Numerical Well Test Analysis of Condensate Dropout Effects in Dual-Permeability Model of Naturally Fractured Gas Condensate Reservoirs: Case Studies in the South of Iran

Naturally fractured reservoirs (NFR) represent an important percentage of worldwide hydrocarbon reserves and production. ,e performance of naturally fractured gas condensate reservoirs would be more complicated regarding both rock and fluid effects. In contrast to the dual-porosity model, dual-porosity/dual-permeability (dual-permeability) model is considered as a modified model, in which flow to the wellbore occurs through both matrix and fracture systems. Fluid flow in gas condensate reservoirs usually demonstrates intricate flow behavior when the flowing bottom-hole pressure falls below the dew point. Accordingly, different regions with different characteristics are formed within the reservoir. ,ese regions can be recognized by pressure transient analysis. Consequently, distinguishing between reservoir effects and fluid effects is challenging in these specific reservoirs and needs numerical simulation. ,e main objective of this paper is to examine the effect of condensate banking on the pressure behavior of lean and rich gas condensate NFRs through a simulation approach. Subsequently, evaluation of early-time characteristics of the pressure transient data is provided through a single well compositional simulation model. ,en, drawdown, buildup, and multirate tests are conducted to establish the condition in which the flowing bottom-hole pressure drops below the dew point causing retrograde condensation. ,e simulation results are confirmed through well test analysis in both Iranian naturally fractured rich and lean gas condensate fields. Interpretations of simulation analysis revealed that the richer gas is more prone to condensation. When the pressure drops below the dew point, the pressure derivative curves in the rich gas system encounter a more shift to the right, and the trough becomes more pronounced as compared to the lean one.


Naturally Fractured Reservoirs.
A substantial amount of worldwide oil and gas was produced by Naturally Fractured Reservoirs (NFR) [1][2][3][4][5][6][7][8]. e recovery factor in NFR varies widely (10 to over 60%). On the one hand, fractures could play the role of the main path of fluid flow, especially in tight formations, and consequently improve the ultimate reservoir recovery [9]. On the other hand, fractures may have the adverse effect on reservoir productivity and cause earlier unwanted water production, erratic sweep patterns, and consequently low recovery factor [9][10][11].
is issue is more critical in naturally fracture water drive gas reservoirs. is water expansion in naturally fractured reservoirs, particularly with low matrix permeability, fills up the fractures, and the free gas will stop flowing.
us, water would be the only fluid passing through the fractures, and early breakthrough happens [12]. Beaver River Field of Canada [13], Dengying gas reservoir in Weiyuan gas Field in China [14], and Aguarague Field in Argentina [15] with the gas recovery of 12, 37, and 34 percent, respectively, are examples of these kinds of reservoirs. us, the effect of fracture on ultimate reservoir recovery could be either positive or negative.
Well test data can offer crucial information about the reservoir properties. e existence of fractures introduces a layer of difficulty in interpreting the pressure response. By now, several idealizations have been suggested to analyze the NFR pressure behavior. e most common idealization is the dual-porosity model proposed by Barenblatt et al. (1960).
is model was firstly applied in reservoir engineering by Warren and Root (1963) by introducing two additional parameters as follows [16].
Storativity ratio (ω): ω � φc t fracture φc t fracture + φc t matrix . (1) in which α is defined as Figure 1 depicts different matrix system elements, n � 1, 2, 3 for slab, cylinder, and cubic matrix blocks, respectively [17]. e dual-porosity model assumes the existence of the two media with large bulk porosity/relatively low permeability (matrix) and very low bulk porosity/relatively high permeability (fracture). In this model, fluid flow from the matrix to the fracture system, and subsequently, through the fracture to the wellbore, leads to hydrocarbon production. e fluid regime between the matrix and interconnected fracture is pseudosteady state flow [18][19][20][21]. e incorporation of the dual-porosity idealization and numerical modeling for large-scale fluid flow application was firstly introduced by Kazemi et al. (1976) [19]. It has been recognized as the standard for NFR modeling and well test interpretation [5,9,[18][19][20][21][22][23][24][25][26][27][28][29]. Figure 2 depicts the pressure transient behavior in the dual-porosity model. Figure 3 represents the different flow behaviors of such reservoirs on log-log pressure derivative response schematically [16,21]. e dual-porosity dual-permeability is another idealization introduced to improve the previous dual-porosity model. is idealization was first suggested by Blaskovich et al. (1983) and Hill and omas (1985). Matrix blocks are connected in this model and participate in the overall fluid flow. In comparison to the dual-porosity model, which is suitable only for intensely fractured reservoirs, the dualporosity/permeability model can be applied for many reservoirs with different fracture degrees [31][32][33][34][35][36]. Figures 4 and 5 show a schematic of fluid flow in these two aforementioned models [37].
Overall, dual-porosity/dual-permeability reservoir includes two homogeneous layers. e cross flow between layers exists, and both can flow into the well. e pressure difference between layers determined the amount of interlayer cross flow. e parameters resulting from well test analysis define the mathematic model to describe this reservoir [38]: Mobility ratio (κ): Storativity ratio (ω): Interlayer cross flow coefficient (λ): At the early time, cross-flow does not occur, and the layers are produced independently. When the interlayer cross flow starts, a transition period is detected by an inflection in the pressure response and a valley in the derivative. After the transition, the reservoir acts as a homogeneous medium, with total kh and storativity. When the mobility ratio is equal to one, the response of the dualporosity and dual-porosity/permeability is the same. In addition, the response tends to a homogeneous reservoir when the mobility ratio decreases, as shown in Figure 6. e different flow behavior, which can be observed in the log-log pressure derivative, is the same as that in Figure 3. e only difference is that, in all cases, the matrix can also produce into the well [21]. Many researches have been conducted on naturally fractured reservoirs applying the dual-porosity and dual-permeability model [36,[38][39][40][41][42][43][44][45][46][47][48][49][50][51].

