Productivity Formulae of an Infinite-Conductivity Hydraulically Fractured Well Producing at Constant Wellbore Pressure Based on Numerical Solutions of a Weakly Singular Integral Equation of the First Kind

Abstract

In order to increase productivity, it is important to study the performance of a hydraulically fractured well producing at constant wellbore pressure. This paper constructs a new productivity formula, which is obtained by solving a weakly singular integral equation of the first kind, for an infinite-conductivity hydraulically fractured well producing at constant pressure. And the two key components of this paper are a weakly singular integral equation of the first kind and a steady-state productivity formula. A new midrectangle algorithm and a Galerkin method are presented in order to solve the weakly singular integral equation. The numerical results of these two methods are in accordance with each other. And then the solutions of the weakly singular integral equation are utilized for the productivity formula of hydraulic fractured wells producing at constant pressure, which provide fast analytical tools to evaluate production performance of infinite-conductivity fractured wells. The paper also shows equipotential threads, which are generated from the numerical results, with different fluid potential values. These threads can be approximately taken as a family of ellipses whose focuses are the two endpoints of the fracture, which is in accordance with the regular assumption in Kuchuk and Brigham, 1979.

1. Introduction

The main objective of hydraulic fracturing for well stimulation is to increase well productivity by creating a highly conductive path some distance away from the skin zone into the formation. The fracture creates more surface area to the wellbore without drilling another well. Since more reservoir area is in direct communication with the wellbore, a greater volume of fluid can be produced into the wellbore per unit time, resulting in an increased production rate. This basic objective has not changed since hydraulic fracturing was introduced in the early 1950s. At the beginning, hydraulic fracturing was considered a good mean to increase productivity of the wells completed in low-permeability reservoirs. Now, it has become an integral part of most well completions.

For hydraulically fractured wells, unsteady-state pressure transient testings are useful tools for evaluating insitu reservoir and wellbore parameters that describe the production characteristics. The use of transient well testing for determining reservoir parameters and productivity of fractured wells has become common, and determination of transient pressure behavior and productivity for fractured wells has aroused considerable interest over the past decades.

Numerous analytic solutions have been presented for the pressure behavior of a fractured well producing at constant flow rate. An extensive literature survey on fractured wells producing at constant flow rates can be found. Guppy et al. present for the first time a technique that analyzes buildup and drawdown data from wells produced at constant pressure with turbulent flow in the fracture [1]. Cinco-Ley and Samaniego-V studied the early-time pressure data for a well intercepted by a finite-conductivity vertical fracture [2]. Ozkan et al. described the characteristics of a well producing at a constant pressure in a natural fractured reservoir [3]. Guppy and Cinco-Ley presented semianalytic solutions for unsteady-state flow behavior of a well intersecting a vertical fracture and analytic solutions for defining certain portions of the early time data for various types of fracture conductivity [4]. Nashawi discussed a semianalytical equation which incorporates the effects of non-Darcy flow in the fracture [5]. Kuchuk and Brigham presented analytical solutions to elliptical flow problems that are applicable to infinite-conductivity vertically fractured wells, elliptically shaped reservoirs, and anisotropic reservoirs producing at constant rate or pressure [6].

Although most well test analysis methods for hydraulically fractured wells assume constant rate production, it must be pointed out that the constant rate production is difficult to maintain, and constant wellbore pressure production conditions are not uncommon. Examples of conditions under which constant pressure is maintained at a well include production into a constant pressure separator or pipeline, open flow to the atmosphere, or production from a low permeability reservoir. It is important to consider wells producing at constant pressure rather than constant rate during large portions of the production life of tight reservoirs [4].

This paper proposes a new productivity model of an infinite-conductive hydraulically fractured well producing at constant pressure. The weakly singular integral equation [7–9] arising from this model has attracted much attention for its numerical solutions [10–12].

In order to solve the problem of constant wellbore pressure production, the numerical solution to a weakly singular integral equation is required, and a new midrectangular algorithm is used to solve the weakly singular integral equation; the corresponding numerical results are compared with those by the Galerkin method.

Furthermore, the solutions of the weakly singular integral equation are used in the productivity formulae of hydraulically fractured wells producing at constant pressure. These formulae provide fast analytical tools to evaluate production performance of infinite-conductivity fractured wells. And equipotential threads with different fluid potential values, which can be approximately taken as a family of ellipses whose focuses are the two endpoints of the fracture, are also shown.

The rest of this paper is organized as follows. In Section 2, the new model is derived. In Section 3, the Galerkin method and the corresponding numerical results are presented. In Section 4, the midrectangle algorithm is constructed, and the corresponding numerical experiments are presented. Then conclusions are reached in Section 5.

