Shut-in Time Research Articles

Effect of Linear Discontinuities on Pressure Build-Up and Drawdown Behavior

Abstract A detailed treatment is given of the transient pressure behavior of a well located near a linear discontinuity. On either side of the discontinuity, the values of permeability, viscosity, compressibility, and porosity are uniform, but may he different from those on the other side of the discontinuity. These results generalize previously published works which assumed the permeability ratio for media on different sides of the discontinuity was zero, one, or infinity. One application of this material would be to facilitate detection of facies changes or fluid-fluid contacts by transient testing. Solutions, presented graphically, show pressure decline for constant rate fluid production. The range of variables examined includes dimensionless time from 0.1 to 100, mobility ratio from 0.001 to 1000 and specific compressibility ratio from 0.001 to 1000. For dimensionless time less than 0.4, the pressure behavior is essentially that of a well in an infinite reservoir. Thus, reservoir properties near the well may be estimated in the usual manner. Using an overlay technique to match an experimental curve with one of the theoretical curves, it is possible to estimate the distance to a discontinuity by substituting the actual time t and the corresponding dimensionless time tp at which a match occurs into the equation where k,/phi mu c is the diffusivity near the well, and may be estimated from data taken at early time. Because of both the complexity of the solutions and limited computer facilities, well interference problems were not examined extensively. However, it was found that well interference may be masked by the discontinuity in some cases. By superposition of the results of constant rate fluid production, the problem of pressure buildup may be treated. Such results show that for early shut-in times correct values for the transmissibility are obtained from conventional semilog plotting. However, erroneous values of the static reservoir pressure are obtained unless data at large shut-in times are taken. Introduction Transient well testing has yielded estimates of such reservoir parameters as static reservoir pressure, permeability, porosity, and well bore effects. In most tests, it was assumed that the reservoirs were homogeneous and isotropic. Anisotropic media and radial discontinuities were considered. Various studies have been made on multiple-layer reservoirs. Linear discontinuities have been described for problems of one-dimensional flow. In the present paper, a mathematical treatment is given for a well located near a linear discontinuity in an otherwise infinite medium (see Fig. 1). An actual reservoir analogy would be a well in a relatively thin reservoir near a fluid-fluid contact, or a well located in a reservoir exhibiting a sudden change in rock and formation characteristics, such as thickness, porosity, or permeability. A sudden change in porosity or permeability may be caused by clay fill or other types of facies changes. Sudden thickness changes may be the result of such geological formations as buried shore lines and channel deposits. In otherwise uniform formations, a linear discontinuity may be caused by fluid-fluid interfaces such as oil-water contacts, gas-oil contacts, or gas-water contacts.

