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Analytical theory of coalbed methane recovery by gas injection


Analytical Theory of Coalbed Methane Recovery by Gas Injection
Jichun Zhu, Kristian Jessen, Anthony R. Kovscek, and Franklin M. Orr Jr., Stanford U.

Summary Injection of eithe

r carbon dioxide (CO2) or nitrogen (N2) enhances recovery of coalbed methane. In this paper, we provide new analytical solutions for the flow of ternary gas mixtures in coalbeds. The adsorption/desorption of gaseous components to/from the coalbed surface is approximated by an extended Langmuir isotherm, and the gas-phase behavior is predicted by the PengRobinson equation of state (EOS). Langmuir isotherm coefficients are used that represent a moist Fruitland coal sample from the San Juan basin (U.S.A.). In these calculations, mobile liquid is not considered. Given constant initial and injection compositions, a self-similar solution consisting of continuous waves and shocks is found. Mixtures of CH4, CO2, and N2 are used to represent coalbed and injection gases. We provide examples for systems where the initial gas is largely CH4, and binary mixtures of CO2 and N2 are injected. Injection of N2-CO2 mixtures rich in N2 leads to relatively fast initial recovery of CH4. Injection of mixtures rich in CO2 gives slower initial recovery, increases breakthrough time, and decreases the injectant needed to sweep out the coalbed. The solutions presented indicate that a coalbed can be used to separate N2 and CO2 chromatographically at the same time coalbed methane (CBM) is recovered. Introduction Coalbeds have large internal surface area and strong affinity for certain gas species such as CH4 and CO2. In CBM reservoirs, most of the total gas exists in an adsorbed state at liquid-like density. Only a small amount of the total gas is in a free phase. Primary recovery using depressurization induces desorption of the CBM by lowering the overall pressure of the reservoir. Primary recovery factors are roughly 50%.1 On the other hand, enhanced recovery of coalbed methane (ECBM) by injecting a second gas maintains the overall reservoir pressure, while lowering the partial pressure of CBM in the free gas. Injectants also sweep desorbed gas through the reservoir. Nitrogen is a natural choice as an injection gas because of its availability. Carbon dioxide is also promising because of the additional benefit of greenhouse gas sequestration. When combusted, methane emits the least amount of CO2 per unit of energy released among all the fossil fuels. Therefore, there is a synergy between CO2 sequestration and production of methane that leads to greater utilization of coalbed resources for both their sequestration ability and energy content. The first application of ECBM by CO2 injection has been carried out in the San Juan Basin.1 One important aspect of ECBM is the adsorption and desorption behavior of gas mixtures on coalbeds. A significant amount of work has been invested on this issue as it is related to coal-mine safety.2–11 However, transport of multicomponent gas mixtures through coalbeds has not been examined in detail. Arri et al.12 studied the primary recovery of a single sorbing component and ECBM by nitrogen injection. In this paper, we extend the analysis to systems with three adsorbing components. Besides CH4, coalbeds may contain significant amounts of CO2, N2, and other gas

species. A coalbed gas representative of San Juan Basin conditions is composed of about 93% CH4, 3% CO2, 3% wet gases, and 1% N2.13 Further, when flue gas is used for ECBM and CO2 sequestration, the injection gas may contain more than two components. The exact composition of flue gas depends on the combustion temperature, oxygen content of the air supply, and fuel moisture content, among other factors.14 Accordingly, a complete spectrum of injection gas conditions is studied. We solve the flow problem analytically using the method of characteristics. Similar approaches have been used for related problems. Rhee et al.15 modeled convective and adsorptive exchange processes from a chromatography point of view. Helfferich16 investigated a similar problem from a different standpoint. A set of rules for coherent waves was developed based on qualitative features of frontal displacement. The topology of the wave pattern was laid out in distance and time, and constructed in a step-bystep fashion. The analyses of Rhee et al.15 and Helfferich16 are extended to the adsorption, desorption, and transport of CBM gases during the ECBM process with gas injection. In a work related to gas injection for enhanced oil recovery, Dindoruk17 coupled equilibrium phase behavior of a multicomponent mixture with multiphase fluid flow through porous media using the method of characteristics. The analysis included the effect of volume change upon mixing and complex phase behavior through an EOS. To develop an analytical model for ECBM, we adopt the analysis of Dindoruk to describe volume change on mixing and solve for flow with adsorption and desorption behavior. Gas properties as a function of composition are described by the Peng-Robinson EOS.18 The following assumptions were made: ? Flow is one-dimensional (1D). ? The adsorption and desorption of gas mixtures on coal is modeled with the extended Langmuir isotherm.19 ? Adsorbed gases occupy negligible volume on coalbed surfaces. ? Flow is single-phase. Any water phase present is immobile as might occur after the coalbed is produced by a primary method. ? The temperature remains constant. ? For the purpose of calculating the adsorption and gas density, the pressure is assumed constant. ? The porosity and permeability of the coalbed are constant and uniformly distributed. ? Frontal advance is rapid enough that diffusion and dispersion along the axial direction are neglected. ? The effects of gravity are negligible. These assumptions simplify the problem while still allowing valuable insights into the subject processes. Importantly, they allow us to begin the formulation of a general analytical theory for multicomponent, two-phase flow in coalbeds. In the following sections, we describe the extended Langmuir isotherm, establish a mathematical model based on material balances, and discuss the solution procedure for systems with up to three components. The analytical solution method is then applied to a few examples, in which we study the effect of injection gas composition on CBM recovery and CO2 storage. Extended Langmuir Isotherm The adsorption behavior is represented by an extended Langmuir isotherm described by Ruthven19 and Yang20 as
371

