Solution and type curves for the seepage model of the water-bearing coalbed with leakage recharge

RESUMEN


Introduction
The coalbed methane (CBM) exists, in the form of adsorbed gas, in the coal reservoir. To recover the CBM, the coalbed is firstly dewatered and depressurized. Only when the coal seam pressure drops to the critical desorption pressure, can the methane be desorbed and diffuse from matrix pores to fracture network. In the Wyodak Anderson coalbed (Onsager and Cox, 2000;Cox and Onsager, 2002) of Fort Union Formation in the Powder River Basin, U.S., the 15# coalbed in Shouyang-Yangquan of the Qinshui Basin, China, and other coal measure strata (Meng 2010), water bodies exist in the upper or lower coalbed, in addition to the coalbed cleat system. The silty or muddy aquitard with a certain thickness of the coalbed and the aquifer has a certain permeability. During the coalbed dewatering, the pressure difference between the aquifer and the coalbed may drive the interlayer leakage recharge, and accordingly water production of the gas well may change, thus impressing the coalbed dewatering and depressurizing.
The scholars in China and abroad have frequently studied interlayer leakage of the reservoir. In the 1960s and 1970s, some scientist elsewhere investigated the reservoir seepage in the case of leakage (Katz and Tek, 1962;Russell and Prats, 1962). In China, Jia Yonglu established the dynamic model of double-layer reservoir with inter-layer leakage in 1997 (Jia 1997). Then proposed the double-layer dual-porosity model, double-layer triple-porosity model and triple-layer confined model with inter-layer leakage (Zhang et al., 2009;Chen et al., 2011;Huo et al., 2006). Sun Hedong, Gao Chengtai, et al. established the dynamic model of infinite triple-layer oil and gas reservoir with inter-layer leakage. Wu Yilu analyzed the causes for the leakage in the multi-layer gas reservoir and established the mathematical seepage model for the multi-layer gas reservoir with leakage [12]. For the models established in previous studies, all layers were perforated for exploitation, and horizontal radial seepage occurred in each layer, except for inter-layer leakage. However, when the coalbed is perforated for production, the aquifers in the roof and floor are selectively avoided, so horizontal radial flow only occurs in the coalbed. Therefore, the existing models for multi-layer reservoir with leakage are not suitable for the analysis of inter-layer leakage in the CBM wells. According to the unsteady seepage theory, the infiltration model for CBM wells considering the inter-layer leakage can be established to obtain the analytical solution of bottom hole flowing pressure under Laplace space. After the solution in true space is obtained by Stehfest numerical inversion, new type curves can be established. Then, these type curves are used to judge whether any leakage recharge exists in the coalbed during dewatering, so as to determine the formation parameters and leakage parameters and analyze the level of leakage recharge.

Physical Model
As shown in Figure 1, for a CBM well in the infinite formation center, only the coalbed is perforated for dewatering and gas production. Compared with the water content in the coalbed, the water body of the aquifer in the roof and floor can be regarded as infinite water body with a great volume (Su and Lin, 2009). Aquitard with low permeability exists between coalbed and water body. Based on the data statistics of the Powder River Basin in the United States, the vertical permeability of such aquitard is 0.003-0.03 md, suggesting a certain impermeable capacity (Onsager and Cox, 2000).
According to the research on the leakage process of the underground water by Hantush and Jacob, et al., the model is based on the following assumptions (Hantush and Jacob, 1955a;Hantush 1955b): 1). Each layer is homogeneous and isotropic, neglecting the effect of capillary pressure and gravity; 2). Each layer has an infinite boundary; 3). Only the coalbed is perforated, and the formation water in the coalbed flows into the wellbore in the form of horizontal radial flow; 4). The formation water in the floor recharges the coalbed through the aquitard in the manner of vertical seepage and then the inter-layer leakage occurs; 5). The compressibility of the rock in the aquitard is not considered; 6). Due to the significant volume, the water body in the aquifer can be regarded as the constant pressure recharge boundary for the coalbed water.

Mathematical Model and Solution
In the coalbed with leakage recharge, the basic mathematic equation for water seepage is expressed as (Streltsova, 1988): (1) The aquifer is regarded as the infinite water body, so the boundary between the aquifer and the aquitard can be treated as the constant pressure boundary. The flow rate recharged into the coalbed by the aquifer through the aquitard is (2) Based on the definition of leakage coefficient , the dimensionless leakage coefficient is defined as: ( 3) Equation (2) is substituted into Equation (1). According to the definitions of dimensionless pressure, dimensionless time and dimensionless radius, the dimensionless partial differential equation is obtained after simplification.
(4) Initial condition: ; Outer boundary condition: ; Inner boundary condition: According to the definition of Laplace transform: V Equation (4) is transformed to ordinary differential equation: Similarly, the expressions of the stablish condition in Laplace space are as follows: (6) (8) Equation (5)

Results and Discussion
In the traditional well testing analysis, the reservoir permeability, skin factor, and other formation parameters can be determined, with the measured data in type curve matching. For the coalbed with leakage recharge, the leakage coefficient should also be determined and the level of interlayer leakage recharge be analyzed. Next, considering the dewatering test with continuous production and water injection test with constant flow rate, the corresponding type curves are derived.

