A mathematical model and semi-analytical solution for transient pressure of vertical fracture with varying conductivity in three crossflow rectangular layers

Abstract

This paper first describes a mathematical model of a vertical fracture with constant conductivity in three crossflow rectangular layers. Then, three forms of vertical fracture (linear, logarithmic, and exponential variations) with varying conductivity are introduced to this mathematical model. A novel mathematical model and its semi-analytical solution of a vertical fracture with varying conductivity intercepting a three-separate-layered crossflow reservoir is developed and executed. Results show that the transient pressures are divided into three stages: the linear-flow phase, the medium unsteady-flow stage, and the later pseudo-steady-flow phase. The parameters of the fracture, reservoir, and the multi-permeability medium directly influence the direction, transition, and shape of the transient pressure. Meanwhile, the fracture conductivity is higher near the well bottom and is smaller at the tip of the fracture for the varying conductivity. Therefore, there are many more differences between varying conductivity and constant conductivity. Varying conductivity can correctly reflect the flow characteristics of a vertical fractured well during well-test analysis.

Keywords

Transient pressure vertical fracture varying conductivity layered reservoir semi-analytical solution

Introduction

For low-permeability reservoirs or damaged formations, hydraulic fracturing is an effective technique to enhance productivity, with a great impact on the performance of a fractured well (Cinco-Ley and Samaniego, 1981; Luo and Tang, 2014, 2015; Ozkan et al., 2011; Yu et al., 2016).

Hydraulic fracturing stimulation usually creates vertical fractures unless it is in shallow layers (Gringarten et al., 1974; Hubbert and Willis, 1955, 1957). There are many multilayered formations due to geologic sedimentary dynamics (Jiao and Zheng, 1998; Patella, 1978; Shu, 2011). During hydraulic fracturing, multilayered formations are fractured and open them (Adachi et al., 2007; Chu et al., 2007; Valko and Economides, 1995). Thus, to establish a mathematical model of the characteristics of a hydraulic fracturing multilayered reservoir has important theoretical directive functions to help optimize fracturing design, analyze fractured-well performance, and provide realistic explanations (Wang et al., 2012, 2013).

Many mathematical models for a non-fractured well in crossflow/no-crossflow multilayered formations have been studied. The methods and equations for calculating transient pressure with wellbore storage and skin effects of a non-fractured well in crossflow/no-crossflow multilayered reservoir were presented (Bourdet, 1985; Kucuk et al., 1986; Lefkovits et al., 1961; Liu and Wang, 1999; Tariq and Ramey, 1978). The transient pressure of a multilayered reservoir and the difference of interlayer communication were performed and analyzed by many researchers (Aly, 1994; Aly and Lee, 1995; Bello et al., 2016; Eisa et al., 2008; Frantz et al., 1992; Gao, 1986; Jatmiko et al., 1996; Jordan and Mattar, 2000; Lu et al., 2000; Nikjoo and Hashemi, 2012; Ryan et al., 1994; Sun et al., 2003; Wang and Wang, 2014a, 2014b; Valdes-Perez et al., 2018; Yuan et al., 2016; Zhang et al., 2001).

The above models for calculating transient pressure mainly focused on a non-fractured well in a multilayered reservoir. Little research on hydraulic fracturing was conducted on multilayered reservoirs. Now that hydraulic fracturing stimulation has been widely and successfully used for economic production in oilfields (Barros-Galvis et al., 2018; Ezulike et al., 2015; Gao and Li, 2015), it is necessary to establish a mathematical model of a hydraulic fracturing well combined with multilayered reservoirs.

When hydraulic fracturing is conducted in a multilayered formation, the vertical hydraulic fracture can be divided into the three categories of uniform-flux, infinite-conductivity, and finite-conductivity fractures based on the dimensionless fracture conductivity (C_FD). C_FD can be defined as the ratio of the ability of the fracture to carry fluids into the well to that of the formation to carry fluids into the fracture (Gringarten et al., 1974, 1975; Liu et al., 2016; Prats et al., 1962).