Copyright ? 2003 Society of Petroleum Engineers This paper (SPE 87338) was revised for publication from paper SPE 75255, prepared for presentation at the 2002 SPE/DOE Improved Oil Recovery Symposium, Tulsa, 13–17 April. Original manuscript received for review 29 May 2002. Revised manuscript received 11 May 2003. Manuscript peer approved 5 September 2003.

December 2003 SPE Journal

ai =

?i ?rVmi pi
1+

?Bp
j=1

nc

, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (1)

j j

where ai ? ??the molar concentration of the adsorbed component i, ?r ? ??the density of the coalbed, ?i ? ??the molar density of component i at standard condition, and Vmi and Bi the Langmuir constants for component i at a given temperature and are normally expressed with units of psi?1 and scf/ton, respectively. The partial pressure, pi, of component i in the free gas phase is pi = zip, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2) where zi ? ??the mole fraction of species i. Mathematical Basis To formulate the problem, we consider a system with nc arbitrary components and a single gaseous phase. A 1D material balance for component i is written as ?Ci ?Fi + = 0, i = 1, . . . , nc, . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3) ?? ?? where Ci ? ??the overall molar concentration of component i, including free and adsorbed species, and Fi ? ??the molar flux of component i. The dimensionless distance, ?, and the dimensionless time, ?, are related to distance and time variables, x and t, as x ? = , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4) L

znc?1, and the local flow velocity u. Between the initial condition at the downstream boundary and the injection condition at the upstream boundary, there are an infinite number of intermediate states. Each state propagates downstream at a specific velocity. Possible solution paths are mapped in composition space from the initial composition to the injection composition. A complete solution path consists of multiple continuous or discontinuous solution segments that satisfy a material balance in differential or integral form separately. The segments along the solution path are subjected to the velocity rule that requires the characteristic wave velocity to increase monotonically from upstream to downstream. For portions of the solution where compositions vary continuously, the differential material balances are converted to an eigenvalue problem that reads
??H? ? ??G?? x = 0, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (10)

where Hif = Gij = x = ?Fi , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (11) ?zj ?Ci , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12) ?zj ?znc ? 1 ?u ?z1 ?z2 , ,..., , ?? ?? ?? ??

?

?

T

. . . . . . . . . . . . . . . . . . . . . . . (13)

?=

qt , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5) ? AL

where L ? ??the characteristic length of the 1D model, A ? ??the cross-sectional area of the 1D model, q ? ??the volumetric flow rate, and ? ? ??the porosity of the coalbed matrix. The concentration and flux are described as Ci = ? ? zi + ?1 ? ?? ai, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6) Fi = u ? z?, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7) where ? ? ??the molar density of the free gas phase, and ? ? ??the porosity of the coalbed, u ? ??the local flow velocity of the free gas phase. We consider also variation in molar density of the free gases in the pore space that results from the mixing of the injected gases and coalbed gases, and from the adsorption and desorption of the gases at coalbed surfaces. The local flow velocity of the free gas, therefore, is no longer constant, and must be found as part of solving the flow problem. The initial and boundary conditions are zi ?x,0? =

The eigenvalue, ?, represents the characteristic wave velocity of the state along continuous solution paths, measured by variable ?. A state is fully described by the dependent variables. The corresponding eigenvector indicates the direction of the continuous solution path in the composition space at the location indicated by the current state. Dindoruk17 introduced a procedure where the local flow velocity is separated from the rest of the dependent variables. The eigenvalue problem is split into two subproblems, one for the unknown variables zi, for i ? ??1 to nc – 1 gas describes the unknown local flow velocity u. Application of a similar procedure for the eigenvalue problem described in Eq. 10 is presented in Appendix A, which also yields two subproblems:
?? ? ?*?? e = 0 , ?* = ?/u . . . . . . . . . . . . . . . . . . . . . . . . . . . . (14)

and
??T ? ?*?T? e +

1 x= 0 u?x,0? = 0 x ? 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (9) Eqs. 8 and 9 state that the initial concentration profile is uniform and the inlet is subject to a step change in concentration and velocity at time zero. The material balances, together with the constant initial and injection conditions, specify a Riemann problem. The overall molar concentration of an individual component is the sum of the free gas phase and adsorbed surface concentrations. Only the components in the free gas phase can be transported by convection. The molar density of the free gas phase is computed from an EOS once its molar composition is known. For all examples in this paper, the Peng-Robinson EOS was used. When the molar composition of the free gas phase varies, its molar density also changes. Therefore, the local flow velocity u is variable and must be found as part of the solution. For a system with nc components, there are nc independent material balance equations. The nc independent variables are the (nc – 1) independent molar compositions such as z1, z2, . . .,
372

?

