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Diagnosis of Water-Influx Locations of Horizontal Well Subject to Bottom-Water Drive through Well-Testing Analysis.

1. Introduction

Horizontal well technology is well established for enhancing well productivity of low-permeability reservoirs, especially for reservoirs with bottom water or gas cap [1-8] and unconventional oil and gas resources [9]. However, oil production decreases sharply when the water front arrives at the horizontal wellbore so that well performance is much worse than expected [10, 11]. How to effectively evaluate the well performance and reservoir performance becomes significant [12-14], including estimating production distribution and diagnosing water-influx segments [15, 16]. On the one hand, production logging has been carried out to obtain fluid-flow profiles and determine water breakthrough sections for many years [17]. Two examples were presented by Al-Behair et al. [18] to diagnose gas entry intervals by advanced production logging. Al-Muthana et al. [19] elaborated flow profiles by using an integrated compact production logging tool. Later, Bawazir et al. [10] presented three field examples on data logging in dead horizontal wells and interpretation to pinpoint water-influx intervals. Aibazarov et al. [20] described how to determine the downhole fluid flow path and further to optimize water shut-off operations by using integrating data from spectral noise logging tools and multisensor production logging. Over the past few years, many researchers pinpointed water entry intervals through production logging [21-36]. On the other hand, another monitoring system can provide vital data (e.g., pressure and temperature) for water-influx detection. Marquardt [27] developed the Levenberg-Marquardt algorithm for inverting temperature and pressure data into desired inflow rates. Later, several interpretation studies were conducted on how to obtain production profiles by matching measurable data (e.g., pressure, temperature, and density) [28-30]. Bui et al. [31] diagnosed inflow behavior in horizontal wells and identified intervals of fluid influx by examining wellbore pressure sensing at multiple points. Based on the variation of the pressure gradient in the wellbore, water influx location can be identified based on trending analysis according to data delivered by sensors [32]. Yoshioka et al. [33-35] presented synthetic and field examples for interpreting the flow profile of single phase by using the inversion model. Li and Zhu [36] improved this model and developed a transient, 3D, multiphase reservoir model to predict pressure and temperature distribution. However, important data (e.g., pressure, temperature, and flow rate) at multiple points along the wellbore cannot be obtained since this monitoring system such as the production logging test is difficult to implement in some situations. Thus, well testing becomes a preferred choice.

Pressure transient analysis (PTA) has been extensively used for evaluating production performance of horizontal wells [5, 37-45]. Kamal et al. [44] pointed out that even if the horizontal well is entirely open or perforated, some of the segments are nonproducing. He et al. [45] and Qin et al. [46] established a PTA model to estimate the effective producing length and identify locations of underperforming horizontal segments. Detailed description about PTA of MSM can be found in He et al. [45]. Additionally, this phenomenon of nonuniform production distribution also exists in multifractured horizontal wells [47-52]. Some works pointed out that the phenomenon is mainly caused by reservoir heterogeneity, formation damage, and selective completion [52, 53].

However, these models are not applicable for an HW in the reservoir subject to bottom-water drive. To fill this gap, this paper develops a novel approach to detect water entry of a horizontal well in a low-permeability bottom-water drive reservoir considering nonuniform production rate distribution through PTA.

2. Model

2.1. Physical Model. The physical model is shown in Figure 1. Some assumptions are needed to derive the practical solutions of this transient-flow model.

(1) There is an MSHW located in the horizontal-slab reservoir with an upper impermeable stratum and a lower bottom-water boundary. The formation is infinite in the horizontal direction. The horizontal-slab reservoir is considered to be anisotropic and homogeneous, and it has a constant thickness (h), permeability ([k.sub.h] = [k.sub.x] = [k.sub.y] and [k.sub.v] = [k.sub.z] in horizontal and vertical directions), porosity ([phi]), initial reservoir pressure ([p.sub.i]), and total compressibility ([C.sub.t]).

(2) The HW is parallel to the x-axis and divided into multiple segments. The length, production rate, and skin factor of the ith segment can be defined as [L.sub.wi], [q.sub.wi], and [S.sub.wi], and the total length of HW is equal to L. The point ([x.sub.wi], [y.sub.wi], and [z.sub.wi]) is the center of the ith segment.

