Assumption Of Radial Flow Research Articles

Evaluation of a Greener Fracturing Fluid for Geothermal Energy Recovery: An Experimental and Simulation Study

The objective of this work is to study the performance of 1wt% StimuFrac fluid [polyallylamine (PAA) in water] in ½-foot size rock samples under thermal and fluid phase conditions representative of enhanced geothermal systems (EGS) and reveal the mechanisms governing the fracturing process. Four fracturing fluids including water, CO2, CO2 with water, and CO2 with aqueous PAA were used. For all the water “only” fracturing tests, the conductivity of the rock fractured is quite low (less than 2 μm3 based on radial flow assumption), regardless of the constant flow rate or discrete pressure increments injection approach. For all three CO2-based fracturing fluids, granite was fractured at higher breakdown pressures, higher transient flow rates, and generated higher-conductivity fractures as compared to water. When partially saturating the rock sample with 1wt% PAA aqueous solution followed by fracturing with CO2, the volume expansion and viscosity increase triggered by CO2-induced cross-linking PAA leads to a faster pressure increase than CO2 in water saturated or initially dry. This faster pressurization rate is caused by a decrease in relative permeability of CO2 in the presence of the high-viscosity cross-linked fluid compared to the un-crosslinked CO2/water. It was also found that CO2 as a fracturing fluid can generate high fracture conductivity only when injected at very high flow rates in the presence of water or in hot dry rock (HDR). However, the conductivity of CO2 fracturing in HDR is highly variable. CO2 injected in the presence of 1wt% aqueous PAA generates fractures with the highest conductivity independently of injection flow rates and using 1/6 of the mass of CO2 as compared to CO2 fracturing in HDR. The results of this study suggest CO2/PAA is the best performing stimulation fluid under the studied P/T conditions. CO2/PAA offers the following three additional advantages over waterless CO2, and CO2/water fracturing fluids: i) it requires significantly lower volumes of CO2 due to the reduced leak off; ii) large fractures can be generated reproducibly at both low and high CO2 injection flow rate, and iii) the reversible (previously reported) viscosity increase could be beneficial to transport proppants when they become available for EGS.

An application of Green’s function technique for computing well inflow without radial flow assumption

Well modeling plays an important role in numerical reservoir simulation. The main difficulty in well modeling is the difference in scale between the wellbore radius and well gridblock dimension used in the simulation. The Peaceman equation is widely used in reservoir simulation to match gridblock pressure to the local solution of the diffusivity equation describing the flow near the well. However, this approach was developed under the assumption of radial flow. At the same time, the well inflow equation can be solved within the Green’s function (GF) formalism which allows the solution to be obtained without the assumption of radial flow. The GF solution can be presented as a series over the eigenvalues of the Laplace differential operator. However, this series converges conditionally and its direct summation is time-consuming. In Posvyanskii et al. (2008), a method for fast summation of such a series was proposed and successfully applied for analyzing the pressure build up curves. In this paper, we adopt the same technique for calculating the well indices for horizontal, slanted and partially penetrated wells. Additionally, the role of different boundary conditions is considered. The semi-analytical expressions for well indices are obtained and compared to the solution of the Peaceman equation. It is shown that in some cases, the difference between these solutions can be significant. The use of the obtained expression in numerical flow simulation allows well inflow to be modeled with high accuracy even on a coarse grid.

Push‐pull tracer tests: Their information content and use for characterizing non‐Fickian, mobile‐immobile behavior