?

1 du = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (15) u d?

ziinjected x = 0 ziinitial x ? 0

i = 1, . . . , nc, . . . . . . . . . . . . . . . . (8)

The subproblem in Eq. 14 gives the normalized eigenvalue, ?*, and its corresponding eigenvector, e ? as solution to the eigenvalue problem in Eq. A-1. Substitution of the normalized eigenvalue and corresponding eigenvector to the subproblem in Eq. 15 gives the local flow velocity u. If a continuous solution violates the velocity rule, a shock must form. Shock solutions are discontinuous and must satisfy a material balance in an integral form. This is a Rankine-Hugoniot condition that reads ?= F iu ? F id C iu ? C id , i = 1, . . . , nc, . . . . . . . . . . . . . . . . . . . . . . . . . . (16)

where ? ? ??the shock velocity. The superscripts u and d denote variables upstream and downstream of the shock. Besides the material balances in integral form, the shock solutions must also satisfy an entropy condition.16,17 This condition states that a shock remains stable, given a finite-sized perturbation. In addition, the solution must satisfy a continuity condition; that is, a small variation in the initial or injection condition should result in only small changes in the solution. The entropy and continuity conditions are used to eliminate nonphysical solutions.
December 2003 SPE Journal

Solution Construction Construction of the analytical solution for ECBM with gas injection differs for systems with respect to the number of components. In this section, we illustrate the solution construction procedure for systems with up to three components, CH4, CO2, and N2. Binary Mixture. Consider first the continuous solution. The composition space is the binary axis because only one of the molar compositions varies independently. The problem for the normalized eigenvalue and the corresponding eigenvector is of first order and yields

creases as CH4 concentration increases. This variation violates the velocity rule. A shock solution must be constructed as indicated by Eq. 20. Similar analysis is applied to CH4-N2 and CO2-N2 systems. It indicates that a shock solution occurs when injection gas is rich in a component with higher affinity for coal than the coalbed gas, for example CO2 displacing CH4. Thus, the breakthrough of strongly adsorbing injection gas components is retarded by the coalbed. Continuous solutions result during injection of gas rich in a weakly adsorbing component, for example N2 displacing CH4. Ternary Mixture. Consider a mixture of CH4, CO2, and N2 as an example for systems with three components. The composition space is described by a ternary diagram that represents the molar compositions. The 2×2 submatrices of the eigenvalue problem are presented in Appendix B. At each point in the ternary composition space, there are two normalized eigenvalues found from the two roots of

?* =

?? + ?1 ? ?? z2
e =

? ?

?

??
?a1 ??2 ? zi ?z1 ??1

?

, . . . . . . . . . . . . . . . . . . . (17)

?z1 = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (18) ??

Substitution of the normalized eigenvalue and eigenvector to the local flow velocity problem yields 1 du =? u d?

?*2 (?11?22 ? ?12?21?
+ ?* ??12?21 + ?21?12 ? ?11?22 ? ?22?11? . . . . . . . . . . . . . . . (21) + ??11?22 ? ?12?21? = 0

??

1 ?? 1 1 ?? 1 1 ? ? 1 ?a2 ? ? ?* ? + ? ?z1 z2 ? ?z1 z2 ? ?z2 ?z1

? ?

??

.

(19) e =

In a continuous solution, the composition varies continuously along the binary axis from the injection gas composition to the initial composition. At each intermediate state, the molar composition is associated with a normalized eigenvalue and local flow velocity. The product of these two is the characteristic wave velocity. If the wave velocity increases monotonically as the solution path is traced from the injection condition upstream to the initial condition downstream, then the velocity rule is satisfied. The solution is a continuous variation from the injection composition to the initial composition. On the other hand, if the continuous solution violates the velocity rule, a shock solution is needed and is obtained by solving ?=
d Fu 1 ? F1

??
dz1 d? dz2 d?

=

?

?12 ? ?*?12 . . . . . . . . . . . . . . . . . . . . . . . (22) ??11 + ?*?11

?

Cu 1

?

Cd 1

=

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (20) d Cu 2 ? C2

d Fu 2 ? F2

The molar compositions at the upstream and downstream sides of the shock are both known. The injection gas flow rate is a given constant. The only unknowns are the local flow velocity on the downstream side of the shock ud and the shock velocity ?. We use the CH4-CO2 system to demonstrate the solution construction procedure. Fig. 1 illustrates the continuous solution. As composition varies in the direction of decreasing CO2 concentration (increasing CH4), both the normalized eigenvalue and local flow velocity decrease. Therefore, if a binary mixture with high CO2 content is injected at a unit rate into a coalbed initially saturated with a binary mixture rich in CH4, the wave velocity de-