(3) The fluid is single-phase, and the total production rate is q.

(4) The effects of capillary and gravity are ignored.

2.2. Mathematical Model. The transient-flow equation can be written as

[mathematical expression not reproducible], (1)

where [z.sup.*] is defined to characterize the permeability heterogeneity. [[eta].sub.h] and [[eta].sub.v] represent the diffusivities in horizontal and vertical directions, respectively.

In the horizontal-slab reservoir, the individual horizontal segment can be considered as a line source with length [L.sub.wi]. By use of the instantaneous-source solutions [54, 55] and the Newman-product principle [56], the pressure drop caused by the i'th segment can be expressed as

[mathematical expression not reproducible], (2)

where

[mathematical expression not reproducible], (3)

where [tau] is the time variable and [h.sup.*] means the reservoir thickness considering the effect of anisotropy.

Dimensionless variables are defined by

[mathematical expression not reproducible]. (4)

The analytical solution of pressure drop of an MSHW subject to bottom water drive can be expressed as

[mathematical expression not reproducible], (5)

where

[mathematical expression not reproducible]. (6)

There exists formation damage around the HW during the process of drilling, completion, production, and other operations [52]. The skin factor (S) was introduced by Van Everdingen [57] to quantitatively characterize the effect of formation damage on transient-pressure behaviors. The dimensionless transient-pressure solution considering the effect of skin factor is

[mathematical expression not reproducible], (7)

where [S.sub.wi] means the skin factor of an individual segment. Van Everdingen and Hurst [58] proposed that the wellbore-storage effect can be incorporated by converting the solutions into the Laplace space. The dimensionless transient-pressure solution considering the wellbore-storage effect can be expressed as

[mathematical expression not reproducible], (8)

where u is the Laplace variable. Therefore, the transient-pressure solution in the Laplace space of an MSHW subject to bottom-water drive can be obtained by considering the skin-factor and wellbore-storage effect by use of Laplace transformation.

[mathematical expression not reproducible], (9)

where [[bar.p].sub.wD] is the dimensionless pressure solution in the Laplace space. Finally, the pressure solution in real space can be obtained through a numerical algorithm [59].

3. Pressure-Transient Analysis

First, type curves are developed and flow regimes of MSHW subject to bottom water drive are discussed. Further, we analyze the effects of main parameters on pressure transient behavior (PTB).

3.1. Flow Regimes of WSM. The pressure response of WSM for different lengths of HW is shown in Figure 2. Compared with the models in the reservoir without bottom-water drive, early flow regimes still can be recognized easily (i.e., wellbore-storage flow, transitional flow, and steady flow regime), while the late pressure behaviors may be covered due to the effect of bottom water. Radial flow occurs between transitional flow and steady flow regimes when the HW is long enough (e.g., [L.sub.D] = 0.9, 1.5, 2.1, 2.7). Furthermore, the duration of radial flow becomes shorter as [L.sub.D] decreases. Finally, the radial flow would be covered when [L.sub.D] is short enough (e.g., [L.sub.D] = 03).

3.2. Flow Regimes of MSM. We assume that the HW is divided into three PS with the same interval, and each PS has the same length, since production rate distribution has a distinct effect on PTB and flows regime. Here, we take [q.sub.wiD] = 01:0.3:0.6 for example to analyze flow regimes of MSM, as shown in Figure 3. In detail, this section can be divided into four cases for further discussion according to the length of HW and PS.

Case 1 (long HW (L = 1200 m, h = 100 m, [[beta].sup.2] = 01))

Case 1-1. For short length of PS ([L.sub.wi] = 133 m), type curves are shown in Figure 3(a). The wellbore storage flow is the first regime, identified by a straight line with unit slope. The second period is transitional flow regime, controlled by a skin factor. After that, fluids flow into HW radially from the formation, which is acknowledged as the early radial flow regime. With pressure further spreading, spherical flow occurs when the predominant flow pattern is towards a point, which is recognized as -0.5 slope on the pressure derivative curve. Since the length of HW is long and the length of PS is quite short, intermediate radial flow appears around each PS. When the effect of the bottom water reaches the HW, the reservoir receives energy supplement and pressure tends to be stable gradually, known as the steady flow regime.