2. Infinite-Conductivity Hydraulically Fractured Well Model Producing at Constant Wellbore Pressure

In this section, we will construct a new productivity model of an infinite-conductivity hydraulically fractured well producing at constant wellbore pressure.

Figure 1 represents an ideal vertical fracture. The usual assumptions apply; that is, the porous medium is isotropic, horizontal, homogeneous, and uniform in thickness and has constant permeability. Also, the fracture fully penetrates the vertical extent of the formation and is the same length on both sides of the well.

Figure 1 

Schematic of ideal fracture.

The porous media domain is as follows: Ω=𝑥,𝑦,𝑧∣𝑥²+𝑦²<𝑅_𝑒²,0<𝑧<𝐻, (2.1) where 𝑅_𝑒 is cylinder radius and Ω is the cylindrical body. Since both the vertical well and the hydraulic fracture are fully penetrating, we may use two-dimensional model to study the pressure behavior. Assume the fracture length is 𝐿,𝐿=2𝑥_𝑓, the fracture is taken as a line sink in the two-dimensional space, and the coordinates of the two ends are (−𝑥_𝑓,0) and (x_𝑓,0). Supposing that point (𝑥^′,0) is on the fracture, in order to obtain the pressure at point (𝑥,𝑦) caused by the point (𝑥^′,0), we have to obtain the basic solution of the following partial differential equation in Ω: 𝐾𝜇𝜕²𝑃𝜕𝑥²+𝐾𝜇𝜕²𝑃𝜕𝑦²=−𝑞𝑥^′𝛿𝑥−𝑥^′𝛿𝑦, (2.2) where 𝑃(𝑥,𝑦) is the pressure at point (𝑥,𝑦). 𝐾 is the formation permeability, 𝜇 is viscosity, 𝑞(𝑥^′) is the flow rate at point (𝑥^′,0), 𝛿(𝑦),𝛿(𝑥−𝑥^′) are Dirac functions and 𝑥^′∈−𝑥_𝑓,𝑥_𝑓, (2.3) that is, the hydraulic fracture is between [−𝑥_𝑓,𝑥_𝑓].

Let 𝑃|_𝑟→∞=𝑃_𝑖>0, (2.4) where 𝑃|_𝑟→∞ is the pressure at infinite distance away from the fracture. Because the hydraulically fractured well is infinite conductivity, the pressure of the fracture denoted by 𝑃₀ is uniform. That is 𝑃(𝑥,0)=𝑃₀=constant,∀𝑥∈−𝑥_𝑓,𝑥_𝑓. (2.5)

Thus, if 𝑞(𝑥^′) is known, the total productivity is given by 𝑄=^𝐻₀^𝑥𝑓_−𝑥𝑓𝑞𝑥^′𝑑𝑥^′𝑑𝑧=𝐻^𝑥𝑓_−𝑥𝑓𝑞𝑥^′𝑑𝑥^′. (2.6)

In order to simplify the previously mentioned equations, we take the following dimensionless transforms: 𝑥_𝐷=𝑥2𝑥_𝑓,𝑦_𝐷=𝑦2𝑥_𝑓,𝑃_𝐷=𝑃_𝑖−𝑃𝑃_𝑖,𝑃_𝐷0=𝑃_𝑖−𝑃₀𝑃_𝑖. (2.7)

Obviously, we obtain 𝜕𝑥_𝐷=1𝜕𝑥2𝑥_𝑓,𝜕𝑃=𝜕𝑥𝜕𝑃𝜕𝑥_𝐷𝜕𝑥_𝐷=1𝜕𝑥2𝑥_𝑓𝜕𝑃𝜕𝑥_𝐷,𝜕²𝑃𝜕𝑥²=𝜕𝜕𝑥𝜕𝑃=1𝜕𝑥4𝑥²_𝑓𝜕²𝑃𝜕𝑥²_𝐷. (2.8)

Since 𝑃_𝐷=𝑃_𝑖−𝑃𝑃_𝑖, (2.9) then 𝜕²𝑃𝜕𝑥²𝑃=−^𝑖4𝑥²_𝑓𝜕²𝑃_𝐷𝜕𝑥²_𝐷. (2.10)

