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Images

**Published:**29 January 2019

Figure 4 Pre- and Post-Frac

**Delta****T**. Note the**delta****T**kicks on the bottom post-frac plot aligned with perforations. Note no changes in DTS, DTC between the pre- and post-frac in two marked perforations clusters. The absence of change (increase in post-frac DT) suggests ineffective or unstimulated MoreImages

**Published:**19 September 2010

Images

**Published:**27 June 2010

Figure 4 A crossplot of the real-time Compressional

**Delta**-**T**result with a processed Compressional**Delta**-**T**shows approximately 1-to-1 correlation and an R-squared value of 0.96. Downhole processing algorithms used to obtain the real-time acoustic log are configured prior to drilling to label MoreImages

**Published:**29 January 2019

Figure 6 Relative changes between Pre- and Post-Frac

**delta****T**Shear horizontal. The higher difference, most likely better cluster efficiency. MoreImages

**Published:**23 January 2012

Figure 1 Overpressure interval is indicated by sonic (DTCO-

**delta**‘**t**’ compressional) data starting at ~14550 ft tvdss in KW field. MoreImages

**Published:**17 May 2017

Images

**Published:**04 March 2014

Figure 10 Using the Acoustic

**delta**-**T**measurement in real-time for pore pressure analysis. Early use of the data established trends and the overburden from the mudline. More
Proceedings Papers

Paper presented at the SPWLA 17th Annual Logging Symposium, June 9–12, 1976

Paper Number: SPWLA-1976-J

...SPWLA SEVENTEENTH ANNUAL LOGGINGSYMPOSIUM,JUNE 9-12,1976 PERMAFROST INVESTIGATIONS BY CRYSTAL CABLE SURVEYS MACKENZIE

**DELTA**, N.W.T. by J.H.D. Walker and A.J. Stuart Imperial Oil Limited Calgary, Alberta ABSTRACT For the past five years, Imperial Oil has run a crystal cable velocity survey...
Abstract

ABSTRACT For the past five years, Imperial Oil has run a crystal cable velocity survey as an integral part of the near-surface borehole logging program in all wells in the Mackenzie Delta. The results have been found to be a useful addition to the data available from conventional logging techniques. The near-surface lithological conditions in the Mackenzie Delta, consisting predominantly of unconsolidated sands, gravels, muds and clays, are such that distinct velocity differences are exhibited by materials in the frozen and unfrozen states. Under these conditions, the crystal cable velocity survey provides a means of obtaining detailed velocity profiles of the permafrost zone with a minimum of time and effort. These profiles can be used to refine the predrill estimates of depths to deeper seismic markers. Interval velocities within the permafrost zone are highly variable, a function of such factors as lithology, porosity, water content and salinity, and temperature. Velocities measured by crystal cable surveys range from 7500 fps to 13,500 fps in the permafrost section, in contrast to velocities in the range of 5000 fps to 6500 fps commonly observed in the non-frozen section. Field examples show that the crystal cable results relate directly to the responses observed on conventional wireline logs, and to properly define a base of permafrost, all available data sources must be used. In general, the base of permafrost defined by log responses does not agree with the 32 degF temperature isotherm, but for the exploration geophysicist, it has a direct bearing on the interpretation of exploration geophysical data.

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Annual Technical Conference and Exhibition, September 27–30, 1987

Paper Number: SPE-16811-MS

... identified. The reason for resorting to a log-log plot [log (pi - pwf) vs. log (

**t**+**delta****t**)] rather than a semilog plot is that an identifiable straight line may not exist on a Horner plot if**delta****t****t**for in such cases (**t**+**delta****t**)/**delta****t**approximates 1. (Under such circumstances, the possibility...
Abstract