The conductivity of fracture proppants and the uniform-flux/infinite-conductivity-mathematical models and solutions for a vertical fracture in multiple layers were evaluated and developed (Cooke, 1973; Gringarten et al., 1974; Ozkan and Raghavan, 1991). The mathematical models for calculating transient pressure for a finite-conductivity fracture were introduced (Bennett et al., 1986; Cinco-Ley and Samaniego, 1981; Gringarten 1974; Hagoort, 2009; Kuchuk et al., 1991; Liu and Wang, 1993; Wang et al., 2005) but they only focused on the infinite-conductivity fracture without considering varying conductivity.

The transient pressure and the flow regimes in detail for a finite-conductivity fracture in a Laplace domain were studied (Cinco-Ley and Samaniego, 1981; Cinco-Ley and Meng, 1981). Over the past 30 years, thousands of papers have focused on the transient pressure analysis for a finite-conductivity fracture (Barros-Galvis et al., 2018; Chen et al., 2017; Cinco-Ley and Samaniego, 1981; Ozkan et al., 2011; Tian et al., 2017; Yuan et al., 2015a, 2015b).

In view of the above research, scholars have often aimed at uniform-flux/infinite-conductivity or finite-conductivity vertical fractures, failing to consider fracture conductivity as a varying parameter at different positions along a fracture. (The pressure drop in a fracture was not considered or was kept constant without considering varying conductivity at different positions along a fracture.) In fact, the conductivity varies at different positions along a fracture, depending on fracture characteristics, such as fracture length or width, penetration, and proppant type, and it also depends on possible damage to the formation surrounding the fracture (Mcguire and Sikora, 1960; Prats et al., 1962; Raghavan et al., 1978; Tinsley et al., 1969). In general, the fracture conductivity is greater near the wellbore.

Luo and Tang (2015) established a semi-analytical solution of a vertical fractured well with varying conductivity. However, they only focused on the transient-pressure behaviors of a varying-conductivity fracture in a one-layered formation and did nothing about the varying fracture conductivity in multi-layered formations. There is still no literature that analyzes or solves a mathematical model of a vertical fracture with varying conductivity in a multi-layered crossflow reservoir.

In this paper, a mathematical model of a vertical fracture with constant conductivity in three rectangular crossflow layers is established. Then, three forms of varying conductivity are introduced (linear, logarithmic, and exponential), a mathematical model of a vertical fracture with varying conductivity intercepting a three-separate-layered crossflow reservoir is developed, and a semi-analytical solution for calculating transient pressure using a Laplace transform and Stehfest numerical inversion is determined. The direct influences of the characteristic parameters of a fracture and three-separate-layered reservoir on the transient pressure are analyzed. Finally, the curves for the above three forms of varying conductivity in three separate crossflow layers are analyzed.

Mathematical model and semi-analytical solution

Mathematical model

Model descriptions

Figure 1 is a schematic of a fractured well with a vertical fracture in the center of a three-layered rectangular reservoir. The basic assumptions of fluid flow in the reservoir and fracture follow.

Figure 1.

Schematic of a fractured vertical well in a three-layered rectangular reservoir.

The three layers are all isotropic, homogeneous, and horizontal in a sealed rectangular drainage area. The three layers respectively have constant thickness h₁, h₂, and h₃; permeability k₁, k₂, and k₃; and porosity $φ_{1}$ , $φ_{2}$ , and $φ_{3}$ . These parameters do not change with time. The total net thickness of pay zones is h (h = h₁+h₂+h₃). Each layer has length 2x_e and width 2y_e.

The fracture is very small compared with the reservoir. The fluid is produced through a vertically fractured well intersected by three fully-penetrating layers. The fracture half-length is x_f, the width is w_f, the fracture permeability is k_f, and the fracture porosity is $φ_{f}$ , and these are also constant. The fracture height is equal to the thickness of the reservoir. The two wings of each fracture are of equal length.

There exists a crossflow interlayer between two layers. λ is the transfer coefficient between the two layers, and ω is the storage coefficient of each layer.