In the same manner, we have 𝜕²𝑃𝜕𝑦²𝑃=−^𝑖4𝑥²_𝑓𝜕²𝑃_𝐷𝜕𝑦²_𝐷,𝛿𝑥−𝑥^′𝛿𝑦=𝛿2𝑥_𝑓𝑥_𝐷−𝑥^′_𝐷𝛿2𝑥_𝑓𝑦_𝐷=2𝑥_𝑓⁻²𝛿𝑥_𝐷−𝑥^′_𝐷𝛿𝑦_𝐷. (2.11) So 𝐾𝜇𝜕²𝑃𝜕𝑥²+𝐾𝜇𝜕²𝑃𝜕𝑦²𝐾=−𝜇𝑃_𝑖4𝑥²_𝑓𝜕²𝑃_𝐷𝜕𝑥²_𝐷+𝜕²𝑃_𝐷𝜕𝑦²_𝐷=−𝑞𝑥_𝑓𝑥^′_𝐷2𝑥_𝑓⁻²𝛿𝑥_𝐷−𝑥^′_𝐷𝛿𝑦_𝐷. (2.12) That is, 𝜕²𝑃_𝐷𝜕𝑥²_𝐷+𝜕²𝑃_𝐷𝜕𝑦²_𝐷=𝑞_𝐷𝑥^′_𝐷𝛿𝑥_𝐷−𝑥^′_𝐷𝛿𝑦_𝐷, (2.13) where 𝑞_𝐷𝑥^′_𝐷=𝑞2𝑥_𝑓𝑥^′_𝐷𝜇𝐾𝑃_𝑖,𝑥^′_𝐷∈−12,12. (2.14)

Note that after taking the dimensionless transforms, producing section [−𝑥_𝑓,𝑥_𝑓] of the fractured well is changed to [−1/2,1/2], and the initial conditions are changed as follows: 𝑃_{𝐷𝑟𝐷→∞}⟶0⁺,𝑞_𝐷𝑥^′_𝐷=0,𝑥^′_𝐷∉−12,12. (2.15)

It is well known that the fundamental solution of two-dimensional Laplace equation is as follows [13, 14]: 𝑔𝑥,𝑦;𝑥^′,𝑦^′=−ln𝑥−𝑥^′2+𝑦−𝑦^′21/2.2𝜋 (2.16)

Thus, if 𝑞_𝐷(𝑥^′_𝐷) is known, the solution to (2.13) is given by 𝑃_𝐷𝑥_𝐷,𝑦_𝐷=^1/2_−1/2𝑔𝑥_𝐷,𝑦_𝐷;𝑥^′_𝐷,0𝑞_𝐷𝑥^′_𝐷𝑑𝑥^′_𝐷. (2.17)

If the hydraulically fractured well is infinite conductivity, then we have 𝑃_𝐷𝑥_𝐷,0=𝑃_𝐷0,𝑥_𝐷∈−12,12, (2.18) where 𝑃_𝐷0 is a constant.

From (2.17), we can obtain the following equation: ^1/2_−1/2𝑔𝑥_𝐷,0;𝑥^′_𝐷,0𝑞_𝐷𝑥^′_𝐷𝑑𝑥^′_𝐷=𝑃_𝐷0, (2.19) and it can be simplified as follows: −12𝜋^1/2_−1/2ln𝑥_𝐷−𝑥^′_𝐷𝑞_𝐷𝑥^′_𝐷𝑑𝑥^′_𝐷=𝑃_𝐷0. (2.20) Let 𝑢𝑥^′_𝐷=𝑞_𝐷𝑥^′_𝐷2𝜋𝑃_𝐷0,𝑥^′_𝐷∈−12,12. (2.21)

Consequently, we obtain the following first kind weakly singular integral equation: ^1/2_−1/2𝑢(𝑥)−1ln𝑥−𝑦𝑑𝑥=−1,∀𝑦∈2,12. (2.22)

Once 𝑢(𝑥) is obtained from the previous equation, we can use it to compute 𝑞_𝐷(𝑥)=2𝜋𝑃_𝐷0𝑢(𝑥)−1,𝑥∈2,12. (2.23)

Using 𝑞_𝐷𝑥^′_𝐷=𝑞2𝑥_𝑓𝑥^′_𝐷𝜇𝐾𝑃_𝑖,𝑥^′_𝐷∈−12,12, (2.24) we can obtain 𝑞2𝑥_𝑓𝑥^′_𝐷=𝐾𝑃_𝑖𝜇𝑞_𝐷𝑥^′_𝐷,𝑥^′_𝐷∈−12,12. (2.25)

The total productivity of the hydraulically fractured well is as follows: 𝑄=𝐻^𝑥𝑓_−𝑥𝑓𝑞𝑥^′𝑑𝑥^′=2𝑥_𝑓𝐻^1/2_−1/2𝑞2𝑥_𝑓𝑥^′_𝐷𝑑𝑥^′_𝐷=2𝑥_𝑓𝐾𝑃_𝑖𝐻𝜇^1/2_−1/2𝑞_𝐷𝑥^′_𝐷𝑑𝑥^′_𝐷=4𝑥_𝑓𝜋𝑃_𝐷0𝐾𝑃_𝑖𝐻𝜇^1/2_−1/2𝑢(𝑥)𝑑𝑥=4𝑥_𝑓𝜋𝑃_𝑖−𝑃₀𝑃_𝑖𝐾𝑃_𝑖𝐻𝜇^1/2_−1/2𝑢(𝑥)𝑑𝑥,𝑄=4𝑥_𝑓𝜋𝑃_𝑖−𝑃₀𝐾𝐻𝜇^1/2_−1/2𝑢(𝑥)𝑑𝑥. (2.26)

Remark 2.1. Equation (2.22) is the first kind weakly singular integral equation, which can be solved with the numerical methods introduced in Sections 3 and 4. And the total productivity can be obtained by (2.26).