There are two different paths. Each is given by the eigenvector of the associated eigenvalue. The various paths form a mesh-like pattern, as illustrated in Fig. 2. To calculate a composition path, we start from a point in the ternary composition space, and assign it an initial local flow velocity. At the start point, the eigenvalue problem yields two normalized eigenvalues with different magnitudes, each with its own eigenvector. Following the direction indicated by the eigenvector corresponding to the larger eigenvalue, we take small steps toward the region with greater CH4 composition. The process is repeated to map out a path. The path is sloped upward to the right (see Fig. 2), and the wave velocity increases monotonically as the path is traced toward the CH4-rich region, as Fig. 3 shows. Similarly, following a path that slopes upward to the left, given by the eigenvector corresponding to the smaller eigenvalue, the wave velocity decreases as the path is traced toward the region with higher CH4 content (Fig. 3). We refer to the paths sloping upward to the left as “slow paths,” and those sloping upward to the right as “fast paths.” Now, consider solutions for CBM processes in which the initial gases are richer in CH4 than the injected gases. Given initial and

Fig. 1—Variation of normalized eigenvalue, local flow velocity, and characteristic wave velocity along CH4-CO2 binary axis. December 2003 SPE Journal

Fig. 2—Continuous solution paths for CH4-CO2-N2 ternary system. 373

This portion violates the velocity rule, and therefore must be replaced by a shock. The shock solution satisfies the material balances in an integral form, and does not necessarily follow the same path for a continuous solution, ?=
d Fu 1 ? F1

Cu 1 ? C1

= d

d Fu 2 ? F2

Cu 2 ? C2

= d

. . . . . . . . . . . . . . . . . . . . . . (23) d Cu 3 ? C3

d Fu 3 ? F3

Fig. 3—Variation of characteristic wave velocity following the fast and slow paths separately toward CH4-rich zone, starting from the same composition.

injection gas compositions, and a fixed injection rate, the displacement is solved by finding a solution path, consisting of either continuous segments or shock segments, that connects the injection and initial gas compositions in the ternary composition space. Each intermediate state along the solution path represents an intermediate molar composition associated with a normalized eigenvalue and local flow velocity. The product of the two is the wave velocity at which the composition wave propagates downstream. Depending on the injected gas composition, the solutions are classified into three types. It is more likely that CO2-rich gases will be used to displace CH4-rich gases than the reverse. Fig. 4 shows the most commonly seen configuration of initial and injection compositions. We name this scenario Type I. Two sets of paths connect the initial composition, represented by point O in the ternary composition space, and injection composition, represented by point I. For generality, the injection gas composition at point I contains three components. A continuous solution path connecting the injection and initial compositions could follow the path I → A → O or path I → B → O. The path I → B → O requires a switch from a fast path to a slow path at point B, as the composition path is traced from upstream to downstream. This switch violates the velocity rule. Therefore, this path configuration is nonphysical and must be discarded. On the other hand, following path IA with a switch to path AO requires a switch from a slow path upstream to a fast path downstream that satisfies the velocity rule. However, as the slow path is traced from I to A, the wave velocity decreases in the downstream direction.

All the variables on the upstream side of the shock are known. The molar composition and local flow velocity are obtained from the Rankine-Hugoniot condition (Eq. 23). The shock landing point is fully determined and must lie on the fast path through the initial composition at point C. Therefore, the final solution consists of a shock between I and C, and a constant state at C where there is a switch to the continuous solution path CO (Fig. 4). As the injection composition is moved closer to the fast path through the initial composition, the upstream shock segment becomes shorter. In the limit when the injection composition falls on the fast path through O, the upstream shock segment completely disappears. The solution is a continuous wave between the initial and injection compositions. On the other hand, when the injection composition moves toward the CO2 vertex, the shock landing point C moves closer to the initial composition O, with a shorter downstream continuous wave segment. The limiting case takes place when point C and O overlap, resulting in complete disappearance of the downstream continuous wave. The solution is a single shock between the initial and injection compositions. The second type of solution (Type II) occurs when the injection composition moves to the left of the fast path through the initial composition, as shown in Fig. 5. Similar analysis using the velocity rule indicates that the solution must be composed of an upstream continuous wave along the slow path from injection composition I to point A, where there is a switch to the fast path and a continuous variation from A to initial composition O. The third type of solution (Type III) occurs when the shock landing point C goes beyond point O, as Fig. 6 shows. Tracing a path from C back to O violates the velocity rule, and therefore a shock is required between C and O. The shock solution does not necessarily occur along paths for continuous solutions. Therefore, a shock point C is found such that shocks CO and CI both satisfy the Rankine-Hugoniot condition. 1D Finite-Difference Simulation A 1D finite-difference scheme was used to simulate the ECBM process by gas injection and to confirm the analytical solutions. With single-point upstream weighting, the finite-difference form of the material balance equation is

Fig. 4—Type I solution paths, composed of upstream shock IC and downstream continuous variation CO. 374

Fig. 5—Type II solution paths, composed of upstream continuous variation IA and downstream continuous variation AO. December 2003 SPE Journal

Fig. 6—Type III solution paths, composed of upstream shock IC and downstream shock CO.

1 n C in,+ k = C i,k ?

