# Productivity-Index Behavior for a Horizontal Well Intercepted by Multiple Finite-Conductivity Fractures Considering Nonlinear Flow Mechanisms under Steady-State Condition

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## Abstract

**:**

## 1. Introduction

## 2. Model Assumption

#### 2.1. Physical Model

- The reservoir is horizontal and of uniform thickness, with impermeable lower and upper boundaries. The pressure on the outer boundary in the x–y plane keeps constant.
- Along the horizontal wellbore, multiple hydraulic fractures are evenly distributed. Hydraulic fractures are considered to be a finite conductivity.
- The wellbore is produced under constant-rate condition.
- Flow in the reservoir is assumed to be single-phase fluid (i.e., pure gas) with slight compressibility and constant viscosity.
- Fluid flowing in the fracture is assumed to be nonlinear.
- Fluids from the reservoir enter the wellbore only through hydraulic fractures, and the pressure loss in the wellbore is neglected.

#### 2.2. Variables Definitions

_{m}is the pressure in the reservoir, Pa; p

_{f}is the pressure in the fracture, Pa; p

_{i}is the pressure on the elliptical boundary, Pa; p

_{ex}is the pressure on the width of inner SRV region, Pa; p

_{ey}is the pressure on the length of inner SRV region, Pa; (x

_{ofm}, y

_{ofm}) is the location of m-th fracture tip, m; x

_{e}is the width of inner SRV region, m; y

_{e}is the length of inner SRV region, m; x

_{R}is the width of the whole drainage region, m

^{3}; h is the reservoir thickness, m; w

_{f}is the fracture width, m; L

_{f}is the fracture length, m; r

_{w}is the radius of wellbore, m; L

_{ref}is the reference length, m; q

_{fm}is the function of inflow distribution along the m-th fracture, m

^{2}/s; q

_{wm}is the production rate, m

^{3}/s; q

_{c}is the influx rate, m

^{3}/s; μ is the viscosity, Pa s; B is the volume factor; γ

_{f}is the permeability modulus, Pa

^{−1}; and β is the Beta factor, m

^{−1}. Here, the subscript D represents dimensionless variables.

## 3. Mathematical Model

#### 3.1. Fluid Flow in the Inner SRV Region

_{f}fractures in the SRV region, the dimensionless length of m-th fracture is L

_{fDm}, and the dimensionless coordinate of m-th fracture tip is (x

_{ofDm}, y

_{ofDm}). The pressure on the outer boundary parallel to the fracture is p

_{eDx}, and the pressure on the outer boundary perpendicular to the fracture is p

_{eDy}.

_{f}fracture is given by

#### 3.2. Fluid Flow in the Outer Region

_{eD}.

_{eDx}and p

_{eDy}(Appendix B), which is written as follows:

_{f}fractures, which is the function with regard to influx distribution along the fracture.

#### 3.3. Fluid Flow in the Fracture

_{Dm}with respect to x

_{Dm}based on Equation (15) twice, the result is given by

#### 3.3.1. Model of a Conductivity Fracture with Non-Darcy Flow

#### 3.3.2. Model of a Conductivity Fracture with Pressure Sensitivity Effect

## 4. Semi-Analytical Solution for Coupled Model

#### 4.1. Dimension Transformation

_{D}is a function of the variable x

_{D}and depends on the distribution of conductivity C

_{fD}(x

_{D}).

#### 4.2. Discretization

_{Di}, as shown in Figure 6a. Note that the equal-length fracture segment would be transformed into an unequal-length segment after using dimension transformation, as shown in Figure 6b.

#### 4.3. Computation Consideration

_{fDn}is the function of pressure p

_{fDn}in the fracture with pressure-dependent conductivity, and the dimensionless conductivity of C

_{fDn}is the function of flowing rate q

_{cDn}in the fracture under non-Darcy flow. In a mathematical context, C

_{fDn}is a function with regard to the spatial variable. At a given k-th step, if the distribution of pressure p

_{fDn}or flowing rate q

_{cDn}was known, C

_{fDn}along the fracture would be obtained. Thus, the nonlinear governing equation for the (k + 1)-th step could be linearized on the assumption of known conductivity distribution on the k-th step.

