Analysis of Pressure/Flow-Rate Data With the Pressure-History-Recovery Method

Summary This study presents a new way to integrate flow rate history and pressure history data in well-test analysis. The method used is an extension of the deconvolution principle to a chosen period of time before the well test. It is based on the recovery of missing parts of the pressure history by taking into account the flow rate history and the available sections of measured pressure history. The method has the advantage of working without making any assumption about the reservoir by using the data itself to define the "model." A real example with 71 pressure points and 70 corresponding flow rate points shows how it is possible to recover the first 20 pressure points correctly considering that only the flow rate history and the last 51 pressure points are known. The main purpose of this procedure is to have an alternative to treat the problem of flow rate variations both before and during the well test. In such a case, neither the usual deconvolution (only applicable for flow rate variations during the well test, when flow rate and pressure are both known), nor the multirate superposition plots (which assume specific reservoir behavior) can be applied properly. In addition, the principle of the method enables us to extend its use to the analysis of well tests with pressure recording errors or missing pressure recording. Examples show that it is possible to make a good analysis of a well test with missing pressure regions when current methods give only parts of the information. Introduction In practical well testing, it is rare to encounter pressure responses generated by only a simple constant step change of rate production. Generally, well tests contain a series of different flow rates or a continuously varying flow rate, and the measured pressure responses are combinations of the pressure transients due to the varying flow rate. Several methods have been developed to perform model recognition when the flow rate pattern is not a perfect drawdown. All these methods integrate, more or less, flow rate data to improve the interpretation. A first set of these methods are those proposed by Horner, 1 Miller et al., 2 and Agarwal3 to treat buildups as drawdowns. With changing flow rate, it is common to resort to the principle of deconvolution (Kuchuk4), thanks to the convenience of the Laplace transform (van Everdingen and Hurst5), when flow rate variations occur during well test, or to handle the convolution with multirate superposition plot techniques (Earlougher 6) when changes of production rate are prior to the pressure recording. However, none of these procedures is completely accurate in handling variations of flow rate both before and during the well test. This paper presents an alternative method to take into account continuously varying flow rate data. This new technique, based on the recovery of missing pressure history (or parts of the pressure history), extends the use of the deconvolution principle to the whole flow rate history. The method is a general procedure, and can be applied to any type of reservoir with any production rate pattern. In fact, the presentation procedure is a simple mathematical treatment using the available data only, that allows us to find an approximation of the missing pressure consistent with the measured pressure and flow rate. After the mathematical background and the description of the method, a simple example, used to demonstrate and refine the algorithm, is presented. Next, a synthetic example shows how the technique can be helpful to solve common well test interpretation problems such as a lack of pressure measurements or obvious pressure recording errors. In such cases, unlike the traditional methods, our procedure enables us to perform a good analysis. Mathematical Preliminaries Dimensionless Variables. Parts of the mathematical treatment of the data require the use of dimensionless variables. Because the permeability, k, is unknown during the procedure, it is impossible to use the usual definitions to calculate the dimensionless variables. Instead, we define characteristic pressure, pref, flow rate, qref, and time, tref from the real data. Thus (1)pD=Δppref; qD=qqref; tD=ttref. We then specify (in consistent units) (2)βp=prefqref=Bμ2πkh (3) and C D = β p q ref t ref . In oilfield units, Eqs. (2) and (3) must be written: (4)βp=141.2Bμkh, CD=qB24βpqreftref. Laplace Transform. If s is the Laplace variable, the definition of the Laplace transform related to the time, t, is (5)L[f(t)]=f¯(s)=∫0∞e−stf(t)dt. Here, f(t) a continuous function of time, can be either the pressure or the flow rate. Therefore, every variable in the Laplace space has a dimension equal to its dimension in real space multiplied by time. For its part, s has the dimension of reciprocal time. So, the dimensionless variables, in Laplace space, are (6)sD=stref; p¯D=p¯preftref; q¯D=q¯qreftref. The integral definition in Eq. (5) can be used for dimensionless variables, but in this case s must be replaced by sD.