This paper presents a study of characteristic value method of well test analysis for horizontal gas well. Owing to the complicated seepage flow mechanism in horizontal gas well and the difficulty in the analysis of transient pressure test data, this paper establishes the mathematical models of well test analysis for horizontal gas well with different inner and outer boundary conditions. On the basis of obtaining the solutions of the mathematical models, several type curves are plotted with Stehfest inversion algorithm. For gas reservoir with closed outer boundary in vertical direction and infinite outer boundary in horizontal direction, while considering the effect of wellbore storage and skin effect, the pseudopressure behavior of the horizontal gas well can manifest four characteristic periods: pure wellbore storage period, early vertical radial flow period, early linear flow period, and late horizontal pseudoradial flow period. For gas reservoir with closed outer boundary both in vertical and horizontal directions, the pseudopressure behavior of the horizontal gas well adds the pseudosteady state flow period which appears after the boundary response. For gas reservoir with closed outer boundary in vertical direction and constant pressure outer boundary in horizontal direction, the pseudopressure behavior of the horizontal gas well adds the steady state flow period which appears after the boundary response. According to the characteristic lines which are manifested by pseudopressure derivative curve of each flow period, formulas are developed to obtain horizontal permeability, vertical permeability, skin factor, reservoir pressure, and pore volume of the gas reservoir, and thus the characteristic value method of well test analysis for horizontal gas well is established. Finally, the example study verifies that the new method is reliable. Characteristic value method of well test analysis for horizontal gas well makes the well test analysis process more simple and the results more accurate.
Recent years have seen the evergrowing application of horizontal wells technology, which aroused considerable interest in the exploration of horizontal well test analysis [
Numerous studies on the pressure transient analysis of horizontal wells have been documented extensively in the literature. Combined with Newman’s product method, Gringarten and Ramey [
Owing to the imperfection of common well test analysis methods including the semilog data plotting analysis technique [
The hypothesis: the formation thickness is
Physical model of horizontal gas well seepage.
Considering the complexity of the seepage flow mechanism of horizontal gas well and in order to make the mathematical model’s solving and calculation more simple, the establishment of mathematical models are divided into two parts: one is to ignore the effect of wellbore storage and skin effect; the other is to consider the effect of wellbore storage and skin effect [
The diffusivity equation is expressed by Ozkan and Raghavan [
Initial condition is
Inner boundary condition is
Infinite outer boundary condition in horizontal direction is
Closed outer boundary condition in horizontal direction is
Constant pressure outer boundary condition in horizontal direction is
Closed outer boundary conditions in vertical direction are
The dimensionless variables are defined as follows:
The defined gas pseudopressure is
According to Duhamel’s principle [
The solutions of the mathematical models [
For gas reservoir with closed outer boundary in vertical direction and infinite outer boundary in horizontal direction, according to (
For gas reservoir with closed outer boundary both in vertical and horizontal direction, according to (
For gas reservoir with closed outer boundary in vertical direction and constant pressure outer boundary in horizontal direction, according to (
Making the Laplace transform to
For gas reservoir with infinite outer boundary in horizontal direction, according to (
Well test analysis type curve of horizontal well in gas reservoir with infinite outer boundary.
As seen from Figure
The characteristic of pure wellbore storage period of horizontal well is the same as vertical well, which is manifested as a 45° straight line segment on the loglog plot of
Expressions of dimensionless bottomhole pseudopressure and pseudopressure derivative during this period can be obtained. This results in
The early vertical radial flow period appears after the effect of wellbore storage; the characteristic of this period is manifested as a horizontal straight line segment on the loglog plot of
The schematic diagram of early vertical radial flow.
Expressions of dimensionless bottomhole pseudopressure and pseudopressure derivative during this period can be obtained. This results in
The early linear flow period appears after the early vertical radial flow period. The characteristic of this period is manifested as a straight line segment with a slope of 0.5 on the loglog plot of
The schematic diagram of early linear flow.
Expressions of dimensionless bottomhole pseudopressure and pseudopressure derivative during this period can be obtained. This results in
The late horizontal pseudoradial flow period appears after the early linear flow period. The characteristic of this period is manifested as a horizontal straight line segment with the value of 0.5 on the loglog plot of
The schematic diagram of late horizontal pseudoradial flow.
Expressions of dimensionless bottomhole pseudopressure and pseudopressure derivative during this period can be obtained. This results in
For gas reservoir with closed outer boundary in horizontal direction, according to (
Well test analysis type curve of horizontal well in gas reservoir with closed outer boundary.
As seen from Figure
Expressions of dimensionless bottomhole pseudopressure and pseudopressure derivative during the pseudosteady state flow period can be obtained. This results in
For gas reservoir with constant pressure outer boundary in horizontal direction, according to (
Well test analysis type curve of horizontal well in gas reservoir with constant pressure outer boundary.
As seen from Figure
The characteristic value method of well test analysis for horizontal gas well can determine the gas reservoir fluid flow parameters according to the characteristic lines which are manifested by pseudopressure derivative curve of each flow period on the loglog plot.
The characteristic of pure wellbore storage period of horizontal well is manifested as a straight line segment with a slope of 1 on the loglog plot of
Equation (
The following can be obtained from the definitions of dimensionless variables:
According to (
According to the definitions of dimensionless variables, the dimensional form of (
The geometric mean permeability of gas reservoir can be determined by (
According to the definitions of dimensionless variables and (
For pressure buildup analysis, when
The initial reservoir pseudopressure can be determined by (
The dimensionless bottomhole pseudopressure derivative curve is manifested as a straight line segment with the slope of 0.5 during the early linear flow period. According to the expression of dimensionless bottomhole pseudopressure derivative during this period and the definitions of dimensionless variables, the following can be obtained:
The horizontal permeability of gas reservoir can be determined by (
Combining (
The dimensionless bottomhole pseudopressure derivative curve is manifested as a horizontal straight line segment with the value of 0.5 during the late horizontal pseudoradial flow period. According to the expression of dimensionless bottomhole pseudopressure derivative during this period and the definitions of dimensionless variables, (
The horizontal permeability of gas reservoir can be determined by (
For pressure buildup analysis, when
The initial reservoir pseudopressure can be determined by (
The dimensionless bottomhole pseudopressure derivative curve is manifested as a straight line segment with the slope of 1 during the pseudosteady flow period. According to the expression of dimensionless bottomhole pseudopressure derivative during this period and the definitions of dimensionless variables, (
The pore volume of gas reservoir can be determined by (
The Longping 1 well is a horizontal development well in JingBian gas field, the well total depth is 4672 m, the drilled formation name is Majiagou group, the middepth of reservoir is 3425.63 m, and the well completion system is screen completion. According to the deliverability test during 26–29 December, 2006, the calculated absolute open flow was 94.26 × 10^{4} m^{3}/d. The commissioning data of Longping 1 well was in 12 May, 2007, the initial formation pressure was 29.39 MPa, before production, and the surface tubing pressure and casing pressure were both 23.90 MPa. The production performance curves of Longping 1 well are shown in Figures
The gas flow rate and water flow rate curve of Longping 1 well.
The tubing head pressure and casing head pressure curve of Longping 1 well.
Longping 1 well has been conducted pressure buildup test during 14 August, 2007, and 23 October, 2007. The gas flow rate of Longping 1 well was 40 × 10^{4} m^{3}/d before the shutin. The bottomhole pressure recovered from 22.38 MPa to 27.83 MPa during the pressure buildup test. Physical parameters of fluid and reservoir are shown in Table
Physical parameters of fluid and reservoir.
Parameter  Value 