Case 1-2. For the long length of PS ([L.sub.wi] = 360 m), type curves are shown in Figure 3(b). Flow regimes still include wellbore storage flow, transitional flow regime, and early radial flow. After that, fluid flows from outer zones and the pressure derivative curve shows a straight line with 0.36 slope. Tiab [60] derived the relationship between ([P'.sub.D] x [t.sub.D]) vs. ([t.sub.D]) and named this flow regime as elliptical flow. Finally, the effect of the bottom water reaches the HW, which leads to a steady flow regime.

Case 2 (short HW (L = 300 m, h = 100 m, [[beta].sup.2] = 01))

Case 2-1. For short length of PS ([L.sub.wi] = 20 m), type curves are shown in Figure 4(a). Different from Case 1, early radial flow disappears since the length of HW is short in Case 2. The pressure response for the short length of PS ([L.sub.wi]) yields four flow regimes: wellbore storage flow, transitional flow, elliptical flow, and steady flow.

Case 2-2. For the long length of PS ([L.sub.wi] = 90), spherical flow with -0.5 slope disappears compared with that when the length of HW is long (e.g., L = 1200 m). As a result, the flow regimes include wellbore storage flow, transitional flow, linear flow, and steady flow regime. Furthermore, the duration of elliptical flow lasts longer compared with Case 2.1, as shown in Figure 4(b).

4. Sensitivity Analysis

The total production rate should be kept the same for different cases. The effect of crucial parameters (e.g., production rate distribution, the length of HW, the length of PS, the number of PS, PS spacing, skin factor, and anisotropy degree) is shown in Figures 5-12.

4.1. Production Rate Distribution

4.1.1. Effect of Production Rate of Constant PS. An HW with a length of 1200 m is divided into four PS with equal length of 300 m. The dimensionless production rate of each PS is defined as [q.sub.wiD]. To discuss the effect of the production rate of a specific PS (i.e., [q.sub.w1D], [q.sub.w2D], [q.sub.w3D], and [q.sub.w4D]) on PTB, the production rate of other PS remain constant and equal to each other (i.e., [q.sub.w2D] = [q.sub.w3D] = [q.sub.w4D], [q.sub.w1D] = [q.sub.w3D] = [q.sub.w4D], [q.sub.w1D] = [q.sub.w2D] = [q.sub.w4D], and [q.sub.w1D] = [q.sub.w2D] = [q.sub.w3D]). Figures 5(a)-5(d) show that [q.sub.w1D] makes a visible influence on type curves. With the increase in [q.sub.w1D], type curves will move up overall. In detail, when the production rate distributes uniformly along the HW (e.g., [q.sub.w1D] = [q.sub.w2D] = [q.sub.w3D] = [q.sub.w4D] = 3/12), the early radial flow starts at [t.sub.D]/[C.sub.D] = 110 and ends at [t.sub.D]/[C.sub.D] = 12,000, which shows the longest period of early radial flow compared with other conditions of production rate distribution. The value of the pressure derivative during this flow period equals to 0.9. Then, with the increase in [q.sub.w1D] (e.g., [q.sub.w1D] = 4/12, [q.sub.w1D] = 6/12, [q.sub.w1D] = 9/12, and [q.sub.w1D] = 12/12), dimensionless pressure and its derivative curves will move up, and vice versa (e.g., [q.sub.w1D] = 2/12, [q.sub.w1D] = 1/12, and [q.sub.w1D] = 0/12). Furthermore, when only one PS produces, as the distance of producing PS to the heel of HW becomes nearer (e.g., [q.sub.wiD] = 0:0:0: 1, [q.sub.wiD] = 0 : 0 : 1: 0, [q.sub.wiD] = 0 : 1 : 0 : 0, and [q.sub.wiD] = 1 : 0 : 0 : 0), transitional flow lasts longer while early radial flow becomes shorter, shown in Figure 5(e). When the well is water-out, the production at water-influx locations will be much higher than that at other locations. Therefore, the water-influx locations cannot be diagnosed if the effect of nonuniform production distribution on pressure response was neglected.

