Interpretation of Well-Block Pressures in Numerical Reservoir Simulation With Nonsquare Grid Blocks and Anisotropic Permeability

1983 ◽  
Vol 23 (03) ◽  
pp. 531-543 ◽  
Author(s):  
D.W. Peaceman
Keyword(s):  
Steady State ◽  
Single Phase ◽  
Numerical Solutions ◽  
Flow Equation ◽  
Anisotropic Permeability ◽  
Equivalent Radius ◽  
Square Grid ◽  
Bottom Hole ◽  
Single Well ◽  
Mathematical Derivation

Abstract Previous work on the interpretation of well-block Previous work on the interpretation of well-block pressure (WBP) for a single isolated well is extended to pressure (WBP) for a single isolated well is extended to the case of non square grid blocks (delta x does not equal delta y). Numerical solutions for the single-phase five-spot problem, involving various grid sizes, show that the effective well-block radius (where the actual flowing pressure equals the numerically calculated WBP) is given by This relationship is verified by a mathematical derivation for a single well in an infinite grid. The exact value of the constant is shown to be c -gamma/4, where gamma is Euler's constant. Finally, the analysis is extended to include anisotropic permeability, and an expression for the effective permeability, and an expression for the effective well-block radius in terms of delta x, delta y, kx, and ky is derived. Introduction In the modeling of a reservoir by numerical methods, it is necessary to use grid blocks whose horizontal dimensions are much larger than the diameter of a well. As a result, the pressure calculated for a block containing a well, po, is greatly different from the flowing bottom hole pressure (BHP) of the well, pwf. In a previous paper, the equivalent radius of a well block, ro, was paper, the equivalent radius of a well block, ro, was defined as that radius at which the steady-state flowing pressure for the actual well is equal to the numerically pressure for the actual well is equal to the numerically calculated pressure for the well block. This definition for ro gives (1) For a square grid (delta x = delta y), careful numerical experiments on a five-spot pattern showed that the ratio of ro to delta x ranges from 0. 1936 (for a 3 × 3 grid to a limit (2) It was also shown that the pressures in the blocks adjacent to a well block approximately satisfy the steady-state radial flow equation (3) By assuming that Eq. 3 is satisfied exactly, one can derive the relation (4) SPEJ P. 531

Pressure Buildup Behavior in a Two-Well Gas-Oil System

1967 ◽  
Vol 7 (02) ◽  
pp. 195-204 ◽  
Author(s):  
R.C. Earlougher ◽  
F.G. Miller ◽  
T.D. Mueller
Keyword(s):  
Steady State ◽  
Single Phase ◽  
Average Pressure ◽  
Two Phase ◽  
Pressure Buildup ◽  
Solution Gas Drive ◽  
Single Well ◽  
Gas Drive ◽  
The Individual ◽  
Solution Gas

Abstract In analyzing pressure buildup tests for field wells producing both oil and gas, the common practice is to use a modification of single-phase flow theory. Validity of such an approximation has been demonstrated for single-well solution-gas-drive systems. This paper indicates that such approximations are also valid for two-well solution-gas-drive systems, which infers that the technique can be used for multiple-well systems. A computer was used to simulate the behavior of a two-well solution-gas-drive reservoir to test the validity of the above type of analysis. Simulation results indicate that pressure buildup tests in such a system can be analyzed within engineering accuracy for formation permeability and pressure. A rule of thumb is given for estimating the length of time a well must be shut in for the pressure in a nearby producing well to increase significantly. To observe such an increase, the shut-in well would have to be left shut in much longer than normal. INTRODUCTION An important technique for obtaining data concerning a producing petroleum reservoir is the pressure buildup test. Such a test, when properly conducted and analyzed, provides information on the average reservoir pressure and the permeability in the major drainage area of a well. The theory upon which the analysis of a pressure buildup test is based assumes that the behavior of the fluid in the reservoir is adequately described by the diffusivity equation.1 Use of the diffusivity equation implies that a single fluid of small and constant compressibility is flowing. To analyze a pressure buildup test in situations which involve more than one fluid phase, the single-fluid analysis is extended. This paper verifies the extension of pressure buildup analyses to two-phase, two-well systems. Miller, Dyes and Hutchinson1 have presented a technique for analyzing pressure buildup tests in circular bounded reservoirs. Their method requires that the reservoir be producing at pseudo-steady state (constant pressure-gradients) prior to shutin. An alternate reservoir model, presented by Horner,2 assumes that the reservoir is infinite. The Homer model does not assume a pseudosteady state prior to shut-in, but the assumption that the reservoir is infinite implies that the average pressure can be determined only if the total production prior to shut-in is small compared to the total fluid originally present. Limitations imposed by the pseudo-steady state and infinite reservoir assumptions are avoided by the technique proposed by Matthews, Brons and Hazebroek.3 Their method can be used to calculate both the permeability and the average pressure in bounded single-well systems which are not necessarily at steady state. They also suggested determining the average pressure of a multi-well reservoir producing from pseudo-steady state by volumetrically averaging the static pressures of the individual wells. Matthews and Lefkovits4 used numerical simulation techniques to verify that this method does produce adequate results for single-phase, multiple-well systems. Perrine5 proposed modifications to the single-phase theory so that pressure buildup tests in multiple-phase systems could be analyzed. He suggested replacing the single-phase mobility (k/µ) by the sum of the mobilities of the individual phases, and replacing the fluid compressibility by an average compressibility weighted by the saturations of the separate phases. By using numerical techniques to solve the two-phase flow problem for a single-well system, he concluded that this approximation gives results within engineering accuracy. More recently, Weller6 has shown that, for a gas-oil system with a single well at the center of a circular reservoir, pressure buildup tests can be analyzed adequately by using the modification proposed by Perrine. Weller's results also indicate that, as the gas saturation increases, this analysis becomes less accurate.