AbstractPath reversibility and radial symmetry are often assumed in push‐pull tracer test analysis. In reality, heterogeneous flow fields mean that both assumptions are idealizations. To understand their impact, we perform a parametric study which quantifies the scattering effects of ambient flow, local‐scale dispersion, and velocity field heterogeneity on push‐pull breakthrough curves and compares them to the effects of mobile‐immobile mass transfer (MIMT) processes including sorption and diffusion into secondary porosity. We identify specific circumstances in which MIMT overwhelmingly determines the breakthrough curve, which may then be considered uninformative about drift and local‐scale dispersion. Assuming path reversibility, we develop a continuous‐time‐random‐walk‐based interpretation framework which is flow‐field‐agnostic and well suited to quantifying MIMT. Adopting this perspective, we show that the radial flow assumption is often harmless: to the extent that solute paths are reversible, the breakthrough curve is uninformative about velocity field heterogeneity. Our interpretation method determines a mapping function (i.e., subordinator) from travel time in the absence of MIMT to travel time in its presence. A mathematical theory allowing this function to be directly “plugged into” an existing Laplace‐domain transport model to incorporate MIMT is presented and demonstrated. Algorithms implementing the calibration are presented and applied to interpretation of data from a push‐pull test performed in a heterogeneous environment. A successful four‐parameter fit is obtained, of comparable fidelity to one obtained using a million‐node 3‐D numerical model. Finally, we demonstrate analytically and numerically how push‐pull tests quantifying MIMT are sensitive to remobilization, but not immobilization, kinetics.

Minimising the spread of residence-time distribution for flat and heaped powders in a wedge-shaped planar hopper

A wedge-shaped planar mass-flow hopper system was modelled using stress-field theory as found in the literature. The authors present governing equations for stress and velocity fields under a radial-flow assumption in a converging hopper. The velocity in the silo above the hopper is modelled as plug flow. Two set-ups are modelled, one where powder layers in the hopper are assumed to be flat, and the second in which the layers are heaped at some characteristic angle. The ejection times and residence-time distributions are calculated and presented for a range of heap angles. For realistic heap angles, the spread of the residence-time distribution decreases with increasing heap angle; in one case, the spread is halved to a well-defined limit. At this limit (the critical heap angle) the geometry of the hopper can be optimised to minimise the spread of the residence-time distribution, and hence to minimise predicted mixing in the system. We present examples of curves for a variety of parameters that minimise the predicted mixing in the hopper–silo system.

Analysis of the radial flow assumption in a converging model silo

An empirical analysis was carried out on the flow parameters in a converging silo model using the data of Sielamowicz, Błoński, and Kowalewski (2006) , obtained using digital particle image velocimetry. The flow was described using the radial flow assumption, and parabolic and Gaussian functions. Using the velocity profiles it was possible to describe vertical velocity values and their dependence on the distance from the outlet. Statistical analysis of the readings taken from velocity profiles was also carried out. The predictions of velocities very close to the outlet of the model hopper were best given by a Gaussian function. Analysis of radial flow indicated that the radial flow assumption did not adequately describe the flow in converging hoppers since the flow did not run radially throughout the whole hopper. Velocity vectors indicated that there was a vertical flow channel. In the upper part of the hopper, there was a combination of both vertical and radial flows and that this form of flow was observed during the whole process of discharge. A modification of the radial flow model was shown to adequately describe velocities only at some distance away from the outlet.

Estimation of REV size and three-dimensional hydraulic conductivity tensor for a fractured rock mass through a single well packer test and discrete fracture fluid flow modeling

Abstract A new methodology is presented to determine the representative elementary volume (REV) size and three-dimensional (3-D) hydraulic conductivity tensor for a fractured rock mass. First, a 3-D stochastic fracture network model was built and validated for a gneissic rock mass based on the fracture data mapped from scanline surveys at the site. This validated fracture network model was combined with the fracture data observed on a borehole to generate a stochastic-deterministic fracture network system in a cubic block around each packer test conducted at a different depth region in the same borehole. Each packer test was simulated numerically applying a developed discrete fracture fluid flow model to estimate the influenced region or effective range for the packer test. A cubic block of size 18 m, with the packer test interval of length about 6.5 m located at the centre of this block, was found to be suitable to represent the influenced region. Using this block size, the average flow rate per unit hydraulic gradient (defined as the transmissivity multiplied by mean width of flow paths) field for fractures was calibrated at different depth regions around the borehole by numerically simulating the packer tests conducted at different depth regions. The average flow rate per unit hydraulic gradient of the fractures that intersect the borehole was considered to be quite different to the average flow rate per unit hydraulic gradient of the fractures that do not intersect the borehole. A relation was developed to quantify the ratio between these two parameters. By studying the directional hydraulic conductivity behaviour of different cubic block sizes having the validated stochastic fracture network and calibrated hydraulic parameters, a REV for the hydraulic behaviour of the rock mass was estimated to be a block size of 15 m. The hydraulic conductivity tensor in 3-D computed through regression analysis using the calculated directional hydraulic conductivity values in many directions was found to be significantly anisotropic. The principal directions of the hydraulic conductivity tensor were found to be agreeable with the existing fracture system of the site. Further, the geometric hydraulic conductivity calculated was found to be comparable to the hydraulic conductivity estimated through the radial flow assumption in continuum porous media.