Remark 2.2. From Tables 1 and 2 in Section 3 or Tables 3 and 4 in Section 4, we can find that ∫^1/2_−1/2𝑢(𝑥)𝑑𝑥 is a constant. So the total productivity formula is as follows: 𝑄=4𝑥_𝑓𝜋𝑃_𝑖−𝑃₀𝐾𝐻𝜇^1/2_−1/2𝑢(𝑥)𝑑𝑥≈4𝑥_𝑓𝜋𝑃_𝑖−𝑃₀𝐾𝐻𝜇ℎ^𝑛−1_𝑗=0𝑢_𝑗≈4𝑥_𝑓𝜋𝑃_𝑖−𝑃₀𝐾𝐻𝜇0.7213=9.0641𝑥_𝑓𝑃_𝑖−𝑃₀𝐾𝐻𝜇𝑃=4.53205^𝑖−𝑃₀𝐾𝐻𝐿𝜇, (2.27) where 𝐿 is the length of the hydraulically fractured well.

Table 1 

Galerkin approximations of (2.22) with ℎ=1/32.

Table 2 

Galerkin approximations of (3.11) with different ℎ.

Table 3 

Mid-rectangle approximations of (2.22) with ℎ=1/32.

Table 4 

Mid-rectangle approximations of (4.16) with different ℎ.

3. Galerkin Method for Steady-State Productivity Computation of the Hydraulically Fractured Well

In this section, we will present the Galerkin method, which has a mature analysis framework, to solve the weakly singular integral equation (2.22). Once 𝑢(𝑥) of (2.22) is known, the approximation of (2.26) is also presented.

Let 1ℎ=𝑛,𝑥_𝑗1=−2,−1+𝑗ℎ,𝑗=0,…,𝑛Γ=2,12,Γ_𝑗=𝑥_𝑗−1,𝑥_𝑗,𝑗=1,…,𝑛. (3.1)

Clearly, Γ=^𝑛_𝑗=1Γ_𝑗. (3.2)

Let 𝑒_𝑗(𝑥)=1,𝑥∈Γ_𝑗,0,𝑥∉Γ_𝑗,𝑆^ℎ=span𝑒_𝑗(𝑥),𝑗=1,…,𝑛. (3.3)

Define the energy inner product [𝑢,𝑣]=(𝐾𝑢,𝑣)=−_Γln𝑥−𝑦𝑢(𝑥)𝑣𝑦𝑑𝑠_𝑥𝑑𝑠_𝑦,∀𝑢,𝑣∈𝐿₂(Γ) (3.4) and the energy norm ‖𝑢‖²=[𝑢,𝑢]. (3.5)

The weak formulation is to find 𝑢∈𝐿₂(Γ) such that [𝑢,𝑣]=𝑔,𝑣,∀𝑣∈𝐿₂(Γ). (3.6)

And the Galerkin formulation is to find 𝑢^ℎ(𝑥)=^𝑛_𝑗=1𝑐_𝑗𝑒_𝑗(𝑥)∈𝑆^ℎ (3.7) such that 𝑢^ℎ,𝑣^ℎ=𝑔,𝑣^ℎ,∀𝑣^ℎ∈𝑆^ℎ. (3.8)

Then the matrix representation of the approximation is 𝐴𝑐=𝑔, (3.9) where 𝐴=𝑎_{𝑖𝑗𝑛×𝑛},𝑎_𝑖𝑗=−_Γ𝑖Γ𝑗ln𝑥−𝑦𝑑𝑠_𝑥𝑑𝑠_𝑦,𝑖,𝑗=1,…,𝑛,𝑐=𝑐₁,𝑐₂,…,𝑐_𝑛^𝑇,𝑔=𝑔₁,𝑔₂,…,𝑔_𝑛^𝑇,𝑔_𝑗=𝑔,𝑒_𝑗=_Γ𝑔(𝑥)𝑒_𝑗𝑦𝑑𝑠_𝑥𝑑𝑠_𝑦,𝑗=1,…,𝑛. (3.10)