4.1.2. Effect of Production Rate Distribution under Different Lengths of HW. Since the length of HW affects flow regimes of MSHW subject to bottom-water drive distinctly, three cases are selected according to the length of HW to eliminate the possibility that the length of HW may disturb the effect of production rate distribution on PTB.

The HW with length of L (i.e., 2100 m, 900 m, and 300 m) consists of three PS. The dimensionless length of each PS equals to 1/3, and the total dimensionless length of PS equals to 1. We discuss the effect of different production rate distributions ([q.sub.wiD] = 0.1 : 0.3 : 0.6, [q.sub.wiD] = 0.1 : 0.6 : 0.3, [q.sub.wiD] = 0.2 : 0.2 : 0.6, [q.sub.wiD] = 0.2 : 0.6 : 0.2, [q.sub.wiD] = 1/3 : 1/3 : 1/3, [q.sub.wiD] = 0.4 = 0.2 : 0.4, [q.sub.wiD] = 0.4 = 0.4 = 0.2, [q.sub.wiD] = 0.6 : 0.1 0.3, and [q.sub.wiD] = 0.6 : 0.3 : 0.1) on PTB under different lengths of HW, as shown in Figure 6.

In general, the following flow regimes (transitional flow, the early radial flow, and steady flow) are influenced by production rate distribution. It is clearly found that the magnitude of influence caused by production rate distribution on pressure drop are shown as (from high to low) [q.sub.w1D] (dimensionless production rate of heel on HW), [q.sub.w2D] (dimensionless production rate of middle section of HW), and [q.sub.w3D] (dimensionless production rate of toe of HW). Furthermore, these nine kinds of production rate distribution can be divided into four categories: [q.sub.w1D] = [q.sub.w2D] = [q.sub.w3D], [q.sub.w1D] = [q.sub.w2D] [not equal to] [q.sub.w3D], [q.sub.w1D] [not equal to] [q.sub.w2D] = [q.sub.w3D], and [q.sub.w1D] [not equal to] [q.sub.w2D] [not equal to] [q.sub.w3D]. Specifically, when the production rate distributes uniformly (e.g., [q.sub.wiD] = 1/3 : 1/3 : 1/3), the duration of early radial flow is the longest. When the production rate distributes nonuniformly, the value of [q.sub.w2D] influences the characteristics of transitional flow between the early radial flow and steady flow ([q.sub.w1D] is a constant). Type curves move up with the increase in [q.sub.w2D] (e.g., [q.sub.wiD] = 0.1 : 0.3 : 0.6, [q.sub.wiD] = 0.1 : 0.6 : 0.3, and [q.sub.wiD] = 0.2 : 0.6 : 0.2), and vice versa (e.g., [q.sub.wiD] = 0.4 : 0.2 : 0.4, [q.sub.wiD] = 0.6 : 01 : 0.3, and [q.sub.wiD] = 0.6 : 0.3 : 0.1). Especially, if [q.sub.w1D] equals to [q.sub.w2D], transitional flow will disappear (e.g., [q.sub.wiD] = 0.2 : 0.2 : 0.6; [q.sub.wiD] = 0.4 : 0.4 : 0.2). Type curves during the early steady flow regime will intersect with each other when [q.sub.w1D] is equal to [q.sub.w3D] (e.g., [q.sub.wiD] = 0.4 : 0.2: 0.4, [q.sub.wiD] = 1/3: 1/3: 1/3, and [q.sub.wiD] = 0.2 : 0.6 : 0.2).

In addition, by comparing the results from Figures 6(a)-6(c), results show that the length of HW only affects the positions of type curves and duration of each flow regime, and it does not influence the typical characteristics of type curves, as shown in Figure 6(d). In these three cases, flow regimes include wellbore storage flow, transitional flow, early radial flow, and elliptical flow which appear before the steady flow regime.

