Applicability of the Arps Rate-Time Relationships for Evaluating Decline Behavior and Ultimate Gas Recovery of Coalbed Methane Wells

This paper presents results of a simulation study designed to evaluate the applicability of Arps' [1945] decline curve methodology for assessing reserves in coalbed methane reservoirs. We simulated various coal properties and well/operational conditions to determine their impact on the production decline behavior as quantified by the Arps decline curve exponent, b. We then evaluated the simulated production with Arps' rate-time equations at specific time periods during the well's production decline period and compared estimated reserves to the "true" value (defined in this paper as the 30-year cumulative production volume). To satisfy requirements for using Arps' models, all simulations were conducted using a constant bottomhole flowing pressure condition in the wellbore. The significant results from our study include: All of the computed values of the long-term decline exponents were within the limits originally defined by Arps, i.e., 0.0 < b < 1.0. Agreement between Arps' recommended b-exponent range and our results using simulated performance data also suggests that, if applied under the correct conditions, the Arps rate-time models are appropriate for assessing reserves in coalbed methane reservoirs; The Arps b-exponents were not constant during the production decline period. For many simulated cases, the early decline behavior (within a few years after reaching the peak production rate) appeared to have exponential decline but eventually became more hyperbolic later in the well's life. Use of Arps' exponential model early in the production history in those wells with long-term hyperbolic decline behavior tended to underestimate gas reserves; The largest reserve estimate errors typically occurred during the first few years after reaching the peak production rate and during the initial production decline period. For those wells exhibiting long-term hyperbolic behavior, the initial reserve estimate errors underestimated reserves by as much as 20 to 30 percent; Heterogeneities in coal properties cause the production declines to deviate from exponential to hyperbolic. Properties having the largest impact on the production decline behavior include the shape of the adsorption isotherm, cleat permeability anisotropies, the shape of cleat gas-water relative permeability curves, stress-dependent cleat permeability and porosity, and layered coal seams with differences in initial reservoir pressures; We also observed a strong influence of well flowing pressure conditions as modeled with a bottomhole flowing pressure constraint. For all other properties and conditions being equal, wells with lower bottomhole flowing pressures exhibited more long-term hyperbolic behavior as defined by higher Arps b-exponents. Introduction Unconventional natural gas resources — tight gas sands, naturally-fractured gas shales, coalbed methane, and deep basincentered gas systems — comprise a significant percentage of our domestic natural gas resource base identified to date and represent an important source for future natural gas production and reserve growth. According to Kawata and Fujita [2001], the coalbed methane (CBM) resource-in-place in North America is estimated to total more than 3,000 Tcf. While the resource base is large, the unique gas storage and flow properties characteristic of CBM reservoirs make efficient and effective gas recovery technically difficult. Of the total resource in place, the total technically recoverable gas is estimated to be 98 Tcf. Those same unique coal properties also cause CBM production profiles to differ in shape from the production profiles for common, more conventional reservoirs. And, these differences in production profiles present unique challenges for the 2 J.A. Rushing, A.D. Perego, and T.A. Blasingame SPE 114514 accurate evaluation of reserves. The typical CBM production profile is an initial inclining production period to some peak or maximum production rate and followed by a declining rate profile. Consequently, assessing CBM reserves using the traditional Arps decline curve models and methodology is only viable during the period when the gas production is declining. However, CBM reserves may be estimated prior to the onset of a declining production profile using various reservoir models as well as numerical simulation. Regardless of the challenges posed by a typical production profile, CBM reserves are routinely assessed using the traditional Arps decline curve methodology and models. The Arps decline curve evaluation methodology consists of plotting the logarithm of production rate against time, history matching the production data using Arps' rate-time equations — i.e., the Arps exponential, hyperbolic or harmonic decline models — and extrapolating the established trend into the future. The long-term decline behavior of the extrapolated trend is typically quantified using Arps' decline exponent, b. The original Arps paper [1945], which was developed and initially applied to conventional reservoirs, indicated the b-exponent should fall between 0 and 1.0 on a semilog production plot. However, the correct b-exponent in unconventional resources like CBM reservoirs is difficult to identify correctly, particularly during the early decline period after reaching the maximum production rate. And, selection of the incorrect b-exponent will have a tremendous impact on the accuracy of CBM reserve estimates. To address these problems with reserve evaluations using the Arps models in CBM reservoirs, we have conducted a series of single-well, parametric simulation studies or "experiments" to develop a better understanding of both the shortand long-term production decline behavior and to identify those parameters affecting the production decline. For this work, our specific study objectives are: To validate the practicality and utility of the Arps rate-time relationships and decline curve methodology for estimating reserves accurately in coalbed methane reservoirs; To develop physical interpretations of Arps' b-exponents in CBM reservoirs — i.e., to determine what coal properties and operational conditions most affect the value of b; and To provide guidelines for the applicable range of b-exponents for reserve evaluations. Description of CBM Model We developed a three-dimensional, two-phase (gas-water) finite-difference model using the Computer Modeling Group's GEM (Generalized Equation-of-State Model) simulator [Reference 9]. GEM is a three-phase, multi-component compositional equation-of-state model that has been modified and adapted to capture all of the storage and flow phenomena characteristic of coalbed methane reservoirs. The model also has the capability of incorporating stress-dependent and sorption-controlled changes in coal porosity and permeability during the gas and water production process. All simulations were conducted for a single-well on a spacing of 80 aces per well. The grid system was constructed with 1521 grids (39 grids in both the xand y-directions). We employed a Cartesian grid geometry so that we could model linear flow geometry and any permeability anisotropy associated with the natural fracture or cleat system in coals. Grid dimensions in the xand y-directions were smaller immediately around the wellbore but increased geometrically away from the wellbore. Dimensions in the vertical direction ranged from three grids in the single-layer case to fifteen grids in the five-layer cases. We should note that we developed a reservoir model that addressed reservoir inflow performance, and other than using a bottomhole flowing pressure constraint, we did not attempt to model well outflow performance. Productive coals are characterized by an extensive, orthogonal set of natural fractures or cleats as illustrated schematically in Fig. 1. The primary cleat system is often referred to as the "face" cleats, while the orthogonal cleats are called "butt" cleats. Typically, the face cleats are better connected and more continuous, while the butt cleats are less well connected and more discontinuous. Spacing between face cleats ranges from tenths of an inch to several inches. Interactions between the natural fracture or cleat system and the coal matrix are modeled with the dual-porosity system (Fig. 1) developed by Gilman and Kazemi [1983], where this model is modified to include all coal storage and flow processes. The majority of coal gas is stored by adsorption (i.e., gas molecules that are physically attached to the coal surfaces) rather than by "free" or unattached gas molecules stored in a matrix porosity structure similar to conventional sandstone or carbonate rocks. Most coal porosity is a combination of a micro-pore structure (pore diameters less than 2 nm) and a meso-pore structure (pore diameters between 2 and 50 nm). Because of the large surface area of the coal particles, significant volumes of gas may be stored in the adsorbed state in the microand meso-pore systems. Although it is usually not considered to be a significant contributor to either gas storage (as "free" gas that is not adsorbed)) or production (by Darcy flow), the matrix does Fig. 1 — Schematic diagram comparing actual coal cleat and matrix system with idealized dual-porosity model used in study. Actual Coal System