SPE Members Abstract The derivative analysis has been shown to be an important tool in identifying flow regimes and also to be of assistance for type curve matching pressure data. This paper extends procedures to analyze pressure data by suggesting new applications of the derivative method to determine formation parameters and initial reservoir pressure. It is shown that specific equalities exist between pressure changes and its derivative and that these relations can be used to analyze buildup data when the effect of producing time on buildup data is important including data following a slug test or an "impulse" test. It is also shown that the derivative method can provide a quantitative measure of the accuracy of the slope of the semilog straight line. Introduction The principal contribution of this paper examines methods to analyze pressure buildup data when producing times are short. The limitations of using drawdown type curves to analyze pressure buildup data were highlighted by Raghavan in 1980. Since that time, several authors have directed their efforts to improve analysis of buildup data following a short flow period. Efforts include methods to analyze data by type curve analysis, identification of the radial or pseudoradial flow period, and log-log methods to analyze data should the radial or pseudoradial flow period exist. The proposal of Soliman (and that of Correa and Ramey) is principally aimed at expanding the time scale so that the data in the radial flow period can be easily identified. The reason for resorting to a log-log plot [log (pi - pwf) vs. log (t + delta t)] rather than a semilog plot is that an identifiable straight line may not exist on a Horner plot if delta t t for in such cases (t + delta t)/delta t approximates 1. (Under such circumstances, the possibility of using Cartesian coordinates after computing log [(t + delta t)/delta t] should not be overlooked.) Theoretically, it can be shown that the pressure buildup response must plot as a straight line on semilog coordinates for the asymptotic approaches suggested by Soliman and Correa and Ramey to be applicable. Fundamentally, Soliman's method represents a long-time approximation to the Horner solution; that is, his idea uses only a portion of the data in the radial flow period should such data exist. Therefore, in certain circumstances, his method may not be applicable although diagnostic procedures may suggest that a radial or pseudoradial flow period exists. Fortunately, it appears that the pressure derivative, d delta p/d ln delta t when plotted along the lines indicated by Soliman can also be used to determine formation flow capacity, kh, and the well condition. In general, for large shut-in times, there is an equivalence between the buildup response and its derivative. The advantages of using the derivative are: knowledge of initial reservoir pressure is not needed, a straight line with slope equal to -1 on log-log coordinates should exist for a longer time span all other things being identical (there are exceptions to this observation and these are noted in the text of this paper), under appropriate circumstances the initial pressure can be determined. MATHEMATICAL MODEL AND ASSUMPTIONS The results presented in this work involve the classic assumptions normally used in well test analysis. We consider the flow of a slightly compressible fluid of constant viscosity in a uniform porous medium wherein gravity effects are assumed to be negligible. In all cases, we consider responses in an infinite porous medium and the well is produced at a constant rate. Several wellbore conditions are considered and specific conditions will be described as needed. P. 613^

Journal Articles

Journal:
Journal of Petroleum Technology

Publisher: Society of Petroleum Engineers (SPE)

*J Pet Technol*24 (09): 1139–1141.

Paper Number: SPE-4145-PA

Published: 01 September 1972

... »

**delta****t**. pwf vs log**t**will also give a straight line if**t**»**delta****t**. When**t**>>**delta****t**, log(**t**+**delta****t**) log**t**and log log**t**. If we let log (**t**+**delta****t**) = log**t**in Eq. 1 and rearrange, the following expression results. ........(3) It is apparent from inspection of Eq. 3 that a plot of Pwf vs log...
Abstract

In 1963, Russell presented a method by which the now familiar two-rate flow test could be analyzed to provide the same information - permeability, skin provide the same information - permeability, skin factor, average reservoir pressure, and distance to a linear barrier to flow as that derived from conventional pressure buildup tests. The two-rate test method offers several practical advantages over conventional pressure buildup tests, but these will not be discussed here. The purpose of this note is to point out two simplifications that can often be used to make the analysis of two-rate flow tests more convenient. General Russell showed that his Eq. 3, ............(1) indicates that a plot of pwf vs will yield a straight line of slope m, given by (2) m = Calculation of is tedious and can often be avoided if we note that a plot of pwf vs log t will also give a straight line if t »delta t. pwf vs log t will also give a straight line if t »delta t. When t >> delta t, log(t + delta t) log t and log log t. If we let log (t + delta t) = log t in Eq. 1 and rearrange, the following expression results. ........(3) It is apparent from inspection of Eq. 3 that a plot of Pwf vs log delta t is a straight line of slope m', given by Pwf vs log delta t is a straight line of slope m', given by .........(4) P. 1139

Journal Articles

Journal:
SPE Formation Evaluation

Publisher: Society of Petroleum Engineers (SPE)

*SPE Form Eval*2 (02): 179–185.

Paper Number: SPE-14313-PA

Published: 01 June 1987

... equals 1. Similarly, the infinite shut-in time corresponds to the time equivalent to tp, the producing time before shut-in, as Eq. 1 indicates: (1) To apply the MBH method in Fig. 1, we simply obtain from the MBH function p*−p for a certain tp and subtract this value from p*-pwf (