_{fDn}and pressure distribution p

_{fDn}along the fracture at the k-th time step would be achieved based on coupled solution. Next, the calculated p

_{fDn}was used to update the fracture conductivity (Case 1); the calculated q

_{cDn}was used to update the fracture conductivity (Case 2). The iterative procedure is repeated until the wellbore-pressure was converged. The calculation procedure is given as follows:

- Step 1: Initial calculation, with k = 0, the fracture is assumed to be uniform (Case 1 and Case 2). By combing through Equation (29) to Equation (32), we can obtain (q
_{fDn})^{k}. The fracture pressure (p_{fDn})^{k}(Case 1) and flowing rate (q_{fDn})^{k}(Case 2) would be then achieved from Equation (28). - Step 2: Calculating the pressure-sensitive conductivity C
_{fDn}[(p_{fDn})^{k}] (Case 1) and C_{fDn}[(q_{cDn})^{k}] (Case 2) along fracture, and then transforming x_{Dn}into ξ_{Dn}based on Equation (26). - Step 3: Solving Equation (33) with the updated C
_{fDn}[(p_{fDn})^{k}] to achieve (p_{fDn})^{k}^{+1}and (p_{wD})^{k}^{+1}in Case 1; solving Equation (34) with the updated C_{fDn}[(q_{cDn})^{k}] to achieve (q_{cDn})^{k+1}and (p_{wD})^{k}^{+1}in Case 2. - Step 4: Terminate the iterative procedure if |(p
_{wD})^{k}^{+1}− (p_{wD})^{k}|/(p_{wD})^{k}< ε; otherwise, update pressure distribution along fracture by setting (p_{fDn})k = (p_{fDn})^{k}^{+1}(Case 1) and flow distribution along fracture by setting (q_{cDn})^{k}= (q_{cDn})^{k}^{+1}(Case 2), and k = k + 1 back to step 2 until the convergence is achieved.

## 5. Results and Sensitivity Analysis

_{A}is the shape factor. Specially, r

_{we}is the effective wellbore radius determined by the geometry of drainage area, well configuration, and nonlinear flow mechanism.

#### 5.1. Influence of Fracture Properties on PI

_{fD}), fracture number (N

_{f}), penetration ratio of fracture length to inner SRV width (I

_{x}= L

_{f}/x

_{e}), penetration ratio of fractured horizontal wellbore to inner SRV length (I

_{y}= D

_{f}/y

_{e}) (D

_{f}is defined as the distance between two outmost fractures), and the drainage ratio of inner drainage to the whole drainage (I

_{e}= x

_{e}/x

_{R}).

_{fD}= 0.1, 0.5, 1, 5, 10, 50, 100, 500, and 1000). As shown in Figure 7a, at a given penetration ratio (I

_{x}), PI is increased with the increase in dimensionless conductivity. However, after a certain dimensionless conductivity, the increase could be miniscule. In other words, beyond a certain dimensionless conductivity, the PI increase with conductivity essentially equals the PI decrease caused by the inner interference within the fracture. Generally, the threshold value is regarded as C

_{fD,threshold}= 300. When the C

_{fD}is larger than 300, there is no significant pressure drop within the fracture; the PI reaches the maximum, and does not increase with the conductivity. Figure 7b shows the PI derivative with regard to ln(C

_{fD}) in the semi-log plot. For each given I

_{x}, the conductivity, corresponding to the maximum PI derivative, is defined as certain conductivity. Taking I

_{x}= 1, for example (J

_{Dmax}= 7.25), when the derivative is at a maximum, and that the certain value is C

_{fD,certain}= 7.5, the corresponding value of PI is J

_{D}= 4.54. This indicates that the ratio of J

_{D}/J

_{Dmax}equals 62.6. This means that PI of MFHW at C

_{fD}= 7.5 could reach 62.6% of the maximum, but PI only increases by 37.4% when the C

_{fD}is further increased from 7.5 to 300. Therefore, the optimum dimensionless conductivity is defined to as the certain value. The optimum conductivity indicates that the inflow from the reservoir could match the outflow of the fracture. As analyzed from Figure 7b, the larger the I

_{x}, the larger the optimum C

_{fD}.

_{f}= 2 to N

_{f}= 3. In comparison, incremental PI would decline to 0.39% when the fracture number is further increased from N

_{f}= 9 to N

_{f}= 10. Overall, the effect of the increased connected area could offset the effect of the increase of fracture interference, which results from the increased fracture number. Besides, the optimum conductivity is decreased with the increase in fracture number, as shown in Figure 8b.