Validation of SAS4A/SASSYS-1 for predicting steady-state single-phase natural circulation

2021 ◽  
Vol 377 ◽  
pp. 111149
Author(s):  
Taiyang Zhang ◽  
Erik R. Smith ◽  
Caleb S. Brooks ◽  
Thomas H. Fanning
Keyword(s):  
Steady State ◽  
Single Phase ◽  
Natural Circulation

Improved Load Sharing Control Techniques for Inverters used in a Microgrid

2009 ◽  
Vol 10 (1) ◽  
Author(s):  
Wei Yao ◽  
Zhaoming Qian
Keyword(s):  
Steady State ◽  
Transient Response ◽  
Single Phase ◽  
Load Sharing ◽  
Power Sharing ◽  
Experimental Results ◽  
Control Technique ◽  
Transient Performance ◽  
Control Techniques ◽  
Control Scheme

In this paper, an improved load sharing control scheme is presented, which is able to improve the transient response and power sharing accuracy of parallel-connected inverters used in microgrid. It also shows how the improved droop method can be easily adapted to account for the operation of parallel-connected inverters, providing good performance under the variation and disturbance of loads, as well as achieving good steady-state objectives and transient performance. Two DSP-based single-phase Microgrid inverters are designed and implemented. Simulation and experimental results are all reported, confirming the validity of the proposed control technique.

The Productivity of Horizontal Wells in an In-Line Development System in Fields with Oil Rims

2021 ◽  
Author(s):  
Dmitriy Alekseevich Samolovov ◽  
Artem Igorevich Varavva ◽  
Vitalij Olegovich Polyakov ◽  
Ekaterina Evgenevna Sandalova
Keyword(s):  
Numerical Experiments ◽  
Single Phase ◽  
Analytical Formula ◽  
Horizontal Wells ◽  
Flow Equation ◽  
Development Pattern ◽  
Phase Flow ◽  
Single Phase Flow ◽  
Limited Applicability ◽  
Good Agreement

Abstract The study proposes an analytical method for calculating the productivity of horizontal wells in a line-drive development pattern in fields with oil rims. The paper presents an analysis of existing techniques and compares them with the results of detailed numerical experiments. It also shows the limited applicability of existing techniques. On the basis of the obtained solution of a single-phase flow equation for a line-drive pattern of horizontal wells, an analytical formula was obtained which more accurately describes the productivity of wells beyond the limits of applicability of existing methods. The resulting formula is in good agreement with the results of a detailed numerical experiment.

The Partial Transformational Decomposition Method for a Hybrid Analytical/Numerical Solution of the 3D Gas-Flow Problem in a Hydraulically Fractured Ultralow-Permeability Reservoir

SPE Journal ◽  
2021 ◽  
pp. 1-28
Author(s):  
George Moridis ◽  
Niwit Anantraksakul ◽  
Thomas A. Blasingame
Keyword(s):  
Differential Equation ◽  
Decomposition Method ◽  
Gas Flow ◽  
Numerical Solutions ◽  
Flow Equation ◽  
Time Variable ◽  
Strong Nonlinearity ◽  
Fully Implicit ◽  
Ultralow Permeability ◽  
Klinkenberg Effects