**delta****t**= 0) to obtain p...
Abstract

Summary. Semilog type-curve matching applied to drawdown, desuperposed drawdown, Miller-Dyes-Hutchinson (MDH), and Matthews-Brons-Hazebroek (MBH) plots can determine drainage shape and the outer boundary conditions of a plots can determine drainage shape and the outer boundary conditions of a well. Because only a single set of type curves for a particular drainage-shape/outer-boundary condition is necessary, the selection of proper well-test model is straightforward. proper well-test model is straightforward. Additionally, the porosity/compressibility/area product and the skin effect can be estimated with a semilog type-curve match. Furthermore, a generalized form of desuperposition has been applied when the duration of the buildup is a multiple of the drawdown period. This method eliminates the use of all drawdown pressures except at the moment of shut-in. Therefore, variable skin effect during drawdown (e.g., cleanup) can be analyzed. Introduction In a constant-rate pressure drawdown, reservoir boundaries are encountered when the flowing pressure departs from the commonly known semilog straight line. In the most obvious case of a single boundary, the classic doubling of the slope results. This period is often referred to as the late-time region. Naturally this is true only when the common well-testing model is assumed--i.e., horizontal radial flow in a homogeneous, unfractured reservoir and negligible wellbore-storage effects. This paper evaluates semilog type-curve matching techniques to determine single, multiple, or entirely enveloping reservoir boundaries. Although semilog type-curve matching has been used on Horner plots, the literature seems to have omitted the application of this method on the most obvious case, the pressure drawdown. When a drawdown is not available or the data are not reliable, an equivalent drawdown can be constructed by temporal desuperposition. This method requires additional information and well conditions, as demonstrated in later examples. This technique has been applied to MDH-shaped curves and/or to its complementary MBH functions when pseudosteady-state flow has been reached. Applications beyond the pressure drawdown should considerably expand the usefulness of this matching technique. Late-Time Analysis To demonstrate best the interrelationship of the type curves used in the proposed matching method, all curves are shown on a comparison graph (Fig. 1). Note that the graph illustrates the actual type curves used for matching the drawdown curve, the MDH-shaped curve, and the MBH curve. The MDH-shaped and MBH curves are used only when pseudosteady-state flow has been reached before shut-in. Equivalent-drawdown-time and Horner curves are also overlaid on this plot. These equivalent-drawdown-time curves are the mirror image of the Horner curves merged laterally to share a common semilog straight line. Because this straight line is also shared by the drawdown curve, it can be helpful to construct a desuperposed drawdown curve. From this important observation it follows that similar information estimated by a Horner analysis (flow capacity, skin factor, average pressure, etc.) can be obtained by the equivalent-drawdown-time method. Because of this observation, it should also be possible to apply the MBH method to the equivalent-time method. To apply the MBH method to a Horner plot, a so-called false pressure, p*, is necessary; this is defined as the pressure pressure, p*, is necessary; this is defined as the pressure on the semilog straight line where the Horner time intercepts the imaginary shut-in time of infinity. Here, the Horner time equals 1. Similarly, the infinite shut-in time corresponds to the time equivalent to tp, the producing time before shut-in, as Eq. 1 indicates: (1) To apply the MBH method in Fig. 1, we simply obtain from the MBH function p*−p for a certain tp and subtract this value from p*-pwf (delta t = 0) to obtain p−pf (delta t = 0). As already mentioned, the semilog straight line remains stationary for different drawdown times in the equivalent-time method, and we make tp a variable. This way, p*−pf(delta t = 0) moves along the straight line and a new p*−pf(delta t = 0) moves along the straight line and a new function can be plotted. This function is shaped like the MDH function; however, it is defined as pws(delta t) -pwf (delta t = 0) rather than p−pws(delta t). SPEFE P. 179

Journal Articles

Journal:
Journal of Petroleum Technology

Publisher: Society of Petroleum Engineers (SPE)

*J Pet Technol*17 (03): 282–286.

Paper Number: SPE-991-PA

Published: 01 March 1965

..., the BHC Sonic (bore hole compensated), has been developed to accomplish these purposes. Previous sonic

**delta****t**logs have had large errors caused by irregular borehole diameter. In rough holes interpretation has been difficult, particularly in thin beds. The new BHC system suppresses these errors...
Abstract