Initial formation pressure 
29.39 
Formation temperature 
95.80 
Formation thickness 
6.31 
Porosity 
7.77 
Initial water saturation 
13.60 
Well radius 
0.0797 
Gas gravity 
0.608 
Gas deviation factor 
0.9738 
Gas viscosity 
0.0222 
The pressure buildup loglog plot of Longping 1 well.
As seen from the contrast between Figure
Using the above characteristic value method of well test analysis for horizontal gas well, well test analysis results of Longping 1 well are shown in Table
Well test analysis results of Longping 1 well.
Parameter  Parameter values 

Wellbore storage coefficient 
1.229 
Horizontal permeability 
7.742 
Vertical permeability 
0.039 
Flow capacity 
48.857 
Skin factor 

Effective horizontal section length 

Reservoir pressure 
28.385 
The pressure buildup semilog plot of Longping 1 well.
The pressure history matching plot of Longping 1 well.
The four main conclusions and summary of this study are as follows.
On the basis of establishing the mathematical models of well test analysis for horizontal gas well and obtaining the solutions of the mathematical models, several type curves which can be used to identify flow regime have been plotted and the seepage characteristic of horizontal gas well has been analyzed.
The expressions of dimensionless bottomhole pseudopressure and pseudopressure derivative during each characteristic period of horizontal gas well have been obtained; formulas have been developed to calculate gas reservoir fluid flow parameters.
The example study verifies that the characteristic value method of well test analysis for horizontal gas well is reliable and practical.
The characteristic value method of well test analysis, which has been included in the well test analysis software at present, has been widely used in vertical well. As long as the characteristic straight line segments which are manifested by pressure derivative curve appear, the reservoir fluid flow parameters can be calculated by the characteristic value method of well test analysis for vertical well. The proposed characteristic value method of well test analysis for horizontal gas well enriches and develops the well test analysis theory and method.
Wellbore storage coefficient, m^{3}/MPa
Dimensionless wellbore storage coefficient
Total compressibility, MPa^{−1}
Reservoir thickness, m
Dimensionless reservoir thickness
Modified Bessel function of first kind of order
Horizontal permeability, mD
Modified Bessel function of second kind of order
Vertical permeability, mD
Horizontal section length, m
Dimensionless horizontal section length
Pseudopressure, MPa^{2}/mPa
Dimensionless pseudopressure
Initial formation pseudopressure, MPa^{2}/mPa
Dimensionless bottomhole pseudopressure
Flowing wellbore pseudopressure, MPa^{2}/mPa
Derivative of
Laplace transform of
Laplace transform of
Pseudopressure difference, MPa
Derivative of
Casing head pressure, MPa
Initial formation pressure, MPa
Reservoir pressure, MPa
Tubing head pressure, MPa
Flowing wellbore pressure at shutin, MPa
Gas flow rate, 10^{4} m^{3}/d
Water production rate, m^{3}/d
Radial distance, m
Dimensionless radial distance
Outer boundary distance, m
Dimensionless outer boundary distance
Wellbore radius, m
Dimensionless wellbore radius
Skin factor
Laplace transform variable
Initial water saturation, fraction
Time, hours
Dimensionless time
Production time, hours
Dimensionless production time
Shutin time, hours
Formation temperature, °C
Vertical distance, m
Dimensionless vertical distance
Horizontal section position, m
Dimensionless horizontal section position
Gas deviation factor
Tiny variable
Porosity, fraction
Gas viscosity, mPa
Gas gravity.
Dimensionless
Early
Early of buildup
Horizontal
Initial
Linear
Late horizontal pseudoradial
Late horizontal pseudoradial of buildup
Pseudosteady flow period
Total
Vertical
Flowing wellbore
Shutin wellbore.
The authors declare that there is no conflict of interests regarding the publication of this paper.