Search citation statements
Paper Sections
Citation Types
Year Published
Publication Types
Relationship
Authors
Journals
Summary. A series of mathematical and numerical developments that simulatethe unsteady-state behavior of unconventional gas reservoirs is reviewed. Fivemajor modules, considered to be unique to the simulation of gas reservoirs, areidentified. The inclusion of these models into gas reservoir simulators isdiscussed in mathematical detail with accompanying assumptions. Introduction More than 35 mathematical models related to the recovery of unconventionalgas resources (predominantly coal seams and Devonian shales) were recentlysurveyed. Of the surveyed models, 14 were two-phase, numerical, finite-difference simulators. Tables 1 through 3 in Ref. 39 (also see Ref. 40)summarize the results of this survey and indicate that numerical simulators forunconventional gas reservoirs have several key features that distinguish themfrom simulators for conventional reservoirs. These key features includeadsorption/desorption kinetics models, various gas transport mechanisms, allowance for composite unconventional/conventional fields, allowance forcurrent exploitation methods, and inclusion of mining activity (for use withminable coals). Refs. 1 and 2 provide an overview of the surveyed models but donot provide the mathematical details used to simulate these key features. Thispaper provides these mathematical details. Adsorption/Desorption Kinetics Because of the pore sizes in tight gas reservoirs, large internal surfaceareas, and thus many potential adsorption sites, exist and adsorbed gas storagebecomes significant. Four important kinetic models currently are used todescribe the desorption and recovery of this gas resource. For tight gas sands(single porosity), it is generally assumed that the adsorbed gas phase is incontact with the gas phase and that these two phases are in a continuous stateof equilibrium. The gas-transport equation used to describe the equilibriumsorption process is similar to that used to model gas flow through conventionalreservoirs, with a minor modification to the storage term to allow foradsorption. The gas-phase mass-balance equation has the form = (/ t)[( ].........(1) and the water-phase mass-balance equation is ).........(2) Eqs. 1 and 2 are coupled through the use of auxiliary relationships relatingsaturations and capillary pressure: Sg + Sw = 1.0,....................................(3) and Pc(Sw) = pg 0 -pw,................................(4) Initial and boundary conditions for Eqs. 1 and 2 are identical to those ofconventional gas reservoirs. We derived Eqs. 1 and 2 assuming that Darcy's lawcould be used to model gas transport. We discuss various modifications andextensions to Darcy's law for use with unconventional gas reservoirs later. Eqs. 1 and 2 further assume that gas adsorption does not affect permeability. In Eq. 1, V eq is an adsorption isotherm that is a function of the free-gaspressure. Three functional relationships are usually used to describe thisisotherm: Henry's law isotherm- V eq = V H pg,....................................(5) Langmuir's isotherm- V eq = V L pg/(pL + pg),..........................(6) and Freundlich's isotherm- V eq = V F pg NF.................................(7) The Langmuir isotherm is generally used to describe monolayer adsorptionbecause with increasing pressure, the volume of adsorbed gas asymptoticallyapproaches the sorptive capacity of the formulation, V L. The Freundlichisotherm is often used to describe multilayer adsorption because it assumesinfinite adsorptive capacity. Under certain conditions, both isotherms collapseto Henry's law. The primary porosity in both coal seams and Devonian shales isrelatively impermeable to gas. The major transport mechanism in theprimary-porosity system of reservoirs is diffusion. In contrast to tight gassands, the adsorbed gas in the primary porosity is not in contact with the freegas in the secondary porosity and only initially (or after long shut-inperiods) are these two gas phases in equilibrium. Therefore, the sorption modelfor these reservoirs must account for the kinetics of gas desorption frominternal surfaces and diffusion through the primary-porosity system. Twoapproaches have been reported for modeling the desorption/diffusion process inunconventional gas reservoirs: the pseuosteady-state approach and theunsteady-state approach. Both pseuosteady-state approach and the unsteady-stateapproach. Both approaches essentially are modified forms of the dual-porositymodels of Warren and Root and de Swaan, respectively. Modifications to thesemodels arise because the reservoir fluid in gas reservoirs is highlycompressible, gas storage in the primary-porosity system is an adsorptionprocess, and gas transport through the primary-porosity system is a process, and gas transport through the primary-porosity system is a diffusion process. The pseudo-steady-state approach is a semiempirical approach that describes gastransport through the primary porosity with a discretized form of Fick's firstlaw. The unsteady-state approach is fully analytical and describes gastransport through the primary porosity with Fick's second law. porosity with Fick's second law. In the pseudo-steady-state approach, gas transport isgoverned by the following set of coupled equations : the secondary-porosity, gas-phase, mass-balance equation, + (),.............(8) and the primary-porosity, gas-phase, mass-balance equation, dV g1/dt = -D1 a(V g1 - V eg),...................(9) where a is the shape factor introduced by Warren and Root. The couplingequation is q't = - Fg (dV g1/dt),............................(10) where Fg is a geometry-dependent prefactor. Table 1 presents values of a and Fg for various primary-porosity matrix geometries. initial and boundaryconditions for Eqs. 2 through 4 and 8 through 10 are given below. pg2 = pg2i over the reservoir volume at t = 0,....(11) Sg2 = Sg2i over the reservoir volume at t = 0,....(12) SPEFE P. 63
