Carbon Dioxide Effects on Wellbore-Pressure Response During Injection/Falloff Test

SPE Journal ◽  
2018 ◽  
Vol 23 (03) ◽  
pp. 919-936 ◽  
Author(s):  
Cíntia G. Machado ◽  
Mohammadreza M. Firoozabad ◽  
Albert C. Reynolds
Keyword(s):  
Carbon Dioxide ◽  
Steady State ◽  
Incompressible Flow ◽  
Analytical Solutions ◽  
Darcy’S Law ◽  
Co2 Concentration ◽  
Darcy's Law ◽  
Pressure Solution ◽  
Wellbore Pressure ◽  
Carbonated Water

Summary We provide analytical solutions for the wellbore pressure during an injection/falloff-test problem under radial-flow conditions in homogeneous porous media where the injected fluid is carbonated water. For both the injection and falloff periods, we assume an isothermal process with thermodynamic equilibrium, a linear adsorption isotherm, and viscosities that depend only on the carbon dioxide (CO2) concentration. We also neglect CO2 diffusion, gravity effects, and capillarity effects. For the injection period, we first determine the saturation and concentration distributions with time in the reservoir by applying the method of characteristics to solve the appropriate system of hyperbolic conservation equations, where we assume incompressible fluids. In solving for water saturation and CO2 concentration in water, we neglect the change in water volume caused by the variation of the CO2 concentration in water. After solving for the saturation and concentration profiles, the pressure solution can be obtained by integrating Darcy's law, from the wellbore radius to infinity, while assuming an infinite-acting reservoir and invoking the Thompson-Reynolds steady-state theory (Thompson and Reynolds 1997b). Because Darcy's law does not assume incompressible flow, the pressure solution generated does not assume incompressible flow. To obtain an analytical expression for the wellbore pressure, however, we do assume that for injection and falloff, the total flow-rate profile in the reservoir is constant in a region from the wellbore to a radius greater than the radius of the flood front. The region within this radius increases with time and it is referred to as the steady-state region or zone (Thompson and Reynolds 1997b). During the falloff stage, it is assumed that there is no change in saturation in the reservoir, which is reasonable because we neglect capillary pressure, the gravity force, and fluid compressibilities when determining the saturation profile. Using these assumptions, we generate analytical solutions for a carbonated-water-injection (CWI)/falloff test and compare these solutions with those obtained with a commercial reservoir simulator using very fine spatial grids and very small timesteps. This comparison suggests that the analytical solutions presented can be used reliably to analyze pressure data obtained during CWI/falloff tests.

An Approximate Method for Non -Darcy Radial Gas Flow

1964 ◽  
Vol 4 (02) ◽  
pp. 96-114 ◽  
Author(s):  
G. Rowan ◽  
M.W. Clegg
Keyword(s):  
Linear Equation ◽  
Flow Rate ◽  
Analytical Solutions ◽  
Gas Flow ◽  
Darcy’S Law ◽  
Darcy's Law ◽  
Non Linear Equation ◽  
Approximate Analytical Solutions ◽  
Non Linear ◽  
Flow Law

Abstract Approximate analytical solutions for non-Darcy radial gas flow are derived for bounded and infinite reservoirs producing at either constant rate or constant pressure. These analytical solutions are compared with published results for non-Darcy flow obtained on digital and analogue computers, and the agreement is shown to be very good. Some observations on the interpretation of gas well tests are made. Introduction The flow of gases in porous media is a problem that has been the subject of much study in recent years, and many methods have been proposed for solving the non-linear equations associated with it. The assumption that the flow satisfies Darcy's Law (1) leads to a non-linear equation of the form (2) in a homogeneous medium, assuming an equation of state(3) It has been observed, however, that the linear relationship between the flow rate and pressure gradient is only approximately valid even at low flow rates, and that as the flow rate increases the deviations from linearity also increase. It has been suggested by a number of authors that Darcy's Law should be replaced by a quadratic flow law of the form (4) This form of equation was first suggested by Forchheimer and, later, Katz and Cornell, and Irmay, developed a similar equation. Houpeurt derived this form of equation using the concept of an idealized pore system in which each channel consists of sequences of truncated cones giving rise to successive restrictive orifices along the channel. This type of representation leads to a quadratic flow law of type, for all fluids, but it is found that the quadratic term is only significant in the case of gas flow. The methods of Houpeurt for solving gas flow problems will be discussed further in another section of this paper. Solutions of the non-linear equation for Darcy gas flow may be classified as either computer (digital and analogue), or approximate analytical ones. The former include the well-known solutions of Bruce et al., and Aronofsky and Jenkins, but the latter solutions, apart from the simple linearization of equation [2] to yield a diffusion equation in p2, are not so well-known. SPEJ P. 96ˆ

scholarly journals Modeling the Movement of Groundwater VOD in a Rectangular Jumper with a Screen