?t n ?F ? F in,k?1?. . . . . . . . . . . . . . . . . . . . . . . . . . (24) ?x i,k

At each timestep, a procedure analogous to a flash calculation was performed. Given an overall molar composition in a cell, the components were distributed between the pore space as free gas and coalbed as adsorbed gas, with the equilibrium compositions determined by the extended Langmuir isotherm for adsorption. The local flow velocity is adjusted according to the molar density of the free phase, and then used for computing the flux of each component into the neighboring cell downstream. Example Solutions The analytical solution procedure was applied to study injection of N2/CO2 mixtures into coalbeds. The thermodynamic properties of these components are summarized in Table 1. The adsorption properties of N2 are estimated, but follow from Ref. 12. The pressure is 1,600 psi, and temperature is 160°F. The initial gas composition is fixed as 96% CH4, 3% CO2, and 1% N2, and the injection gases are binary mixtures composed of CO2 and N2. Five injection compositions were considered containing 0, 25, 50, 75, and 100% mole fraction of CO2, with the remainder N2. These compositions yield analytical solutions of the three different types. A summary of the solution paths for these examples is presented in Fig. 7. For each of these sample cases, a 1D finite-difference simulation was run to confirm the analytical solutions. The number of gridblocks was 5,000, and ? t / ? x was set to 0.1. Fig. 8 shows the Type II solution resulting from the injection of pure N2. The solution includes an upstream continuous variation from the injection gas composition to the path switch point A, and a downstream continuous wave along the fast path from A to the initial composition O. The analytical solution profiles are compared with those obtained from finite-difference simulation. Excellent agreement between the two solutions is observed, with only a small difference caused by numerical dispersion in the finitedifference solution. A similar level of agreement between the analytical and numerical solutions was obtained for the remaining

examples, but is not shown individually. At the producing end in Fig. 8, the CH4 concentration starts to decline after about 0.91 pore volume (PV) has been injected. This coincides with the arrival of the leading edge of the continuous solution for N2. About 2.08 PV of gas is injected to sweep out completely the CH4 originally in place. A Type I solution occurs for an injection gas consisting of 75% N2 and 25% CO2, as shown in Fig. 9. An upstream shock occurs along the CO2-N2 axis on Fig. 7 and ends at point A. A constant state exists at point A. The solution then switches to the fast path and follows a continuous variation to the initial composition O. The affinity of CO2 for the coal surfaces is greatest among the three components, and, consequently, CO2 moves through the coalbed most slowly. The CO2 adsorbs strongly and sweeps out both N2 and CH4, forming a trailing step change. At the downstream end, the interaction of N2 and CH4 adsorption/desorption is evidenced by a continuous variation between ?/? equal to 0.63 and 0.81. A bank of N2 forms between the leading continuous variation and trailing step change. At the producing end, the CH4 concentration starts to decline after 0.99 PV has been injected. After 1.90 PV of gas has been injected, the CH4 originally in place is recovered completely. The example shown in Fig. 9 indicates that a coalbed is useful to separate CO2 and N2. Because CO2 adsorbs more strongly on the coal, it propagates through the coalbed more slowly. Therefore, a coalbed can be used as an adsorption chromatograph in CO2 sequestration operations to separate CO2 from flue gas. While a relatively expensive surface separation of CO2 from the stack gas would be avoided, it would be replaced, at least partially by the cost of the compression of the larger volume of flue gas over that for CO2 alone. The solution structure remains rather similar for the next two examples. The injection gas consists of 50% N2 and 50% CO2, or 25% N2 and 75% CO2. When 50% N2 and 50% CO2 is used as injection gas, the trailing shock occurs along the N2-CO2 axis, whereas the trailing shock goes to the interior of the ternary composition space for the latter case. For the purpose of comparison, the solution profile of the latter case is presented in Fig. 10. It is seen that with 25% N2 and 75% CO2 in the injection gas, the leading front is slower and the trailing shock becomes faster, resulting in a more compressed solution profile. A general observation is that for Type I solutions, as the CO2 fraction in the injection gas increases, the trailing shock is “stronger” (i.e., spans greater difference in gas composition), and the leading continuous variation becomes “weaker.” A stronger shock propagates faster, but has lower local flow velocity at the downstream side. This is the result of volume change as CO2 adsorbs. The loss of CO2 volume is greater than the volume of the CH4 released from the coalbed surfaces. Along a downstream continuous variation, the local flow velocity increases, but less strongly with a weaker continuous variation. Therefore, the wave velocity farthest downstream is a combined effect of the trailing shock and the leading continuous variation. As the CO2 content in the injection gas increases, the trailing shock dominates and the leading continuous variation becomes less significant, resulting in a slower moving leading front and faster moving trailing shock, as seen by comparing the CO2 profiles in Figs. 9 and 10. In the last example, pure CO2 is injected. The solution is of Type III, as shown in Fig. 11. It consists of two discontinuous waves. Starting from the injection gas I5, the trailing shock occurs along the CH4-CO2 axis, ending at point C in Fig. 7. After a

December 2003 SPE Journal

375

Fig. 8—Solution profile for example solution with an injection gas of 100% N2.