Type Curves of the Bottom hole Pressure in the Dewatering with Constant Production
The drawdown well test is conducted to measure the change of the bottom hole fluid pressure with time in the process of dewatering with constant production. According to the ideas of Ramey, Gringarten, and Bourdet to make type curves (Liu, 2003). New type curves applicable to dewatering test analysis with constant production can be derived by establishing a new combination of parameters.
The dynamic change of the bottom-hole pressure tested is always susceptible to the wellbore storage effect. Even before the wellbore storage effect ends, the leakage effect of the aquifer has superseded the influence of after flow. To avoid the interference of wellbore storage effect, the well should be opened or shut in by using downhole control valve in the test process (Yang et al., 2015). The wellbore storage coefficient is minuscle and can be ignored, i.e. 0 C D = .

Let
, then . Equation (11) can be simplified as: It is proved below that is the Laplace transformation result of D p based on .
Equation (12) can be finally written as: (14) The solution in right space can be calculated by using Stehfest numerical inversion. The basic equation is [19] : In the log-log coordinate system, a group of dimensionless pressure type curves ( Figure 2) are prepared, with as ordinate against as abscissa and , the model only has radial seepage, and the derivative curve of the bottom-hole pressure is a horizontal line, which conforms to the study results of Gringarten, Bourdet, et al. (Yang et al., 2015).
According to the definition of D b , the effect of aquitard thickness conf h and vertical permeability conf k , v on the formation water leakage is comprehensively considered. From the underground fluid seepage law, we can see that the smaller the aquitard thickness conf h is, or the bigger the vertical permeability conf k , v of the aquitard is, the weaker the barrier action of the aquitard on the vertical flow of the formation water is and the stronger the effect of the formation water leakage on coalbed seepage is. Figures 2-3 indicates that the bigger the value of D b is (i.e. the bigger the value of -2S D e b is), the earlier the leakage affects the coalbed water seepage and the earlier the coalbed radial flow stage ends.
In the actual dewatering and production of the coalbed, it is also proved that when the surrounding rock of the coalbed roof and floor has high permeability and strong leakage effect, desorption of CBM can only occur in the small range near the borehole, which is manifested as the deficiency of gas supply (Zhang et al., 2013).
According to the definition of dimensionless pressure It is shown that the smaller D p is, the higher the bottom-hole flowing pressure is. In other words, the inter-layer leakage may increase the bottomhole pressure and affect desorption of the methane. According to the characteristics of the type curves, the existence of leakage recharge in the aquifer during testing can be qualitatively judged.  Then, the formation parameters can be obtained by using the type curves shown in Figures 2-3.

Formation Parameters Matching by Using the Type Curves
First, the actual testing data obtained from the well site is processed. On the log-log coordinate with the same size of type curves, the curve is drawn with actual differential pressure as ordinate and actual dewatering time t as abscissa and then matched with the type curves in Figure 2.
Permeability can be obtained through the pressure matching value.
The skin coefficient can be obtained by time matching value.
The leakage coefficient can be obtained by curve matching value. (20)

Formation Parameters Matching by Using the Type Curves of Pressure Derivative
Similarly, on the log-log coordinate with the same size of pressure derivative type curves, the curve is drawn with the product of actually measured differential pressure derivative and time t as ordinate against actual dewatering time t as abscissa, and then matched with the interpretation chart.
Permeability can be obtained through the pressure derivative matching value.
By time matching, the skin coefficient can be obtained and its calculation equation is same with Equation (19). By curve value matching, the leakage coefficient can be obtained and its calculation equation is same with Equation (20).

Type Curves of the Bottom-hole Pressure in the Water Injection Test with Constant Production
In the process of dewatering test, the bottom-hole pressure may drop below the critical desorption pressure, resulting in the gas and water (two-phase) flow, which may interfere the well test interpretation. In order to prevent such two-phase flow, the coalbed can be tested by using the method of water injection with constant flow rate. In the process of water injection, the added value of the bottom-hole pressure is (22) The differential pressure is transformed to , then all the parameters may be determined and the effect of leakage on gas well production can be analyzed by using the type curves in Figure.2 and Figure.3 directly and matching of Equations (18)-(21).

Conclusions
(1) The mathematical model for inter-layer leakage in CBM wells is established. Under bottom hole open and shut-in condition, the true spatial solution of the mathematical model is obtained by using Laplace transformation and Stehfest reverse inversion.
(2) New combination of parameters is developed under the precondition of dewatering or water injection with constant production, respectively. On the log-log coordinate, the type curves of and , and are drawn, respectively. So the existence of leakage recharge in the aquifer in the process of testing can be judged. The characteristics of the type curves show that the bigger the leakage efficient is, the earlier the leakage recharge is and the higher the bottom-hole pressure is.
(3) By type curve matching, the coalbed permeability, skin factor, and leakage coefficient can be determined. The curve matching contributes to quantitatively evaluating the level of leakage recharge, and judging the effect of the aquifer on the coalbed dewatering and production, so that subsequent dewatering and production can be adjusted timely and properly. indicates the dimensionless pressure derivative, dimensionless; q indicates daily water production, m3/d; B indicates the water volume factor, m 3 /m 3 ;

Explanation of Symbols
indicates the producing pressure drop, MPa; i p indicates the initial formation pressure, MPa; indicates the bottom-hole flowing pressure, MPa; indicates the drawdown pressure derivative.