Experience in the field of sonic logging has shown the necessity for more accurate logs under difficult borehole conditions and has indicated the desirability of automatically recording new acoustic parameters. A completely new logging system, the BHC Sonic (bore hole compensated), has been developed to accomplish these purposes. Previous sonic delta t logs have had large errors caused by irregular borehole diameter. In rough holes interpretation has been difficult, particularly in thin beds. The new BHC system suppresses these errors on the delta t measurement. The system uses one transmitter above and one transmitter below the sonic receivers. The effect of hole-size changes on the upper half of the array, consisting of a transmitter and two receivers, is in opposite direction to the effect on the lower half. An average of delta t measurements from the two halves automatically cancels the error. This arrangement also substantially eliminates the delta t errors caused by sonde tilt, i.e., axis of sonde not parallel to axis of borehole. Introduction Sonic logging has grown rapidly since its introduction in 1957. Over 10,000 logs are now run each year to investigate acoustic properties of formations. These logs are widely used by the industry for the determination of formation porosity, for correlation and lithologic determination, for fracture location, and as an aid to geophysical interpretation. Present sonic equipment, effective as it is, has some limitations. Changes in borehole size cause undesirable anomalies on the transit time curve. The resulting distortions interfere with porosity determinations in thin beds, and with the use of the logs for correlation. In addition, tilting of the sonde with respect to the axis of the hole causes errors in transit time measurements. The recently introduced borehole compensated sonic system, henceforth referred to as the BHC, overcomes the deficiencies outlined above. In addition, accuracy and stability in transit time measurement is ensured by the use of digital techniques in the equipment. A high degree of flexibility is provided in the system for recording travel times of later arrivals, amplitudes, time durations of selected portions of the wave train, and other acoustic variables. Provision is made for photographing the sonic signals in various modes, for detailed study of signal character. The purpose of this paper is to describe the main features of the BHC system and to present examples of field logs. Sonic Time Measurements The Sonic log normally includes a curve of interval transit time delta t and a presentation of the total travel time along the borehole. A measurement of single receiver travel time is often recorded with sonic amplitude curves. The means by which these measurements are improved over the conventional system will be described in some detail. Compensation for Borehole-Size Change EffectThe principal innovation of the BHC system is the use of two transmitters for the measurement of interval transit time delta t. This arrangement eliminates the "horns" resulting from changes in hole size that have been common to all single-transmitter two-receiver arrays. How the new transducer array compensates for hole effect will be readily understood from a study of Fig. 1. At the right is shown the conventional array with an upper transmitter and two receivers. The "point of measurement" is halfway between the receivers. Consider what happens when the sonde enters a cave on the way up the hole. When the sonde is in uniform-sized hole, as shown in the lowermost position, the mud paths to the receivers are equal and the delta t reading between receivers is correct. As the transmitter first enters the cave, the time required for the sound to reach both receivers is increased by the same amount so that there is no effect on the log. Then, the upper receiver passes into the region of enlarged hole, and the signal to this receiver is delayed by the additional mud path. JPT P. 282ˆ

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Gas Technology Symposium, June 7–9, 1978

Paper Number: SPE-7162-MS

.... The procedure, as shown in Figure 1, consists of flowing the well at a constant rate, q, for a time,

**t**, and then shutting-in the well for a time,**delta****t**, while measuring the bottom-hole pressure during the shut-in period. pressure during the shut-in period. There are a substantial number of papers written...
Abstract

Abstract When pressure buildup data are plotted on semilogarithmic coordinates, several straight lines can be obtained, even though theoretical considerations indicate that only one straight line should appear. Thus, an engineer is faced with the problem of choosing one of these straight lines to estimate formation permeability, average reservoir pressure, and flow efficiency. During the past few years, methods have been suggested whereby an engineer may extricate himself/herself from this quandary. In this paper we consider a few field examples which demonstrate the correct procedure one may follow to choose a straight line. Methods to identify after flow, the presence of a fracture, and the existence of boundaries are discussed. The advantages and limitations of the various methods are also discussed. Introduction The pressure buildup test is the most common of transient well tests. The procedure, as shown in Figure 1, consists of flowing the well at a constant rate, q, for a time, t, and then shutting-in the well for a time, delta t, while measuring the bottom-hole pressure during the shut-in period. pressure during the shut-in period. There are a substantial number of papers written on the subject of pressure transient analysis. The objective of this paper is to promote the combined and simultaneous use of the traditional semilogarithmic techniques with the newer log-log method. The two best approaches of pressure buildup analysis are the Horner and the Miller, Dyes, Hutchinson methods. The Horner method involves plotting the bottom-hole shut-in pressure, VS. plotting the bottom-hole shut-in pressure, VS. the logarithm of the time ratio (tp + delta t)/delta t, while the Miller-Dyes-Hutchinson (MDH) procedure involves plotting pws vs. the logarithm of delta t. Here, tp is plotting pws vs. the logarithm of delta t. Here, tp is the producing time prior to shut-in and delta t is the shut-in time. These methods show that such a graph should yield a straight line, whose slope is inversely proportional to the permeability-thickness product, proportional to the permeability-thickness product, kh, as illustrated in Figure 2. Other parameters such as wellbore damage or stimulation, average reservoir pressure, and distance to the nearest boundary can be pressure, and distance to the nearest boundary can be obtained from a Horner or MDH graph. The main problem in analyzing pressure buildup data is that, often, when buildup data are plotted on semilogarithmic coordinates, several straight lines can be obtained, even though theoretical considerations indicate that only one straight line should appear. Thus, the engineer is faced with the problem of choosing one of these straight lines for analysis, or concluding that the reservoir is heterogeneous; in the latter case, the conventional procedures suggested in the literature are not applicable. The appearance of several straight lines, or even a smooth curve, may be due to near wellbore effects such as afterflow, and/or fractures intersecting the wellbore. This paper is concerned with the identification of the proper straight line, if such a straight line exists. The methods suggested here should also be helpful in answering such questions as: Has the test run long enough to get the straight line needed to obtain formation permeability, skin factor, and average reservoir pressure? Is the reservoir heterogeneous? Is a more complex procedure or reservoir simulator (computer approach) needed to analyze the data? What special precautions should be taken or what improvements can be made when the test is rerun at a later date? PRELIMINARY CONSIDERATIONS PRELIMINARY CONSIDERATIONS To establish a basis for discussion, let us consider two gas well tests shown in Figure 3 where buildup data have been plotted as suggested by Horner. Since these are gas wells, we use p2, rather than p. From Figure 3, we see two similarities between the two graphs. First, two well-defined straight lines can be seen on both tests—a straight line with a shallow slope, followed by a second straight line with a much steeper slope. Either line on each test could be used to estimate formation permeability. Secondly, on both tests the slope of the second straight line is twice that of the first.

