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. 2024 Mar 8;9(11):12789–12800. doi: 10.1021/acsomega.3c08594

Proppant Settlement and Long-Term Conductivity Calculation in Complex Fractures

Xiaopeng Wang †,‡, Xiaocheng Zhang †,‡, Ming Zhang †,‡, Qilong Zhang †,‡, Pinghua Dong ‡, Haibo Ding ‡, Xiaoqiang Liu §,∥,*
PMCID: PMC10955708  PMID: 38524481

Abstract

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The current research on fracture conductivity ignores the placement of the proppant in fractures and relies on single-fracture conductivity testing and calculation, which cannot represent the overall conductivity of complex fracture systems. This research proposes a calculation method for the long-term conductivity of complex fractures based on proppant placement. This method considers fracture morphology, proppant placement, proppant embedment, and deformation under high closing pressure. The research results show that fracture conductivity decreases with increasing time, which can be divided into three stages: the embedding stage, the creep stage, and the stabilization stage. The long-term conductivity of the main fracture is higher than that of the branching fracture. With increasing closing pressure, the conductivities of both the main fracture and the branching fracture decrease. This is because increasing closure stress accelerates proppant embedment and creep, compressing the fluid flow space and further reducing fracture conductivity. Fracture conductivity is related to the placement of the proppant and sand concentration. Increasing the sand ratio can significantly increase the placement of the proppant in the main fracture and branching fractures, thereby improving fracture conductivity. Increasing the fracturing fluid viscosity can increase its proppant migration capacity. The proppant does not easily settle prematurely in high-viscosity fracturing fluid and can enter more into branching fractures, thereby improving their conductivity.

1. Introduction

Hydraulic fracturing is widely used for exploiting low-permeability oil and gas reservoirs.1−3 A proppant is injected to support the fracture and prevent the fracture from closing under in situ stress. The placement of a proppant in a hydraulic fracture will directly affect the fracture conductivity and thus affect the stimulation effect of hydraulic fracturing.4−6

Experimental testing is the most common method for studying fracture conductivity.7−9 The experimental approach primarily involves conducting a single-fracture conductivity test between two parallel rock plates under closing pressure conditions, based on Darcy’s law. The effect of proppant type, particle size, sand concentration, closing pressure, and time on fracture conductivity can be analyzed through experiments. However, an experimental test takes a long time, and the obtained test results can help characterize the fracture conductivity only under specific conditions.

Based on the experimental test, a theoretical calculation model of fracture conductivity is further established. Yan et al.10 established a mathematical model of fluid flow in propping fractures and analyzed the influence of the proppant diameter and shape on fracture conductivity. Bolintineanu et al.11 created network fractures by connecting randomly placed points with splines and investigated the fracture conductivity under various closure stresses. Cooke12 analyzed the variations in fracture conductivity with different fracture widths. Li et al.13 proposed a method to correct for lateral pressure errors in determining the supported fracture conductivity, improving the precision of calculations. Shen et al.14 conducted a theoretical study on the effects of closing pressure, proppant particle, and concentration on fracture conductivity.

At present, the study on fracture conductivity ignores the placement of a proppant in complex fractures and directly uses two parallel rock slabs to test single-fracture conductivity or proposes a theoretical calculation based on a parallel rock slab test. However, the placement of proppants varies with the complex fracture morphology, leading to varying levels of fracture conductivities at different hierarchies. Therefore, the conductivity of a complex fracture system cannot be simply represented by the results of a parallel slab experimental test and model calculations. In addition, the high closing pressure in geological formation also has an impact on the proppant, causing the proppant to embed, break, or creep. The deformation of the proppant will affect the fracture width and permeability, thus influencing fracture conductivity.

The Computational fluid dynamics—discrete element method (CFD-DEM) coupling method is used to simulate proppant migration in fractures, and the simulation results can intuitively reflect the placement of proppants in fractures. Tomac and Gutierrez15 used the CFD-DEM coupling method to simulate proppant transport in fractures and found that the size ratio between the proppant diameter and fracture width significantly affects proppant migration. Wilson et al.16 simulated the migration of proppant carried by slippery water and explored the influence of the proppant density, particle size, and pumping rate on proppant placement. Blyton17 used the CFD-DEM method to establish a 3D hydraulic fracturing model to conduct numerical simulations of the proppant in planar vertical fractures. The effects of the proppant particle size, proppant concentration, and fracture width on proppant placement were studied. Zeng et al.18 used the CFD-DEM coupling model to study the interaction forces between proppant particles during proppant migration and found that these forces could affect the formation process of sand dikes. The above studies prove that the CFD-DEM method can reveal proppant placement in complex fractures.

