3 /hr•m 2 ; T is the temperature, • C; T a is the ambient temperature, • C; T c is the peak temperature, • C; x is the distance from the burning front, cm; U is the overall heat conduction coefficient through the annular insulation, kJ/hr • m  Substituting The conceptual representation of air injection process along with the combustion tube counterpart is illustrated in Figure 1.Air injection process involves injection of high pressure air into a reservoir to promote in situ oil oxidation.Oxidation is a slow process in which O 2 present in the injected high-pressure air reacts with hydrocarbons in the oil.When this happens, reservoir temperature in the vicinity of the air-oil contact rises and, eventually, oil autoignites resulting in the in situ generation of flue gases and steam 13, 23 .Reservoir oil will be mobilized towards producing wells through a combination of several complex mechanisms.Combustion is an exothermic process, which is due to the exothermic reaction between the fuel in the porous solid and oxidizer contained in the gas flowing through the solid 21 .In addition, the reservoir often creates better adiabatic conditions, and that can create a more favourable condition for heat conduction 24 .Hence, after oil autoignition, reservoir temperature profile can steadily move to a far distance, and the frontal oil properties will soon be affected.It seems matching with the interpretation of Figure 1 that the complex recovery mechanisms such as flue gas driven, oil swell, "stripping" of light volatiles, and reservoir repressurized, are only dominated in the vicinity of the injection well zone, 1, 2, and 3 the unswept reservoir zone 4 will receive steady thermal energy to reduce oil viscosity.
In this study, the indicated oil production contributed by thermal effects is based on oil physical viscosity reduction as mentioned in Introduction.For application of HPAI in light oil  reservoirs, low temperature oxidation LTO is the main concerned aspect 25 ; in this stage, bond-scission reactions typically occur at temperature in the range of 150-350 • C 26, 27 .Hence, crude oil will experience more complex chemical reactions and oil-viscosity reduction will not exist in LTO stage.Whereas crude oil often does not experience bond-scission reactions below 150 • C 26 , only simple physical phenomena of viscosity reduction will be exhibited.For the next discussion, we choose 150 • C as the peak temperature for reducing oil viscosity and oil can be mobilized by thermal effects in the temperature decreasing profile.The legend of thermal effects affected zone in Figure 1 vividly shows that extra oil is expelled out below the upper limit temperature 150 • C zone due to viscosity reduction.Correspondingly, the value of combustion front temperature T C described in Penberthy and Ramey's model is equal to upper limit temperature 150 • C .

Physical Model and Assumptions
We choose a long combustion tube as the basic physical model.The assumptions are listed as follows.
1 The combustion tube is heterogeneous and prior saturated with oil only.The inlet and the outlet act as the injection well and produce well, respectively.Backpressure is employed as the reservoir pressure, while injection pressure keeps constant to make oil producing pressure drop in a constant value in whole process.
2 The conceptual combustion tube is long enough and air injection rate is as low as possible so that premature gas breakthrough would not happen.Therefore, oil phase located at a far distance unswept zone ahead of peak temperature still displays single-phase porous flow and oil permeability is calculated by Darcy's law.It was assumed that the capillary pressure is zero.This is a reasonable assumption for crushed sands which generally have large pore sizes 28 .
3 As mentioned above Section 2 , the unswept reservoir will receive steady thermal energy to reduce oil viscosity and some other recovery mechanisms i.e., oil swelling, reservoir repressurized, etc. do not exist; it should be noted that oil phase located at the far distance is extracted out under the constant producing pressure gradient; in other words, it will exhibit the same pressure gradient below the upper limit temperature 150 • C zone.