Pressure Build-Up Analysis, Variable-Rate Case

Abstract A second-order approximation to the exact solution of the diffusivity equation corresponding to the pressure build-up of a well producing at a variable rate is derived. This approximation is applicable when the well's shut-in time is larger than the total time elapsed since the well was first produced. The resulting equations are compact in form and easy to use. Thus, the need for Horner's theoretically precise but rather laborious solution to the above problem is eliminated. In addition, these equations apply where the use of Horner's widely known approximate method is questionable.From a practical point of view, the reported method is best suited for analysis of drill-stem tests and short production tests conducted on new wells. Introduction The utility of drill-stem and short production tests in reservoir studies has long been recognized by the reservoir engineer. If interpreted correctly they could lead to a wealth of information upon which may depend the success or failure of reservoirs' analyses. Initial reservoir pressure and the average flow capacity are two quantities that are normally sought from a drillstem and/or a short production test analysis. Pressures are the most valuable and useful data in reservoir engineering. Directly or indirectly, they enter into all phases of reservoir engineering calculations. Therefore, their accurate determination is of utmost importance. The flow capacity kh of the reservoir is indicative of its commercial capability. In addition, it can indicate the presence of a damaged zone around the wellbore and, thus, the necessity for remedial measures. Of the several methods used to analyze drill-stem and short production tests, Horner's' method is by and large the most common. It applies to an infinite reservoir and/ or a limited reservoir where the effect of production has not been felt by the boundary. Horner's method makes use of the so-called "point- source" solution of the diffusivity equation. The point-source solution is approximated by a logarithmic function and the superposition theorem is utilized to give the familiar pressure build-up equation where theta is the shut-in time, q is in reservoir barrels per day and the rest of the symbols conform with AIME nomenclature.Eq. 1 was derived for a well which produced at a constant rate q from time zero to time t and was then shut in. In actuality, such a constant rate of production does not normally obtain. Therefore, a correction must be applied to Eq. 1 to account for the varying rates of production. Horner suggested two methods. The first, which results in a theoretically accurate solution, is rather lengthy and laborious and, thus, it is not suited for routine analysis. The second which has been termed a "good working approximation" is the one used by the majority of the reservoir engineers. In the second method, Eq. 1 is modified by simply introducing a corrected time t, and writing (2) where q is the last established production rate prior to shut-in, and t. is obtained by dividing the total cumulative production by the last established rate. Horner's original paper does not give any indication that this method of correction is based on any theoretical justification. In addition, there is a question as to what constitutes the last established rate. In case of a drill-stem test some engineers use the average rate obtained by dividing the total fluid produced by the total flow time, while others calculate the average rate by dividing the total fluid produced by the last flow-period time. Obviously, different results obtain for the different flow rates used.Because of this, a simple method to the varying-rate case was developed which is theoretically sound and which defines clearly the flow rate and its associated time to be used in the calculations. The final equation arrived at is(3) where q* and t* are a modified rate and time, respectively, and can be easily calculated. In addition, it is shown theoretically that Horner's approximate method, if used for a variable-rate case, gives the correct pressure but would not be expected to give the correct flow capacity. MATHEMATICAL ANALYSIS The general equation governing the flow of slightly compressible fluid in porous media may be written as (4) JPT P. 790^

Calculation of Formation Temperature Disturbances Caused by Mud Circulation

Abstract Quantitative interpretation of electric logs requires knowledge of formation temperature. In this paper, methods are developed for computing changes in formation temperature caused by circulation of mud during drilling operations. The basis of the method is the mathematical solution of the differential equation of heat conduction. The solution of this equation is presented in a series of graphs. These graphs are used to determine formation temperature disturbance at various radii for arbitrary mud circulation histories. Example comparisons with field results show reasonable agreement. It is concluded that, in general, the temperature disturbances caused by circulating mud are small beyond 10 ft from the wellbore but are quite significant near the wellbore. Introduction Quantitative interpretation of electric logs requires knowledge of the formation temperature in order to establish the resistivity of the formation water with accuracy. To determine the formation temperature, the temperature disturbances produced by circulating drilling mud must be evaluated. The objective of this investigation was to develop a method for the numerical determination of these temperature disturbances at any distance from the wellbore as a function of time. The first step was to solve the differential equation describing the temperature behavior in the formation during and after mud circulation. This solution was used to calculate a series of curves relating temperature disturbance to shut-in time for various circulating periods. An "exact" method for computing formation temperatures with the use of these temperature-disturbance plots is described, and the results of an application of this method to a well in Montana are presented. Procedures for approximating formation temperatures are also discussed. BASIC EQUATIONS AND ASSUMPTIONS With the assumptions thatcylindrical symmetry exists, with the borehole as the axis,heat flow is due only to conductivity andvertical heat flow in the formation is zero, the temperature distribution around a wellbore is defined by the following differential equation. ...........................(1) where T = temperature, r = radius, Cp = specific heat capacity of formation rock, p = density of formation rock, K = thermal conductivity of formation rock, and t = time. In terms of dimensionless time, and dimensionless radius, the constants in Eq. 1 disappear, and it becomes ...........................(2) Other simplifying assumptions made in the treatment of the temperature problem are as follows.The formation can be treated as though it is radially infinite and homogeneous in extent, as regards heat flow.The presence of mud cake can be disregarded.The effect of heat generated by bit action is negligible.After mud circulation ceases, the rate of radial heat flow at the wellbore is negligible and is assumed zero for purposes of calculating temperature change with time. These assumptions are believed to be reasonable because of the agreement achieved between calculated and measured borehole temperature in a well, to be discussed later. Further, the agreement at the wellbore indicates that temperatures in the formation were also approximately correct, since it is the formation temperatures that determine wellbore temperature after ceasing circulations. JPT P. 416^

