A Two-Phase Numerical Model of Well Test Analysis to Characterize Formation Damage in Near-Well Regions of Injection Wells

Formation damage usually occurs in near-well regions for injection wells completed in offshore oilfields under the development of line drive patterns. However, current works on characterizing the damage by well test analysis were basically focused on using single-phase analogy to solve two-phase flow issues, resulting in errors on the diagnosis and interpretation of transient pressure data. In this paper, we developed a two-phase model to simulate the pressure transient behavior of a water injection well in a multiwell system. To solve the model more efficiently, we used the finite volume method to discretize partially differential flow equations in a hybrid grid system, including both Cartesian and radial meshes. The fully implicit Newton-Raphson method was also employed to solve the equations in our model. With this methodology, we compared the resulting solutions with a commercial simulator. Our results keep a good agreement with the solutions from the simulator. We then graphed the solutions on a log-log plot and concluded that the effects of transitional zone and interwell interference can be individually identified by analyzing specific flow regimes on the plot. Further, seven scenarios were raised to understand the parameters which dominate the pressure transient behavior of these flow regimes. Finally, we showed a workflow and verified the applicability of our model by demonstrating a case study in a Chinese offshore oilfield. Our model provides a useful tool to reduce errors in the interpretation of pressure transient data derived from injection wells located in a line drive pattern.


Introduction
Currently, offshore oilfields have assumed a significant role in petroleum production. To improve oil recovery, line drive patterns have been widely employed in offshore oilfields, as shown in Figure 1. However, formation damage is quite common in near-well regions of injection wells in offshore oilfields due to interaction between injection water and clay minerals [1][2][3][4][5][6][7][8][9][10][11][12]. To characterize the formation damage and estimate reservoir properties, transient pressure data observed from the testing wells are usually interpreted by traditional pressure transient analysis models [13][14][15][16][17][18][19][20][21][22], which are simplified single-phase models divided into two or three regions according to the properties of formation and fluids.
However, the flow behaviors in such wells are much complicated than the situations assumed in the traditional composite model. As shown in Figure 2, saturation gradients are generated from water injection due to the different properties of injected water and reservoir fluids. There are saturation discontinuities between the region where the injected water completely floods and the region where the original fluids predominate. Hence, the results derived by the single-phase flow model will lead to large errors. To solve this problem, numerical well test techniques were developed to simulate the two-phase flow in the water-flooding system. Weinstein [23] presented a numerical model of two-phase flow to analyze the pressure data of falloff tests. However, he ignored the effect of the transitional zone between the zone completely flooded by water and the unflooded zone. After that, Sosa et al. [24] investigated the effect of saturation gradients on the pressure response of the falloff test by considering relative permeability characteristics with the aid of a two-phase numerical model, in which the influence of relative permeability characteristics, injection rate, and injection time on saturation gradients was analyzed. Yeh and Agarwal [25] developed a systematic method to analyze the pressure data from an injection well in a water-flooding system with multiple fluid banks with the assistance of a two-phase twodimensional model and discussed the characteristics of flow regimes of the zone completely flooded by water and the unflooded zone. Later, Warren [26] applied numerical solu-tions in pressure transient analysis and discussed the technical advantages of the numerical model in pressure transient analysis of two-phase flow. Liao and Liu [27] found that the pressure transient behavior of multilayer two-phase flow was similar to that of the single-phase flow in a dualporosity reservoir with a three-dimensional model of twophase flow in PEBI grids. Xiang et al. [28] studied the effect of different sets of oil relative permeabilities on the pressure transient behavior of a production well with a numerical model of two-phase flow in PEBI grids. Xu [29] presented an approach based on a theoretical developed by Thompson [30] to analyze the pressure response in the transition region of two-phase flow with the aid of a commercial simulator. Furthermore, two-phase numerical models applicable to formation damage were proposed. Jian and Xin [31] analyzed the influence of interwell heterogeneity on water flooding effectiveness with a numerical model of two-phase flow in a composite reservoir, which was divided according to permeability and water saturation distribution. Li et al. [32] estimated the formation damage with a numerical model of multiphase flow in a CO 2 flooding system. Ouyang et al. [33] investigated the characteristics of a formation damaged zone with a numerical model of multiphase flow. However, the characteristics of the flow regime of the transitional zone were ignored in the above models of two-phase flow, which might be helpful in history matching.
Although the above models are satisfactory in the pressure transient analysis of single-well systems, the interference effect from adjacent injection wells can obscure or distort the pressure transient behavior of the testing well in multiwell systems, which is particularly pronounced in line drive patterns of offshore oilfields. Therefore, the interference should be considered for pressure transient analysis in multiwell systems, and different models were proposed in the past few decades. Onur et al. [34] investigated a single-phase flow model to analyze the pressure buildup response of a multiwell system containing producers. On this basis, Lin [35] raised a novel method for pressure transient behavior in a multiwell system consisting of both producers and injectors. More and more single-phase flow models for analyzing pressure transient behavior in multiwell systems have been developed [36][37][38]. However, these single-phase flow models are developed with idealized assumptions and not suitable for complicated cases. Hence, some scholars proposed numerical models for pressure transient analysis in multiwell systems. Zhang et al. [39] developed a numerical model of twophase flow in a multiwell system containing producers and investigated the effect of adjacent producers on pressure buildup behavior. Al-Wehaibi et al. [40] presented a numerical model for a multiwell system consisting of producers to match the pressure data obtained from an intelligent field system. Although these numerical models can be used in a multiwell system containing injectors, the effect of saturation gradients on the pressure transient behavior of water injection wells in a multiwell system has not been investigated.
In conclusion, the ignorance of the effect of the transitional zone or the interference of adjacent injectors in a multiwell system will lead to errors in transient pressure analysis. To solve this problem, a numerical model of two-phase flow Injector Producer Figure 1: Schematic of the line drive pattern widely employed in offshore oilfields. The water injection wells are usually arranged in a straight line parallel to the production wells. In the schematic, the residual oil saturation is 0.2; the irreducible water saturation is 0.2. The injection well is at the center of the circle. The dark blue is a zone completely flooded by water. The orange is an unflooded zone. The light blue is the transitional zones between the unflooded zone and the zone completely flooded by water.