Model Derivation
We introduce here a basic solution in an extremely simple case, because it provides a guideline for the interpretation of more detailed simulation results.Most of the following is classical.In addition, the assumptions are not required for establishing the main global results.Hence, we remain here at a rather coarse level, which is sufficient for our present purpose.Complex recovery mechanisms take place in the travelling reaction zone, but we do not need to detail them at this stage.In the following section, we formulate the mathematical model and the physical conditions to which it corresponds.Oil flow rate is based on Darcy's law and can be expressed as follows 20 : where q is the oil flow rate in cm 3 /s, K o is the oil permeability in μm 2 , A is the cross-sectional area of combustion tube available for flow in cm 2 , μ o is the oil viscosity in mPa • s, ρ o is the oil density in kg/m 3 , g is acceleration due to gravity in 10 7 m/s 2 , P is pressure in a certain point in combustion tube in 10 5 Pa, x is the distance from burning front in cm.θ represents the reservoir dip angle in • ; "±" indicates that the gravity can be acted as driving force and resistance for oil flow, respectively.When the producing pressure gradient keeps constant, 2.7 becomes where G is the producing pressure gradient across the combustion tube in 10 5 Pa/cm and is calculated through the expression G ΔP/L, L is the combustion tube length in cm, and ΔP is the producing pressure difference across the combustion tube in 10 5 Pa.
If the reservoir dip is equal to zero, 2.8 is written as The regressed oil-viscosity temperature curve usually can be written by where λ and ω are correlation coefficients in the regressed equations in mPa • s and • C. T is temperature in • C. Before oil autoignition, oil flow is at reservoir temperature T ν , implying that the constant oil viscosity μ ν at T ν can provide a constant oil flow rate q ν which can be written as Defining the parameter Θ K o AG/λ, 2.11 becomes q Θ e ωT .

2.12
From 2.12 , temperature is given as follows The ambient temperature T a in the aforementioned model is equal to the initial reservoir temperature T ν .After substitution and rearrangement, 2.6 becomes In combination with 2.2 -2.4 and 2.13 , we rearrange the initial and boundary conditions.
The initial condition is written as follows: The boundary conditions are When burning front formed, the internal boundary in the moving x-coordinate always can provide a maximum oil flow rate of q c at upper limit temperature T C .It should be noted that oil viscosity becomes μ c , correspondingly.whereas in the external boundary L , oil flow rate will not be affected by thermal effects with time passing and still keep at q ν .The limitation of heat conduction ability in a long enough combustion tube is responsible for this.Then, 2.17 can be rewritten as follows: A basic solution in the simplest case with local thermal equilibrium is given as a reference for the discussion of subsequent results, based on remarkably limited knowledge consisting of very global balance arguments and disregarding all the details of the mechanisms in the reaction zone.Then, the local governing equations are formulated in a dimensionless form; 2.14 can be solved by defining the following parameters and dimensionless groups. Let

2.19
Finally, the expression of dimensionless oil flow rate ahead of burning front is where C is dimensionless heat loss constant; q D is the dimensionless oil flow rate; x D is the dimensionless distance; L D is the dimensionless combustion tube length; t D is the dimensionless time.
And then, the initial and boundary conditions can be presented as below: Similar scenario is that 2.23 becomes, It should be noted that q D 0 means that a certain place is not swept by thermal effects and oil flow rate keeps constant at q ν .

Discretization
Penberthy and Ramey 17 gave an analytical solution for the heat conduction model.For the development of this model, other authors use analytical method 18-20 .In this paper, we solve the governing PDEs by finite difference method.Dimensionless oil flow rate q D in 1D combustion tube was modeled.The principle of this method is illustrated in Figure 2.For most field upscaling, combustion can be included in a limited fashion in simulations at the expense of extra computational time and complexity.A fine grid can capture the combustion front propagation 14, 29, 30 .In this study, a fine grid block in 1D simulations is employed.The combustion tube length L is divided into I space intervals Δx D .The time is divided into J time intervals Δt D for all of the simulations we have chosen: I 100 and J 20000; Δx D 0.01 and Δt D 0.01, therefore the total dimensionless length and dimensionless time are 1 and 200, resp. .Every approximate value of q D above t 0 jΔt D time axis can be independently calculated from the value in t 0 j − 1 Δt D time axis.More specifically, q D for a new time step t D t 0 Δt D can be calculated, when q D is known at the previous time step t D t 0 .q D at the outer grid points i 0 v i I v j 0 v j J for the new time step t D t 0 Δt D is calculated using the boundary conditions.At time t D 0, q D is given by the initial condition; hence, The backward difference method is employed in this study.Discretization of 2.20 can be written as follows; Δx D C q i Dj−1 q c − q ν q ν q c − q ν ln q ν q i Dj−1 q c − q ν q ν . 3.1

Parameters Determination
The dimensionless oil flow rate modeling and simulation require a fit to experiment data to determine model parameters.In this study, most of model parameters are according to Keke Ya reservoir conditions Tarim oilfield, branch of CNPC , which are stated in previous studies 11, 12, 31 .The viscosity of crude oil under different temperature was measured using Brookfield viscometer DV-III.The viscosity versus temperature curve is shown in Figure 3.The experimental date can be perfectly exponential regressed with expression μ 4.70844e −0.0094T .Hence, the values of μ ν and μ c are obtained.This study has different types of parameters as shown in Table 1.

