Transient Analysis of Acoustically Derived Pressure and Rate Data c.s. Kablr*, SPE, Schlumberger Perforating & Testing Center F.J. Kuchuk, SPE, Schiumberger-Doll Research A.R. Hasan, SPE, U. of North Dakota Summary. A pressure-buildup test conducted on a sucker-rod pumping well is often distorted by long-duration wellbore storage. In fact, this distortion could be so severe that even a week's shut-in period may not allow a semilog analysis. A longer shut-in period becomes economically discouraging because of lost production. Low energy and low transmissivity in the reservoir, coupled with increased fluid compressibility, contribute to this long-duration storage phenomenon. One way of reducing the storage effect clearly lies in the simultaneous analysis of downhole pressure and flow rate, estimated from casinghead pressure and rising annular liquid-level measurement made by acoustic well sounding (AWS). Ascertaining the quality of the indirectly measured pressure and rate data constitutes one of the objectives of this study. Several methods exist to translate the A WS measurement to downhole pressure and rate data for the subsequent transient analysis. We show that even an empirical hydrodynamic correlation provides satisfactory transient-pressurelflow-rate data for convolution and deconvolution analyses for moderate pumping-liquid columns. When long annular liquid columns are encountered, translating the A WS measurement with a mechanistically based hydrodynamic model appears to be a prudent approach. Interpretation of several transient tests show that automated convolved-type-curve or history matching of field data is a powerful tool for reservoir-parameter (total mobility, skin, fracture half-length, and storage coefficient) estimation. Use of downhole rate for the convolved type-curve matching reduces the variable storage coefficient by several orders of magnitude, thereby enhancing the quality of the estimates. Deconvolution of downhole pressure and rate data and the pressure-derivative approach significantly aided in well/reservoir flow-model identification. A simple algorithm for computing the Laplace transform of the wellbore pressure for an infinite-conductivity vertically fractured well in an infinite reservoir is developed in this work for a rapid, iterative-type computation used in automated convolved-type- curve analysis. Introduction Testing pumping wells presents an interesting challenge because of the need to deal with indirect measurements of downhole pres- sure and flow rate. Because of economic considerations, direct pres- sure measurements, after the tubular hardware is removed, is not commonly practiced. Thus questions often arise regarding the va- lidity of indirect measurements and their subsequent interpretation- leading to estimation of permeability, skin, average pressure, frac- ture half-length, etc. There are two approaches for estimating downhole pressure and flow rate. The first method involves continuous measurement of casinghead pressure together with tracking the movement of the gaslliquid interface during a transient test, using an A WS device. 1-3 Translation of these measurements to downhole pres- sure and rate is made with a wellbore hydrodynamic model/corre- lation. 1,4-6 The second approach uses the mass-balance principle 7 to accomplish the same task. Both methods have certain limitations. For example, the A WS method is not particularly suitable in a foaming annulus. The major apparent uncertainty in the A WS method stems from the use of hydrodynamic model/correlation, however. On the other hand, any leak in the system (e.g., a wellhead leak) or wellbore crossflow during well shut-in can cause problems in the second method. In a typical pumping well, one is confronted with a long test be- cause of long-duration wellbore-storage distortion period. Partic- ularly, low reservoir transmissivity and high system compressibility , precipitated by high gas saturation in the reservoir and/or in the wellbore vicinity, contribute to this long-duration storage problem. The problem is compounded further by low reservoir energy as- sociated with pumping wells. In general, if the wellbore-storage coefficient remains constant, type-curve analysis can be applied to pressure data directly. Un- fortunately, in pumping wells, the early-time data reflect a chang- ing storage coefficient because of the changing density in the liquid • Now with Schlumberger Well Services. Copyright 1988 Society of Petroleum Engineers SPE Formation Evaluation, September 1988 column (gas percolation effect) and compression of the gas column by the liquid column. The varying wellbore-storage problem can be handled in two ways. One approach is to analyze the transient-pressure and flow- rate data 8 - IO simultaneously with convolution and deconvolution approaches. The alternative is to adapt the unconventional test method II and to perform interpretation with either pressure and rate data simultaneously or pressure data alone. The other alternative approaches that we explored are deconvo- lution with constraints l2 and the automated type-curve analysis. Recently, the automated type-curve approach was used for homogeneous reservoirs 13 and naturally fractured reservoirs 14 without considering the downhole-rate data. In other words, down- hole rate is predicted by the constant-storage model. Subsequent- ly, a number of studies used the automated convolved-type-curve matching to interpret tests from homogeneous reservoirs, 15 fault- ed reservoirs, 16 low-transmissivity reservoirs, II and layered reser- voirs,17 involving the downhole flow-rate data. The purpose of this study is two-fold: to search for a suitable wellbore hydrodynamic model to generate the downhole pressure data and to search for an appropriate reservoir flow model (p D model) for interpreting the test data. In addition, we wanted to ex- plore the quality of indirectly measured flow-rate data through deconvolution analysis because the technology for direct measure- ment oflow flow rates (less than 100 RB/D [16 m 3 /dD, prevalent in pumping wells, is not readily available. Mathematical Models Continuously-Varying-Rate Case. The relationship between in- directly measured downhole transient-pressure and flow-rate data in a pumping well can be given by the convolution integra1 18 r tD PwbD=J qb(T)[PD(tD-T)+S]dr, ................... (1) o provided that q D(O + ) = O. 607