Once the solution of (3.9) is obtained, we can utilize it to obtain the total productivity of the hydraulically fractured well by the following formula: 𝑄(ℎ)=ℎ^𝑛_𝑗=1𝑐_𝑗. (3.11)

In fact𝑄(ℎ)=^1/2_−1/2𝑢^ℎ(𝑥)=𝑑𝑥_{𝑛1/2−1/2𝑗=1}𝑐_𝑗𝑒_𝑗(𝑥)=𝑑𝑥_𝑛𝑗=1𝑐_𝑗^1/2_−1/2𝑒_𝑗(𝑥)=𝑑𝑥_𝑛𝑗=1𝑐_𝑗ℎ, (3.12) which completes the proof of Formula (3.11).

In the following, we will present the error estimate of the Galerkin approximation. Since 𝑆^ℎ⊂𝐿²(Γ), then (3.6) and (3.8) lead to 𝑢^ℎ−𝑢,𝑣=0,∀𝑣∈𝑆^ℎ. (3.13)

Hence 𝑢^ℎ=𝑃^ℎ𝑢, (3.14) where 𝑃^ℎ∶𝐿₂(Γ)↦𝑆^ℎ is the orthogonal projection with respect to the energy inner product. Consequently, there holds 𝑢^ℎ−𝑢=inf_{𝑣∈𝑆ℎ}‖𝑢−𝑣‖. (3.15) Using (3.15), the following theorem, which is recalled from [7, 8], is valid.

Theorem 3.1 (see [7, 8]). If Γ is sufficiently smooth, g is a smooth function on Γ, and 𝑆^ℎ is a subspace which is construct of piecewise smooth constant functions, then there is a constant 𝑐>0 such that 𝑢^ℎ−𝑢≤𝑐ℎ^2/3,𝑢^ℎ−𝑢,𝑔≤𝑐ℎ³. (3.16)

Now we turn to the implementation issue of (3.9). Clearly, lim_ℎ→0ℎlnℎ=0. (3.17)

Using (3.17), we can obtain 𝑎_𝑖𝑗=−_Γ𝑖Γ𝑗ln𝑥−𝑦𝑑𝑠_𝑥𝑑𝑠_𝑦=12𝑗−𝑖−1²ℎ²ln𝑗−𝑖−1ℎ−𝑗−𝑖²ℎ²ℎ−3ln𝑗−𝑖2ℎ²+12𝑗−𝑖+1²ℎ²ℎ,𝑔ln𝑗−𝑖+1_𝑗=𝑔,𝑒_𝑗=_Γ𝑔(𝑥)𝑒_𝑗𝑦𝑑𝑠_𝑥𝑑𝑠_𝑦=_Γ𝑔(𝑥)𝑑𝑠_𝑥Γ𝑒_𝑗𝑦𝑑𝑠_𝑦=_Γ𝑔(𝑥)𝑑𝑠_𝑥ℎ. (3.18)

In the numerical experiment, we choose 𝑔(𝑥)=−1 and Γ=[−1/2,1/2] in the previous algorithm in order to obtain the numerical solutions of (2.22). Table 1 shows the Galerkin approximations with ℎ=1/32, from which we can see that the solution is symmetric, that is, 𝑐_𝑖=𝑐_n+1−𝑖,𝑖=1,…,𝑛. And Table 2 shows the total productivity of fractured well with different ℎ, which is convergent when ℎ→0. More observations and conclusions from the numerical results will be presented at the end of Section 4.

4. Midrectangle Algorithm for Steady-State Productivity Computation of the Hydraulically Fractured Well

In this section, we will construct and analyze a midrectangle algorithm, which is much more convenient to implement but still has similar accuracy to that of the Galerkin method.

We first prove the following theorem, which will lead to the algorithm and provide the error estimate of the method.

Theorem 4.1. For any 𝑔(𝑥)∈𝐶²_{[𝑡−𝑎,𝑡+𝑎]}, there holds ^𝑡+𝑎_𝑡−𝑎𝑔(𝑥)ln|𝑥−𝑡|𝑑𝑥=2𝑔(𝑡)𝑎(ln𝑎−1)+𝑂𝑎³ln𝑎. (4.1)