Results and Discussion
Several variables can affect the oil flow rate contributed by thermal effects.In this section, the influences of two main variables t D , x D on oil flow rate and resulting guidelines will be studied.It should be noted that no systematic experimental results of process variation were Representation of grid blocks used in the simulation: calculation of the q D at a new time step from the q D at the previous time step.Pa q ν 0.089 cm 3 /s r t 2.5 cm q c 0.172 cm 3 /s Note: G, q ν , and q c are calculated from the above parameters.
available for the oil flow rate of this study.Hence, simulation results will be used to quantify trends caused by process variations.Systematical experimental studies will be conducted in comparison with the predicted trend in the forthcoming paper.

Effect of t D
Figure 4 shows the profiles of the relationships of q D in different x D 0, 0.01, 0.03, 0.05, and 0.07, shown in the legend with the increasing of t D .For x D 0, q D has the maximum value of 1.0; it indicates that the formed peak temperaure T c in the initial space can provide the steady highest oil flow ability.In the space adjacent to T c i.e., x D 0.01 , q D initially shows linear increasing characteristic with high positive slopes dq D /dt D > 0 , and the slope value sharply increases to 0.4 in less than t D 4, and then followed by a smooth increasing stage until to a plateau.The final q D value is around 0.95 which is slight below the highest oil flow rate value of 1.0 at the initial place x D 0 .It reflects the temperature profiles ahead of combustion front have smooth characteristic.Several studies also demonstrated that temperature profiles both from simulation run or laboratory experiments will show the same law of smooth turning to a plateau after the peak temerature 32 .
The simulation result reveals that the closest space to T c can provide high oil flow rate in a short time.because high temperature can reduce oil viscosity to a high extent for its better mobilization.The high positive slope in the earlier stage means that temperature in x D 0.01 rapidly elevates to maximum value due to the propagation of combustion front from the initial place; when the steady combustion front formed, oil mobilization ability will become constant and show around 0.95 in later period from t D 160 to 200.For x D 0.03, 0.05, 0.07, the corresponding curves show the similar trend as the former x D 0.01.Most importantly, the three curves have three important characteristics: 1 a q D 0 period was found and delayed with the increasing of x D ; 2 the positive slopes in the early stage are apparently decreased with the increasing of x D ; 3 the smooth increasing period of q D has been largely extended in this given total time interval.
The specified "q D 0 period" means that oil flow rate is at the original level and is not affected by the thermal effects.The relative far dimensionless distance shows a longer q D 0 period; for instance, in x D 0.03, 0.05, 0.07, the q D 0 periods are around t D 0.5, 2, 3, respectively.
Furthermore, q D at terminating time t D 200 shows greater difference in each x D interval.Compared to the value of 0.95 at x D 0.01 at t D 200, q D in the other three x D intervals reduces to 0.84, 0.74, and 0.65, respectively.In addition, we found that q D decreasing gradient is nearly the same for the same x D intervals i.e., the D value of q D keeps at around 0.1 for the same space interval increments of 0.02Δx D .It denotes that temperature in each place along the combustion tube will become constant with the increasing of time.Hence, smooth q D curves can be detected due to the stable lowered oil viscosity.However, q D decreasing gradient at the same t D does not show the same value in the early stage as obviously illustrated by the arrowheads.The discrepancy may be ascribed to the sharply decreasing of temperature in the relative high temperature T c approached zone.For instance, a rapid temperature decreasing followed by a smooth temperature decreasing tendency is often observed in combustion tube experiments.
Figure 5 shows that a longer q D 0 period widely exists in further distances and also prolongs with increasing of distance.For x D 0.15 to 0.45, the q D 0 period ranges from t D 9 to 65.However, in x D 0.60, q D values are kept constant at zero for all the time.It implies that the dimensionless distance is not affected by the combustion front and oil is mobilized only under the initial oil viscosity at reservoir temperature.On the other hand, the endpoint values of the curve clusters are sharply decreased compared to the distance approach to the upper limit temperature, it reveals that the ability of thermal effects on oil flow rate is restricted in a short distance.For a far distance, temperature gradually tends to reservoir temperature, which corresponde to the illustration in Figure 1.And hence, thermal effects cannot efficiently sweep in this area.For heavy oil reservoirs, crude oil located at a far distance could be swept by thermal effects because a slight temperature increasing can efficiently reduce oil viscosity for improving oil production.But it is not the same case for light oil reservoirs due to the nature of low viscosity of light crude oil because the decreasing amplitude degree of light crude oil viscosity is rather limited.The rapid q D decreasing also relates to the thin combustion front 13, 23 .The available literature has shown that a combustion front inside a porous medium could have a length of 2 to 5 mm 33 .
A smooth increasing trend is obviously detected for all the curves.However, these curves will not unlimitedly increase and are bound to reach a plateau with low q D values as time passes.On the other hand, it seems that high oil flow rate can be achieved by extending production time in an HPAI process.It is proposed that extending air injection project cycle is desired to achieve high ultimate oil recovery.Nevertheless, all the results are based on the assumption that combustion front is steady propagating that can sweep to any far distance provided that injection time is infinite.However, in actual reservoirs, the combustion front may not favourably propagate because of the complex lithology and liquid properties under reservoir conditions.