Gas Condensate Reservoirs and Pressure Transient
Analysis. A gas condensate is a naturally occurring hydrocarbon mixture found between the cricondentherm and critical reservoir temperature [52][53][54][55]. It can be categorized into rich and lean gas condensate reservoirs. Below the dew point, rich and lean gas condensate generates 100 and more than 150 STB/MMSCF of liquid, respectively [56]. Due to the complex phase and fluid flow behavior of these reservoirs, their evaluation becomes a challenging issue. With condensate accumulation around the well, the effective permeability is reduced substantially [38,[57][58][59][60][61][62][63][64][65][66][67][68]. e first investigation on condensate blockage was directed by Muskat (1949) [69]. O'Dell and Miller (1967) suggested a pseudopressure equation to analyze this phenomenon. is equation is limited to reservoirs over the dew point pressure and when the radius of condensate banking is not significant. In 1973, Fussel developed the compositional model, which was applicable to analysis gas condensate reservoirs behavior below the dew point pressure [70]. Later, Jones, and Raghavan (1988) used buildup and drawdown pressure tests to study the gas condensate reservoirs [71]. e concept three-flow region in gas condensate reservoirs was proposed by Fevang and Whitson (1996) and Penuela and Civan (2000) [59,72]. Properties of these regions can be summarized as follows: Region (1): this is the farthest region around the well with pressure over the dew point. e system contains a single gas-phase including the initial liquid saturation. Region (2): this region is near the first region toward the well. It is generated with pressure reduction below the dew point, and where the liquid saturation increases rapidly. It should be noted that, in this region, the liquid phase is immobile (SL < SLC). Region (3): this region is the nearest region around the well. e liquid saturation is higher than the critical saturation of condensate, so, gas and oil phases are mobile.
In some references, the existence of another region in the vicinity of wells was reported. is fourth region was firstly reported by Gringarten et al. (2000) [30].
One of the approved methods in investigating the gas condensate reservoir behavior is well-test analysis (pressure transient tests). Figure 7 shows the regions around the wellbore by applying a single-phase pseudopressure function for gas condensate reservoir.
Due to the high conductivity of fractures, condensate flow in fractures is simpler, and consequently, condensate saturation near the wellbore can be higher with respect to other regions in reservoirs. In opposition, the residual condensate saturation in the matrix is relatively large; hence, it could remain immobile for quite sometimes.
In gas condensate NFRs, these regions are usually formed in the fracture system because of the high fracture mobility and minor change in matrix gas saturation. Figure 8 illustrates the profile of gas saturation in naturally fracture reservoirs [37].    Interaction of competing viscous, capillary, and inertial forces as the pressure change during reservoir life can affect the well deliverability of gas condensate reservoirs. e physical processes that reduce productivity are condensate banking, and inertial effect, which leads to the reduction of k rg as the velocity increases. Conversely, positive coupling or velocity stripping effect (capillary number effect) results in the increase of gas relative permeability and to some extent that of liquid, as velocity increases and/or IFT decreases. e overall balance of these forces specifies the well deliverability [30].

Approach
Since there is not any analytical composite model for gas condensate NFRs, a numerical compositional model is a suitable choice to assess the pressure behavior in these types of reservoirs.
In the current study, the effect of condensate banking on the pressure behavior of lean and rich gas condensate NFRs is investigated through a simulation approach. e necessary simulation data (fluid and rock properties) was acquired from a real gas condensate field. Multirate tests (drawdown and buildup) were conducted to create a condition in which the flowing bottomhole pressure went below the dew point, and consequently, the condensate was formed. In the next step, the results of the numerical model were analyzed by standard well tests analysis to derive the results.

Simulation Studies.
e characteristics of the proposed model for dual-porosity/dual permeability reservoirs will be discussed. Two types of gases (lean and rich) were used for validating the presented model.

