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The bottom-hole pressure response which can reflect the gas flow characteristics is important to study. A mathematical model for description of gas from porous coalbed methane (CBM) reservoirs with complex boundary conditions flowing into horizontal wells has been developed. Meanwhile, basic solution of boundary elements has been acquired by combination of Lord Kelvin point source solution, the integral of Bessel function, and Poisson superimpose formula for CBM horizontal wells with complex boundary conditions. Using this model, type curves of dimensionless pressure and pressure derivative are obtained, and flow characteristics of horizontal wells in complex boundary reservoirs and relevant factors are accordingly analyzed.

Coalbed methane (CBM) is a kind of green and clean energy. The development and utilization of coalbed methane could not only relieve the tense situation of conventional oil and gas in supply, but also reduce the atmospheric environment pollution.

Different from conventional gas reservoirs, the migration mechanism of gas in coal is more complex and diverse [

In order to improve the production, a large number of horizontal wells have been used in the CBM reservoirs. Sung and Ertekin [

Generally, the theories of calculating reservoir and bottom-hole pressures which can reflect the gas flow characteristics are mostly based on homogeneous reservoirs and regular geometry, such as infinite boundary or circular boundary. Outer boundary conditions of a reservoir have also been simplified; it is generally regarded as simple situations as constant pressure or closed boundary. However, influenced by characteristics of geological structures, the true reservoirs usually have complex and diversiform boundaries. In this case, the conventional flow theories and solving methods could do nothing to calculate reservoir or bottom-hole pressures with mixed boundary conditions.

The boundary element method (BEM) is a numerical computational method of solving linear partial differential equations developed after the better-known finite element method (FEM) and finite difference method (FDM). The BEM could be able to reduce dimension and save computer memory and running time. Coupling with the BEM [

In this paper, a mathematical model is developed to describe gas flow in horizontal wells in CBM reservoirs based on the theory of fluid flow through porous. The type curves of pressure derivative, characteristics of gas flow with complex external boundary, and relevant affecting factors are analyzed.

Coal reservoir is a dual porous medium composed of matrix and fracture. The matrix is the main reservoir space of coalbed methane (Figure

Pore characteristics of coal ((a) micropore; (b) fracture).

According to the reservoir dual pore structure of coal, a physical model is set up for gas flow. The migration of gas in coal is shown in Figure

Desorption and diffusion in the CBM reservoirs.

To facilitate the derivation, it is assumed that the length of a horizontal well is

Horizontal wells with different boundaries ((a) closed boundary; (b) constant pressure boundary; (c) mixed boundary).

The fundamental assumptions are as follows:

CBM diffuses directly from matrix to fracture, and the process of diffusion is unsteady.

Gas flow in fracture is radial laminar flow, in agreement with Darcy’s law.

Only single-phase gas flow exists in coal.

The effects of gravity and capillary force are negligible.

The effects of temperature are negligible.

CBM isothermal adsorption process is in line with the Langmuir isotherm adsorption law, and the initial state conforms to the isothermal adsorption curve.

Radius of the gas well is regarded infinitesimal, and gas well production is constant.

In combination with the mass conservation equation with the second Fick’s diffusion law, the change of gas concentration with time is given by

The planar radial flow equation is as follows:

The dimensionless equation of gas diffusion in matrix is

In the process of diffusion of CBM, the concentration often changes. Therefore, unsteady diffusion equation which accords with actual situation is used.

Free gas concentration is as follows:

Concentration of the adsorbed gas is

Gas concentration in coal is

Volume of gas desorption from coal is

Density of gas is defined as

Spherical matrix has the following relationship:

Combining (

Based on the basic principle of material balance, the governing equations of gas flow in fracture system are given by

Equation (

Diffusion and governing equations of gas flow in fracture are combined as follows:

Dimensionless initial and boundary conditions are given as follows.