Fig. 7—Solution paths for example solutions. The initial composition is fixed at 96% CH4, 3% CO2, and 1% N2.

constant state at C, there is a downstream shock, and a jump to the initial composition O. The leading shock is relatively insignificant compared to the trailing shock. It arrives at the exit after 1.16 PV has been injected, followed by a short period of production of CH4-CO2 binary mixture that has slightly higher CH4 content than the initial gas. Sweepout of CH4 occurs when the trailing shock arrives at 1.42 PV. The arrival times of the leading front and trailing shocks for the example solutions are summarized in Table 2. The recovery curves for CH4 are shown in Fig. 12. Fig. 12 shows that with increasing CO2 content in the injection gas, the initial recovery rate of CH4 decreases, the breakthrough of CO2 occurs later, and the total recovery of CH4 is achieved at an earlier time. The range of prediction illustrated in Fig. 12, however, is remarkably narrow despite input parameters that yield CO2 adsorption; by mass, that is about four times that of N2. Volume change on mixing and the modest difference between the free-phase molar density of CO2 that is adsorbed and that of the N2 desorbed conspire to give fairly similar behavior among all injection gases. The greatest differences among cases as a result of the injection gas composition are found in the times for complete sweepout, as shown in Table 2. Nevertheless, Fig. 12 is consistent with current ECBM practice where N2 is the injectant of choice. Enhanced CH4 production occurs earlier when N2 is injected, and the enhancement in recovery rate is greater. Because gases are injected at a fixed volumetric rate, for a solution of Type II, the local flow velocity increases along the upstream continuous path and also the downstream con-

tinuous path, resulting in a high local flow velocity at the leading front. Therefore, the recovery curve of the initial gas is steeper. By the time the leading front arrives at the producing end, and the CH4 composition starts to decline, most of the CH4 originally in place has been recovered. The remaining CH4 is produced as its concentration in the produced gas declines continuously. Therefore, it takes an extended period to produce the remaining CH4 and also requires separation of the CH4 from the produced gas. In a Type I solution, the local flow velocity shows a dramatic decline across the upstream shock. Although the flow velocity increases along the fast path towards downstream, the local flow velocity at the leading front is lower than it is in a Type II solution. Within this solution type, the greater CO2 composition in the injection gas results in a stronger upstream shock, and weaker downstream continuous variation, a slower leading front, and a faster upstream shock. Therefore, if there is more CO2 in the injection gas, the recovery of gas is initially slower, but the total recovery of the CH4 originally in place occurs earlier. Separation of CH4 from the produced gas is still required, because the CH4 composition decreases for an extended period until the upstream shock arrives. When an injection gas very rich in CO2 is used, a Type III solution occurs. The upstream shock spans a large difference of concentration. The downstream shock, on the other hand, is relatively insignificant, as can be seen from the sample solution on Fig. 11. There is a fast-moving upstream shock but low local flow velocity at the leading front, resulting in a lower production rate of the initial gas, compared to the Type I and Type II examples. The duration for initial gas production is longer, and it takes less time to recover all of the CH4 originally in place. Additionally, the theory predicts that much less separation of injection gas from methane is required.

Fig. 9—Solution profile for example solution with an injection gas of 25% CO2 and 75% N2. 376

Fig. 10—Solution profile for example solution with an injection gas of 75% CO2 and 25% N2. December 2003 SPE Journal

Fig. 11—Solution profile for example solution with an injection gas of 100% CO2.

Conclusions We describe an analytical theory of multicomponent flow for ECBM processes in which adsorption and desorption of gas components on coal play important roles. The theory and examples presented lead to the following observations and conclusions: 1. Gas injection ECBM recovery significantly. Local displacement efficiency approaches 100%. 2. In binary systems, shock solutions occur when an injected gas component adsorbs more strongly than initial gas components. Continuous variation occurs when the injection gas contains less strongly adsorbing components than the initial gases. 3. In ternary systems, CO2 moves through coal in a plug-like fashion, whereas N2 propagates more rapidly than CO2. Therefore, coalbeds appear capable of chromatographic separation of CO2 and N2. CO2 can be separated from N2 in a coalbed, at the cost of compression of the injected CO2/N2 mixtures and separation of produced N2/CH4 mixtures. 4. A two-component model should not be used to predict the behavior of mixed CO2/N2 injection gases. The two-component model does not describe the separation of components addressed in Eq. 3, and the two-component model does not describe, in even an approximate sense, the concentration of the produced gas after breakthrough. These differences are neither subtle nor minor. 5. In ternary systems, the composition of injection gas has a relatively modest effect on the amount of time to recover all CH4 originally in place. The differences among gas molar densities partially offset the differences in adsorption. However, injection gas composition has significant effect on the produced gas composition and the time to breakthrough of the injection gas. 6. An injection gas rich in N2 yields a greater initial recovery rate but earlier breakthrough time of injected N2. Thus, N2 must be separated from produced gas for a substantial period of time. 7. Injection of a gas rich in CO2 yields recovery of CH4-rich gas for a greater period of time, reducing the amount of separation that is required.