In this paper, a calculation method for the long-term conductivity of complex fractures based on the CFD-DEM proppant placement is proposed. The method takes into account the fracture morphology, proppant placement, proppant embedment, and deformation under high closing pressure. By extracting the information on proppant placement in complex fractures with CFD-DEM, the conductivity of proppant placement at different fracture locations can be calculated according to the amount of proppant. This method fundamentally addresses the limitation of the current research that neglects the placement of proppants in fractures and relies on single-fracture conductivity testing and calculations using parallel rock plates, which cannot represent the overall conductivity of complex fracture systems.

2. Calculation Method

The calculation flow of long-term fracture conductivity for fractured porous media based on proppant placement morphology is shown in Figure 1: (a) The proppant placement morphology in complex fractures is simulated based on CFD-DEM, and the proppant position is derived according to simulation results. (b) The fracture surface is meshed to account for the number of proppants and proppant layers. (c) The fracture width is calculated considering the effect of proppant deformation caused by embedment, creep, and breakage on fracture width. (d) The fracture permeability is calculated according to the porosity, tortuosity, and pore diameter. (e) The conductivity of each grid is calculated based on the fracture permeability and fracture width, and complex fracture conductivity is further calculated based on harmonic averaging.

Figure 1.

Figure 1

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Calculation flow of long-term fracture conductivity.

2.1. Derivation of Proppant Information from CFD-DEM

Based on the CFD-DEM method in the Euler–Lagrange model, the interaction between proppants and the fracturing fluid is considered to establish the proppant coupling model in the fracturing fluid flow. The results of proppant migration and placement in fractures under different conditions of proppant parameters, fracturing fluid parameters, fracturing pumping process, rough or smooth fracture wall, with or without branch fractures, and branch fracture parameters can be obtained, and the position coordinates of proppant particles at each time node is derived.

2.2. Number of Proppant Layers and Meshing

Assume that the proppant is distributed in a diamond shape in the fracture, as shown in Figure 2 (from left to right is single-layer, double-layer, and multilayer). In order to characterize the arrangement of proppant particles, each of the 3 layers of proppant particles is used as a unit to calculate the embedment and deformation of the proppant.

Figure 2.

Figure 2

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Diamond-shaped displacement diagram of a proppant.

The total number of proppant particles in a fracture and the number of proppants embedded in the fracture wall can be calculated as follows:19,20

2.2. 1

in which Nij is the total number of proppant particles (dimensionless), nij is the number of proppant layers (dimensionless), HF is the fracture height (m), Dp is the proppant particle diameter (mm), CA is a coefficient related to the number of proppant layers (dimensionless) (if nij = 3m – 2, CA = 2(m – 1); if nij = 3m – 1, CA = 2(m – 1); if nij = 3m, CA = 2m – 1), L is the fracture length (m), and ()int stands for integer.

2.2. 2

in which Nij is the number of proppants embedded in the fracture wall (dimensionless), and CB is a coefficient related to the number of proppant layers (dimensionless) (if nij = 3m – 2, CB = 0; if nij = 3m – 1, CB = 1; if nij = 3m, CB = 1).

Orthogonal meshes are divided on the fracture surface, and each lattice size is dx × dy (as shown in Figure 3). According to the proppant position coordinates derived at different times, the number of proppant particles falling into each grid is counted. The number of proppant layers is estimated based on the number of proppant particles in each grid, as shown in eq 3. Based on this, the number of proppant layers in different fracture locations at different times can be obtained:

2.2. 3

in which i represents the ith grid in the X direction, j represents the jth grid in the Y direction, dx is the grid length in the X direction (m), dy is the grid length in the Y direction (m), nij(t) is the number of proppant layers in row i and column j grid at time t, Nij(t) is the number of proppant particles in row i and column j grid at time t, and Dp is the proppant particle diameter (m).

Figure 3.

Figure 3

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Diagram of the proppant layer and meshing.

2.3. Dynamic Fracture Width

The fracture width changes due to deformation and embedment between proppant particles and between proppant particles and fracture walls under closing pressure.

The embedment of proppant particles can be expressed as follows:21

2.3. 4

The deformation of proppant particles can be written as follows:

2.3. 5

in which t is time (s), Lin(t) is the proppant embedment at time t (m), Ld(t) is the proppant deformation at time t (m), Ck is the proppant particle spacing coefficient (dimensionless), σf is the closing pressure (MPa), Ep and Ef are elastic models of the proppant particle and the fracture, respectively (MPa), vp and vf are Poisson’s ratio of proppant particles and the fracture, respectively (dimensionless), Df is the thickness of the fracture wall (m), and Sp and Sf are the viscosity coefficients of proppant particles and the fracture, respectively (dimensionless).