Proof. Let 𝑎ℎ=𝑛,𝑥_𝑖=𝑡−𝑎+𝑖ℎ,(𝑖=0,1,…,𝑛−1,𝑛,𝑛+1,…,2𝑛). (4.2) Obviously, 𝑥_𝑛=𝑡,𝑥_𝑖+𝑥_2𝑛−𝑖=2𝑡. (4.3) By Taylor expansion, we can obtain 𝑔𝑥_𝑖+𝑔𝑥_2𝑛−𝑖(𝑡)+𝑔=2𝑔^′′𝜁_𝑖2𝑥_𝑖−𝑡². (4.4) That is, 𝑔𝑥_𝑖+𝑔𝑥_2𝑛−𝑖(𝑡)+𝑔=2𝑔^′′𝜁_𝑖2((𝑛−𝑖)ℎ)². (4.5) Hence ^𝑡+𝑎_𝑡−𝑎𝑔(𝑥)ln|𝑥−𝑡|𝑑𝑥=lim_ℎ→0ℎ^𝑛−1_𝑖=0𝑔𝑥_𝑖ln𝑥_𝑖−𝑡+ℎ^2𝑛_𝑖=𝑛+1𝑔𝑥_𝑖ln𝑥_𝑖−𝑡=lim_ℎ→0ℎ^𝑛−1_𝑖=0𝑔𝑥_𝑖ln(𝑎−𝑖ℎ)+ℎ^2𝑛_𝑖=𝑛+1𝑔𝑥_𝑖ln(𝑖ℎ−𝑎)=lim_ℎ→0ℎ^𝑛−1_𝑖=0𝑔𝑥_𝑖ln(𝑎−𝑖ℎ)+ℎ^{2𝑛−2𝑛}_{𝑖=𝑛+1−2𝑛}𝑔𝑥_𝑖+2𝑛ln((𝑖+2𝑛)ℎ−𝑎)=lim_ℎ→0ℎ^𝑛−1_𝑖=0𝑔𝑥_𝑖ln(𝑎−𝑖ℎ)+ℎ⁰_{𝑖=1−𝑛}𝑔𝑥_𝑖+2𝑛ln((𝑖+2𝑛)ℎ−𝑎)=lim_ℎ→0ℎ^𝑛−1_𝑖=0𝑔𝑥_𝑖ln(𝑎−𝑖ℎ)+ℎ⁻⁰_{()𝑖=−1−𝑛}𝑔𝑥_{−𝑖+2𝑛}ln((−𝑖+2𝑛)ℎ−𝑎)=lim_ℎ→0ℎ^𝑛−1_𝑖=0𝑔𝑥_𝑖ln(𝑎−𝑖ℎ)+ℎ⁰_{𝑖=𝑛−1}𝑔𝑥_{−𝑖+2𝑛}ln((−𝑖+2𝑛)ℎ−𝑎)=lim_ℎ→0ℎ^𝑛−1_𝑖=0𝑔𝑥_𝑖ln(𝑎−𝑖ℎ)+ℎ^𝑛−1_𝑖=0𝑔𝑥_{−𝑖+2𝑛}ln((−𝑖+2𝑛)ℎ−𝑎)=lim_ℎ→0ℎ^𝑛−1_𝑖=0𝑔𝑥_𝑖ln(𝑎−𝑖ℎ)+ℎ^𝑛−1_𝑖=0𝑔𝑥_2𝑛−𝑖ln(−𝑖ℎ+2𝑛ℎ−𝑎)=lim_ℎ→0ℎ^𝑛−1_𝑖=0𝑔𝑥_𝑖ln(𝑎−𝑖ℎ)+ℎ^𝑛−1_𝑖=0𝑔𝑥_2𝑛−𝑖ln(𝑎−𝑖ℎ)=lim_ℎ→0ℎ^𝑛−1_𝑖=0𝑔𝑥_𝑖+𝑔𝑥_2𝑛−𝑖ln(𝑎−𝑖ℎ)=lim_ℎ→0ℎ^𝑛−1_𝑖=0(𝑡)+𝑔2𝑔^′′𝜁_𝑖2(𝑛−𝑖)ℎ)²ln(𝑎−𝑖ℎ)=lim_ℎ→0ℎ^𝑛−1_𝑖=02𝑔(𝑡)ln(𝑎−𝑖ℎ)+lim_ℎ→0ℎ^𝑛−1_𝑖=0𝑔^′′𝜁_𝑖2((𝑛−𝑖)ℎ)²ln(𝑎−𝑖ℎ)=lim_ℎ→0ℎ^𝑛−1_𝑖=02𝑔(𝑡)ln((𝑛−𝑖)ℎ)+lim_ℎ→0ℎ^𝑛−1_𝑖=0𝑔^′′𝜁_𝑖2((𝑛−𝑖)ℎ)²ln((𝑛−𝑖)ℎ)=𝐼₁+𝐼₂. (4.6) Here 𝐼₁=lim_ℎ→0ℎ^𝑛−1_𝑖=0(𝑡))ℎ(𝑡)2𝑔ln((𝑛−𝑖)=2𝑔lim_ℎ→0ℎ^𝑛−1_𝑖=0()ℎ(𝑡)ln((𝑛−𝑖)=2𝑔lim_ℎ→0ℎ^𝑛_𝑖=1((𝑡)ln𝑖ℎ)=2𝑔^𝑎₀(𝑡ln𝑥𝑑𝑥=2𝑔)[𝑥ln𝑥−𝑥]^𝑎₀=2𝑔(𝑡)[𝑎ln𝑎−𝑎], (4.7) where we have used the result of lim_𝑥→0𝑥ln𝑥=0. Now we consider 𝐼₂. Let 𝑚≤𝑔^′′(𝜁_𝑖)≤𝑀,(𝑖=0,1,…,𝑛−1). There exists such a constant 𝑘 that 𝐼₂=lim_ℎ→0ℎ^𝑛−1_𝑖=0𝑔^′′𝜁_𝑖2((𝑛−𝑖)ℎ)²)ℎ=𝑘ln((𝑛−𝑖)2lim_ℎ→0ℎ^𝑛−1_𝑖=0((𝑛−𝑖)ℎ)²)ℎ=𝑘ln((𝑛−𝑖)2lim_ℎ→0ℎ^𝑛_𝑖=1(𝑖ℎ)²(=𝑘ln𝑖ℎ)2^𝑎₀𝑥²=𝑘ln𝑥𝑑𝑥2𝑥³3𝑥ln𝑥−³9^𝑎₀=𝑘2𝑎³3𝑎ln𝑎−³9. (4.8)
Plugging 𝐼₁ and 𝐼₂ into (4.6), the proof is completed.

