1. Introduction
Whether CO
2 geological sequestration, disposal of high-level radioactive waste, or exploitation of geothermal and hydrocarbon resources, all of them are associated with fractured reservoirs [
1,
2,
3]. Therefore, it is increasingly significant to understand fluid flow behavior through naturally fractured reservoir in hydrology, environment, and petroleum engineering. Especially, with the development of natural gas exploration and exploitation in the world, the proven reserves and production of naturally fractured gas reservoirs are increasing year by year [
4].
The majority of naturally fractured gas reservoirs exhibit abnormally high-pressured as a result of disequilibrium compaction [
5]. In China, Kela-2 gas field in Tarim Basin and Moxi gas field in Sichuan Basin which have been developed for several years are both abnormally high-pressured fractured gas reservoirs. Due to the characteristics of abnormal pressure, the rock and fluid properties of naturally fractured gas reservoirs are commonly pressure-dependent.
With microseismic monitoring and transient pressure analysis, it is demonstrated that some naturally fractured gas reservoirs may exhibit composite properties in the radial direction. That is, a large number of natural fractures exist near the wellbore area while the outer region far away from well has fewer ones [
6]. Therefore, a composite model is appropriate to analyze the production performance of the well. Composite reservoir models were originally targeted to model special reservoir configuration consisting of two or more concentric zones with different diffusivities or fluid properties. As early as 1960, Hurst [
7] developed the analytical solution for radially composite reservoirs in which the storativity ratio between two regions are identical. Two decades later, Olarewaju and Lee [
8,
9] proposed an analytical solution to analyze pressure transient behavior for wells in finite composite reservoir system. Besides the composite formation with single porosity media in inner and outer regions, many researchers [
10,
11,
12,
13,
14] also developed the mathematical model for composite dual-porosity reservoirs. Subsequently, Kuchuk and Habashy [
15] solved fluid flow problems in composite reservoirs by reflection and transmission theory of electromagnetics. In recent years, composite model is applied to describe stimulated region volume around well in unconventional reservoirs [
16,
17,
18].
Unfortunately, most literature aimed at studying fully penetrated wells in composite reservoirs. Nevertheless, some wells in actual fields were partially penetrated especially in thick formations. For the research of partially penetrated wells, extensive works were carried out in hydrology and petroleum engineering. Muskat [
19] first investigated steady flow through porous media for partially penetrated vertical well. Nisle [
20] and Brons and Marting [
21] studied the effect of partial penetration on the pressure drawdown and buildup of oil wells in an isotropic layer with no-flow boundary conditions at the top and bottom of the layer. Odeh [
22] established the steady-state flow model of partially penetrated well by Fourier transforms. Kazemi and Seth [
23] studied the combined effect of anisotropy and stratification with crossflow on pressure response for partially penetrated well. With Laplace and Hankel transforms, Streltsova [
24,
25,
26] solved the unsteady-state flow problems for partially penetrated well. The Laplace transform was also used by Dougherty and Babu [
27] to analyze hydraulics problem for a partially penetrated well in a dual-porosity reservoir. Bui et al. [
28] developed an analytical solution to analyze transient pressure behavior of partially penetrated wells in naturally fractured reservoirs with infinite radial extent. Based on the solution of Bui et al. [
28], a set of pressure derivative type curves were generated by Slimani and Tiab [
29]. Recently, Mishra et al. [
30] studied the radial flow to a partially penetrated well with storage in an anisotropic confined aquifer. Biryukov and Kuchuk [
31] presented an analytical pressure transient solution for a limited-thickness open cylindrical interval on a nonpermeable cylindrical wellbore. In their model, the mixed boundary value problems were investigated. Javandel [
32] proposed a semianalytical solution for partial penetration in two layer aquifers. Dejam et al. [
33] studied the effect of a constant top pressure on the pressure transient analysis of a partially penetrated well in an infinite-acting fractured reservoir with wellbore storage and skin factor effects. However, most of them merely investigated transient pressure behavior for partially penetrated well in oil reservoirs or aquifers. Despite the great efforts presented in the aforementioned literature, the rate decline analysis model for partially penetrated wells in composite gas reservoir is lacking, particularly in stress-sensitive composite naturally fractured gas reservoir.