Proceedings Papers

Paper presented at the SPWLA 57th Annual Logging Symposium, June 25–29, 2016

Paper Number: SPWLA-2016-ZZ

... fundamental mode typically begins at approximately 3 kHz, and it has an infinite number of higher order modes, similar to the situation encountered with a dipole flexural mode. Many existing algorithms that determine

**delta**-**T**compressional (DTC) do not use the low-frequency dispersion characteristics...
Abstract

Abstract This paper presents an improved methodology to determine compressional velocity in soft formations using the low-frequency monopole firings from a wireline tool. Traditional methods used to obtain compressional velocity assume a non-dispersive mode pick from a waveform acquired at a relatively high frequency (above 8 kHz). In soft formations, this high-frequency compressional velocity pick does not provide an accurate representation of the true velocity. The reservoir section just below sea level is often soft, which yields a corresponding slow compressional velocity (i.e., large slowness). In such slow formations, the compressional wave packet is dispersive and is known as the P-Leaky mode. The characteristics of the P-Leaky mode is that it propagates at the formation P-wave velocity at low frequencies and asymptotically approaches the mud velocity at high frequencies. The P-Leaky fundamental mode typically begins at approximately 3 kHz, and it has an infinite number of higher order modes, similar to the situation encountered with a dipole flexural mode. Many existing algorithms that determine delta-T compressional (DTC) do not use the low-frequency dispersion characteristics associated with the P-Leaky mode in soft formations. The delta-T compressional picks using traditional methods may generate a velocity value slower than the true formation value as a result of including high-frequency dispersive content. This pick error affects the seismic time-to-depth correlation, placing reservoir sections at incorrect shallower depths. To address the challenge of measuring delta-T compressional in slow formations, the P-Leaky mode dispersion characteristics are used to accurately measure the compressional slowness at a low frequency. A low-frequency monopole firing generates a P-Leaky mode. The dipole firing may also generate a P-Leaky mode (in addition to the dipole flexure mode), which can be enhanced by summing the dipole waveforms. Advanced mode measurement techniques are applied that are sensitive and isolate the weak, yet desired, portion of P-Leaky that corresponds to the compressional velocity. The method and solutions described in this paper are demonstrated with field data from south Texas. The results deliver a true formation compressional velocity that provides critical inputs to the seismic time-to-depth velocity model in the deepwater environment.

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Gas Technology Symposium, June 7–9, 1989

Paper Number: SPE-19098-MS

... of Pwf decreases; for pressure buildup conditions a graph of mD(1,

**delta****t**aD)/qD(**delta****t**aD=0) vs (**t**aD+**delta****t**aD)/**delta****t**aD shows values of this slope within 1% of the 1.1513 value. The maximum error was in the rate performance simulated cases that included high-velocity flow, being less than 13...
Abstract