Calculation Of Flowing Well Pressures In Reservoir Simulation Using Nine-Point Differencing

Abstract A quantity of considerable importance in flow rate predictions in reservoir simulation is the equivalent well block radius (r o) because it directly relates the numerically calculated grid block pressure to the actual well flowing pressure. Simulation of steam and miscible floods generally requires nine-point differencing. Previous work on approximating ro for nine-point differencing has either severe limitations or can give erroneous results and, hence, incorrect rate predictions. This paper presents, for the first time, a rigorous, closed form expression for ro for an isolated well, valid for any nine-point method. This expression is based on mathematical analysis and confirmed by careful numerical testing. It identically reduces to the expression for five-point differencing (obtained earlier by Peaceman) as a special case. Extension is made to the anisotropic case and confirmed numerically. Results for some additional well-geometries are also presented for the case of nine-point differencing. These include: arbitrary location of a well in the well block, multiple wells within a single block, a well in an edge or comer block. Finally, effects of heterogeneity on predicting flow rates are considered. Introduction In the numerical simulation of reservoir flow, it is usually necessary to use grid blocks whose areal dimensions are much larger than the diameter of a well. Consequently, the numerically calculated well block pressure, po is substantially different from the actual flowing bottomhole pressure of the well, pwf) as shown in Figure 1. Well inflow models are used to relate the well block pressure to the actual wellbore pressure and flow rate through the concept of an equivalent well block radius(1) (ro)), defined by Equation (1) Available In Full Paper. Equation (1) can be expressed as a rate prediction equation: Equation (2) Available In Full Paper. Equation (1) indicates that ro, may be interpreted as the distance from the well at which the steady-state flowing pressure equals the numerically calculated well block pressure, as shown in Figure 1. Peaceman(1) showed that for an isolated vertical well centred in its grid block in a homogeneous and isotropic medium, Equation (3) Available In Full Paper. when conventional five-point differencing is employed over a square grid. Peaceman obtained Equation (3) in two different ways. The first was by numerical experiments; the second by an approximate analytical approach that assumes that the pressures in the grid blocks immediately adjacent to the well block all satisfy the exact radial flow solution. For the square grid case, the first method yielded: Equation (4) Available In Full Paper. while the second method, which is henceforth referred to as the radial flow method, yielded Equation (5) Available In Full Paper. In a second paper, Peaceman(2) pointed out that the radial flow assumption is not generally valid and can result in significant error. For non-square (i.e. rectangular) grids, the radial flow method was found to predict: Equation (6) Available In Full Paper. whereas numerical experimentation showed that the correct expression is Equation (7) Available In Full Paper.

Asymmetric envelope expansion of supernova 1987A

Analytic models for the spherically symmetric expansion of SN 1987 A are discussed. The SN model includes a bend in the expanded density profiles at a velocity of about 4000 km/s and employs an asymmetric initial density distribution or energy deposition on the assumption of radial flow. The calculations show that there is a tendency for the flow to become more spherical during the expansion phase, and that the sense of density asymmetry changes across the bend in the density profile.

Analytical Well Models for Reservoir Simulation

Abstract The computation of flowing-well bottomhole pressure from the pressure of the block containing the well or of well flow rate when the flowing bottomhole pressure is specified are important considerations in reservoir simulation. While this problem has been addressed by several authors, some important aspects of the problem are not treated adequately in the literature. We present an analytical method for computing the well-block factors (constants of the PI) for a well located anywhere in a square or rectangular block (aspect ratio between ½ and 2). Equations for well geometric factors and well fraction constants are given for gridblocks of various types, containing a single well, encountered in reservoir simulation studies. The equations given in this paper can be used for both block-centered and point-distributed grids in five- and nine-point two-dimensional (2D), finite-difference formulations. The radial flow assumption used in deriving the equations in this paper is not always strictly valid; however, for most practical situations it provides an adequate approximation for near-well flow.