Geofluids
is urgently needed to interpret the pressure data obtained from water injection wells in line drive patterns of offshore oilfields instead of oversimplified single-phase models. The objective of this paper is to propose a numerical model of two-phase flow for well test interpretation in a multiwell system consisting of injectors, which takes into account the formation damage in the near-well zone of the testing injector. The model was validated by comparing with a commercial simulator. Additionally, although commercial simulators can simulate two-phase flow in reservoirs, they fail to take into account the influence of wellbore storage on the pressure transient response. Therefore, our model is still urgently needed for pressure transient analysis. After validation, we discussed the influence of the saturation gradients and interference from adjacent injectors on the pressure transient behavior of the testing injector. To demonstrate the practicability of the model, we interpreted the pressure data of a water injection well in a line drive pattern of a Chinese offshore oilfield.

Model Development
In this section, we present a numerical model of two-phase flow for well test interpretation in a multiwell system firstly.
When it comes to the differentiation of the partially differential flow equations, finite difference method [41][42][43][44][45][46][47][48] and finite volume method [49][50][51][52][53][54][55][56] are both commonly employed in solving fluid flow equation systems. However, in unstructured grids, formulating the finite volume method is much easier. Therefore, we adopt finite volume method to discretize the flow equations in a hybrid grid system consisting of both Cartesian and radial meshes. After that, we solve the discrete equations with the fully implicit Newton-Raphson method.