Abstract This work shows the results of a systematic study of transient pressure analysis of gas flow under either constant bottom-hole pressure conditions or the constant wellhead pressure conditions. The effects of formation damage, wellbore storage and high-velocity flow are included in the model. The analysis of simulated well tests showed that the interpretation methods used for liquid flow are generally accurate when the m(p) is used. For these conditions, a graph of 1/qD vs log tD to presents gradually lower values than the exact conventional value of liquid flow of 1.1513 as the value of Pwf decreases; for pressure buildup conditions a graph of mD(1,delta t aD)/qD(delta t aD=0) vs (t aD+delta t aD)/delta t aD shows values of this slope within 1% of the 1.1513 value. The maximum error was in the rate performance simulated cases that included high-velocity flow, being less than 13%. This upper limit occurs when the formation has a relatively "high" permeability (around 1md) and the rate performance test is permeability (around 1md) and the rate performance test is affected by high-velocity flow. It was found that pressure buildup tests are superior to rate performance tests pressure buildup tests are superior to rate performance tests because high-velocity flow does not affect the slope of the straight line portion of the buildup curve. Introduction Most well tests analysis methods assume constant-rate production, but constant-pressure production conditions production, but constant-pressure production conditions are not uncommon. Examples of conditions under which constant-pressure production is maintained at a well include production into a constant-pressure separator or pipeline, open flow to the atmosphere or production from pipeline, open flow to the atmosphere or production from a low permeability reservoir (k less than 1 md). An analysis of the literature indicates that a majority of papers dealing with the flow of fluids through porous media consider constant rate production porous media consider constant rate production conditions. Well tests conducted in low permeability reservoirs show approximately flow conditions of constant pressure. It has been concluded that vast gas reserves pressure. It has been concluded that vast gas reserves are found in these reservoirs. van Everdingen and Hurst published analytical solutions for radial flow under published analytical solutions for radial flow under constant pressure conditions. Jacob and Lohman derived as analytical solution in terms of the dimensionless flow rate, qD, for a well that produces under constant-pressure conditions. Tables of dimensionless flow rate vs dimensionless time were presented by Ferris et al. for unbounded systems and by Tsarevich and Kuranov for the closed-boundary radial reservoir. van Poollen also discussed pressure analysis techniques for liquid flow under constant pressure conditions. Samaniego and Cinco studied the case of constant pressure production in reservoirs with pressure sensitive rock and production in reservoirs with pressure sensitive rock and fluid properties. More recently Ehlig-Economides and Ramey and Uraiet and Raghavan considered drawdown and buildup tests for constant-pressure production. With regard to gas flow under constant production. With regard to gas flow under constant pressure, the work of Greene and that of Carter pressure, the work of Greene and that of Carter oriented to production decline analysis can be mentioned. Recently, Camacho has discussed the transient flow in solution gas-drive reservoirs producing under constant pressure conditions. There seems to be for constant pressure gas flow no systematic analysis similar to that of Wattenbarger and Ramey for constant rate gas flow. The purpose of this work is to present the results of a systematic study of transient pressure analysis of gas flow, under constant bottomhole pressure conditions. A radial reservoir flow model was coupled with a vertical tubing flow model. The effects of formation damage, wellbore storage and high velocity flow are included in the reservoir model. The results of this study are useful for prefracture testing analysis. GAS FLOW MODELS Reservoir Model To formulate the mathematical model, the following assumptions usually made in well testing are applied. The reservoir has radial geometry with a closed outer boundary and is composed of a horizontal porous formation that has uniform and isotropic rock properties and constant thickness. P. 407

Journal Articles

Journal:
SPE Formation Evaluation

Publisher: Society of Petroleum Engineers (SPE)

*SPE Form Eval*2 (04): 609–610.

Paper Number: SPE-13733-PA

Published: 01 December 1987

.... Extension to Two-Rate Tests If the rate at the active well is changed from q1 to q2 at time

**t**, it can be shown that (5) where (6) Eq. 5 is similar to Eq. 2 and yields a straight line if**Delta****t**(p'+C)q1/(q1 -q2) is plotted vs. 1/At on a semilog plot. In this format, the slope and intercept of the straight...
Abstract