_{e}. Here the value that I

_{e}= 1 means that the area of inner SRV equals to the whole drainage area. Compared with the PI that I

_{e}= 1, the more approaching unity the value is, the larger the PI is, which is caused by the decrease of the proportion of inner SRV. Moreover, the tendency is more noticeable in the condition of high conductivity. When C

_{fD}= 0.1, the PI at I

_{e}= 0.5 is that J

_{[email protected]}

_{=0.5}= 0.45, while the PI at I

_{e}= 1 is that J

_{[email protected]}

_{=1}= 0.65. Hence, the ratio of I

_{e}= 1 to I

_{e}= 0.5 in PI value is 1.44. As a comparison, when C

_{fD}= 1000, the PI at I

_{e}= 0.5 is that J

_{[email protected]}

_{=0.5}= 0.72, while the PI at I

_{e}= 1 is that J

_{[email protected]}

_{=1}= 1.95. The ratio in PI is 2.7.

#### 5.2. Influence of Non-Darcy Flow on PI

_{fD}= 1. In this figure, the conductivity is set to be relatively low. It is shown that the influx density, which is defined as the influx rate per length along fracture face, is concentrated around the wellbore and fracture tips. It is noted that the concentration region of the influx density is controlled by a larger pressure drop/depletion. The variable of (q

_{DND})

_{f}, defined in Equation (2), is the Forchheimer number for a given inertial factor β. When the (q

_{DND})

_{f}is increased, the influx density is increasingly more concentrated towards wellbore. Put another way, an extra pressure drop is required to offset the effect of non-Darcy flow caused by the inertial force. Figure 11 presents the effect of fracture conductivity on influx-density distribution along the fracture. Without the non-Darcy effect (Forchheimer number equals zero) shown in Figure 11a, the increase in conductivity makes the distribution of fluid influx more gentle. When C

_{fD}> 300 (i.e., infinite conductivity), which indicates that the pressure on any point in fracture panel is same as the pressure on the wellbore, the inflow flux is concentrated on the fracture tips. With the non-Darcy effect (Figure 11b), when C

_{fD}> 1, more volume of influx fluid is concentrated on the nearby region of wellbore. The phenomenon is consistent with the characteristics given in Figure 11a. As the conductivity increases, the effect of non-Darcy flow on the distribution of inflow flux becomes weak, until the distribution of inflow flux is independent of the non-Darcy flow effect.

_{DND})

_{f}= 0. PI increases with the increase of C

_{fD}, until it reaches the maximum value (J

_{Dmax}) at C

_{fD}= 300. However, the increasing rate of PI is gradually slowed down with C

_{fD}; likewise, the maximum PI is independent of (q

_{DND})

_{f}. In other words, non-Darcy flow plays a negative role in determining PI for the range of low- and intermediate-conductivity (C

_{fD}< 300), but has no effect on the maximum PI, which corresponds to the infinite conductivity. The relationship between PI and the Forchheimer number is further shown in Figure 12b. It is shown that, although PI declined with the Forchheimer number, the decreasing rate slows down.

_{f}= 2 and (q

_{DND})

_{f}= 0, PI is that J

_{D}= 0.96; when N

_{f}= 2 and (q

_{DND})

_{f}= 10, PI is that J

_{D}= 0.75. The relative loss in PI is up to 22%, which is defined as 1 − J

_{[email protected](qDND)f}

_{=10}/J

_{[email protected](qDND)f}

_{=0}. In comparison, the relative loss in PI is only 9%. Figure 13b shows the penetration ratio of fracture length with respect to the inner SRV width (I

_{x}). The effect of Forchheimer number is weaker when the I

_{x}is relatively small, but its effect would be amplified when the I

_{x}is relatively small. Figure 13c shows the penetration ratio of the inner SRV region with regard to whole drainage (I

_{e}). In the small range (I

_{e}< 0.9), the relationship between I

_{e}and PI exhibits an approximate linear behavior. When I

_{e}> 0.9, PI is increased rapidly with the I

_{e}.

#### 5.3. Influence of the Pressure-Sensitivity Effect on PI

_{fD}. As shown in Figure 15a, the effect of pressure sensitivity on PI becomes weak with the increase of the conductivity. When the magnitude of the conductivity reaches the level of infinite conductivity, the effect of pressure sensitivity is almost neglected. As expected, as presented in Figure 15b, PI declines until the minimum with the increasing γ

_{fD}, and corresponding γ

_{fD}is defined as the threshold. Meanwhile, the decreasing rate slows down. For example, if C

_{fD}= 1, PI reaches the minimum J

_{Dmin}when γ

_{fDthreshold}= 1.2; if C

_{fD}= 10, PI reaches the minimum J

_{Dmin}when γ

_{fDthreshold}= 2.7. Thus, a larger conductivity contributes to a larger γ

_{fDthreshold}.