Model Description.
During the falloff test, the rate of the testing injector keeps constant before shut-in. Rates of adjacent injectors remain constant throughout the whole test. In this paper, we only focus on the interference of adjacent injectors in the line drive pattern because the rates of the injectors are commonly much higher than those of the producers in the pattern. Figure 3 is the schematic of a physical model in which the testing well is a water injection well interfered by two adjacent injectors. Formation is divided into two regions. The area of formation damage is the inner zone (green); the remaining part is the outer zone (orange).
Our other basic assumptions are as follows: (1) The reservoir is homogenous in each zone; the initial pressure and water saturation of the reservoir are distributed uniformly. External boundary of the reservoir is sealed (2) Fluid flow is a two-phase flow in which the relative permeability of oil and water phase is taken into account. Capillary pressure is ignored in our model due to its quite limited effect on the pressure transient response in moderate-permeability or high-permeability offshore oilfields with water flooding With the above assumptions, we formulate the governing differential flow equations for the multiwell flow system [56]; the general flow equation for the oil/water phase can be written as where ϕ is the rock porosity, b α is the reciprocal of the oil/water formation volume factor, S α is the oil/water saturation, v α is the velocity of oil/water phase in porous media, and Q α is the well flux of oil/water phase. The fluid flow in the reservoir is assumed to follow the Darcy's law. Given that a two-phase flow issue is involved in this study, relative permeability should be taken into account in the flow equations. Therefore, the velocity can be replaced by the following equation.
where r m is the radius of the inner region in the schematic of Figure 3, K 1 is the permeability of the inner zone (formation  3 Geofluids damage zone), K 2 is the permeability of the outer zone, p α is the pressure of oil/water phase in reservoir, k rα is the relative permeability of oil/water phase, μ α is the viscosity of oil/water phase, ρ α is the density of oil/water phase, g is the gravity acceleration, and z is the depth in gravity direction.
Since the capillary pressure is ignored in our model, the relation of oil-phase pressure and water-phase pressure is Total rates of wells are known quantities in pressure transient analysis. However, oil flow rate and water flow rate in wellbores still need to be calculated.
where Q s is the total well flux of oil and water phase, λ α is the ratio of relative permeability to viscosity for oil/water phase, h is the thickness of formation, r e is the equivalent radius in well flux equation, r w is the wellbore radius, S is the skin factor, p wf is the bottom-hole pressure, C is the wellbore storage constant, and t is the time.

Differentiation and Solution.
We need to craft a grid system before the differential process. To compute the discrete equations more efficiently, we used a hybrid grid system consisting of both Cartesian and radial meshes ( Figure 4). The unstructured refined radial meshes are distributed in the near-well region to improve the computing accuracy. The large Cartesian meshes are located far from the wellbore to reduce the computational time. The basic properties of each grid include cell (c), face (f ), and node. In our discretization process, we employ the discrete operators div and grad to represent divergence and gradient operators, respectively [57]. The div operator maps from face to cell. Subsequently, we will employ boldfaced letters to denote arrays of discrete quantities. For instance, we denote that q is the array of discrete fluxes, then q½ f is the discrete where F ðcÞ is the set of all faces that make up cell c.
The discrete operator grad is a linear mapping from cell to face. If we define that p is the array of discrete pressures, then the pressure gradient restricted to face f is Additionally, we designate that T is the array of discrete transmissibilities across faces; then, T½ f is the transmissibility of face f . Figure 5 illustrates an example of two neighboring cells, which are used for introducing the procedure to calculate transmissibility in the discretized equations. More specifically, the one-sided transmissibility from cell i to cell k in finite volume discretization should be [57,58] Similarly, the one-sided transmissibility from cell k to cell i is where A i,k is the area of the interior face between the two neighboring cells in Figure 5(a), n ! i,k is the normal from the centroid of the interior face to cell k, c ! i,k is the vector from the centroid of cell i to the centroid of the interior face, K i is the permeability tensor of cell i, n ! k,i is the vector from the centroid of cell k to the centroid of the interior face, c  4 Geofluids is the normal from the centroid of the interior face to cell i, and K k is the permeability tensor of cell k.
The transmissibility of the interior face between the two adjacent cells in Figure 5(a) is defined based on the above one-sided transmissibilities.
Therefore, the flux passing through the face f is According to the above discrete operators, the discrete form of equation (1) is where V is the array of volumes of grids, n is the time step, and l is the iterating step of each time step. Again, note that we employ boldfaced letters to define arrays of discrete quantities in discrete equations. v α ½ f in equation (11) can be replaced by the following discrete expression The discrete form of the well equation is The discrete expression of oil/water flow rate in equation (13) is Coupling with equations (11)- (14), we obtain where X is the array of primary variables. Fully implicit Newton-Raphson method is a standard and common method to solve the above nonlinear system. The iterative scheme of equation (15) is To ensure the precision of the Newton iteration, we   Figure 7: The relative permeability curves. 5 Geofluids employ three constraint conditions to judge the convergence of the solution [57]. The first measure is the relative incremental error, which is developed based on the increment of the primary variables for each iteration step.
The second measure is the relative mass-balance error, which follows the mass balance.
where Φ α is the array of quality of oil/water phase in cells. The third measure is the maximum normalized residual, which reads as follows We can compute the bottom-hole pressure at each time step to analyze the pressure transient behavior of testing wells by the above procedure.