Dynamic fractures considering proppant deformation and embedment are expressed as follows:

2.3. 6

in which wini is the initial fracture width (m), and w(t) is the dynamic fracture width at time t (m).

2.4. Dynamic Permeability

According to the Carman–Kozeny model, fracture permeability is a function of the porosity, aperture diameter, and tortuosity (Formula 7), which should be dynamically corrected due to proppant embedment and deformation:22

2.4. 7

in which k is permeability (μm2), ϕ is the porosity (dimensionless), r is the aperture diameter (m), and τ is the tortuosity (dimensionless).

The porosity considering proppant deformation is expressed as follows:

2.4. 8

in which ϕij(t) is the porosity of the grid in row i and column j at time t (dimensionless), and D is the fracture width when the closing pressure is 0 and the fracture is just closed and pressurized onto the proppant (m).

The tortuosity is calculated based on the following equation:

2.4. 9

in which τij(t) is the tortuosity of the grid in row i and column j at time t (dimensionless).

The aperture diameter equation is expressed as follows

2.4. 10

in which rij(t) is the aperture diameter at the grid of row i and column j at time t (m), and Nin is the number of proppant particles embedded in fracture at the grid of row i and column j.

By incorporating formulas 8 and 10 into formula 7, the dynamic permeability considering proppant embedding and deformation can be obtained as follows:

2.4. 11

2.5. Long-Term Fracture Conductivity

The fracture conductivity is defined as the product of the fracture permeability and width. The long-term fracture conductivity can be obtained by combining formulas 6 and 11:

2.5. 12

in which FCDij(t) is the fracture conductivity at the grid of row i and column j at time t (μm2 cm), and c0 is the correction factor (dimensionless).

Based on the calculated fracture conductivity at each grid, linear interpolation is carried out to obtain the conductivity at any position of the fracture, and the distribution map of fracture conductivity is drawn. By harmonically averaging the conductivity of each grid, we calculated the conductivity of the whole complex fracture:

2.5. 13

3. Verification

In this section, the conductivity experiment is performed using a custom-made conductivity experimental apparatus, and the experimental result is compared with the calculated result under the same conditions to verify the accuracy of the calculation method proposed in this paper.

The apparatus used in this experiment is a fracture conductivity system designed according to API standards that can evaluate the conductivity of different types of proppants under different conditions. Three commonly used proppants (quartz sand, ceramite, and laminate sand) were tested for 48 h at different closing pressures using shale slabs. The experimental conditions are as follows: proppant particle size, 40/70 mesh; sand placement concentration, 3 kg/m2; fracturing fluid is slick water with a viscosity of 1 mPa s; and the injection rate is 1.47 mL/min. The test period is 48 h. The calculation method adopts the same parameters as the experiment, and a comparison between the calculated results and the experimental results is shown in Figure 4.

Figure 4.

Figure 4

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Comparison between the calculated conductivity and the tested conductivity.

Both the calculated and tested results show that fracture conductivity decreases gradually with increasing test time, and it decreases quickly in the early stage and slowly in the later stage. This is because, in the early stage, the proppant will quickly embed under high closing pressure, resulting in a rapid decrease in fracture conductivity. In the later stage, the proppant will slowly creep, causing a slow decrease in the fracture conductivity.

Under three closing pressures of 30, 50, and 70 MPa, quartz sand has the lowest conductivity among the three proppants. This is because the compressive strength of quartz sand is the lowest, and quartz sand will break under high closing pressure, resulting in a significant reduction in fracture conductivity. When the closing pressure is less than 50 MPa, ceramite has the highest fracture conductivity. When the closing pressure is greater than 50 MPa, coated sand has the highest fracture conductivity. As can be seen in Figure 4, the calculated results of long-term fracture conductivity are in good agreement with the experimental results, which proves the accuracy of the calculation method proposed in this paper.

4. Result Analysis and Discussion

In this section, a basic model with one main fracture and three branching fractures is established, as shown in Figure 5. The fracturing fluid carrying the proppant is injected from the inlet of the main fracture and flows out from the outlet of the main fracture and branching fractures. The initial boundary conditions for both the inlet and outlet are open boundary conditions. The proppant migrates and settles in fractures along with the fracturing fluid. Based on the placement of the proppant in fractures under different conditions, the fracture conductivity is further calculated. The basic parameters used in this model are presented in Table 1, and a detailed result analysis is presented in the following sections.

Figure 5.

Figure 5

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Basic model.