4.1.3. Effect of Different Production Distributions of PS. In this part, an HW with the length of 1200 m is divided into four PS. Figure 7 indicates that pressure and its derivative curves are upper with a higher production rate of the first PS (e.g., [q.sub.w1D] increases from 0.1 to 0.5) during flow regimes including transitional flow, the early radial flow, and steady flow regimes. As pressure further spreads, type curves will bifurcate when [q.sub.w2D] is unequal during the early radial flow regime (e.g. [q.sub.w1D] = 0.1, [q.sub.w2D] = 0.1, and [q.sub.w2D] = 0.4; [q.sub.w1D] = 0.3, [q.sub.w2D] = 0.2, and [q.sub.w2D] = 0.3 and [q.sub.w1D] = 0.5, [q.sub.w2D] = 0.0, and [q.sub.w2D] = 0.5), based on Figure 7(a). Additionally, type curves are lower when [q.sub.w2D] is smaller than [q.sub.w1D] (e.g., [q.sub.w2D] = 0.0 < 0.5 = [q.sub.w1D]; [q.sub.w2D] = 0.2 < 0.3 = [q.sub.w1D]), and type curves are higher when [q.sub.w2D] is bigger than [q.sub.w1D] (e.g., [q.sub.w2D] = 0.4 > 0.1 = [q.sub.w1D]) since a higher production rate leads to a bigger pressure drop. Apparently, Figures 7(b) and 7(c) demonstrate that the early radial flow is distinct when the production rate distribution pattern behaves as dumbbell-shaped (e.g., [q.sub.wiD] = 0.5 : 0.0 : 0.0 : 0.5; [q.sub.wiD] = 0.4 : 0.1 : 0.1 : 0.4) and spindle-shaped (e.g., [q.sub.wiD] = 0.1 : 0.4 : 0.4 : 0.1). However, due to the strong interference of bottom-water, the value of the horizontal line does not equal to 0.5 during the early radial flow regime. Besides, according to Figure 7(c), pressure derivative curves will intersect at the early steady flow regime when the production rate of the first and last PS are equal (e.g., [q.sub.wiD] = 0.1 : 0.4 : 0.4: 0.1, [q.sub.wiD] = 0.4 : 0.1 : 0.1 : 0.4, and [q.sub.wiD] = 1/4 : 1/4 : 1/4 : 1/4) compared with the ladder shape (e.g., [q.sub.wiD] = 0.1 : 0.2 : 0.3 : 0.4; [q.sub.wiD] = 04 : 0.3 : 0.2 : 0.1). Finally, we discussed the situation when only two segments are producing. Figure 7(d) shows that the early radial flow lasts longer while the linear flow with 0.5 slope appears later as the second PS is far from the heel of HW (e.g., [q.sub.wiD] = 0.1 : 0.9 : 0.0 : 0.0, [q.sub.wiD] = 0.1 : 0.0 : 0.9 : 0.0, and [q.sub.wiD] = 0.1 : 0.0 : 0.0 : 0.9).

4.2. Number of PS. In this section, the number of PS (N) ranges from 2 to 5. For further discussion, this section is divided into two parts.

4.2.1. Constant Total Length of PS. In this part, the total length of PS equals to 0.36 L/N for MSM, while it is equal to L for WSM. Parameters (e.g., length, production rate, and skin factor of each PS) distribute uniformly along HW, and the total production rate remains constant for different cases. As shown in Figure 8(a), obvious differences exist among type curves of WSM and MSM except for the wellbore storage flow and steady flow regime. The effect of the number of PS on type curves for MSM can be observed during early radial flow and transitional flow regime. As the number of PS increases, early radial flow lasts shorter and pressure drop decreases during the transitional flow regime. When the number of PS is big enough, the early radial flow will disappear and may be masked by transitional flow before the steady flow regime. Furthermore, the pressure derivative during the early radial flow regime remains constant for a different number of PS (e.g., N =1, 2, 3, 4).