Model Validation
The next consideration is the accuracy of our model. We craft an example to verify the numerical solution with a commercial simulator. Figure 6 is the schematic of a 1 000 m × 1 000 m × 20 m grid block reservoir model with hybrid grids (i.e., regular Cartesian and radial grids). The simulator to be compared just uses simple Cartesian grids with the same reservoir size. We set an injector and a producer in the two models, as shown in Figure 6, in which W1 refers to the injection well and W2 is the production well. Well spacing between the two wells is 100 m. The basic reservoir and production parameters used in this validation process are listed in Table 1. In addition, the relative permeability curves are shown in Figure 7. The flow rates of the two wells are both 100 m 3 /d. With the production of 100 d, we compute bottom-hole pressures by the two approaches. Note that our goal in this validation should not focus on the effect of wellbore storage; hence, we ignore this effect in our validation.
As shown in Figure 8, the comparison results show that the bottom-hole pressures computed from the two approaches

Results and Discussion
In this section, we introduce the type curve computed by our model and flow regimes of the pressure transient behavior of a falloff test with interference from two adjacent injectors firstly. Then, our models, model of single-phase flow in a single-well system and model of two-phase flow in a singlewell system, are compared to illustrate the novelty of our model. After that, seven cases are crafted to investigate the effect of model parameters on pressure transient behavior.
More specifically, five parameters influencing the transitional zone (i.e., duration of water injection, injection rate of testing well, initial water saturation, type of relative permeability, and radius of inner region) and two parameters of adjacent injectors (i.e., rate of adjacent injector and well spacing) are analyzed. Figure 9 illustrates the type curve of a falloff test with the interference of two adjacent injectors, which is generated from our two-phase numerical model. The rates of the two adjacent injectors remain constant during the whole procedure of the falloff test. We divide the type curve into seven flow regimes according to its characteristics. After the flow regime of wellbore storage and damage, the duration of first radial flow regime is predominated by the flow of the zone completely flooded by water. Then, the flow regime of the transitional zone depends on the saturation gradients derived from water injection prior to shut-in. The average water saturation of the transitional zone is smaller than that of the zone completely flooded by water, resulting in smaller water relative permeability and larger oil relative permeability than that of the zone completely flooded by water. In this scenario, the average water saturation of the transitional zone is larger than the average oil saturation due to water injection before shut-in. Hence, the water relative permeability has a greater impact on the total mobility of the transitional zone, and the total mobility of the transitional zone is smaller than that of the zone completely flooded by water, leading to the upward moving of pressure derivative curve. After that, the second radial flow regime reflects the flow of the unflooded zone. Subsequently, the transitional flow regime of formation damage is generated from the permeability difference of the inner zone and outer zone. The radial flow regime of the outer zone is covered by the flow regime of interwell interference, in which two adjacent injectors start to affect the pressure response of the testing injector by replenishing the formation energy, leading to the sharp downward movement of the pressure derivative curve.