Summary The analysis of gas production from fractured ultralow-permeability (ULP) reservoirs is most often accomplished using numerical simulation, which requires large 3D grids, many inputs, and typically long execution times. We propose a new hybrid analytical/numerical method that reduces the 3D equation of gas flow into either a simple ordinary-differential equation (ODE) in time or a 1D partial-differential equation (PDE) in space and time without compromising the strong nonlinearity of the gas-flow relation, thus vastly decreasing the size of the simulation problem and the execution time. We first expand the concept of pseudopressure of Al-Hussainy et al. (1966) to account for the pressure dependence of permeability and Klinkenberg effects, and we also expand the corresponding gas-flow equation to account for Langmuir sorption. In the proposed hybrid partial transformational decomposition method (TDM) (PTDM), successive finite cosine transforms (FCTs) are applied to the expanded, pseudopressure-based 3D diffusivity equation of gas flow, leading to the elimination of the corresponding physical dimensions. For production under a constant- or time-variable rate (q) regime, three levels of FCTs yield a first-order ODE in time. For production under a constant- or time-variable pressure (pwf) regime, two levels of FCTs lead to a 1D second-order PDE in space and time. The fully implicit numerical solutions for the FCT-based equations in the multitransformed spaces are inverted, providing solutions that are analytical in 2D or 3D and account for the nonlinearity of gas flow. The PTDM solution was coded in a FORTRAN95 program that used the Laplace-transform (LT) analytical solution for the q-problem and a finite-difference method for the pwf problem in their respective multitransformed spaces. Using a 3D stencil (the minimum repeatable element in the horizontal well and hydraulically fractured system), solutions over an extended production time and a substantial pressure drop were obtained for a range of isotropic and anisotropic matrix and fracture properties, constant and time-variableQ and pwf production schemes, combinations of stimulated-reservoir-volume (SRV) and non-SRV subdomains, sorbing and nonsorbing gases of different compositions and at different temperatures, Klinkenberg effects, and the dependence of matrix permeability on porosity. The limits of applicability of PTDM were also explored. The results were compared with the numerical solutions from a widely used, fully implicit 3D simulator that involved a finely discretized (high-definition) 3D domain involving 220,000 elements and show that the PTDM solutions can provide accurate results for long times for large well drawdowns even under challenging conditions. Of the two versions of PTDM, the PTD-1D was by far the better option and its solutions were shown to be in very good agreement with the full numerical solutions, while requiring a fraction of the memory and orders-of-magnitude lower execution times because these solutions require discretization of only the time domain and a single axis (instead of three). The PTD-0D method was slower than PTD-1D (but still much faster than the numerical solution), and although its solutions were accurate for t < 6 months, these solutions deteriorated beyond that point. The PTDM is an entirely new approach to the analysis of gas flow in hydraulically fractured ULP reservoirs. The PTDM solutions preserve the strong nonlinearity of the gas-flow equation and are analytical in 2D or 3D. This being a semianalytical approach, it needs very limited input data and requires computer storage and computational times that are orders-of-magnitude smaller than those in conventional (numerical) simulators because its discretization is limited to time and (possibly) a single spatial dimension.

scholarly journals Source shape and data analysis procedure effects on hydraulic conductivity of a sandy-loam soil determined by ponding infiltration runs

2017 ◽  
Vol 48 (2) ◽  
pp. 71 ◽  
Author(s):  
Vincenzo Bagarello ◽  
Andrea De Santis ◽  
Giuseppe Giordano ◽  
Massimo Iovino
Keyword(s):  
Steady State ◽  
Hydraulic Conductivity ◽  
Sandy Loam ◽  
Analysis Procedure ◽  
Sandy Loam Soil ◽  
Second Phase ◽  
General Validity ◽  
Equivalent Radius ◽  
Loam Soil ◽  
Ks Values

Performing ponding infiltration runs with non-circular sources could represent a good means to sample completely an area of interest. Regardless of the shape of the source, predicting the expected reliability of the collected data by infiltrometers should facilitate soil hydraulic characterisation and also allow a more conscious use of the field data. The influence of the shape of the infiltration source (i.e., circular or square) and the analysis procedure of the steady-state infiltration data on the saturated hydraulic conductivity, Ks, of a sandy-loam soil was tested in this investigation. Circular and square surfaces sampled with the pressure infiltrometer (PI) yielded similar estimates of Ks (i.e., differing by a factor of 1.05-1.16, depending on the calculation method) when an equivalent radius was considered to geometrically describe the square source. With the simplified falling head (SFH) technique, the shape of the source was irrelevant (i.e., circular and square sources yielding Ks values that differed by a factor of 1.19), as theoretically expected. For the steady-state PI experiment, the twoponding depth approach yielded two times smaller Ks values than the one-ponding depth (OPD) approach, probably due to lower steady-state flow rates than those expected for the second phase of the two-level run. The conclusions were that: i) simple infiltrometer experiments (PI, SFH) can be carried out with square sources; and ii) the simplest PI run (OPD approach) is expected to yield the most reliable predictions of Ks. Sampling other soils is advisable in an attempt to make these conclusions of general validity.

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