The initial pressure is p₀. The drainage area contains a slightly compressible single-phase fluid of constant compressibility c_t and constant viscosity μ. No flow is allowed through the fracture tips, and fluid flow is linear in the fracture. The flow is assumed to obey Darcy’s law in the reservoir and fracture. The total production of a vertically fractured well in multiple layers has a constant flow rate of Q.

The conductivity varies at different positions along a fracture. Varying fracture conductivity is introduced to describe this phenomenon.

Dimensionless definitions

For the sake of simplicity, the dimensionless parameters are defined as follows

x_{D} = \frac{x}{x_{f}}, x_{e D} = \frac{x_{e}}{x_{f}}, y_{D} = \frac{y}{x_{f}}, y_{e D} = \frac{y_{e}}{x_{f}}, w_{D} = \frac{w}{x_{f}}, C_{F D} = \frac{k_{f} w}{k x_{f}}

p_{j D} = \frac{(k_{1} h_{1} + k_{2} h_{2} + k_{3} h_{3})}{1.842 \times 10^{- 3} Q μ B} (p_{i} - p_{j}), j = 1, 2, 3

t_{D} = \frac{3.6 \times (k_{1} h_{1} + k_{2} h_{2} + k_{3} h_{3}) t}{[{(φ c_{t} h)}_{1} + {(φ c_{t} h)}_{2} + {(φ c_{t} h)}_{3}] μ x_{f}^{2}}

ω_{j} = \frac{{(φ c_{t} h)}_{j}}{[{(φ c_{t} h)}_{1} + {(φ c_{t} h)}_{2} + {(φ c_{t} h)}_{3}]}, j = 1, 2, 3, ω_{1} + ω_{2} + ω_{3} = 1

λ_{j} = α_{j} \frac{k_{1} h_{1}}{(k_{1} h_{1} + k_{2} h_{2} + k_{3} h_{3})} x_{f}^{2}, j = 1, 2

κ_{j} = \frac{{(k h)}_{j}}{(k_{1} h_{1} + k_{2} h_{2} + k_{3} h_{3})}, j = 1, 2, 3; κ_{1} + κ_{2} + κ_{3} = 1

q_{j D} = \frac{q_{j}}{Q}, j = 1, 2, 3; q_{1 D} + q_{2 D} + q_{3 D} = 1

where x is the distance in the x-direction, m; y is the distance in the y-direction, m; w_f is the fracture width, m; x_f is half the fracture length, m; x_e is the length of the drainage volume, m; y_e is the width of the drainage volume, m; φ_j is the porosity of the jth zone, decimal; C_FD is the dimensionless fracture conductivity; μ is fluid viscosity, mPa·s; ω_j is the storability ratio, fraction; λ_j is the transfer coefficient between the two layers (λ₁ stands for the transfer coefficient between layer-1and layer-2; λ₂ stands for the transfer coefficient between layers-1and layer-3), fraction;

κ_{j}

is the formation capacity ratio, fraction; α is the shape factor between the above two layers,1/m²; q_j is the flow rate of the jth zone, m³/d; Q is the surface production of the well, m³/d; c_tj is the total compressibility of the jth zone, 1/MPa; h_j is the total height of the zones, m; h_j is the pay height of the jth zone, m; k_j is the permeability of the jth zone, um²; p_j is the pressure of the jth zone, MPa; B is a volumetric factor, fraction; t is the time, hour; p₀ is initial pressure, MPa; p_w is the wellbore pressure, MPa; and p_f is the fracture pressure, MPa. Special subscripts: D is dimensionless; f is the fracture property; w is the wellbore property; and j is the jth zone.

Flow model in reservoir

A single-phase fluid flows from the three-layered sealed rectangular reservoir to a vertical fracture with constant conductivity at a constant flow rate Q, as presented by equations (1)–(8) (Liu et al., 2016; Wang et al., 2005)

κ_{1} (\frac{\partial^{2} p_{1 D}}{\partial x_{D}^{2}} + \frac{\partial^{2} p_{1 D}}{\partial y_{D}^{2}}) + λ_{1} (p_{2 D} - p_{1 D}) + λ_{2} (p_{3 D} - p_{1 D}) = ω_{1} \frac{\partial p_{1 D}}{\partial t_{D f}}

(1)