Let 1ℎ=𝑛,𝑥_𝑗1=−2,𝑦+𝑗ℎ,𝑗=0,1,…,𝑛_𝑗1=−2+1𝑗+2ℎ,𝑗=0,1,…,𝑛−1. (4.9)

Clearly, 𝑦_𝑗=12𝑥_𝑗+𝑥_𝑗+1,𝑗=0,1,…,𝑛−1. (4.10)

Let 𝑢_𝑗 be the approximation of 𝑢(𝑦_𝑗), and let 𝑦_𝑖=(1/2)(𝑥_𝑖+𝑥_𝑖+1),(𝑖=0,1,…,𝑛−1) be the singular points in (2.22). Then ^1/2_−1/2𝑢(𝑥)ln𝑥−𝑦_𝑖𝑑𝑥=^𝑛−1_𝑗=0^𝑥𝑗+1_𝑥𝑗𝑢(𝑥)ln𝑥−𝑦_𝑖=𝑑𝑥_{𝑖−1𝑗=0𝑥𝑗+1𝑥𝑗}𝑢(𝑥)ln𝑥−𝑦_𝑖𝑑𝑥+^𝑥𝑖+1_𝑥𝑖𝑢(𝑥)ln𝑥−𝑦_𝑖+𝑑𝑥_{𝑛−1𝑗=𝑖+1𝑥𝑗+1𝑥𝑗}𝑢(𝑥)ln𝑥−𝑦_𝑖=𝑑𝑥_{𝑖−1𝑗=0}ℎ𝑢𝑦_𝑗ln𝑦_𝑗−𝑦_𝑖+𝑂ℎ²+2𝑢𝑦_𝑖ℎ2ℎln2ℎ−1+𝑂2³ℎln2+^𝑛−1_𝑗=𝑖+1ℎ𝑢𝑦_𝑗ln𝑦_𝑗−𝑦_𝑖+𝑂ℎ²=^𝑛−1_{𝑗=0,𝑗≠𝑖}ℎ𝑢𝑦_𝑗ln𝑦_𝑗−𝑦_𝑖+ℎ𝑢𝑦_𝑖ℎlnℎ2𝑒+𝑂². (4.11) Hence we obtain the following theorem.

Theorem 4.2. Forall𝑢(𝑥)∈𝐶^′′_[−1/2,1/2],ℎ=1/𝑛,𝑥_𝑗=(−1/2)+𝑗ℎ,𝑗=0,1,…,𝑛,  𝑦_𝑗=−1/2+(𝑗+1/2)ℎ,(𝑗=0,1,…,𝑛−1), there holds ^1/2_−1/2𝑢(𝑥)ln𝑥−𝑦_𝑖𝑑𝑥=^𝑛−1_{𝑗=0,𝑗≠𝑖}ℎ𝑢𝑦_𝑗ln𝑦_𝑗−𝑦_𝑖+𝑢𝑦_𝑖ℎℎlnℎ2𝑒+𝑂²,(𝑖=0,1,…,𝑛−1). (4.12)