The objective of this paper is to develop a semianalytical model for rate decline analysis of partially penetrated well in abnormally high-pressured composite naturally fractured gas reservoirs. By the definition of pseudopressure function and pseudotime factor, fluid and rock pressure-dependent properties were taken into consideration. Laplace and finite Fourier cosine transforms were used to solve the 2D diffusivity equation. Based on the proposed model, the effects of prevailing factors on production performance were investigated. Finally, some remarkable conclusions are drawn.
3. Model Verification
Since there is no relevant literature on partially penetrated well in composite naturally fractured gas reservoir, to validate the accuracy of the proposed model, the comparison of this model with an analytical solution of completely penetrated well in composite reservoir is implemented.
Although Prado and Da [
10] mainly aimed at analyzing transient pressure, the dimensionless production rate solution can also be obtained based on the relationship of Equation (14). Therefore, we consider the circumstance of
hw =
h in our model, which means the well is penetrated completely. We compare the results obtained from our model with Prado and Da’s. The dimensionless parameters used for validation are presented in
Table 1. All of the dimensionless parameters are defined in
Appendix A.
Figure 3 shows the comparison of production rate curves. The lines represent the results obtained from the proposed model, and the dots represent the results obtained from Prado and Da’s method. As we can see, there is a good agreement between these two models, which indicates that our proposed model for partially penetrated well in composite naturally fractured reservoir is accurate.
5. Field Data Matching
In order to prove the practical application of the proposed model, the production data of an actual case from Moxi gas field in Sichuan Basin was used. The well was partially penetrated and a schedule of constant well bottom-hole pressure was adopted at the initial production stage. Thus, the gas flow rate was characterized by a decreasing trend for a long time. The following equation was used to describe the relationship between porosity and reservoir pressure:
where,
φi and
φ denote porosity at initial and current reservoir pressure, respectively, following Equation (3);
cφ denotes formation compressibility, and it can be calculated by the following equation:
where,
cm and
cf denote matrix and fracture system compressibility, respectively, following Equation (4);
φm and
φf denote matrix and fracture system compressibility, respectively.
All input parameters are listed in
Table 3.
Figure 14 illustrates a comparison between the field production rate data and the matching results obtained from the proposed model. In
Figure 14, the solid line represents the matching result considering formation compressibility. The dash line represents the matching result not considering formation compressibility. That is, the formation compressibility equals to zero and the porosity is constant. As we can see from
Figure 14, the real field data represented by the red circles is in good agreement with the result considering formation compressibility. The matching data is not good for the model without considering formation compressibility at late time. Therefore, it is necessary to consider formation compressibility and porosity stress-sensitivity in abnormally high-pressure gas reservoir production performance analysis.
6. Conclusions
Through the modified definition of pseudopressure and pseudotime factor, the 2D diffusivity equations of gas flow in abnormally high-pressured composite naturally fractured reservoirs are established. The semianalytical solution was obtained by the utility of Laplace and finite Fourier cosine transforms. The presented model can account for the pressure-dependent fluid and rock properties, permeability anisotropy, and limited-entry well phenomena. Real gas field data matching demonstrates that the proposed model has good applicability. Based on this work, the following conclusions can be drawn:
(1) Porosity stress-sensitivity is considered due to the high formation energy in abnormally high-pressured reservoirs and the classical Warren–Root model is improved to account for permeability anisotropy.
(2) The inner region radius and mobility ratio between outer and inner region exhibit the same characteristics of the effect on production performance. Larger the value of them, the higher production rate at initial stage while rapid reduction in late stage due to weak supplement.
(3) The fracture permeability anisotropy factor mainly affects production rate in early stage. In late stage, the effect of it is small. Permeability stress-sensitivity has a notably negative effect on production rate. To avoid the rapid decrease of fracture permeability, reasonable well bottom-hole pressure should be set up in field.