κ_{2} (\frac{\partial^{2} p_{2 D}}{\partial x_{D}^{2}} + \frac{\partial^{2} p_{2 D}}{\partial y_{D}^{2}}) - λ_{1} (p_{2 D} - p_{1 D}) = ω_{2} \frac{\partial p_{2 D}}{\partial t_{D f}}

(2)

κ_{3} (\frac{\partial^{2} p_{3 D}}{\partial x_{D}^{2}} + \frac{\partial^{2} p_{3 D}}{\partial y_{D}^{2}}) - λ_{2} (P_{3 D} - P_{1 D}) = ω_{3} \frac{\partial p_{3 D}}{\partial t_{D f}}

(3)

p_{j D} (x_{D}, y_{D}, 0) = 0, j = 1, 2, 3

(4)

\frac{\partial p_{j D} (x_{D}, y_{D}, t_{D})}{\partial x_{D}} = 0, x_{D} = x_{e D} or x_{D} = 0 o r y_{D} = y_{e D}

(5-7)

\lim_{y_{D} \to 0} (κ_{1} \frac{\partial p_{1 D}}{\partial y_{D}} + κ_{2} \frac{\partial p_{2 D}}{\partial y_{D}} + κ_{3} \frac{\partial p_{3 D}}{\partial y_{D}}) = {\begin{matrix} - \frac{π}{2} \cdot \frac{1}{x_{F D}}, & 0 \leq x_{D} \leq 1 \\ 0, & 1 < x_{D} \leq x_{e D} \end{matrix}

(8)

First, the Laplace transforms of equations (1)–(8) are determined

κ_{1} (\frac{\partial^{2} {\tilde{p}}_{1 D}}{\partial x_{D}^{2}} + \frac{\partial^{2} {\tilde{p}}_{1 D}}{\partial y_{D}^{2}}) + λ_{1} ({\tilde{p}}_{2 D} - {\tilde{p}}_{1 D}) + λ_{2} ({\tilde{p}}_{3 D} - {\tilde{p}}_{1 D}) = ω_{1} s {\tilde{p}}_{1 D}

(9)

κ_{2} (\frac{\partial^{2} {\tilde{p}}_{2 D}}{\partial x_{D}^{2}} + \frac{\partial^{2} {\tilde{p}}_{2 D}}{\partial y_{D}^{2}}) - λ_{1} ({\tilde{p}}_{2 D} - {\tilde{p}}_{1 D}) = ω_{2} s {\tilde{p}}_{2 D}

(10)

κ_{3} (\frac{\partial^{2} {\tilde{p}}_{3 D}}{\partial x_{D}^{2}} + \frac{\partial^{2} {\tilde{p}}_{3 D}}{\partial y_{D}^{2}}) - λ_{2} ({\tilde{p}}_{3 D} - {\tilde{p}}_{1 D}) = ω_{3} s {\tilde{p}}_{3 D}

(11)

\frac{\partial {\tilde{p}}_{j D} (x_{e D}, y_{D}, s)}{\partial x_{D}} = 0; \frac{\partial {\tilde{p}}_{j D} (0, y_{D}, s)}{\partial x_{D}} = 0; \frac{\partial {\tilde{p}}_{j D} (x_{D}, y_{e D}, s)}{\partial y_{D}} = 0

(12-14)

\lim_{y_{D} \to 0} (κ_{1} \frac{\partial {\tilde{p}}_{1 D}}{\partial y_{D}} + κ_{2} \frac{\partial {\tilde{p}}_{2 D}}{\partial y_{D}} + κ_{3} \frac{\partial {\tilde{p}}_{3 D}}{\partial y_{D}}) = {\begin{matrix} - \frac{π}{2 s} \cdot \frac{1}{x_{F D}}, & 0 \leq x_{D} \leq 1 \\ 0 & 1 < x_{D} \leq x_{e D} \end{matrix}

(15)

where s is a dimensionless time variable in the Laplace domain. So, the bottomhole pressure equation of a vertical fracture with constant conductivity in a three-layered sealed rectangular crossflow reservoir can be obtained as

{\tilde{p}}_{w D} (s) = \sum_{j = 1}^{3} c_{j} {\bar{R}}_{D} (σ_{j})

(16)

The derivation of equation (16) is shown in Appendix 1.