A Simple Method for Correcting Spot Pressure Readings

Abstract Pressure information for use in material-balance calculations is obtained, where possible, from pressure build-up surveys in shut-in wells. Using proper extrapolation methods, static pressures are obtained which, by averaging, give the field static pressure. Often, particularly in fields with a large number of wells, only spot pressure readings are available. A graphical method is presented here which enables the determination of the corrections which must be applied to the spot readings to give an estimate of the static pressure. Although the static pressures thus obtained are not so reliable as those from build-up surveys, they are more nearly correct than the spot readings themselves for use in material-balance calculations. Introduction The average static pressure in a bounded reservoir is determined from a number of individual-well pressure measurements. Where pressure build-up curves are available, extrapolation methods such as described in Ref. 3 may be used to determine the static pressure in the drainage area of each individual well. The possibility of using pressure build-up curves presupposes that each well being surveyed is closed in during a certain period and that, during this time, a pressure recorder is left suspended in the hole. For surveys in fields with a large number of wells, this is usually not feasible for economic and other reasons. Either a large number of pressure recorders would have to be used simultaneously or, using a limited number of recorders, the time interval needed to complete a field-wide survey would be too long. The data obtained for this incomplete coverage are insufficient to permit reliable averaging of the few calculated static pressures over the entire reservoir. In many such cases another type of pressure information is available, i.e., spot pressure readings. During this type of pressure survey all wells are closed-in simultaneously and, after a reasonable shut-in time (e.g., three days), pressure readings are taken by running the recorder in and out of each successive well. These surveys, which are less expensive, are often taken at regular intervals, say once or twice a year.

A Low Cost Method of Production Stimulation

Abstract A simple treatment has been developed by which water-block is removed from oil wells by lowering oil-water interfacial tension to a very low value. The treatment is made by injecting a crude oil solution of a carefully selected surface-active agent into the water-blocked formation. From 1,000 to 4,000 gals. of crude oil containing 1 per cent surface active agent is used. Shut-in time following treatment is usually from 12 to 24 hours. The treatment has been used in 90 wells with successful results obtained in 39 wells representing seven formations in nine fields. The successful treatments have resulted in added productivity of 1,217 BOPD. Treatment cost is unusually low, compared with other types of well treatments and workovers normally done for improvement of productivity. Introduction The Problem Although water-block removal has received considerable attention in the past, there has remained a need for an inexpensive, effective remedial treatment for water-blocked wells. This paper discusses the development of a remedial treatment utilizing a surface active agent which has shown considerable ability to increase production rates from wells in which some water-blocking was known or suspected to exist. The need for a remedial treatment to remove waterblock has been most clearly observed when established producing wells have shown marked productivity loss following exposure of the producing formations to water during workover operations involving water-base muds and cement. The need for such a treatment has not been so clearly evident in acidized wells. Partial water-block in such wells may occur undetected because of the complexity of operations and particularly when such wells are capable of making a maximum allowable after workover. Field tests reported in this paper show that the incidence of such water-blocking may be fairly high in at least one formation previously subjected to acidization.