From Theorem 4.2, we can construct the following midrectangle algorithm for (2.22): −1=^1/2_−1/2u(𝑥)ln𝑥−𝑦_𝑖≈𝑑𝑥_{𝑛−1𝑗=0,𝑗≠𝑖}ℎln𝑦_𝑗−𝑦_𝑖𝑢_𝑗+ℎℎln𝑢2𝑒^𝑖,(𝑖=0,1,…,𝑛−1). (4.13)

The error estimate of this method is given by Theorem 4.2. And the matrix representation of (4.13) is 𝐴𝑢=−𝐼, (4.14) where 𝑢=𝑢₀,𝑢₁,…,𝑢_𝑛−1^𝑇,𝐼=[1,1,…,1]^𝑇,𝐴=𝑎_𝑖𝑗^𝑛−1_𝑖,𝑗=0,𝑎_𝑖,𝑗=ℎℎlnℎ2𝑒,.𝑖=𝑗ℎln𝑗−𝑖𝑖≠𝑗𝑖,𝑗=0,1,…,𝑛−1 (4.15)

After 𝑢 in (4.14) is obtained, the total productivity of the hydraulically fractured well can be approximated as follows: 𝑄^′(ℎ)=ℎ^𝑛−1_𝑗=0𝑢_𝑗. (4.16)

Table 3 shows the midrectangle approximations with ℎ=1/32. And Table 4 shows the total productivity of fractured well with different ℎ. The numerical results perform similarly to those of the Galerkin method presented in Tables 1 and 2.

From Tables 1 and 3, we can find that when a hydraulically fractured well is producing at constant wellbore pressure, the point convergence intensity at endpoint is bigger than that of midpoint, and the point convergence intensity is at monotone decreasing trend from endpoint to midpoint.

From Tables 2 and 4, we can see that the numerical results converge to 0.7213≈^1/2_−1/2𝑢(𝑥)𝑑𝑥≈𝑄(ℎ)≈𝑄^′(ℎ). (4.17)

Furthermore, this constant is independent of well length, permeability, viscosity, and other fluid properties. Hence this result can be used to calculate flow rates for many realistic cases as follows: 𝑄=0.7213×2𝜋×𝐿×𝐾𝐻𝜇×𝑃_𝑖−𝑃₀. (4.18)

Here 𝐿 is the length of the hydraulically fractured well, 𝐾 is permeability, 𝜇 is viscosity 𝑃_𝑖 is pressure at an infinite distance from the well, and 𝑃₀ is wellbore pressure of the hydraulically fractured well. From (4.18), we can see that when a hydraulically fractured well is producing at constant wellbore pressure, its flow rate is directly proportional to well length, permeability, and pressure difference.

If (𝑃_𝑖−𝑃₀) is a constant, the productivity index is given by 𝐽_𝑣=0.7213×2𝜋×𝐿×𝐾𝐻𝜇=4.53205×𝐾𝐻𝐿𝜇. (4.19)

Using (2.17) and the results in Table 1, we can calculate the pressure at a point (𝑥,𝑦) as follows: 𝑃≈ℎ𝑥,𝑦2𝜋³¹_𝑖=0𝑢_𝑖ln𝑥−𝑥_𝑖+0.5²+𝑦². (4.20)

And the points, which satisfy 𝑃≈ℎ𝑥,𝑦2𝜋³¹_𝑖=0𝑢_𝑖ln𝑥−𝑥_𝑖+0.5²+𝑦²=constant (4.21) form equipotential threads when the well is producing at constant wellbore pressure. Figure 2 shows equipotential threads with different fluid potential values. It can be found that the equipotential threads can be approximately taken as a family of ellipses whose focuses are the two endpoints of the hydraulic fractured well. This conclusion is in accordance with the regular assumption in [6].

Figure 2 

Profile of equipotential threads of a hydraulically fractured well.

5. Conclusions

This paper proposes a new model using a weakly singular integral equation for the productivity of infinite-conductivity hydraulically fractured wells producing at constant wellbore pressure. A Galerkin method and a midrectangle algorithm are constructed to solve this integral equation. And their numerical results are similar to each other. Then the numerical solutions of this equation are utilized in the productivity formula, which provide fast analytical tools to evaluate production performance of infinite-conductivity fractured wells. Furthermore, the following conclusions are obtained.(1)The numerical results for the weakly singular integral equation are convergent. Moreover, the point convergence intensity is symmetric. And the point convergence intensity is monotonically decreasing from the endpoints to the midpoint when a hydraulically fractured well is producing at constant wellbore pressure. (2)The numerical approximation for the productivity is convergent when ℎ→0.(3)The equipotential threads, which are generated from the numerical results, can be approximately taken as a family of ellipses whose focuses are the two endpoints of the hydraulically fractured well. This conclusion is in accordance with the regular assumption in [6].