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Recent laboratory studies and analyses (Lai et al. Presented at the 2009 RockyMountain Petroleum Technology Conference, 14-16 April, Denver, CO, 2009) have shown that the Barree and Conway model is able to describe the entire range of relationships between flow rate and potential gradient from low-to high-flow rates through porous media. A Buckley and Leverett type analytical solution is derived for non-Darcy displacement of immiscible fluids in porous media, in which non-Darcy flow is described using the Barree and Conway model. The comparison between Forchheimer and Barree and Conway non-Darcy models is discussed. We also present a general mathematical and numerical model for incorporating the Barree and Conway model in a general reservoir simulator to simulate multiphase nonDarcy flow in porous media. As an application example, we use the analytical solution to verify the numerical solution for and to obtain some insight into one-dimensional non-Darcy displacement of two immiscible fluids with the Barree and Conway model. The results show how non-Darcy displacement is controlled not only by relative permeability, but also by non-Darcy coefficients, characteristic length, and injection rates. Overall, this study provides an analysis approach for modeling multiphase non-Darcy flow in reservoirs according to the Barree and Conway model. List of SymbolsA Cross-section area of flow, m 2 A i j Common interface area between the connected blocks or nodes i and j, m 2 C β non-Darcy constant, m 0.25 D Depth from a datum, m D i Distance from the center of block i to the common interface of blocks i and j f β Fractional flow of phase β, fraction flow β mass flux of fluid β, kg/s g Gravitational acceleration constant, m/s 2 k d Darcy permeability, m 2 k min Minimum permeability at high rate, m 2 k mr Minimum permeability ratio, relative to Darcy permeability, fraction k rβ Relative permeability to fluid β, fraction N Total number of nodes/elements/gridblocks of the grid P β Pressure of fluid β, Pa P cgo Gas-oil capillary pressure, Pa P cgw Gas-water capillary pressure, Pa P cow Oil-water capillary pressure, Pa ∇ P Pressure gradient, Pa/m q Injection rate, m 3 /sec q β Mass sink/source per unit volume for the fluid β, kg/m 3 Q Fluid volumetric flow rate, m 3 /s Q βi Mass sink/source term at element i, for the fluid β, kg/m 3 R i Residue term of mass balance at element i, kg S β Saturation of fluid β, fraction S β Average saturation of fluid β, fraction t Time step size, s V i Volume of block i, m 3 v β Velocity of fluid β, m/s x sw Location of the specific saturation, m Greeks α Angle from horizontal plane, Degree β Non-Darcy coefficient, 1/m ρ β Fluid density of fluid β, kg/m 3 μ β Viscosity of fluid β, Pa s τ Characteristic length, 1/m φ Effective porosity of the medium, fraction ∇ Flow potential gradient, Pa/m

SUMMARYAnalysis of large deformation of geomaterials subjected to time-varying load poses a very difficult problem for the geotechnical profession. Conventional finite element schemes using the updated Lagrangian formulation may suffer from serious numerical difficulties when the deformation of geomaterials is significantly large such that the discretized elements are severely distorted. In this paper, an operator-split arbitrary Lagrangian-Eulerian (ALE) finite element model is proposed for large deformation analysis of a soil mass subjected to either static or dynamic loading, where the soil is modelled as a saturated porous material with solid-fluid coupling and strong material non-linearity. Each time step of the operator-split ALE algorithm consists of a Lagrangian step and an Eulerian step. In the Lagrangian step, the equilibrium equation and continuity equation of the saturated soil are solved by the updated Lagrangian method. In the Eulerian step, mesh smoothing is performed for the deformed body and the state variables obtained in the updated Lagrangian step are then transferred to the new mesh system. The accuracy and efficiency of the proposed ALE method are verified by comparison of its results with the results produced by an analytical solution for one-dimensional finite elastic consolidation of a soil column and with the results from the small strain finite element analysis and the updated Lagrangian analysis. Its performance is further illustrated by simulation of a complex problem involving the transient response of an embankment subjected to earthquake loading.

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Small pore sizes on the order of nanometers in the shale gas and tight oil reservoir formations can lead to a large capillary pressure. The presence of capillary pressure significantly affects both the thermodynamic behavior of fluid mixtures and the fluid flow process. Although there have been some attempts to study the effect of capillary pressure on phase behavior, it has not been clearly understood in the application considering multiple components in tight oil reservoirs. In this work, we present a methodology to calculate the phase behavior of CO2/hydrocarbon systems in the presence of capillary pressure. We modify the Peng-Robinson equation of state considering inequalities of hydrocarbon liquid and vapor pressures. The criterion of Gibbs free energy minimization and Rachford-Rice flash calculation are applied in the phase equilibrium calculation. The Young-Laplace equation is utilized to calculate capillary pressure. The Newton-Raphson method is used to solve the nonlinear phase equilibrium equations. We validate the methodology against two experimental measurements and a published numerical model. Subsequently, binary mixture and one typical fluid from the Bakken Formation are used to study the influence of capillarity in the unconventional reservoir. The simulation results indicate that capillary pressure plays an important role in the phase equilibrium calculation when pore size is less than 50 nm. Additionally, the bubble-point pressure of Bakken oil reduces nearby 500 psi when the nano-pore size is 10 nm. The developed method can address the thermodynamics governing unconventional reservoirs and provide better understanding of the phase behavior of CO2/hydrocarbon systems in the case of CO2 injection into unconventional reservoirs.

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Summary Carbon dioxide (CO2) injection is an effective enhanced-oil-recovery (EOR) method in unconventional oil reservoirs. However, investigation of the CO2 huff ’n’ puff process in tight oil reservoirs with nanopore confinement is lacking in the petroleum industry. The conventional models need to be modified to consider nanopore confinement in both phase equilibrium and fluid transport. Hence, we develop an efficient model to fill this gap and apply to the field production of the Bakken tight oil reservoir. Complex-fracture geometries are also handled in this model. First, we revised the phase equilibrium calculation and evaluated the fluid properties with nanopore confinement. An excellent agreement between this proposed model and the experimental data is obtained considering nanopore confinement. Afterward, we verified the calculated minimum miscibility pressure (MMP) using this model against the experimental data from a rising-bubble apparatus (RBA). We analyzed the MMP and well performance of CO2 EOR in the Bakken tight oil reservoir. On the basis of the prediction of the field data, the MMP is 450 psi lower than the MMP with bulk fluid when the pore size reduces to 10 nm. Subsequently, we examined the effects of key parameters such as matrix permeability and CO2 molecular diffusion on the CO2 huff ’n’ puff process. Results show that both CO2-diffusion and capillary pressure effects improve the oil recovery factor from tight oil reservoirs, which should be correctly implemented in the simulation model. Finally, we analyzed well performance of a field-scale horizontal well from the Bakken Formation with nonplanar fractures and natural fractures. Contributions of CO2-diffusion and capillary pressure effects are also examined in depth in field scale with complex-fracture geometries. The oil recovery factor of the CO2 huff ’n’ puff process with both CO2-diffusion and capillary pressure effects increases by as much as 5.1% in the 20-year period compared with the case without these factors. This work efficiently analyzes the CO2 huff ’n’ puff process with complex-fracture geometries considering CO2 diffusion and nanopore confinement in the field production from the Bakken tight oil reservoir. This model can provide a strong basis for accurately predicting the long-term production with complex-fracture geometries in tight oil reservoirs.