4.2.2. Variable Total Length of PS (Constant Length of Each PS). In this part, the length of each PS equals to 0.12 L for MSM so that the total length of PS is variable for MSM. The length of PS for WSM still equals to L. The total skin factor and total production rates are kept constant for different cases. As shown in Figure 8(b), it is obvious that there exist obvious differences during early radial flow and transitional flow. Particularly, unlike the results from case one (total length of PS is constant), the duration of early radial flow does not change as the number of PS increases, while early radial flow and transitional flow regime appear earlier with the increase in the number of PS. Furthermore, since the total length of PS is variable, WSM is a specific situation of MSM when the number of PS is close to eight. In this case, transition flow will be covered by the steady flow.

4.3. Length of PS. The length of PS distribution can be classified into two categories: uniformly and nonuniformly. First, we discuss the effect of uniformly distributed length of PS (UDLPS) on type curves in this section. An HW is divided into three PS with equal length (e.g., [L.sub.wiD] = 0.05, 0.10, 0.20, 0.25, 0.30), as shown in Figure 9(a). It is clearly observed that only wellbore storage flow and steady flow regimes are not influenced by the length of PS. Particularly, when [L.sub.wiD] equals to 0.05, spherical flow appears since the predominant flow pattern in the reservoir is toward a point. Then, as [L.sub.wiD] increases gradually, spherical flow disappears while elliptical flow with 0.36 slope appears following early radial flow.

For further discussion, we investigate the effect of nonuniformly distributed length of PS (NDLPS) (e.g., [L.sub.wiD] = 0.24 : 0.04 : 0.02, 0.14:0.02:0.14, 0.10:0.10:0.10, 0.05 :0.20 :0.05, and 0.02 :0.04:0.24) on pressure drop. As shown in Figure 9(b), obvious distinctions can be observed during flow regimes including early radial flow and transitional flow regimes which are different from those of UDLPS. Furthermore, slopes of the transitional flow regime on pressure derivative curves are equal in this situation.

4.4. PS Spacing. A horizontal well with the length of 1000 m consists of two PS with the length of 200 m, one of which is located at the heel of HW. The dimensionless spacing between PS is defined as [DELTA][x.sub.D] = ([x.sub.w2] - [x.sub.w1])/L. Different dimensionless PS spacings ([DELTA][x.sub.D] = 0.20, 0.35, 0.50, 0.65, and 0.80) are taken into account for analyzing their effects on type curves, shown in Figure 10. In the following cases, other parameters (e.g., total production rate, the number of PS, and skin factor of each PS) are constant. There exist distinct differences during the early radial flow, transitional flow, and early steady flow regime when PS spacing varies from 0.20 to 0.80. As [DELTA][x.sub.D] increases, early radial flow lasts longer and pressure drop decreases during the transitional flow regime. In detail, when [t.sub.D]/[C.sub.D] equals to 10,000, the value of the pressure derivative changes from 0.35 to 0.9 as [DELTA][x.sub.D] changes from 0.20 to 0.80.

4.5. Skin Factor. A horizontal well with the length of 1000 m consists of four segments with the same length of PS ([L.sub.wiD] = 0.1) and PS spacing ([DELTA][x.sub.D]). For further discussion, this section is divided into two cases.

4.5.1. Variable Total Skin Factor of PS (Equal Skin Factor of Each PS). In this situation, the skin factor of each PS is equal so that the total skin factor of PS is different for different situations. With the decrease in total skin factor ([S.sub.i] = 8 : 8 : 8 : 8, [S.sub.i] = 4 : 4 : 4 : 4, [S.sub.i] = 2 : 2 : 2 : 2, [S.sub.i] = 1 : 1 : 1 : 1, and [S.sub.i] = 0.5 : 0.5 : 0.5 : 0.5), type curves during the transitional flow regime move down as shown in Figure 11(a).

4.5.2. Constant Total Skin Factor of PS. In this case, the total skin factor is constant. If the length and production rate of each PS are the same, the effect of the nonuniform distribution of the skin factor on PTB is unapparent. This is because total pressure drop caused by skin effect depends on the length, production rate, and skin factor of each PS. When the length and production rate of each PS are different, slight distinctions can be observed on type curves, shown in Figure 11(b).