Nomenclature ai ? ??adsorbed concentration of individual components A ? ??cross-sectional area B ? ??Langmuir constant Ci ? ??overall molar concentration of individual components e ? ? ??eigen vector Fi ? ??overall molar flux of individual components G ? ??matrix in the eigenvalue problem H ? ??matrix in the eigenvalue problem L ? ??characteristic length Mw ? ??molecular weight nc ? ??number of components in mixture p ? ??pressure pc ? ??critical pressure q ? ??volumetric flow rate t ? ??time Tc ? ??critical temperature u ? ??local flow velocity Vm ? ??Langmuir constant x ? ??distance zi ? ??molar composition ?i ? ??component molar density ? ? ??submatrices in the decoupled eigenvalue problem ? ? ??distance variable along continuous solution paths ? ? ??fractional surface coverage of individual components ? ? ??binary interaction coefficient for EOS ? ? ??eigenvalue, charactistic continuous wave velocity ?* ? ??normalized eigenvalue ? ? ??shock velocity ? ? ??dimensionless distance ? ? ??molar density of the free gas phase ?r ? ??coal density ?I ? ??molar density of individual components at standard condition ? ? ??dimensionless time ? ? ??coal porosity ? ? ??submatrices in the decoupled eigenvalue problem ? ? ??accentric factor for EOS Acknowledgments Support for this work was provided by the Assistant Secretary for Fossil Energy, Office of Coal and Power Systems, through the Natl. Energy Technology Laboratory and the GEO-SEQ project of the Lawrence Berkeley Natl. Laboratory. References
1. Stevens, S.H., Spector, D., and Riemer, P.: “Enhanced Coalbed Methane Recovery Using CO2 Injection: Worldwide Resource and CO2 Sequestration Potential,” paper SPE 48881 presented at the 1998 SPE International Oil & Gas Conference and Exhibition in China, Beijing, 2–6 November. 2. Joubert, J.I., Grein, C.T., and Bienstock, D.: “Sorption of Methane in Moist Coal,” Fuel (July 1973) 181.

December 2003 SPE Journal

377

20. Yang, R.T.: “Gas Separation by Adsorption Processes,” Butterworths Publishers, Johannesburg, South Africa (1987).

Appendix A—Decoupling Local Flow Velocity Starting with the eigenvalue problem for gas injection with adsorption as presented in Eq. 10, and completing a derivation similar to that presented by Dindoruk17 for enhanced oil recovery by gas injection, we have

?
and
Fig. 12—Total recovery of CH4 for the example solutions with different injection gas compositions. 3. Ruppel, T.C., Grein, C.T., and Bienstock, D.: “Adsorption of Methane on Dry Coal at Elevated Pressure,” Fuel (July 1974) 152. 4. Fulton, P.F. et al.: “A Laboratory Investigation of Enhanced Recovery of Methane from Coal by Carbon Dioxide Injection,” paper SPE/DOE 8930 presented at the 1980 SPE/DOE Symposium on Unconventional Gas Recovery, Pittsburgh, Pennsylvania, 18–21 May. 5. Reznik, A.A., Singh, P.K., and Foley, W.L.: “Analysis of the Effect of CO2 Injection on the Recovery of In-Situ Methane from Bituminous Coal: An Experimental Simulation,” SPEJ (October 1984) 521. 6. Yang, R.T. and Saunders, J.T.: “Adsorption of Gases on Coals and Heat-Treated Coals at Elevated-Temperature and Pressure. 1. Adsorption from Hydrogen and Methane as Single Gases,” Fuel (1985) 64, No. 5, 616. 7. Bell, G.J. and Rakop, K.C.: “Hysteresis of Methane/Coal Sorption Isotherms,” paper SPE 15454 presented at the 1986 SPE Annual Technical Conference and Exhibition, New Orleans, 5–8 October. 8. Stevenson, M.D. et al.: “Adsorption/Desorption of Multicomponent Gas Mixtures at In-Seam Conditions,” paper SPE 23026 presented at the 1991 SPE Asia-Pacific Conference, Perth, Western Australia, 4–7 November. 9. DeGance, A.E., Morgan, W.D., and Yee, D.: “High-Pressure Adsorption of Methane, Nitrogen and Carbon-Dioxide on Coal Substrates,” Fluid Phase Equilibria (February 1993) 215. 10. Greaves, K.H. et al.: “Multi-Component Gas Adsorption-Desorption Behavior of Coal,” Proc., International Coalbed Methane Symposium, Birmingham, Alabama, Vol. 1, 197 (1993). 11. Chaback, J.J., Morgan W.D., and Yee, D.: “Sorption of Nitrogen, Methane, Carbon-Dioxide and Their Mixtures on Bituminous Coals at In-Situ Conditions,” Fluid Phase Equilibria (March 1996) 289. 12. Arri, L.E. et al.: “Modeling Coalbed Methane Production with Binary Gas Sorption,” paper SPE 24363 presented at the 1992 SPE Rocky Mountain Regional Meeting, Casper, Wyoming, 18–21 May. 13. Scott, A.R.: “Composition of Coalbed Gases,” In Situ (1994) 18, No. 2, 185. 14. Saxena, S.C. and Thomas, L.A.: “An Equibrium-Model for Predicting Flue-Gas Composition of an Incinerator,” International Journal of Energy Research (June 1995) 317. 15. Rhee, H.K., Aris, R., and Amundson, N.R.: “First-Order Partial Differential Equations: Volume II—Theory and Application of Hyperbolic Systems of Quasilinear Equations,” International Series in the Physical and Chemical Engineering Sciences, Prentice-Hall, Englewood Cliffs, New Jersey (1986). 16. Helfferich, F.G.: “Non-Linear Waves in Chromatography. 3. Multicomponent Langmuir and Langmuir-Like Systems,” Journal of Chromatography A (April 1997) 169. 17. Dindoruk, B.: “Analytical Theory of Multiphase, Multicomponent Flow in Porous Media,” PhD dissertation, Stanford U., Palo Alto, California (1992). 18. Peng, D.Y. and Robinson, D.B.: “A New Two-Constant Equation of State,” Ind. Eng. Chem. Fund. (1976) 15, 59. 19. Ruthven, D.M.: “Principles of Adsorption and Adsorption Processes,” John Wiley & Sons, Inc., New York City (1984). 378