Analysis of Pressure Build-Up Data

Abstract Several methods of analyzing pressure build-up data in wells have beenpresented by various authors. This paper reviews the theory and method of D. R.Horner and presents example calculations performed on data obtained by testingseveral different types of wells. These calculations include,graphicalestimation of final static pressure,determination of the productivecapacity of the pay away from the well bore and,the degree to which theformation adjacent to the well bore has been damaged by completion or othercauses. The methods of testing and precautions which should be taken to assurethe best data possible are discussed. Limitations and reliability of calculatedresults are also treated. Introduction Pressure testing of wells is generally limited to the determination ofproducing and static mean formation pressures. The so-called "static" pressure determinations, along with PVT, electric log and production data, enable the reservoir engineer to determine, within reasonable limits, the drivemechanism of the reservoir and in some cases, the amount of edge waterencroachment. Producing pressure tests enable calculation of productivityindices and allow the engineer to plan the systematic production of a poolfor optimum conservation of subsurface energy. The radial flow formula advanced by Muskat has been based on the assumptionof incompressible radial fluid flow. It has been known that reservoir fluids donot behave in an ideally incompressible manner. For example, incompressibleflow theory indicates a simple logarithmic relationship between the differenceof the instantaneous and static well pressures when plotted against time. Thelatter stages of this type of plot of pressure build-up data generally show amarked deviation from the earlier straight line trend, which deviation may beshown to be due to the compressible flow of fluids toward the well bore.Shut-in times of 24, 48, 72, or at most 96 hours are currently in wide use fordetermining so-called "static" reservoir pressure. Due to thecontinuation of compressible flow of fluids into the well bore long after thisarbitrarily taken shut-in time, the determination of static pressures hasalmost invariably resulted in lower than equilibrium values. Materials balancecalculations made early in the life of a reservoir often result in a calculatedreserve which later observations prove to be too low. Failure to obtainreliable "static" reservoir pressures within the prescribed 24 or48-hour build-up period has undoubtedly been a major factor in obtaining theselow estimates. Comparison of theoretical and actual productivity performance has indicated that formation damage and not being able to attaintrue static pressures have been partially responsible for the observeddiscrepancies. T.P. 3542

Geothermal Gradients in Mid-Continent and Gulf Coast Oil Fields

The purpose of this paper is to present a contour map that shows thegeographical variation in geothermal gradients in the Mid-Continent and GulfCoast oil fields. Introduction Frequently there arises the need for an estimation of the temperature to beexpected at a predetermined depth in working with a wildcat or in deepening oldproduction into zones that are relatively unknown. The study of geothermalgradients in oil fields has been made previously by many investigators. One ofthe pioneers in this work was C. E. Van Orstrand, who, as well as most of theother investigators, was primarily interested in the relation betweentemperature and depth in a given locality. Between 1921 and 1929, AmericanPetroleum Institute Research Project No. 25 fostered a thorough study of earthtemperatures. R. W. French did some work in California, the results of whichwere published in 1939.3 In most of these studies, an attempt was made tocorrelate temperatures in a rather localized area. This paper will show theresults of an attempt to correlate temperature gradients over rather largeportions of the Mid-Continent and Gulf Coast area. Method of Obtaining Data It has been found that for complete thermal equilibrium to be established in awell, the well must stand undisturbed for a period of several hours to severalweeks, depending upon the way the well was produced prior to the shut-inperiod. The data in this paper were obtained in conjunction with running pressures inwells or in taking samples of reservoir fluid. Glass maximum recordingthermometers were used in all but a few instances. The wells had been shut infrom one hour up to 10 or 12 days. Since the wells were normal producing wells, some of the shut-in times undoubtedly were insufficient for complete thermalequilibrium to have been established and the temperatures measured do notnecessarily represent the true earth temperatures. They do represent thetemperature conditions that will? be encountered in a well that is producingoil. The data, therefore, can be useful in working with or in estimating thetemperatures to be expected in producing reservoirs. In obtaining a thermal gradient by dividing the difference in temperature ofthe formation and the mean annual surface temperature by the depth of theformation, one is assuming the depth-temperature line to be straight.Depth-temperature curves are almost straight lines. However, on closeexamination of accurate data, a slight curvature usually is observed. Mostwells yield curves with slopes increasing with increasing depth. These curvesare believed to be from wells that are in and are underlain by sedimentarymaterial. Some few curves have slopes that decrease with increasing depth.These curves are believed to be from wells that are underlain rather closely bygranite or some other igneous rock. T.P. 2114