The initial condition is

The boundary condition is

The infinite outer boundary condition is

The constant pressure boundary condition is

The closed boundary condition is

When

The Laplace transform of (

The general solution of (

The coefficients

Combining the dimensionless definition

So,

Laplace transform of (

The initial conditions are

The inner boundary conditions are

The outer boundary conditions are

Substituting the dimensionless definition

Substituting

Defining

Substituting (

Applying the theory of boundary element, previous equation can be solved from the integral transformation of the governing equation based on the expression of the fundamental solutions and differential equation of fluid flow through porous medium:

According to the properties of

The boundary

It is crucial to find its fundamental solution when the horizontal flow problem in complex reservoirs is resolved using the boundary element method. According to the properties of the boundary element and the mathematic equation governing pressure transmission in porous medium, the fundamental solution must satisfy the modified Helmholtz operator. The equation is given by

With Lord Kelvin point source solution, the fundamental solution of (

With mirror image method of fluid mechanics in porous medium, the transient point source fundamental solution of closed boundary is

With Poisson superposition formula, (

The fundamental boundary element solution of horizontal wells in a reservoir with closed top and bottom boundaries is

If

The integral boundary equation (

The unknown variable in the integral boundary equation is

Once the unknown variables are acquired, we can solve

In order to validate the gas flow model proposed in this paper, we test the adsorption and desorption data of coal. Then, we compare the boundary element model with those published works.

The adsorption and desorption data of coal are tested in the laboratory; the pressure continues to drop as the coal continues to adsorb methane. We deal with the experimental data in log-log coordinates as shown in Figure

Double logarithmic curve of pressure drop of test data.

A pressure buildup test example of a horizontal gas well in tight gas reservoir has been presented by Han [

Figure

Comparison of typical curves between horizontal wells in CBM and conventional gas reservoirs.

Type curves for horizontal wells in a CBM reservoir with complex boundary calculated by the BEM in this paper are shown in Figure

Typical curve of CBM horizontal well with complex boundary conditions (

Figure

Influence of diffusion coefficient

Figure

Influence of

Figure

Influence of

Figure

Influence of

Figure

Influence of

To demonstrate the application of the model proposed in this paper, a horizontal well ZX-12 in a CBM reservoir is studied. The vertical depth and horizontal length of this well are 689 m and 536 m, respectively, with well radius ^{3}/t and the Langmuir pressure

Fitted curves of test data from an actual well.

According to the actual field data, some of reservoir parameters are fitted as follows: the porosity is 0.04 and the permeability is 3 mD. The stage of desorption and boundary response can be clearly seen in Figure

The mathematic model of gas flowing into horizontal wells in CBM reservoirs with complex boundary conditions is derived based on the percolation theory. The curves of bottom-hole pressure and pressure derivative are obtained by using boundary element method and Laplace transform. The conclusions are as follows.

The fundamental boundary element solutions for transient pressure response of horizontal wells in CBM reservoirs with complex boundary conditions could be obtained by using Lord Kelvin’s point source solution, point source function theory, and Poisson’s summation formula.

Comparison of the typical curves of flow characteristics between horizontal wells in CBM and conventional gas reservoirs shows that there is an additional radial flow which reflects gas flow in fracture.

Comparing the typical curves of flow characteristics with complex boundary conditions, the pressure and pressure derivative curves with closed boundary would be upward after pressure transmitting to the boundary. The pressure derivative curves with constant pressure boundary and mixed boundary would be fallen, but the falling range of pressure derivative curve with mixed boundary is less.

Radius, m

Permeability, mD

Porosity, fraction

Diffusion flux,

Laplace transform variable

Thickness, m

Volume factor, fraction

Influx into the wellbore,

Half length of horizontal well, m

Viscosity, mPa·s

Concentration of coalbed methane,

Initial pressure, MPa

Production of well,

Langmuir volume constant, m^{3}/ton

Volume of coal matrix,

Pseudopressure

Fracture storage ratio, fraction.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This study is supported by National Natural Science Foundation of China (Grant no. 51304032), National Major Science and Technology Special Project of China (Grant no. 2011ZX05037-003), and Research Fund for the Doctoral Program of Higher Education of China (Grant no. 20125122110017).