Type Curve and Model Comparison.
To further illustrate the specialty of our model, three scenarios are crafted for comparison. Note that the relative permeability and viscosity of the single-phase flow in a single-well system are the average values calculated according to the parameters of the two-phase flow in Figure 10(a). As illustrated in Figure 10(a), the radial flow regime of the inner zone of the two-phase flow is divided into two parts by the flow regime of the transitional zone. Additionally, the water saturation in the near-well region of the twophase flow increases significantly due to water injection before shut-in, leading to the increase of water relative permeability and the decrease of oil relative permeability. The water relative permeability has a greater effect on the total mobility of the zone flooded by water due to high water saturation of this zone, leading to a smaller total mobility of two-phase flow than that of single-phase flow in the zone flooded by water. Therefore, the pressure drop of the twophase flow is smaller than that of the single-phase flow before the radial flow regime of the outer zone. Figure 10(b) illustrates that the pressure derivative curve moves downward drastically after the transitional flow The comparisons in Figure 10 further demonstrate that it would lead to errors in well test interpretation if the effect of saturation gradients or the impact caused by adjacent injectors is ignored.

Transitional Zone.
To investigate the characteristics of the flow regime of the transitional zone, the effect of five parameters on the characteristics is analyzed, including the duration of water injection, injection rate of testing well, initial water saturation, type of relative permeability, and radius of inner region. In these cases, a testing injector and two neighboring injectors are located in a 40000 m × 40000 m × 10 m reservoir model with hybrid grids (i.e., regular Cartesian and radial grids); the radius of each wellbore grid is 2.8 m, and well spacings (L 1 , L 2 ) are 500 m. The injection rates of the two adjacent injectors remain constant during the falloff test. Shut-in time for the testing well (t 2 ) is 42 d. The permeability of the inner region (K 1 ) is 0.1 D, the permeability of the outer region (K 2 ) is 0.3 D, and wellbore storage constant is 0.1 m 3 /MPa. Some basic input parameters are listed in Table 1.

Duration of Water Injection.
In this case, the injection rate of the testing well (q w ) before shut-in is 200 m 3 /d; the injection rates of the two adjacent injectors (q 1 , q 2 ) are both 400 m 3 /d. The initial water saturation (S wi ) is 0.5. Relative permeability curves are shown in Figure 7. Radius of inner region (r) is 50.47 m. Figure 11 illustrates that the characteristics of the flow regime of the transitional zone are more significant with the extension of injection time because of the expansion of the transitional zone. In addition, the characteristics of this regime are recognizable when the water injection time (t 1 ) reaches 10 d, indicating that the flow regime of the transitional zone is common in the pressure transient behavior of injectors located in oilfields of moderate permeability.

Injection
Rate of Testing Well. In this scenario, q 1 = q 2 = 400 m 3 /d, r = 50:47 m, S wi = 0:5, and t 1 = 50 d. Relative permeability curves are shown in Figure 7. Figure 12 illustrates that the characteristics of the flow regime of the transitional zone become more obvious with the increase of injection rate because of the expansion of the transitional zone. Additionally, the characteristics of this regime are distinct when the injection rate (q w ) reaches 50 m 3 /d, which illustrates that the flow regime of the transitional zone is common in the pressure transient response of injectors in oilfields of moderate permeability.     Figure 13: Effect of initial water saturation on pressure transient response. S wi is the initial water saturation. 8 Geofluids

Initial Water Saturation.
It is necessary to analyze the initial water saturation because the relative permeability is a function of water saturation. Most of the parameters of this case are the same as those of the scenario in Section 4.2.2. The injection rate (q w ) of the testing well is 200 m 3 /d. Figure 13 shows that the characteristics of the flow regime of the transitional zone are less obvious with the increase of initial water saturation. This is because the increase of water saturation is limited during water injection when the initial water saturation is close to the maximum value of the oilfield, resulting in the shrinking of the transitional zone.

Type of Relative Permeability.
We designate three different types of relative permeability to analyze their influence on the characteristics of the flow regime of the transitional zone, which are shown in Figure 14(b). In this scenario, q w = 200 m 3 /d, q 1 = q 2 = 400 m 3 /d, r = 50:47 m, S wi = 0:5, and t 1 = 50 d. As shown in Figure 14(a), with the three different types of relative permeability, the characteristics of the flow regime of the transitional zone are significant.