?? ? ?*??
T

0

1 ?? ? ?*? ? u

?? ? ? ?
e du d? = 0 0

. . . . . . . . . . . . . . (A-1)

[?? =
??? =

??i ?i ??nc ? , ? ????zi, . . . . . . . . . . . . . . . . . . . . . . . . . (A-2) ?zj ?nc ?zj i ?Ci ?i ?Cnc ? , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-3) ?zj ?nc ?zj 1 ??nc , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-4) ?nc ?zj

?T = ?T = e =

1 ?Cnc , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-5) ?nc ?zj

dzj , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-6) d?

? ?* = , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-7) u
?ai = vmi?i?rBip ?zj

?

1 1+

?
k=1

nc

?zi ? ?zj Bk pk

zi

?

? B p ?z
k=1 k

nc

?zk
j

1+

?
k=1

nc

Bkpk

??
2

, . . . . . . . . . . . . . (A-8)

where the subscripts i and j vary from 1 to nc – 1. Appendix B—Ternary Mixture For a ternary mixture composed of CH4, CO2, and N2, the subproblems in Eqs. 10 and 11 have the following elements: ?11 = ? 1 +

? ?

z1 , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (B-1) z3

z1 ?12 = ? , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (B-2) z3 z2 ?21 = ? , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (B-3) z3 ?22 = ? 1 + ?11 = ? 1 + ?12 = ? ?21 = ?

? ? ? ? ? ?

z2 , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (B-4) z3

z1 1 ? ? ?a1 z1 ?a3 ? + z3 ? ?z1 z3 ?z1

z1 1 ? ? ?a1 z1 ?a3 + ? z3 ? ?z2 z3 ?z2 z2 1 ? ? ?a2 z2 ?a3 + ? z3 ? ?z1 z3 ?z1

? ?

? ?

? ?

? ?

. . . . . . . . . . . . . . (B-5)

. . . . . . . . . . . . . . . . . . . . . (B-6) . . . . . . . . . . . . . . . . . . . . . (B-7) . . . . . . . . . . . . . . (B-8)

?22 = ? 1 +

z2 1 ? ? ?a2 z2 ?a3 + ? z3 ? ?z2 z3 ?z2

December 2003 SPE Journal

?T =

?T =

? ? ?
1 ?? 1 ? ? ?z2 z3

1 ?? 1 ? ? ?z1 z3

, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (B-9)

1 ?? 1 1 ? ? 1 ?a3 ? + ? ?z1 z3 ? ?z3 ?z1 1 ?? 1 1 ? ? 1 ?a3 ? + ? ?z2 z3 ? ?z3 ?z1

?

. . . . . . . . . . . . . . . . . . (B-10)

Jichun Zhu is currently a reservoir engineer with Object Reservoir, Inc. e-mail: jichun@pangea.stanford.edu. He holds a BS degree in mechanical engineering from the U. of Science and Technology of China, and a PhD degree in petroleum engineering from Stanford U. Kristian Jessen is currently Acting Assistant Professor at the Petroleum Engineering Dept., Stanford U. e-mail: krisj@pangea.stanford.edu. He holds a BS degree in chemical engineering from the Danish Engineering Academy,

and MS and PhD degrees, both in chemical engineering, from the Technical U. of Denmark. Tony Kovscek is Associate Professor of Petroleum Engineering at Stanford U. e-mail: kovscek@pangea.stanford.edu. His research interests include the physics of oil recovery from porous media, interfacial phenomena, and reservoir definition techniques. Kovscek holds BS and PhD degrees from the U. of Washington and U. of California at Berkeley, respectively. He is the Stanford U. SPE Student Chapter faculty adviser and a member of the Continuing Education Committee. Franklin M. Orr Jr. is the Beal Professor of Petroleum Engineering at Stanford U. e-mail: fmorr@pangea.stanford.edu. He served as Dean of Stanford’s School of Earth Sciences from 1994–2002. Previously, he was head of the miscible flooding section at the New Mexico Petroleum Recovery Research Center, and Adjunct Associate Professor of Petroleum Engineering at New Mexico Inst. of Mining and Technology. Before that, he was a research engineer at the Shell Development Co., Bellaire Research Center. He holds a BS degree from Stanford U. and a PhD degree from the U. of Minnesota, both in chemical engineering.

December 2003 SPE Journal

379


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