2021 ◽  
Vol 2 (2) ◽  
pp. 069-073
Author(s):  
EN Bereslavsky
Keyword(s):  
Free Surface ◽  
Steady State ◽  
Analytical Solution ◽  
Incompressible Fluid ◽  
Darcy’S Law ◽  
Exact Analytical Solution ◽  
Darcy's Law

Within the framework of planar steady-state filtration of incompressible fluid according to Darcy’s law, an exact analytical solution of the problem of flow in a rectangular cofferdam with a screen in the presence of evaporation from the free surface of groundwater is given. The limiting cases of the considered motion - filtration in unconfined reservoir to imperfect gallery, as well as the flow in the absence of evaporation - are noted.

scholarly journals On the validity of modeling concepts for (the simulation of) groundwater flow in lowland peat areas – case study at the Zegveld experimental field

2011 ◽  
Vol 8 (1) ◽  
pp. 2065-2101
Author(s):  
P. Trambauer ◽  
J. Nonner ◽  
J. Heijkers ◽  
S. Uhlenbrook
Keyword(s):  
Steady State ◽  
Groundwater Flow ◽  
Darcy’S Law ◽  
Single Layer ◽  
Darcy's Law ◽  
Layer Model ◽  
Peat Soils ◽  
Slug Tests ◽  
Flow Models ◽  
Humified Peat

Abstract. The groundwater flow models currently used in the western part of The Netherlands and in other similar peaty areas are thought to be a too simplified representation of the hydrological reality. One of the reasons is that due to the schematization of the subsoil, its heterogeneity cannot be represented adequately. Moreover, the applicability of Darcy's law in these types of soils has been questioned, but this law forms the basis of most groundwater flow models. With the purpose of assessing the typical heterogeneity of the subsoil and to verify the applicability of Darcy's law fieldwork was completed at a research site in the western part of The Netherlands. The assessments were carried for the so called Complex Confining Layer (CCL), which is the Holocene peaty to clayey layer overlying Pleistocene sandy deposits. Borehole drilling through the CCL with a hand auger was completed and revealed the typical heterogeneous character of this layer showing a dominance of muddy, humified peat which is alternated with fresher peat and clay. Slug tests were carried out to study the applicability of Darcy's law given that previous studies suggested the non validity for humified peat soils given by a variable hydraulic conductivity K with the hydraulic gradient. For higher humification degrees, the experiments indeed suggested a variable K, but this seems to be the result of the inappropriate use of steady-state formulae for transient experiments in peaty environments. The muddy peat sampled has a rather plastic nature, and the high compressibility of this material leads to transient behavior. However, using transient formulae, the slug tests conducted for different initial hydraulic heads showed that there was hardly any evidence of a variation of the hydraulic conductivity with the hydraulic gradient. Therefore, Darcy's law can be used for peat soils. The heterogeneity of the subsoil and the apparent applicability of Darcy's law were taking into account for the detailed heterogeneous model that was prepared for the research site. A MODFLOW model consisting of 13 layers in which 4 layers represent the heterogeneous CCL was set up for an average year assuming steady state conditions and for the winter of 2009 to 2010 adopting transient conditions. The transient model was then extended for a whole hydrological year and for an eight year period with the objective of visualizing the flowpaths through the CCL. The results from these models were compared with a 10 layer model whereby the CCL is represented by a single layer assuming homogeneity. From the comparison of the two model types the conclusion could be drawn that a single layer schematization of the CCL produces flowpath patterns which are not the same but still quite similar to a 4 layer representation of the CCL. However, the single layer schematization results in a considerable underestimation of the flow velocity, and subsequently a longer travel time, through the CCL. Therefore, a single layer model of the CCL seems quite appropriate to represent the flow behavior of the shallow groundwater system, but would be inappropriate for transport modeling through the CCL.