## 6. Conclusions

- PI increases with the increase of C
_{fD}, until it reaches the maximum (J_{Dmax}) at C_{fD}= 300. - PI is deteriorated under the influence of nonlinear flow mechanisms. With the consideration of non-Darcy flow, for the small range of the penetration ratio of the inner SRV region with regard to whole drainage (I
_{e}< 0.9), the relationship between I_{e}and PI exhibits an approximately linear behavior. When I_{e}> 0.9, PI is increased rapidly with the I_{e}. - With the consideration of pressure sensitivity, the apparent fracture is degraded from the initial conductivity to the minimal conductivity, which is caused by fracture closure. As a result, an extra pressure drop is acquired to offset the conductivity degradation, and PI would be declined to the minimal PI.
- The disadvantage dimensions of fracture (such as small conductivity, penetration ratio, and less fractures) contribute to severe pressure depletion, while, in turn, the severe pressure depletion will strengthen the effect of the nonlinear flow mechanism on PI behavior.
- If the conductivity in the fracture reaches the level of infinite conductivity, the influence of nonlinear flow mechanism on PI could be neglected.

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Analytical Solution for the Inner SRV Region

## Appendix B. Analytical Solution for the Outer Region

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**Figure 1.**The flow regime characteristics of MHFW, where (

**a**) sequence of flow-regimes for a MFHW, completed in a tight sand gas reservoir modified from Clarkson [4]. Lines represent isopotential lines; arrows represent streamlines. Flow-regime 1 corresponds to linear flow; flow-regime 2 corresponds to elliptical flow; flow-regime 3 corresponds to fracture interference; flow-regime 4 corresponds to compound linear, and flow-regime 5 corresponds to compound elliptical flow. (

**b**) simulation of flow regimes for a stimulated MFHW in a tight sand gas reservoir. (simulated through numerical simulator).

**Figure 2.**Schematic of the compound elliptical-flow model representing four contiguous regions for a multiple-fractured horizontal well (MFHW). SRV, stimulated rock volume.

**Figure 7.**Productivity-index behavior of multiple fractures, including (

**a**) the productivity index vs. dimensionless conductivity, and (

**b**) the derivative of productivity index vs. dimensionless conductivity.

**Figure 8.**Influence of number of fractures on PI behavior, including (

**a**) the productivity index vs. dimensionless conductivity, and (

**b**) the derivative of productivity index vs. dimensionless conductivity.

**Figure 9.**Influence of penetration ratio of inner SRV region with regard to the whole region on productivity-index (PI) behavior.

**Figure 11.**Influx-flow distribution along the fracture under the influence of conductivity (

**a**) without and (

**b**) with non-Darcy flow effect.

**Figure 12.**(

**a**) Influence of fracture conductivity on PI behavior under the effect of non-Darcy flow, (

**b**) influence of Forchheimer number on PI behavior with the consideration of finite conductivity.

**Figure 13.**Influence of dimensions of fracture on PI behavior in the consideration of non-Darcy flow, including (

**a**) the influence of the number of fractures, (

**b**) the influence of penetration ratio of fracture, and (

**c**) the influence of penetration ratio of horizontal well.

**Figure 14.**Influence of dimensionless conductivity on influx-flow distribution (

**a**) with and (

**b**) without the pressure-sensitivity effect.

**Figure 15.**(

**a**) Influence of fracture conductivity on PI behavior with the pressure-sensitivity effect, and (

**b**) influence of permeability modulus on PI behavior with consideration of finite conductivity.

**Figure 16.**Influence of dimensions of the fracture on PI behavior in the consideration of pressure sensitivity, including (

**a**) number of fracture, (

**b**) penetration index of fracture, and (

**c**) penetration of horizontal wellbore.

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**MDPI and ACS Style**

Cao, M.; Xiao, H.; Wang, C. Productivity-Index Behavior for a Horizontal Well Intercepted by Multiple Finite-Conductivity Fractures Considering Nonlinear Flow Mechanisms under Steady-State Condition. *Energies* **2020**, *13*, 2015.
https://doi.org/10.3390/en13082015

**AMA Style**

Cao M, Xiao H, Wang C. Productivity-Index Behavior for a Horizontal Well Intercepted by Multiple Finite-Conductivity Fractures Considering Nonlinear Flow Mechanisms under Steady-State Condition. *Energies*. 2020; 13(8):2015.
https://doi.org/10.3390/en13082015

**Chicago/Turabian Style**

Cao, Maojun, Hong Xiao, and Caizhi Wang. 2020. "Productivity-Index Behavior for a Horizontal Well Intercepted by Multiple Finite-Conductivity Fractures Considering Nonlinear Flow Mechanisms under Steady-State Condition" *Energies* 13, no. 8: 2015.
https://doi.org/10.3390/en13082015