4.6. Anisotropy Degree. The HW with the length of 800 m consists of two segments located at the heel and toe of HW, respectively, as shown in Figure 12. Other parameters of each PS are kept the same (e.g., production rate, length, and skin factor). The anisotropy degree (AD) can be characterized using the ratio of vertical permeability to horizontal permeability (AD = [k.sub.v]/[k.sub.h]). When [k.sub.h] remains at 1000 mD, AD can be changed by altering the value of [k.sub.v]. The bottom water may rise faster when AD becomes bigger, and the well can receive energy supply earlier so that the late steady flow regime will occur early. Additionally, the early-radial flow regime will disappear as the AD increases, and the typical characteristic of the MSHW model will be covered when [k.sub.v] is higher than 500 mD, which is adverse for diagnosis. The strong influence of the bottom water will increase the multiplicity of well-testing interpretation, so that this model is available when [k.sub.v] is lower than 500 mD.

5. Limitations and Future Work

The strong influence caused by bottom water will increase the multiplicity of well-testing interpretation, so that this model is available when [k.sub.v] is lower than 500 mD. Since only single-phase fluid is incorporated in the model, the proposed model is mainly used to solve the interpretation of those wells whose water cut are close to 100%.

Therefore, the multiphase model of MSHW may be established to better distinguish the water and oil production profile based on field test data. Furthermore, novel methods need to be developed to reduce the nonuniqueness of pressure interpretation with strong bottom water drive.

6. Conclusions

A novel approach was presented for determining nonuniform production distribution as well as detecting the location of water entry. First, HW is divided into multiple segments with arbitrary parameters (e.g., production, length, location, and skin factor). Then, new pressure transient solutions for HW are derived in the anisotropic reservoir with bottom-water drive. Clear distinctions can be observed between type curves of MSM and WSM. As a result, neglecting nonuniform production distribution along the wellbore could lead to erroneous results for detecting the location of water entry. The early radial flow regime of MSM appears later than WSM. Spherical flow and elliptical flow regimes appear on type curves of MSM which do not exist on that of WSM.

Sensitivity analysis shows that anisotropy degree and production rate distribution play important roles in pressure response, followed by length, number, spacing, and skin factor of PS. When the total skin factor remains constant, the effect of the nonuniform distribution of skin factor on pressure response can be ignored. Additionally, the late stable flow appears earlier with increasing vertical permeability, which covers up the early radial flow and is bad for parameter estimation. Besides, water controlling for water-out segments and stimulation treatments for nonproducing segments can be carried out to improve oil recovery based on interpretation results.
Nomenclature

C: Wellbore storage coefficient, [atm.sup.-1]

[C.sub.D]: Dimensionless wellbore storage coefficient

[C.sub.t]: Total compressibility, [atm.sup.-1]

h: Formation thickness, cm

[h.sup.*]: Formation thickness considering
permeability anisotropy, cm

[h.sub.D.sup.*]: Dimensionless formation thickness
considering permeability anisotropy

[k.sub.h]: Horizontal permeability, D

[k.sub.v]: Vertical permeability, D

L: Total length of HW, cm

[L.sub.D]: Dimensionless length of HW

[L.sub.Di]: Ratio of the ith segment length to formation
thickness considering anisotropy

[L.sub.wi]: Length of the ith segment, cm

[L.sub.wiD]: Ratio of the ith segment length to total
length of HW

N: Number of PS, dimensionless

p: Pressure, atm

[p.sub.i]: Initial reservoir pressure, atm

[p.sub.D]: Dimensionless pressure drop

[p.sub.SD]: Dimensionless transient pressure solution
considering the effect of skin factor