Table 1. Basic Parameters.

type parameter value
proppant density 2500 kg m–3
size 40/70 mesh (average diameter of 0.3185 mm)
sand ratio 10, 15, and 20%
fracturing fluid viscosity 10 and 20 mPa s
injection rate 0.3 m s–1
fracture length of major fracture 50 cm
length of branching fracture 15, 20, and 25 cm
height of major fracture 6 cm
height of branching fracture 6 cm
width of major fracture 5 mm
width of branching fracture 3 mm
roughness 2.5
angle of branching fracture 30, 45, and 90°
closing pressure 20, 30, and 40 MPa
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4.1. Sand Ratio

The effect of the sand ratio (10, 15, and 20%) on proppant migration and fracture conductivity is analyzed. The fracturing fluid is 10 mPa s, and other parameters are given in Table 1.

Figure 6 shows the proppant placement in fractures with different sand ratios. As the sand ratio increases, the amount of proppant settled in fractures in the same volume of fracturing fluid increases. The interaction between proppants accelerates the settlement of the proppant near the inlet to form a sandbank. As the proppant continues to enter the fracture, the height of the sandbank increases rapidly until it reaches the equilibrium flow rate under the experimental conditions.

Figure 6.

Figure 6

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Proppant placement in fractures with different sand ratios.

Figure 7 shows the fracture conductivity with sand ratios of 10, 15, and 20% after 1000 days. It is clear that the fracture conductivity is related to the placement of the proppant and sand concentration. An increasing sand ratio can significantly increase the placement of the proppant in the main fracture and branching fractures, thereby improving fracture conductivity. Especially when the sand ratio is higher than 15%, the conductivity of branching fractures with angles of 30 and 45° is greatly improved. However, when the sand ratio is 10%, the effective conductivity supported by the proppant in branching fractures is very small, and the branch fracture cannot provide effective conductivity, which can be regarded as an invalid fracture.

Figure 7.

Figure 7

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Fracture conductivity with different sand ratios (time: 1000th day).

Figure 8 shows the long-term fracture conductivity with different sand ratios. The fracture conductivity decreases with an increase in time, which can be divided into three stages (embedding stage, creep stage, and stabilization stage). The decrease in fracture conductivity is low in the first 10 days. During this period, the decrease in fracture conductivity is mainly caused by the proppant embedding into the fracture wall, which is the embedding stage. From almost 20 to 50 days, the rate of decline in fracture conductivity increases. During this period, proppant embedment is basically finished, and fracture conductivity decreases mainly due to proppant creep caused by closing pressure, which is the creep stage. After 50 days, the proppant does not deform after creep to a certain extent under closure stress. During this period, the fracture conductivity basically remains stable, which is the stabilization stage.

Figure 8.

Figure 8

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Long-term fracture conductivity with different sand ratios.

In addition, the long-term conductivity of the main fracture is higher than that of the branching fracture. Moreover, with increasing closing pressure, the conductivity of both the main fracture and the branching fracture decreases obviously. This is because increasing closure stress accelerates proppant embedment and creep, compressing the fluid flow space and further reducing fracture conductivity.

4.2. Fracturing Fluid Viscosity

On the basis of 15% sand ratio, proppant migration, placement, and fracture conductivity with fracturing fluid viscosities of 10 and 20 mPa s are analyzed.

Figure 9 shows that increasing the fracturing fluid viscosity (from 10 to 20 mPa s) can increase its proppant migration capacity. The proppant migrates farther with a fracturing fluid viscosity of 20 mPa s than that with 10 mPa s, but the fracture height decreases. Especially in branching fractures with an angle of 90°, the proppant is not easy to settle prematurely in high-viscosity fracturing fluids and can enter more into branching fracture, thereby improving the conductivity of the branching fracture.

Figure 9.

Figure 9

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Proppant placement (left) and fracture conductivity (right) with different fracturing fluid viscosities.

At low closure stress, increasing the fracturing fluid viscosity can enhance the initial conductivity of the main fracture and the branching fracture with an angle of 90°. However, it has little effect on the conductivity of the branching fracture with angles of 30 and 45°. With an increase in the closing pressure, the influence of the fracturing fluid viscosity on the fracture conductivity gradually weakens. During the stabilization stage, the effect of the fracturing fluid viscosity on the main fracture conductivity is offset by deformation processes of proppant embedding and creep. The main fracture conductivity at 10 and 20 mPa s tends to be similar to that at high closure stress (Figure 10).

Figure 10.

Figure 10

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Long-term fracture conductivity with different fracturing fluid viscosities.

4.3. Angle of the Branching Fracture

In this section, the ability of the proppant to enter branching fractures with different angles is analyzed. The placement of a proppant and the conductivity in the main fracture and the branching fractures with angles of 30, 45, and 60° are simulated when the sand ratio is 15%.