Radius of Inner Region.
In this scenario, most of the parameters of this case are the same as those of the scenario in Section 4.2.4. Relative permeability curves are shown in Figure 7. Figure 15 shows that the characteristics of the flow regime of the transitional zone are obvious when the radius of the inner region reaches 50.47 m. The characteristics could be covered by the transitional flow regime generated from formation damage when the radius of the inner region is small. Hence, the characteristics may be ignored in pressure transient analysis. However, the ignorance of the characteristics would lead to erroneous results in well test interpretation.

Adjacent Injectors.
Two scenarios are crafted to investigate the impact of the rate of the adjacent injector and well spacing on the pressure transient behavior of the testing well.
The difference of well spacings in the flood patterns is usually not sufficient to distinguish which well the interference is coming from in pressure transient analysis. Therefore, we just discuss the scenarios that well spacings (L 1 = L 2 ) or rates of adjacent injectors (q 1 = q 2 ) are distributed uniformly. The reservoir models are the same as those of Section 4.2.    Figure 7. Figure 16 shows that the flow regime of interwell interference appears later as the rates of the two adjacent injectors decrease. The transitional flow regime of formation damage is difficult to be covered up because the well spacing is usually much larger than the radius of the formation damage zone in offshore oilfields.

Well Spacing.
Most of the parameters of this scenario are the same as those of the scenario in Section 4.2.5; the rates of adjacent injectors are 400 m 3 /d. As shown in Figure 17, the flow regime of interwell interference appears earlier as well spacings decrease.

Field Application
To obtain accurate results in a well test interpretation, we must choose an appropriate model for history matching according to the geological information and characteristics of flow regimes in pressure and pressure derivative curves. Researchers mainly focused on models for single-phase flow in single-well systems in the past. Now, it has been demonstrated that the characteristics of the flow regimes of the transitional zone and the interference from adjacent injectors cannot be ignored in pressure transient analysis.
This section is to illustrate that our two-phase numerical model for multiwell systems is accurate and practical. Therefore, we interpret the pressure data from a testing injector in a Chinese offshore oilfield with two traditional models and our model.

Basic Information of a Testing Injector in a Chinese
Offshore Oilfield. The basic parameters of a testing injector in a line drive pattern of a Chinese offshore oilfield are shown in Table 2. The relative permeability curves are illustrated in Figure 7. The injection time of the testing well is 1200 h; shutin time is 106.43 h. Rates of two adjacent injectors remain constant during the falloff test. Acidizing treatment was employed to remove near-well plugging every year because the injection pressure of this well increased abnormally.

Well Test
Interpretation with Traditional Models. Two traditional models of a single-well system are used for history   10 Geofluids matching in this section, including a single-phase composite model and a two-phase composite model. Figure 18(a) illustrates the pressure curves of the single-phase composite model can only match with the pressure data during the flow regime of wellbore storage and the transitional flow regime of formation damage, which are unable to match the curves of the flow regime of transitional zone and the flow regime of interwell interference. As shown in Figure 18(b), the results computed by the two-phase composite model match well with the pressure data except for the flow regime of interference.

Well Test Interpretation with Our
Model. For our model in this paper, the matching procedure includes the following three steps.
Step 1. Input the pressure data of the testing injector and basic parameters listed in Section 5.1 into the program that we compiled based on our model.
Step 2. Set the range of unknown parameters. To reduce the matching time, we can limit the range of certain parameters based on field geologic and dynamic information. For instance, the permeabilities of two zones should change from 0 to 0.285 D in this interpretation according to the permeability derived from well log. The permeability of the inner zone (formation damage zone) should be smaller than that of the outer zone due to formation damage in the nearwell zone.
Step 3. Adjust the unknown parameters according to the features discussed in the sensitivity analysis. For instance, the characteristics of the flow regimes of the transitional zone and interference from adjacent injectors are significant and recognizable. Hence, the uncertain parameters can be adjusted according to the features of different flow regimes to match the pressure data from the testing well.
As shown in Figure 19, the pressure curves computed by our model match well with the pressure data of the testing well, which verifies the applicability of our model. The interpretation results are shown in Table 3. The radius of formation damage zone is 40.21 m; the permeability of formation damage zone is 0.035 D.  Bottom-hole pressure of a testing well Two-phase flow in a multi-well system Figure 19: History matching results computed by our model.