[p.sub.wD]: Dimensionless transient pressure solution
considering wellbore-storage effect

q: Total production rate, [cm.sup.3]/s

[q.sub.wi]: Production rate of the ith segment, [cm.sup.3]/s

[q.sub.wiD]: Dimensionless production rate of the ith
segment, [cm.sup.3]/s

[r.sub.w]: Wellbore radius, cm

[r.sub.wD]: Dimensionless wellbore radius

S: Skin factor

[S.sub.wi]: Skin factor of the ith segment

t: Time, s

[t.sub.D]: Dimensionless time

u: Laplace transform variable

x, y, z: Cartesian coordinates

[x.sub.D], [y.sub.D], [z.sub.D]: Dimensionless Cartesian coordinate

[x.sub.wi], [y.sub.wi], [z.sub.wi]: Coordinates of the center of the
ith segment

[x.sub.wiD], [y.sub.wiD], [z.sub.wiD]: Dimensionless coordinates of
the center of the ith segment

[z.sup.*]: z-Coordinate considering permeability
anisotropy, cm

[phi]: Porosity, fraction

[[eta].sub.h]: Diffusivity in horizontal direction, [cm.sup.3]/s

[[eta].sub.v]: Diffusivity in vertical direction, [cm.sup.3]/s

[mu]: Fluid viscosity, cP

[beta]: Anisotropy coefficient

[tau]: Time variable

[DELTA]p: Pressure drop caused by the ith segment

[[bar.p].sub.wD]: Dimensionless pressure solution in
Laplace space.


https://doi.org/10.1155/2018/6385252

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The authors would like to thank the National Natural Science Foundation of China (No. U1762101) and National Science and Technology Major Projects (No. 2017ZX05009-003).

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Jiazheng Qin (iD), Shiqing Cheng (iD), Youwei He, Dingyi Li, Le Luo, Xinzhe Shen, and Haiyang Yu (iD)

State Key Laboratory of Petroleum Resources and Prospecting, China University of Petroleum, Beijing 102249, China

Correspondence should be addressed to Shiqing Cheng; chengsq@cup.edu.cn

Received 16 October 2017; Revised 4 January 2018; Accepted 10 September 2018; Published 25 November 2018

Guest Editor: Zhien Zhang

Caption: Figure 1: Physical model of MSHW subject to bottom water drive.

Caption: Figure 2: Type curves of WSM of HW subject to bottom water drive.

Caption: Figure 3: Type curves of MSM when the length of HW is long: (a) with short length of PS and (b) with long length of PS.

Caption: Figure 4: Type curves of MSM when the length of HW is short: (a) with short length of PS and (b) with long length of PS.

Caption: Figure 5: Effect of production rate of specific PS on PTB: (a) [q.sub.w1D] [not equal to] [q.sub.w2D] = [q.sub.w3D] = [q.sub.wiD], (b) [q.sub.w2D] [not equal to] [q.sub.w1D] = [q.sub.w3D] = [q.sub.w1D], (c) [q.sub.w3D] [not equal to] [q.sub.w1D] = [q.sub.w2D] = [q.sub.w4D], (d) [q.sub.w4D] [not equal to] [q.sub.w1D] = [q.sub.w2D] = [q.sub.w3D], and (e) comparison between the above four cases.

Caption: Figure 6: Effect of production rate distribution on PTB under different lengths of HW: (a) L = 2100 m, (b) L = 900 m, (c) L = 300 m, and (d) comparison between above three cases.

Caption: Figure 7: Effect of distribution pattern with four PS: (a) effect of [q.sub.w2D], (b) dumbbell-shaped/spindle-shaped distribution, (c) ladder shape distribution, and (d) with different location of the second PS.

Caption: Figure 8: Effect of number of PS on PTB when the total length of PS is (a) constant and (b) variable.

Caption: Figure 9: Effect of length of PS on PTB: (a) UDLPS and (b) NDLPS.

Caption: Figure 10: Effect of PS spacing on PTB.

Caption: Figure 11: Effect of skin factor on PTB: (a) with variable total skin factor of PS and (b) with constant total skin factor of PS.

Caption: Figure 12: Effect of AD on PTB.
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Title Annotation:Research Article
Author:Qin, Jiazheng; Cheng, Shiqing; He, Youwei; Li, Dingyi; Luo, Le; Shen, Xinzhe; Yu, Haiyang
Publication:Geofluids
Geographic Code:1U7TX
Date:Jan 1, 2018
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