Effect of x D
q D profiles as a function of x D are shown in Figure 6.q D distribution curves varied in different t D t D 0, 20, 60, 100, and 200, shown in the legend and all curves show the similar decreasing trend with the increase of x D .The horizontal line used for comparison shows that in initial time oil flow rate has not been affected by thermal effects in any place of the combustion tube.When combustion front formed, q D shows the maximum value 1.0 all the time in the initial place.In the earlier stage, q D profiles display a high negative slope dq D /dx D < 0 and sharply decrease along the combustion tube, and then followed by a smooth decreasing in a very short distance until they become zero.For instance, q D profile in t D 20 depicts a high oil flow rate decline stage and the influences of thermal effects on oil flow rate are only restricted within x D 0.2.It means that oil flow rate in subsequent distance is not affected by thermal effects.However, oil flow decreasing rate is mitigated with the increasing of dimensionless time.The swept distances done by thermal effects are x D 0.2, 0.3, 0.4 and 0.6 at t D 20, 60, 100 and 200, respectively.It indicates that a longer time is needed for heat conduction to a far distance when combustion front velocity v b keeps constant.In t D 200, a long transitional period exists in q D profile; we can see that a decline stage apparently shows from x D 0.4 to 0.6, which is much longer than the earlier stage.It gives us an important enlightenment that oil production contributed by thermal effects can be enhanced through reducing well spacing or extending air injection cycle.That is to say, for 100 m x D 0.1 well spacing test wells can achieve more oil production than that of 200 m x D 0.2 well spacing test wells whilst air injection cycle is the same.
It should be noted that increasing air injection cycle can maintain the successive combustion front and heat conduction ability, because a stable-persistent combustion front strongly relies on the addition of enough oxygen 5 .But it will increase the extra operation cost.

Conclusions
We show how to derive mathematical models for studying the space-time distributions of oil flow rate caused by thermal effects and the finite difference method is employed to solve the governing equations.The theory of "thermal effects" is described by the previous discussion and reveals that the role of combustion front seems like a bulldozer to move the oil ahead of it, and the very adjacent place to the combustion front upper limit temperature can provide high oil flow rate, whereas a far distance will be swept by thermal effects provided time increases.Sensitive analysis on oil flow rate proposes that such methods as reducing well spacing and extending air injection cycle shoud be taken into consideration to achieve high cumulative oil production contributed by thermal effects, and further studies should be addressed to reappraise its contribution to oil production by way of quantitative analysis.The influences of thermal effects on oil production should not be treated as an episode.Our findings may give a new avenue for the study of thermal effects on oil production in HPAI process in light oil reservoirs.However, there is still a lot of work that should be loaded, because the combustion process is very complex.We hope that the results of our analysis encourage such investigations.

Figure 1 :
Figure 1: Compared diagrams of conceptual air injection process and combustion tube experiment.

R 2 =Figure 3 :
Figure 3: The evolution of oil viscosity at different temperatures.

Figure 4 :
Figure 4:Results from dynamic simulation of the relationship of q D versus t D in different x D The selected x D is approach to the upper limit temperature .

Figure 5 :
Figure 5: Results from dynamic simulation of the relationship of q D versus t D in different x D The selected x D is far away from the upper limit temperature .

1Figure 6 :
Figure 6: Results from dynamic simulation of the relationship of q D distribution in x D . 2 • • C; r t is the radius of the combustion tube, cm; t is time, hr; v b is the velocity of the burning front, m/hr.It is noted that 2.2 and 2.4 specify that the initial temperature in the pack and the surrounding temperature are different.

Table 1 :
Model parameters and properties.