Summary. The line-source solution of the diffusivity equation is differentiated and rearranged so that a linear relation between the variables is obtained. A plot of log tp' vs. 1/t yields a straight line whose intercept and slope are used to estimate the transmissivity, k h/mu, and the storativity, h phi ct, respectively. The method is extended to handle two-rate tests, including buildup tests, and can he used for the analysis of the combined data of the two periods. Introduction In interference test analysis, the semilog plotting techniques are inadequate because of the invalidity of the logarithmic approximation of the exponential integral function at large times. Usually type-curve matching is used. Recently, analysis methods based on the pressure derivative, p', were introduced. Tiab and Kumar used the maximum value of p' and the time at that point to estimate the transmissivity and storativity of the reservoir. Bourdet et al. introduced type-curve matching methods that involve both pressure-drop and pressure-lerivative matching. Clark and van Golf-Racht extended the pressure-lerivative matching. Clark and van Golf-Racht extended the pressure-derivative methods to variable-rate testing pressure-derivative methods to variable-rate testing using a superposition time function. In this work, a derivative method that yields a straight-line plot is introduced. More details can be found in Ref. 4. Method The derivative of the pressure, with respect to time at a radial distance Ar from an active well as obtained from the line-source solution, is (1) Multiplying Eq. 1 by -t and taking logarithms of both sides, we get (2) The constants A and b are related to the transmissivity, T, and storativity, S, by the following equations: (3) and (4) It is clear from Eq. 2 that a plot of tp' vs. 1/t on a semilog graph gives a straight line. The intercept and slope can be used to estimate the transmissivity and storativity of the reservoir. The intercept, A, is read directly on the logarithmic scale as the value of tp' at 1/t=0. The slope in cycles/hr would be -b/2.303. Extension to Two-Rate Tests If the rate at the active well is changed from q1 to q2 at time t, it can be shown that (5) where (6) Eq. 5 is similar to Eq. 2 and yields a straight line if Delta t(p'+C)q1/(q1 -q2) is plotted vs. 1/At on a semilog plot. In this format, the slope and intercept of the straight line are the same as those of Eq. 2. This means that data points from the first rate region (q = q1) calculated according to Eq. 2 can be combined with data points from the second rate region (q = q2) calculated according to Eq. 5 and analyzed together to obtain the values of T and S that fit the data points in the two regions. In pressure-buildup testing, q2 --0 and a semilog plot of Delta t(p'+ C) vs. 1/Delta t would give a straight line with the same slope and intercept as those of Eqs. 2 and 5. Because C is not known in advance and depends on A and b in addition to At, an iterative procedure must be used in the analysis. An initial value of C = 0 may be used, and the constants A and b are estimated either graphically or by linear regression. The values of A and b obtained are used to update C according to Eq. 6. The iteration is continued until successive values of A and b become constant within a prescribed limit. The final values of A and b are then used to estimate the transmissivity and storativity from Eqs. 3 and 4, respectively. Illustrative Example The developed method is applied to the interference test data of a gas well reported by Ramey et al. Table 1 shows the test data, calculations, and final results. The term is approximated by Deltap/Delta 1n at the geometric average time . The method of least squares was used to find the constants A and b. A graphic presentation of the data is shown in Fig. 1. Type-curve matching presentation of the data is shown in Fig. 1. Type-curve matching results reported by Ramey et al. are also shown in Table 1. Comparison with results obtained by this method indicates a difference of about 3% in the transmissivity and 0.6% in the storativity, indicating the accuracy of the proposed method. Conclusions A new approach for interference test analysis is introduced. A semilog plot of tp vs. 1/t gives a straight line from its intercept, and slope reservoir parameters can be estimated. The method can also be applied to two-rate interference tests for which an iterative procedure is used. Data points from the two regions may be analyzed procedure is used. Data points from the two regions may be analyzed separately or combined. SPEFE P. 609

Journal Articles

Journal:
SPE Production & Operations

Publisher: Society of Petroleum Engineers (SPE)

*SPE Prod & Oper*3 (02): 273–279.

Paper Number: SPE-13602-PA

Published: 01 May 1988

... to be zero. A mass balance of solvent in the element A

**delta**× in a time**delta****t**yields Divide by**delta****t****delta**× and let**delta****t**,**delta**x-O. In the limit, Experimental observations indicate that once a stable flow rate has been attained with water, the injection of solvent is not followed by any appreciable...
Abstract

Summary. Solvents may be injected into a formation once communication has been achieved between injection and production wells to increase the size of the communication path. However, the solvent disperses into the formation and bitumen flows into the communication path, increasing the viscosity of the flowing phase. As a result, no solvent may ever appear at the producing well. In this paper, the problem is modeled for the one-dimensional (ID) case and an analytic solution is obtained from which break-through time, outlet concentration of solvent. and other significant parameters can be estimated. Introduction It is a common practice to fracture reservoirs having little injectivity with high-pressure, high-flow cold water, followed by steam or steam additive. For unconsolidated media, however, the fracture may heal during the switch from water to steam injection. A program involving a combination of experiments was carried out to determine whether communication could be enhanced during fracturing so that injectivity would be maintained during the initial steaming. It was found that the injected solvent would in some cases take an infinitely long time to appear at the production well. Hence, a laboratory investigation involving theoretical and experimental work was carried out to determine the conditions at which the solvent should be injected. We describe here the simplified model developed, the experimental results obtained in the laboratory, and the theoretical considerations that allow extrapolation to the field. The performance of a solvent slug injected in a communication path was analyzed, taking into consideration the two main results path was analyzed, taking into consideration the two main results that affect its economical application-viscosity reduction and solvent losses to the formation-by assuming that all parameters of no interest for the present objective were constant. Further sim-plification was obtained assuming a linear pressure drop and an empirical functional expression for the viscosity dependence, leading to a nonlinear first-order partial-differential equation. The mathe-matical model was then used to explain the experimental resultand to obtain the mass-transfer coefficient. Finally, a hypothetical field case was considered. Derivation of the Equation Consider a ID element of volume in the communication path, as shown in Fig. 1. Assume that bitumen and solvent are miscible in all proportions and call the resultant phase "oil phase," or simply "oil." The amount of solvent in an element of volume A delta × is where phi = porosity, phi = porosity, So = oil saturation, po = oil density, and po = oil density, and fw = weight fraction of solvent in oil phase. As a first approximation, the solvent is transported by convection of the oil phase. Assuming Darcy's law, the transport of solvent by convection only is where qo = volumetric flow rate, k = absolute permeability, kro = relative permeability, and mu(fw)= viscosity of oil phase as function of weight fraction of solvent. Then the net transport (i.e., in -out) of solvent into the element A delta × is The amount of solvent lost to the formation will be accounted for through a mass-transfer coefficient, K. Hence, the solvent loss, Vst is where K = mass-transfer coefficient based on unit area between communication path and formation, P = perimeter of fracture in direction perpendicular to P = perimeter of fracture in direction perpendicular to flow, fw = weight fraction of solvent in fracture, and fwoo = weight fraction of solvent in formation at large distance from fracture, assumed to be zero. A mass balance of solvent in the element A delta × in a time delta t yields Divide by delta t delta × and let delta t, delta x-O. In the limit, Experimental observations indicate that once a stable flow rate has been attained with water, the injection of solvent is not followed by any appreciable change in pressure drop; i.e., the flow behaviors of solvent and of water are not significantly different. SPEPE P. 273