The angle of the branching fracture mainly affects the ability of the proppant to enter it. The proppant is more likely to enter a branching fracture with an angle of 45° than those with 30 and 60° angles under the same conditions (Figure 11). However, the effect of the angle on fracture conductivity is not obvious. The main fracture conductivity with an angle of 45° is a bit smaller than those at 30° and 60°. This is because more proppant enters the branching fracture with an angle of 45°, causing an increase in the branching fracture conductivity and a decrease in the main fracture conductivity. However, the conductivity difference in the main fracture and the branching fracture is quite small, especially under high closing pressure conditions (Figure 12).

Figure 11.

Figure 11

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Proppant placement (left) and fracture conductivity (right) with different angles of the branching fracture.

Figure 12.

Figure 12

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Long-term fracture conductivity with different angles of the branching fracture.

5. Conclusions

A calculation method for the long-term conductivity of complex fractures based on CFD-DEM proppant placement is proposed. Some conclusions obtained are as follows:

  • (1)

    Fracture conductivity decreases with increasing time, which can be divided into three stages: embedding stage, creep stage, and stabilization stage. During the embedding stage, a decrease in the fracture conductivity is mainly caused by the proppant embedding into the fracture wall. From almost 10 to 50 days, the rate of decline in fracture conductivity increases. During the creep stage, proppant embedment is basically completed, and fracture conductivity decreases mainly due to proppant creep caused by closing pressure. During the stabilization stage, the proppant does not deform after creep to a certain extent under closure stress, and the fracture conductivity basically remains stable.

  • (2)

    The long-term conductivity of the main fracture is higher than that of the branching fracture. With increasing closing pressure, the conductivities of both the main fracture and the branching fracture decrease. This is because increasing closure stress accelerates proppant embedment and creep, compressing the fluid flow space and further reducing the fracture conductivity.

  • (3)

    Fracture conductivity is related to the placement of the proppant and the sand concentration. Increasing the sand ratio can significantly increase the placement of the proppant in the main fracture and branching fractures, thereby improving the fracture conductivity. Increasing the fracturing fluid viscosity can increase the proppant migration capacity carried by the fracturing fluid. The proppant does not easily settle prematurely in a high-viscosity fracturing fluid and can enter more into branching fractures, thereby improving the conductivity of the branching fracture.

Acknowledgments

This study was sponsored by the National Natural Science Foundation of China (Grand No. 52204024), the CNPC Innovation Fund (Grant No. 2021DQ02-1006), and the State and Key Laboratory of Offshore Oil Exploitation.

Glossary

Nomenclature

CA

coefficient related to the number of proppant layers (dimensionless)

c0

correction factor (dimensionless)

D

fracture width when the fracture is just closed and pressurized onto the proppant (m)

Df

thickness of the fracture wall (m)

dy

grid length in the Y direction (m)

Ep

elastic model of the proppant particle (MPa)

HF

fracture height (m)

j

jth grid in the Y direction

Lin(t)

proppant embedment at time t (m)

Nij

total number of proppant particles (dimensionless)

Nin

number of proppant particles embedded in the fracture at the grid of row i and column j

nij(t)

number of proppant layers in row i and column j grid at time t

Sp

viscosity coefficients of proppant particles (dimensionless)

Sf

viscosity coefficients of a fracture (dimensionless)

vp

Poisson’s ratio of proppant particles (dimensionless)

w(t)

dynamic fracture width at time t (m)

ϕ

porosity (dimensionless)

τ

tortuosity (dimensionless)

σf

closing pressure (MPa)

CB

coefficient related to the number of proppant layers (dimensionless)

Ck

proppant particle spacing coefficient (dimensionless)

Dp

proppant particle diameter (m)

dx

grid length in the X direction (m)

Ef

elastic model of a fracture (MPa)

FCDij(t)

fracture conductivity at the grid of row i and column j at time t (μm2 cm)

i

ith grid in the X direction

k

permeability (μm2)

Ld(t)

proppant deformation at time t (m)

nij

number of proppant layers (dimensionless)

Nij(t)

number of proppant particles in row i and column j grid at time t

r

aperture diameter (m)

rij(t)

aperture diameter at the grid of row i and column j at time t (m)

t

time (s)

vf

Poisson’s ratio of fracture (dimensionless)

wini

initial fracture width (m)

ϕij(t)

porosity of the grid in row i and column j at time t (dimensionless)

τij(t)

tortuosity of the grid in row i and column j at time t (dimensionless)

The authors declare no competing financial interest.

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