Flow model in fracture

For a vertical fracture with finite conductivity, the governing equation describing fluid flow in a fracture is strictly described as

\frac{\partial^{2} p_{f}}{\partial x^{2}} + \frac{\partial^{2} p_{f}}{\partial y^{2}} = \frac{φ_{f} μ c_{f t}}{k_{f}} \frac{\partial p_{f}}{\partial t}, 0 < x < x_{f}, 0 < y < w_{f} / 2

(17)

Since the fracture size is relatively small compared to the drainage area of a well in the reservoir, the above equation can be simplified by means of the integral average in the fracture

\frac{\partial^{2} p_{f}}{\partial x^{2}} + \frac{2 k}{w k_{f}} {\frac{\partial p}{\partial y} |}_{y = \frac{w}{2}} = \frac{φ_{f} μ c_{f t}}{k_{f}} \frac{\partial p_{f}}{\partial t}, 0 < x < x_{f}

(18)

No flow is allowed into the fracture through the fracture tips, and the flow of the fluid is linear in the fracture. The fluid flow is steady, and the flow model based on dimensionless variables can be described by

\frac{d^{2} p_{f D}}{d x_{D}^{2}} + \frac{2}{C_{F D}} {\frac{d p_{D}}{d y_{D}} |}_{y_{D} = \frac{w_{D}}{2}} = 0, 0 < x_{D} < 1

(19)

Outer boundary condition

\frac{d p_{f D} (1)}{d x_{D}} = 0

(20)

Inner boundary condition

\frac{d p_{f D} (0)}{d x_{D}} = - \frac{π}{C_{F D}}

(21)

The flow correlation formula at the surface of a fracture

q_{f D} = - \frac{2}{π} \cdot {\frac{\partial p_{D}}{\partial y_{D}} |}_{y_{D} = \frac{w_{D}}{2}}

(22)

Combining equations (19)–(22), the solutions can be obtained as

p_{w D} - p_{f D} (x_{D}) = \frac{π}{C_{F D}} (x_{D} - \int_{0}^{x_{D}} \int_{0}^{v} q_{f D} (u) d u d v)

(23)

The Laplace transformation of equation (23) yields

{\tilde{p}}_{w D} - {\tilde{p}}_{f D} (x_{D}) = \frac{π}{s \cdot C_{F D}} (x_{D} - s \cdot \int_{0}^{x_{D}} \int_{0}^{v} \tilde{q} (u) d u d v)

(24)

Connection condition between reservoir and fracture

The reservoir pressure is the same as the fracture pressure at the surface of the fracture

{\tilde{p}}_{D} (x_{D}, 0) = {\tilde{p}}_{f D} (x_{D}), {\tilde{p}}_{w D} = {\tilde{p}}_{f D} (0)

(25)

Normalization condition of flow rate in fracture

There is a normalized relation along a fracture at the constant flow rate of a hydraulic fracturing well

\int_{0}^{1} x_{D} \tilde{q} (α, s) d x_{D} = \frac{1}{s}

(26)

Combining equations (16), (24), (25), and (26), the simultaneous solution yields

{\tilde{p}}_{w D} (s) - \sum_{j = 1}^{3} c_{j} {\bar{R}}_{D} (σ_{j}) = \frac{π}{s . C_{F D}} (x_{D} - s \cdot \int_{0}^{x_{D}} \int_{0}^{v} \tilde{q} (u) d u d v)

(27)

Model of varying-conductivity fracture

The varying conductivity cannot be accurately described. Generally speaking, the varying conductivity of a vertical fracture has three trends, namely, linear variation, logarithmic variation, and exponential variation, along a fracture (Mou and Fan, 2006).

Linear variation : C_{F D} (x_{D}) = - a x_{D} + C_{F D 0}

(28a)

Exponential variation : C_{F D} (x_{D}) = C_{F D 0} \cdot \exp (- b x_{D})

(28b)

Logarithmic variation : C_{F D} (x_{D}) = C_{F D 0} [1 - c \log (x + 1)]

(28c)

where C_FD0 is the fracture conductivity in the wellbore; C_FD (x_D) is the varying conductivity at any position x_D along a vertical fracture; and a, b, and c are undetermined coefficients.