Ammonium application mitigates the effects of elevated carbon dioxide on the carbon/nitrogen balance of Phoebe bournei seedlings

2021 ◽  
Author(s):  
Xiao Wang ◽  
Xiaoli Wei ◽  
Gaoyin Wu ◽  
Shengqun Chen
Keyword(s):  
Carbon Dioxide ◽  
Enzyme Activities ◽  
Atmospheric Carbon Dioxide ◽  
Global Climate ◽  
Co2 Concentration ◽  
Rubisco Activase ◽  
Plant Responses ◽  
N Metabolism ◽  
C And N ◽  
Phoebe Bournei

Abstract The study of plant responses to increases in atmospheric carbon dioxide (CO2) concentration is crucial to understand and to predict the effect of future global climate change on plant adaptation and evolution. Increasing amount of nitrogen (N) can promote the positive effect of CO2, while how N forms would modify the degree of CO2 effect is rarely studied. The aim of this study was to determine whether the amount and form of nitrogen (N) could mitigate the effects of elevated CO2 (eCO2) on enzyme activities related to carbon (C) and N metabolism, the C/N ratio, and growth of Phoebe bournei (Hemsl.) Y.C. Yang. One-year-old P. bournei seedlings were grown in an open-top air chamber under either an ambient CO2 (aCO2) (350 ± 70 μmol•mol−1) or an eCO2 (700 ± 10 μmol•mol−1) concentration and cultivated in soil treated with either moderate (0.8 g per seedling) or high applications (1.2 g per seedling) of nitrate or ammonium. In seedlings treated with a moderate level of nitrate, the activities of key enzymes involved in C and N metabolism (i.e., Rubisco, Rubisco activase and glutamine synthetase) were lower under eCO2 than under aCO2. By contrast, key enzyme activities (except GS) in seedlings treated with high nitrate or ammonium were not significantly different between aCO2 and eCO2 or higher under eCO2 than under aCO2. The C/N ratio of seedlings treated with moderate or high nitrate under eCO2was significantly changed compared with the seedlings grown under aCO2, whereas the C/N ratio of seedlings treated with ammonium was not significantly different between aCO2 and eCO2. Therefore, under eCO2, application of ammonium can be beneficial C and N metabolism and mitigate effects on the C/N ratio.

scholarly journals An extension of Darcy’s law incorporating dynamic length scales

2019 ◽  
Vol 129 ◽  
pp. 70-79 ◽  
Author(s):  
Yuhang Wang ◽  
Saman A. Aryana ◽  
Myron B. Allen
Keyword(s):  
Darcy’S Law ◽  
Darcy's Law ◽  
Length Scales ◽  
Dynamic Length

scholarly journals The Use of Nonlinear Constitutive Equations to Evaluate Draw Resistance and Filter Ventilation

2001 ◽  
Vol 19 (4) ◽  
pp. 177-188
Author(s):  
B Eitzinger ◽  
G Ederer
Keyword(s):  
Pressure Drop ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Constitutive Equations ◽  
Darcy’S Law ◽  
Stationary Flow ◽  
Darcy's Law ◽  
Nonlinear Behaviour ◽  
Cigarette Paper ◽  
Drop Flow

AbstractThis study investigates by nonlinear constitutive equations the influence of tipping paper, cigarette paper, filter, and tobacco rod on the degree of filter ventilation and draw resistance. Starting from the laws of conservation, the path to the theory of fluid dynamics in porous media and Darcy's law is reviewed and, as an extension to Darcy's law, two different nonlinear pressure drop-flow relations are proposed. It is proven that these relations are valid constitutive equations and the partial differential equations for the stationary flow in an unlit cigarette covering anisotropic, inhomogeneous and nonlinear behaviour are derived. From these equations a system of ordinary differential equations for the one-dimensional flow in the cigarette is derived by averaging pressure and velocity over the cross section of the cigarette. By further integration, the concept of an electrical analog is reached and discussed in the light of nonlinear pressure drop-flow relations. By numerical calculations based on the system of ordinary differential equations, it is shown that the influence of nonlinearities cannot be neglected because variations in the degree of filter ventilation can reach up to 20% of its nominal value.

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