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Annual Technical Conference and Exhibition, September 21–24, 1980

Paper Number: SPE-9289-MS

... pressure buildup data is not as bad as it may first appear. As long as the producing time,

**t**, prior to shut-in is sufficiently long compared to the shut-in time,**Delta****t**[that is (**t**+**Delta****t**)/**t**1], for liquid systems, it is reasonable to analyze pressure buildup data using drawdown type curves. However...
Abstract

Abstract Currently, type curve analysis methods are being commonly used in conjunction with the conventional methods to obtain better interpretation of well test data- Although the majority of published type curves are based on pressure drawdown solutions, they are often applied indiscriminately to analyze both pressure drawdown and buildup data. Moreover, the limitations of drawdown type curves, to analyze pressure buildup data collected after short producing times, are not well understood by the practicing engineers. This may often result in an erroneous interpretation of such buildup tests. While analyzing buildup data by the conventional semi-log method, the Horner method takes into account the effect of producing time. On the otherhand, for type curve analysis of the same set of buildup data, it is customary to ignore producing time effects and utilize the existing drawdown type curves. This causes discrepancies in results obtained by the Horner method and type curve methods. Although a few buildup type curves which account for the effect of producing times have appeared in the petroleum literature, they are either limited in scope or somewhat difficult to use. In view of the preceding, a novel but simple method has been developed which eliminates the dependence on producing time effects and allows the user to utilize the existing drawdown type curves for analyzing pressure buildup data. This method may also be used to analyze two-rate, multiple-rate and other kinds of tests by type curve methods as well as the conventional methods. The method appears to work for both unfractured and fractured wells. Wellbore effects such as storage and/or damage may be taken into account except in certain cases. The purpose of this paper is to present the new method and demonstrate its utility and application by means of example problems. Introduction Type curves have appeared in the petroleum literature since 1970 to analyze pressure transient(pressure drawdown and pressure buildup) tests taken on both unfractured and fractured wells. The majority of type curves which have been developed and published to date were generated using data obtained from pressure drawdown solutions and obviously are most suited to analyze pressure drawdown tests. These drawdown type curves are also commonly used to analyze pressure buildup data. The application of drawdown type curves in analyzing pressure buildup data is not as bad as it may first appear. As long as the producing time, t, prior to shut-in is sufficiently long compared to the shut-in time, Delta t [that is (t +Delta t)/t 1], for liquid systems, it is reasonable to analyze pressure buildup data using drawdown type curves. However, for cases where producing times prior to pressure buildup tests are of the same magnitude or only slightly larger than the shut-in times [that is, (t + Delta t)/t »1], the drawdown type curves may not be used to analyze data from pressure buildup tests. The above requirement on the duration of producing times is the same for the conventional semi-log analysis. If pressure buildup data obtained after short producing pressure buildup data obtained after short producing time are to be analyzed, the Horner methodic is recommended over the MDH (Miller-Dyes-Hutchinson)method. The MDH method is generally used to analyze buildup data collected after long producing times, whereas the Horner method is used for those obtained after relatively short producing times. Although pressure buildup tests with short producing times may occur often under any situation, they are rather more common in the case of drill stem tests and prefracturing tests on low permeability gas wells. Thus, there is a need for generating buildup type curves, which account for the effects of producing time. Some limited work has been done in producing time. Some limited work has been done in this regard. McKinley has published type curves for analyzing buildup data for a radial flow system.

Images

**Published:**01 December 2020

Figure 13

**Delta**pressure P 1 (**t**)-P 0 *(**t**) calculation and**t**01 detection time determination by tangent method – W7/W8 interference MoreImages

**Published:**01 December 2020

Figure 18

**Delta**pressure P 2 (**t**)-P 1 *(**t**) calculation and**t**02 detection time determination by tangent method – W5/W8 interference MoreAdvertisement