A Unique Approach in Modelling Flow in Tight Oil Unconventional Reservoirs With Viscous, Inertial and Diffusion Forces Contributions

A model for single-phase fluid flow in tight UCRs was previously produced by modifying the flow Forchheimer’s equation. The new modification addresses the fluid transport phenomena into three scales incorporating a diffusion term. In this study, a new liner model, numerically solved, has been developed and deployed for a gas huff and puff project. The new model has been numerically validated and verified using synthetic data and huff and puff case study. Ideally, the new model suits fluid flow in tight UCRs. The modified Forchheimer’s model presented is solved using the MATLAB numerical method for linear multiphase flow. For the huff & puff case, very simple profiles and flow dynamics of the main flow parameters have been established and a thorough parametric analysis and verifications were performed. It has been observed that the diffusion system becomes more prominent in regulating flow velocity with low permeability of the formation rock and low viscosity of the flowing fluid. The findings indicate a behavioral alignment with a previous hypothesis that matches actual reservoir behavior.

Prediction of pressure drop through packed beds of granular materials: influence of moisture reduction and air velocity

The influence of the moisture content of the solids on the pressure drop through packed beds of soy, barley, lentils and oats was investigated. The properties of the grains and the packed beds were determined varying the moisture contents of the solids between the equilibrium moisture content and 0.24 g g-1 (mass of water per mass of dry solid). The pressure drops were measured as a function of the aeration velocity under different moisture contents of the solids. It was observed that a reduction in the moisture content caused a decrease in the particle dimensions but did not affect their shape. Due to the reduction in size caused by the moisture removal, the packed beds become denser and less permeable to the airflow, resulting in an increase in the pressure drop. In the moisture range investigated, the pressure drops increased by more than 39% for lentils, soy and barley, and by about 24% for oats, indicating that the energy consumption during aeration could rise significantly. The parameters of the Forchheimer’s equation were modified to take into account the influence of the moisture content of the solids on the pressure drops. For each particle, empirical equations were proposed and shown to be adequate to accurately predict the pressure drop of the packed beds as a function of the moisture content and aeration velocity.

Effect of fluid-porous interface conditions on steady flow around and through a porous circular cylinder

Purpose – The purpose of this paper is to implement the numerical analysis based on finite volume method to compare the effects of stress-jump (SJ) and stress-continuity (SC) conditions on flow structure around and through a porous circular cylinder. Design/methodology/approach – In this study, a steady flow of a viscous, incompressible fluid around and through a porous circular cylinder of diameter “D,” using Darcy-Brinkman-Forchheimer’s equation in the porous region, is discussed. The SJ condition proposed by Ochoa-Tapia and Whitaker is applied at the porous-fluid interface and compared with the traditional interfacial condition based on the SC condition in fluid and porous media. Equations with the relevant boundary conditions are numerically solved using a finite volume approach. In this study, Reynolds and Darcy numbers are varied within the ranges of 1 < Re < 40 and 10-7 < Da < 10-2, respectively, and the porosities are e=0.45, 0.7 and 0.95. Findings – Results show that the SJ condition leads to a much smaller boundary layer within porous medium near the interface as compared to the SC condition. Two interfacial conditions yield similar results with decrease in porosity. Originality/value – There is no published research in the literature about the effects of important parameters, such as Porosity and Darcy numbers on different fluid-porous interface conditions for a porous cylinder and comparison the effects of SJ and SC conditions on flow structure around and through a porous circular cylinder.