Model Setup.
e compositional simulator (Elipse-300) was utilized to set up a single vertical model in radial coordinates. e thickness of the synthetic multilayer model proposed for the dual-porosity/dual-permeability reservoir is 100 ft. For attaining a better resolution near the wellbore to detect condensate behavior, the gird blocks were adjusted, so that their numbers increased logarithmically toward the well. e reservoir simulation input, initial reservoir saturations, rock properties, and rate schedules are presented in Tables 1-4, respectively.

PVT Modeling.
e reservoir fluid behavior is an essential part of reservoir simulation. e PVTi package is utilized to simulate the reservoir fluid properties. e results of all PVT experiments are matched simultaneously to determine a representative equation of state (EOS) for the reservoir fluid. 3-parameter Peng-Robinson (PR3) with lumped 8 and 13 components is selected as EOS input into the simulator in lean and rich gas, respectively. Tables 5 and  6 show the composition of lean and rich gas, and Table 7 summarizes detailed results of PVT matching.

Relative Permeability.
e feed relative permeability data in the simulator were taken from laboratory core analysis. Since the relative permeability curve in gas condensate reservoirs depends mainly on velocity and IFT, Eclipse-300 interpolates the objective curve between a base and a miscible curve. e base curve is measured when IFT is at the highest level, and velocity is at the lowest possible level. e miscible curve accounts for inertial effects. e interpolation is weighted by capillary number dependent functions.

Model Validation.
e output of the well test model was used to validate the accuracy of the proposed numerical model. To compare the results of real field (input data of simulator) and simulations, the buildup test was conducted. Tables 8 and 9 represent the comparison results above the dew point pressure.

Simulation Results.
e proposed models (lean and rich gas condensate) are representative of dual-porosity/ dual-permeability NFR, and the main objective is the investigation of condensate blockage. Simulation scenarios and their characteristics are presented in Table 10.
In addition, the log-log plot of buildup data for lean and rich gas reservoir model at pressures above and below dew point is shown in Figures 9 and 10.

Discussion
As can be seen in both Figures 9 and 10, the effect of condensate accumulation is superimposed on dual-porosity/ dual-permeability flow behavior. e observed fluid flow behavior can be categorized into early transition and late time zones.

Early Time Behavior.
Derivative curves indicate that, below the dew point, reduced mobility region is formed near the wellbore. is behavior occurs after an upward shift in the early time derivative stabilization. Although more liquid saturation is formed in the case of the rich gas condensate system compared to the lean one, there is no sharp difference in the upward shift between these two different gas systems.
is phenomenon is observed because condensate is mainly developed in the high mobility of the fracture systems.

Transition Behavior.
Fluid flow between the matrix and fractures in the reservoir starts at the transition zone. Any phenomenon that restricted this communication results in the expansion of this period. Interpretations of log-log curves reveal that the surface of matrix blocks can be damaged by condensate formation. is damage is interpolated by an additional interporosity skin factor. is particular skin factor causes a delay in fluid flow from the matrix blocks to the fractures. Since the rich gas is more prone to condensation with respect to the lean gas, the skin factor is higher in the rich gas system. Consequently, the reduction in the value of interlayer cross-flow coefficientλ is more. is fact is illustrated in Figures 9 and 10, in which the derivative curve is shifted to the right in the transition period.

Late Time Behavior.
As the bottom-hole pressure decreases, more condensate is formed and accumulated. If the entire reservoir pressure drops below the dew point, the

Rock properties Matrix
Fracture 1E-4 Mathematical Problems in Engineering      Mathematical Problems in Engineering composite behavior can appear at the late times. is situation happens more in the lean gas system, in which condensate can be formed in the entire reservoir. On the contrary, in the rich gas system, the falling of reservoir pressure below the dew point is less probable; thus, the probability of condensate formation in the entire reservoir, especially in far regions from the wellbore, is less. In this situation, late time stabilization is the same for both below and above dew point cases ( Figure 10). Similar to the early times periods, liquid condensation causes a reduction ink rg , which is characterized with an upward shift in the late time derivative stabilization (Figure 9).

Conclusion
e outcomes of the proposed simulator for detecting the condensate formation in lean and rich gas reservoirs lead to the following conclusion: (1) During the well test analysis of gas condensate naturally fractured reservoirs, condensate creation adds more complexity to the analysis.
(2) Since external edges of the matrix blocks will be the first to host condensate in the system, condensation deposition creates an impediment layer at surface of the matrix blocks, which results in delaying matrix feeding to fractures. is phenomenon shifts pressure derivative curve to the right and causes deeper transition period. (3) e richer gas is more prone to condensation. When the pressure drops below the dew point, the pressure derivative curves in rich gas system encounter a more shift to the right, and the trough becomes more prominent as compared to the lean one.

Data Availability
e data used to support the findings of this study are included within the article.