Pseudosteady-State Flow Capacity of Oil Wells With Limited Entry and an Altered Zone Around the Wellbore

Abstract An analytic solution for pseudosteady-state flow of an oil well with limited entry and with an altered zone is derived. Wells that are open to flow along a traction of their productive interval, and that have a wellbore surrounded by a zone whose physical characteristics have been altered because of damage or improvement, are termed wells with limited entry and an altered zone. The finite cosine transform was used to solve the partial differential equations that describe the behavior of the physical system. The solution was programmed for the CDC 6400 computer, and many programmed for the CDC 6400 computer, and many computer runs covering a wide range of variables were made. The results were used to reduce the complicated analytical solution to a simple approximate solution that permits the engineer to account easily for the combined effect of limited entry and altered zone on the pseudosteady-state flow behavior of an oil well. Introduction This paper reports the derivation of an analytical relation between drawdown and flow rate for a well producing under pseudosteady-state flow with producing under pseudosteady-state flow with restricted entry to flow and with an altered zone around the wellbore. "Altered zone" refers to a zone whose permeability and/or porosity values have been changed because of mud damage, acidizing, or sand consolidation, for example. The effect of an altered zone around the wellbore on the calculation of the productive capacity of a well is normally accounted for by introducing a skin factor, s, in the proper flow equations. Hawkins developed an algebraic expression relating the wellbore radius, the altered-zone radius and permeability, and the reservoir permeability to s. Hawkins' equation assumes radial flow into the wellbore. This implies that the well is open to flow along the total length of the productive interval. Flow into the wellbore when only a fraction of the productive interval is open to flow, with or without an altered zone, is not radial. The common practice of using Hawkins' equation in such cases practice of using Hawkins' equation in such cases to calculate s could lead to highly erroneous results. For example, the flow rate calculated for a well producing under pseudosteady state, with restricted producing under pseudosteady state, with restricted entry and with an altered zone and using a skin factor calculated by Hawkins' equation, can be too high by more than 100 percent. Rowland and Jones and Watts introduced a correction factor into Hawkins' equation to account for the effect of nonradial flow behavior around the wellbore. Hawkins' skin value is multiplied by ht/hp, where ht is the total thickness of the productive interval and hp is the length of the productive interval and hp is the length of the interval open to flow at the wellbore. The ht/hp correction factor is based on the assumption that the flow into the wellbore is radial in the altered zone opposite the open interval. This implies that the nonradial components of flow into the wellbore are negligible. Using model studies, Jones and Watts showed that the error introduced in the calculated skin values by neglecting the nonradial components of flow is less than 20 percent in the worst case. Furthermore, they proposed a modified equation to eliminate this error. The suitability of the radial flow assumption upon which the factor ht/hp is based depends on the values of many variables such as the radius of the altered zone and its permeability, and the length of the open interval. Because of their discretized nature, numerical models at times do not accurately represent the physical flow pattern around the wellbore. Most models assume the flow to be radial between the wellbore and the adjacent cells, regardless of whether part or all of the productive interval is open to flow. Therefore, results from model studies may not be adequate to define the limits of applicability of the modification to the radial flow formula, such as the ht/hp correction factor. One purpose of the work reported in this paper is to define these limits. paper is to define these limits. The analytic solution is very complex and is not suited for engineering work. Because of this, it was programmed for the CDC 6400 computer, and many computer runs were made. SPEJ P. 271

Graphical solution and interpretation of a new drain-spacing formula

A drain-spacing formula of one of the authors (Kirkham) which is based on rigorous mathematical procedures and which has been tested in the field is now presented in graphical and tabulated form. The formula is useful for computing the removal of steady excess rainfall or steady excess irrigation water. A special case of the formula is compared with a formula derived by Hooghoudt for which he utilized the assumption of radial flow. Since Hooghoudt showed that his formula applied either to the removal of steady surface-applied recharge or to artesian water originating from an aquifer at great depth, we conclude that Kirkham's special formula applies also to surface recharge or to artesian water seeping upward from great depth.