Estimation of Turbulence Coefficient Based on Field Observations

Summary Isochronal testing is commonly used to evaluate the performance of gas wells. This paper proposes a new technique to estimate the value of the turbulence coefficient based on isochronal tests. The proposed method is easy to apply and evaluate. Further, the method also provides a value of bg under stabilized conditions which can be used to predict the performance of gas wells under stabilized conditions. The proposed method is validated using field data under a variety of operating conditions. The values of the turbulence coefficient based on the field data can differ significantly compared to the literature correlations. This further shows the importance of obtaining appropriate reservoir parameters based on the field rather than the laboratory data. Introduction The use of isochronal or modified isochronal testing is well established in the gas industry. These tests are common for gas wells which take a long time to reach a stabilized rate. A common example would be a low permeability, fractured reservoir. Instead of testing these wells until a stabilized rate is reached, the wells are tested for a fixed period of time and the bottomhole pressure is measured. For isochronal testing, the well is then shut in until it reaches a stabilized pressure and the procedure is repeated for a different rate. For modified isochronal testing, the well is shut in for a fixed period of time, and the shut-in pressure is measured at the end of that period. The procedure is then repeated at other rates. By repeating this procedure for different time intervals, we can gather information about rate vs. pressure drop in the formation for these time intervals. Ultimately, using this information, our goal is to establish an appropriate rate vs. pressure drop relationship under stabilized conditions. Two procedures are commonly used to establish the equation for rate vs. pressure drop. An empirical method states that q g = C ( p ￣ 2 − p w f 2 ) n . ( 1 ) We can write a simpler equation in terms of pseudo-real pressures as q g = C [ m ( p ￣ ) − m ( p w f ) ] n . ( 2 ) Under transient conditions, the value of C is not constant. Instead, we can write Eq. 2 as q g = C ( t ) [ m ( p ￣ ) − m ( p w f ) ] n , ( 3 ) where C(t) represents a term which is a function of isochronal interval t. In the literature, methods are proposed to estimate the value of C corresponding to the stabilized rate based on the transient state information ?C(t) For example, Hinchman et al.1 propose that 1/C(t)1/n be plotted as a function of log t, and the line be extrapolated until t is equal to the time it takes to reach the stabilized state period. In their method, they assume that n is constant, where n is an inverse of slope when log[m(p¯)−m(pwf)] is plotted as a function of qg. Although we get different straight lines corresponding to different t, the authors assume that the slopes are approximately constant. Another commonly used approach in analyzing isochronal tests is to use an equation, m ( p ￣ ) − m ( p w f ) = a g q g + b g q g 2 . ( 4 ) A similar equation can also be written in terms of pressure squared terms. Eq. 4 is derived starting from Forchheimer's equation. Under transient conditions, we can rewrite Eq. 4 as m ( p ￣ ) − m ( p w f ) = a g ( t ) q g + b g q g 2 , ( 5 ) where ag(t) is a function of isochronal interval, and bg is assumed to be constant. A commonly used technique is to plot ag(t) vs. log (t) and extrapolate ag(t) corresponding to a value of t which represents the time required to reach a stabilized rate.2–4 In using both Eqs. 3 and 5, we have assumed that the contribution due to the non-Darcy effect is not affected during the transient conditions. For example, in applying Eq. 3, we assume that n is constant during the transient period, and in applying Eq. 5, we assume that bg is constant during the transient period. Both n and bg represent the relative contributions of the non-Darcy flow. n will approach 0.5 as the non-Darcy effect becomes dominant, and bg becomes larger as the non-Darcy effect becomes significant. However, by assuming that n and bg are constant during the transient periods, we are ignoring the changes in the relative contributions due to the Darcy and non-Darcy terms. In this article, we extend the previous analysis to account for changes in the non-Darcy term during the transient period. Further, by proper analysis, we propose a method to estimate the value of the turbulence coefficient based on the evaluation of the transient period data. Approach In our approach, instead of using the empirical equation (Eq. 3), we will begin with Forchheimer's equation, where the pressure gradient in a radial reservoir is calculated by ∂ p ∂ r = μ g k v + β ρ g v 2 . ( 6 ) The permeability (k) of the reservoir may be established based on well test data or core information. The turbulence coefficient is difficult to estimate. Although literature correlations5,6 exist to calculate the value of ? based on the laboratory experiments, field evidence7 indicates that the ? values in the field are significantly greater than the laboratory experiments.

Resistance to the Flow of Fluids Through Simple and Complex Porous Media Whose Matrices Are Composed of Randomly Packed Spheres

Experimental data relating to the flow of fluids through simple and complex porous media whose matrices are composed of randomly packed spheres have been obtained. In this context the term “simple” refers to porous media whose matrices are composed of spheres of uniform diameter, while “complex” refers to matrices composed of spheres having different diameters. It was found that Darcy’s law is valid for simple media within a range of the Reynolds number, Re, whose upper bound is 2.3. The upper bounds of Darcy flow for complex media were found to be consistent with this value. It is shown that the resistance to flow in the Darcy regime can be characterized by taking the Kozeny-Carman constant equal to 5.34 if the characteristic dimension is taken equal to the weighted harmonic mean diameter of the spheres that comprise the matrix. Forchheimer’s equation was found to be valid for simple media within the range 5 ≤ Re ≤ 80. The corresponding bounds for complex media were found to be consistent with this range. It is shown that the resistance to flow in the Forchheimer regime for both simple and complex media can be characterized by adopting the following values of the Ergun constants: A = 182 and B = 1.92. Finally, it is shown that fully developed turbulent flow exists when Re > 120 and that the resistance to flow in the turbulent regime can be calculated using Forchheimer’s equation by adopting the following values of the Ergun constants: A′ = 225 and B′ = 1.61. A simple method for characterizing the behavior of porous media in the transition regions between Darcy and Forchheimer and between Forchheimer and turbulent flow is presented.

Transient Fluid Flow in Porous Media: Inertia-Dominated to Viscous-Dominated Transition

A one-dimensional isothermal flow is induced by a step change in the pressure at the boundary of a semi-infinite medium. The early flow is inertia-dominated, in accordance with Ergun’s equation, and is self-similar in the variable x/t3. The late flow is viscous-dominated, in accordance with Darcy’s law, and is self-similar in the variable x/t. Comprehensive numerical results are presented for both of these asymptotic regimes and also for the intermediate transition period which is governed by Forchheimer’s equation. The only explicit parameter is the pressure ratio, N, which is varied from N → ∞ (strong gas-compression), through N → 1 (constant compressibility liquid), to N → 0 (strong gas-rarefaction). The solution procedure is based on a generalized separation-of-variables approach which should also be useful in other problems which possess self-similar asymptotic solutions both at early times and at late times.