Different types of vertical fractures have different proppant supporting laws, which leads to a different fracture heterogeneity and different fracture closure law. The parameters a, b, and c in equation (28) have specific values. To determine the specific expression of variable conductivity of a vertical fracture in equation (28), laboratory experiments often are conducted based on the hydraulic fracturing operations combined with the studies on proppant type, reservoir characteristics, and the law of fluid flow in a fracture.

The regression relationships between variable conductivity and the distance along a vertical fracture were obtained in sandstone reservoirs according to a large number of laboratory conductivity measurements for a vertical hydraulic fracture filled with Yixing, Chengdu, Oriental, Tengfei, Weifang, and Cabo ceramsite or fracturing proppants in China (Yu, 1987; Zhang and Zhou, 2000).

Figure 2 displays the relationships between varying conductivity and the distance along a fracture for the three trends of linear, logarithmic, and exponential variation, which can well express the varying conductivity from well bottom to the tip of the fracture.

Figure 2.

Relationships of three trends between variable conductivity and the distance along a vertical fracture.

C_FD (x_D) is introduced and substituted in equation (28) for C_FD in equation (27), to obtain

{\tilde{p}}_{w D} (s) - \sum_{j = 1}^{3} c_{j} {\bar{R}}_{D} (σ_{j}) = \frac{π}{s . C_{F D} (x_{D})} (x_{D} - s \cdot \int_{0}^{x_{D}} \int_{0}^{v} \tilde{q} (u) d u d v)

(29)

Equation (29) is the mathematical model in a Laplace domain of a vertical fracture with varying conductivity in three rectangular crossflow layers. Note that p_wD is a function of the variable x_D that depends on the distribution of varying conductivity C_FD (x_D).

Semi-analytical solution of the mathematical model

Equation (29) is a Fredholm-type integral equation (Fredholm, 1903; Wazwaz,2011). The transient–pressure solution can be semi-analytically solved by dividing the half fracture into N segments with equal length Δx_Di, where α is an arbitrary dimensionless length along the fracture (as shown in Figure 3).

Figure 3.

Fracture divided into N equal segments along the half fracture.

Then, equation (26) can be discretized as

Δ x_{D i} \sum_{i = 1}^{N} \tilde{q} (α, s) = \frac{1}{s}

(30)

Discretizing equation (29), a system of linear equations of order N + 1 can be expressed as

[A_{i j}] \cdot [\begin{array}{l} \tilde{q} (u) \\ {\tilde{p}}_{w D} (s) \end{array}] = [B_{j}] (1 \leq j \leq N; j \leq i \leq N)

(31)

Equation (31) can be easily solved by Gaussian elimination. The wellbore pressure and flow rate can be obtained in the Laplace domain (Luo et al., 2016). Then, the solutions in a real domain can be further inverted by Stehfest numerical inversion (Stehfest, 1970).

Results and discussion

Parameter analysis of constant finite-conductivity fracture

Many parameters can influence the transient pressure of a vertical fracture with a constant finite conductivity in a closed-rectangular three-separate-layered crossflow reservoir. This paper only discusses the influences of the fracture half-length x_FD, drainage boundary y_eD, formation capacity ratio ω₁, and interporosity flow ratio ratio λ₁ and λ₂.

Figures 4 and 5 display the effects of parameters (ω₁ and λ₁, λ₂) on the transient pressure of a vertical fractured well in a three-separate-layered reservoir. Figures 6 and 7 respectively display the effects of parameters x_FD and Y_eD on the type curves of a vertical fracture in a three-separate-layered reservoir.

Figure 4.

Effects of parameter ω₁ on transient pressure of a vertical fractured well in three separate layers.

Figure 5.

Effects of parameter (λ_1, and λ₂) on pressure curves of a vertical fractured well in three-separate layers. (a) Effects of parameter λ₁ on pressure curves. (b). Effects of parameter λ₂ on pressure curves.