Conclusions
To characterize the formation damage in near-well regions of injection wells, we developed a two-phase model to simulate the pressure transient behavior of a water injection well in the multiwell system. A hybrid grid system consisting of both Cartesian and radial meshes was employed in discretizing partially differential flow equations to compute our model more efficiently. The model was validated by comparing with a commercial simulator. To demonstrate the applicability of the model, we interpreted the pressure data derived from a testing injector with formation damage in the near-well region, which is located in the line drive pattern of a Chinese offshore oilfield. The investigation of our model led to the following conclusions: (1) New flow regimes can be observed on the type curve, including the flow regime of the transitional zone and flow regime of interwell interference. The pressure derivative curve moves upward during the flow regime of the transitional zone and moves downward during the flow regime of interwell interference. The pressure drop of our model is smaller than that of the single-phase model before the radial flow regime of the outer zone (2) Duration of water injection, injection rate of testing well, initial water saturation, and radius of the inner region have a great influence on the characteristics of the flow regime of transitional zone established by water injection. Duration of water injection, injection rate of testing well, and initial water saturation can affect the expansion of the transitional zone. The flow regime of the transitional zone will be covered by the flow regime of formation damage when the radius of the inner region is small enough (3) Ignoring the effects of saturation gradients and interference from adjacent injectors on the pressure response of testing injectors located in the line drive pattern will cause errors in well test interpretation Nomenclature ϕ: Rock porosity, fraction α: Oil or water phase o: Oil phase w: Water phase S α : Oil or water saturation, fraction b α : Reciprocal of the oil/water formation volume factor, m 3 /m 3 p α : Oil or water pressure in reservoir, MPa v α : Velocity of oil or water phase in porous media, m 3 /s Q α : Well flux of phase α, m 3 /d K: Absolute permeability of reservoir, D k rα : Relative permeability of phase α, fraction μ α : Viscosity of phase α, mPa·s ρ α : Density of phase α, kg/m 3 g: Gravity acceleration, m/s 2 x, y, z: Coordinates in different directions Q s : Total well flux of oil and water, m 3 /d λ α : Ratio of relative permeability to viscosity for phase α h: Thickness of formation, m r e : Equivalent radius in well flux equation, m r w : Wellbore radius, m S: Skin factor, unitless p wf : Bottom-hole pressure, MPa C: Wellbore storage constant, m 3 /MPa t: Time, s div: Divergence operator grad: Gradient operator q: Array of discrete fluxes, m 3 /d f : Face c: Cell C 1 ð f Þ: Cell including face f C 1 ð f Þ: Cell including face f p: Array of discrete pressures T: Array of discrete transmissibilities across faces A i,k : Area of face, m 2 n Normal from the centroid of interior face to cell i K k : Permeability tensor of cell k T ik : Transmissibility of interior face T k,i : One-sided transmissibility from cell k to cell i T i,k : One-sided transmissibility from cell i to cell k n: Time step l: Iterating step R n+1,l α : Array of discrete form of conservation equation of phase α at the lth iterating step of n + 1 time step R n+1,l q : Array of discrete form of well equation of phase α at the lth iterating step of n + 1 time step V: Array of volumes of grids X n+1,l : Array of primary variables at the lth iterating step of n + 1 time step ΔX n+1,l : Array of increment of primary variables at the l th iterating step of n +1 time step ϕ: Array of rock porosity S α : Array of oil or water saturation b α : Array of inverse formation-volume factor of oil/water Table 3: Interpretation results generated from our model.

Interpreting parameters Values
Wellbore storage constant of injector, m 3 /MPa 0.8 Skin factor of injector, unitless -0.8 Radius of inner zone (formation damage zone), m 40.21 Permeability of inner zone (formation damage zone), D 0.035 Permeability of outer zone, D 0.115