Summary. A series of mathematical and numerical developments that simulatethe unsteady-state behavior of unconventional gas reservoirs is reviewed. Fivemajor modules, considered to be unique to the simulation of gas reservoirs, areidentified. The inclusion of these models into gas reservoir simulators isdiscussed in mathematical detail with accompanying assumptions. Introduction More than 35 mathematical models related to the recovery of unconventionalgas resources (predominantly coal seams and Devonian shales) were recentlysurveyed. Of the surveyed models, 14 were two-phase, numerical, finite-difference simulators. Tables 1 through 3 in Ref. 39 (also see Ref. 40)summarize the results of this survey and indicate that numerical simulators forunconventional gas reservoirs have several key features that distinguish themfrom simulators for conventional reservoirs. These key features includeadsorption/desorption kinetics models, various gas transport mechanisms, allowance for composite unconventional/conventional fields, allowance forcurrent exploitation methods, and inclusion of mining activity (for use withminable coals). Refs. 1 and 2 provide an overview of the surveyed models but donot provide the mathematical details used to simulate these key features. Thispaper provides these mathematical details. Adsorption/Desorption Kinetics Because of the pore sizes in tight gas reservoirs, large internal surfaceareas, and thus many potential adsorption sites, exist and adsorbed gas storagebecomes significant. Four important kinetic models currently are used todescribe the desorption and recovery of this gas resource. For tight gas sands(single porosity), it is generally assumed that the adsorbed gas phase is incontact with the gas phase and that these two phases are in a continuous stateof equilibrium. The gas-transport equation used to describe the equilibriumsorption process is similar to that used to model gas flow through conventionalreservoirs, with a minor modification to the storage term to allow foradsorption. The gas-phase mass-balance equation has the form = (/ t)[( ].........(1) and the water-phase mass-balance equation is ).........(2) Eqs. 1 and 2 are coupled through the use of auxiliary relationships relatingsaturations and capillary pressure: Sg + Sw = 1.0,....................................(3) and Pc(Sw) = pg 0 -pw,................................(4) Initial and boundary conditions for Eqs. 1 and 2 are identical to those ofconventional gas reservoirs. We derived Eqs. 1 and 2 assuming that Darcy's lawcould be used to model gas transport. We discuss various modifications andextensions to Darcy's law for use with unconventional gas reservoirs later. Eqs. 1 and 2 further assume that gas adsorption does not affect permeability. In Eq. 1, V eq is an adsorption isotherm that is a function of the free-gaspressure. Three functional relationships are usually used to describe thisisotherm: Henry's law isotherm- V eq = V H pg,....................................(5) Langmuir's isotherm- V eq = V L pg/(pL + pg),..........................(6) and Freundlich's isotherm- V eq = V F pg NF.................................(7) The Langmuir isotherm is generally used to describe monolayer adsorptionbecause with increasing pressure, the volume of adsorbed gas asymptoticallyapproaches the sorptive capacity of the formulation, V L. The Freundlichisotherm is often used to describe multilayer adsorption because it assumesinfinite adsorptive capacity. Under certain conditions, both isotherms collapseto Henry's law. The primary porosity in both coal seams and Devonian shales isrelatively impermeable to gas. The major transport mechanism in theprimary-porosity system of reservoirs is diffusion. In contrast to tight gassands, the adsorbed gas in the primary porosity is not in contact with the freegas in the secondary porosity and only initially (or after long shut-inperiods) are these two gas phases in equilibrium. Therefore, the sorption modelfor these reservoirs must account for the kinetics of gas desorption frominternal surfaces and diffusion through the primary-porosity system. Twoapproaches have been reported for modeling the desorption/diffusion process inunconventional gas reservoirs: the pseuosteady-state approach and theunsteady-state approach. Both pseuosteady-state approach and the unsteady-stateapproach. Both approaches essentially are modified forms of the dual-porositymodels of Warren and Root and de Swaan, respectively. Modifications to thesemodels arise because the reservoir fluid in gas reservoirs is highlycompressible, gas storage in the primary-porosity system is an adsorptionprocess, and gas transport through the primary-porosity system is a process, and gas transport through the primary-porosity system is a diffusion process. The pseudo-steady-state approach is a semiempirical approach that describes gastransport through the primary porosity with a discretized form of Fick's firstlaw. The unsteady-state approach is fully analytical and describes gastransport through the primary porosity with Fick's second law. porosity with Fick's second law. In the pseudo-steady-state approach, gas transport isgoverned by the following set of coupled equations : the secondary-porosity, gas-phase, mass-balance equation, + (),.............(8) and the primary-porosity, gas-phase, mass-balance equation, dV g1/dt = -D1 a(V g1 - V eg),...................(9) where a is the shape factor introduced by Warren and Root. The couplingequation is q't = - Fg (dV g1/dt),............................(10) where Fg is a geometry-dependent prefactor. Table 1 presents values of a and Fg for various primary-porosity matrix geometries. initial and boundaryconditions for Eqs. 2 through 4 and 8 through 10 are given below. pg2 = pg2i over the reservoir volume at t = 0,....(11) Sg2 = Sg2i over the reservoir volume at t = 0,....(12) SPEFE P. 63
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
customersupport@researchsolutions.com
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
This site is protected by reCAPTCHA and the Google Privacy Policy and Terms of Service apply.
Copyright © 2025 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.