Figure 6.

Effects of parameter x_FD on type curves of a vertical fractured well in three separate layers.

Figure 7.

Effects of parameter Y_eD on type curves of a vertical fractured well in three separate layers.

From Figures 4 to 7, the transient pressures are divided into three stages: the linear-flow phase (the early slope 0.5), the medium unsteady-flow stage, and the later pseudo-steady-flow phase. The parameters of the fracture, the three-separate-layered rectangular reservoir, and the multi-permeability medium (such as X_eD, Y_eD, X_FD, k₁, k₂, k₃, ω₁, ω₂, ω₃, λ₁, λ₂, and λ₃) directly influence the direction, transition, and shape of the transient pressure during the three stages of a vertical hydraulic fracture.

Parameter analysis of varying-conductivity fracture

Figures 8 –10 display the type curves (dimensionless pressure and dimensionless pressure derivative) of a vertical fractured well in three-separate layers with varying conductivity by linear variation (Figure 8), logarithmic variation (Figure 9), and exponential variation (Figure 10), where the undetermined coefficients a = 10.0, 8.0, 5.0, 3.0, b = 2.0, 1.5, 1.0, 0.5, or c = 3.0, 2.0, 1.0, 0.5 are displayed from top to bottom followed. The bottom line is the constant conductivity for each.

Figure 8.

Type curves of a vertical fractured well in three separate layers (linear variation).

Figure 9.

Type curves of a vertical fractured well in three separate layers (logarithmic variation).

Figure 10.

Type curves of a vertical fractured well in three separate layers (exponential variation).

From Figures 8 to 10, the change trend of varying conductivity is greater if the coefficients of a, b, and c are bigger, which also illustrates that the fracture conductivity is higher near the well bottom and is smaller at the tip of the fracture. During the process of a vertical hydraulic fracture, a fracture near the well bottom was filled with more ceramsites or fracturing proppants than at a long distance from the well bottom, resulting in the fracture having a greater width near the well bottom.

As shown in Figures 8 –10, the dimensionless pressure and dimensionless pressure derivative perform better with varying conductivity than with constant conductivity. Therefore, there are many more differences between varying conductivity and constant conductivity during well-test analysis. Varying conductivity can correctly reflect the flow characteristics of a vertical fractured well, which can provide a theoretical basis for well-test analysis.

Specially note there is not a real field case to validate the presented model. But, the proposed model can provide a theoretical basis for well-test analysis. If the well testing data are available in three cross flow rectangular layers during the process of hydraulic fracturing in the future, the proposed model can conduct the well testing interpretation.

Conclusions

A mathematical model of a vertical fracture with constant conductivity was established in three separate rectangular crossflow layers.

Three forms of varying conductivity were introduced (linear, logarithmic, and exponential). A new mathematical model of a vertical fracture with varying conductivity intercepting a three-separate-layered crossflow reservoir was developed, and its semi-analytical solution was achieved by the methods of Laplace transform, Fourier cosine transform, and Stehfest inversion.

The transient pressures are divided into three stages: the linear-flow phase, the medium unsteady-flow stage, and the later pseudo-steady-flow phase. The parameters of the fracture and reservoir directly influence the direction, transition, and shape of the dimensionless pressure and dimensionless pressure derivative during the three stages of a vertical fractured well.

The type curves of a vertical fractured well in three-separate layers were analyzed and diagnosed with three forms of varying conductivity. Results show that the fracture conductivity is higher near the well bottom and is smaller at the tip of the fracture for the varying conductivity. The dimensionless pressure and dimensionless pressure derivative performed better when considering varying conductivity than with constant conductivity. Varying conductivity can correctly reflect the flow characteristics of a vertical fractured well during well-test analysis.

The novel mathematical model and its semi-analytical solution of a vertical fracture in three separate layers with varying conductivity may form a theoretical basis for well-test analysis.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was supported by the National Natural Science Foundation of China (No. 51774256), Fundamental Research Funds for the Central Universities of China (2–9-2017–310, 53200759268). and Science and Technology Special Funds of China for 2016ZX05015–002.

Appendix 1

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