estimation of permeability and permeability anisotropy from
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Estimation of Permeability andPermeability Anisotropy From
Straddle-Packer Formation-TesterMeasurements Based on the Physics of
Two-Phase Immiscible Flow and InvasionRenzo Angeles, Carlos Torres-Verdn, Hee-Jae Lee, and Faruk O. Alpak, University of Texas at Austin; and
James Sheng,* Baker Atlas
SummaryWe describe the successful application of a new method to esti-mate permeability and permeability anisotropy from transient mea-surements of pressure acquired with a wireline straddle-packerformation tester. Unlike standard algorithms used for the interpre-tation of formation-tester measurements, the method developed inthis paper incorporates the physics of two-phase immiscible flowas well as the processes of mudcake buildup and invasion.
An efficient 2D (cylindrical coordinates) implicit-pressure ex-plicit-saturation finite-difference algorithm is used to simulateboth the process of invasion and the pressure measurements ac-quired with the straddle-packer formation tester. Initial conditionsfor the simulation of formation-tester measurements are deter-mined by the spatial distributions of pressure and fluid satura-tion resulting from mud-filtrate invasion. Inversion is performedwith a Levenberg-Marquardt nonlinear minimization algorithm.Sensitivity analyses are conducted to assess nonuniqueness andthe impact of explicit assumptions made about fluid viscosity,capillary pressure, relative permeability, mudcake growth, andtime of invasion on the estimated values of permeability and per-meability anisotropy.
Applications of the inversion method to noisy synthetic mea-surements include homogeneous, anisotropic, single- and multi-layer formations for cases of low- and high-permeability rocks. Wealso study the effect of unaccounted impermeable bed boundarieson inverted formation properties. For cases where a priori infor-mation can be sufficiently constrained, our inversion methodologyprovides reliable and accurate estimates of permeability and per-meability anisotropy. In addition, we show that estimation errorsof permeability inversion procedures that neglect the physics oftwo-phase immiscible fluid flow and mud-filtrate invasion can beas high as 100%.
IntroductionModular and multiprobe formation testers have proved advanta-geous in the determination of permeability at intermediate-scalelengths because of the increased distance between the observationand sink probes (Pop et al. 1993; Badaam et al. 1998; Proett et al.2000). Moreover, the use of dual-packer or straddle-packer mod-ules over point-probe modules is known to improve the interpre-tation of pressure transient measurements when testing laminated,shaly, fractured, vuggy, unconsolidated, and low-permeability for-mations (Ayan et al. 2001). Several papers have been published todescribe interpretation techniques and applications of these newformation-testing approaches (Kuchuk 1998; Hurst et al. 2000;Onur et al. 2004).
The new method introduced in this paper interprets formation-tester measurements acquired with wireline straddle-packer tools.It incorporates the physics of two-phase, axisymmetric, immisciblefluid flow to simulate the measurements, and it is combined witha nonlinear minimization algorithm for history-matching purposes.Comparable inversion approaches have been documented in theopen technical literature (Proett et al. 2000; Xian et al. 2004;Jackson et al. 2003) but they assumed single-phase fluid flow.Recently, Zeybek et al. (2001) introduced a multiphase flowmethod to integrate formation-tester pressure and fractional flowmeasurements with the objective of refining relative permeabilityvalues estimated from openhole resistivity logs. The same authorsconsidered the manual inversion of radial invasion profiles, hori-zontal permeability, and permeability anisotropy but did not assessthe uncertainty of their estimations introduced by a priori assump-tions about multiphase flow parameters. By contrast, the develop-ments reported in this paper integrate the flow simulator with adynamically coupled mudcake growth and mud-filtrate invasionalgorithm (Wu et al. 2002), which improves the physical consis-tency and reliability of the quantitative estimation of both perme-ability and permeability anisotropy.
MethodThere are three main components in the workflow developed inthis paper:
1. Mud-filtrate invasion algorithm2. Two-phase axisymmetric simulator3. Nonlinear minimization algorithmTransient measurements of pressure and flow rate are com-
pared to the outputs of a two-phase axisymmetric simulator toyield new model parameters through nonlinear minimization. Theinvasion algorithm makes use of these parameters (permeabilityand permeability anisotropy), in addition to pressure overbalance,invasion geometry, mudcake properties, and other rock-formationproperties, to simulate the process of mud-filtrate invasion. Sub-sequently, the calculated spatial distributions of pressure and fluidsaturation resulting from mud-filtrate invasion are used as initialconditions for pressure-transient tests. To reduce the time requiredby the inversion, this last step could be approximated with aninvariant mud-filtrate invasion profile calculated only once duringthe minimization. However, such a strategy is not recommendedfor supercharged formations where updates of initial conditionsduring minimization can drastically impact inversion results. Forthe synthetic case examples considered in this paper, we constantlyupdate the initial conditions during minimization. The geometry ofthe formation model (e.g., multilayer formations, impermeable bedshoulders) is fixed when entered into the flow simulator. Conse-quently, field measurements, well-log measurements, and otherindependent sources of information are needed to define the geo-metrical properties of the rock formation model. To complete theestimation, the fluid-flow simulator yields pressure transients to becompared against actual measurements. This process repeats itselfuntil the quadratic norm of the residuals between simulations andmeasurements decreases to a predefined value. When the latter
* Currently with TOTAL.
Copyright 2007 Society of Petroleum Engineers
This paper (SPE 95897) was first presented at the 2005 SPE Annual Technical Conferenceand Exhibition, Dallas, 912 October, and revised for publication. Original manuscript re-ceived for review 14 July 2005. Revised manuscript received 22 May 2007. Paper peerapproved 29 May 2007.
339September 2007 SPE Journal
condition is met, the inversion algorithm outputs the estimatedvalues of permeability and permeability anisotropy.
Systematic description of the inversion methodology on syn-thetic measurements requires the use of a base case model (de-scribed later), which includes petrophysical variables and geo-metrical properties that can reproduce arbitrary formation modelsand tool configurations. Fig. 1 shows the measurement configu-ration for the assumed straddle-packer wireline formation tester.Dimensions of the base case tool were chosen according totypical physical dimensions of wireline formation testers designedfor interval pressure transient tests (IPTTs): two vertical observa-tion probes and one packer flow area that acts as a sink. Thesimulator used in this paper was developed by the FormationEvaluation Research Program of the University of Texas at Austin.
Special considerations about skin factor, tool storage, and totalcompressibility are discussed in a separate section of the paper.Although not investigated here, skin damage can be readily imple-mented in the inversion method by using a similar approach to themultilayer formation example presented in this work. In addition,even though we ignore tool-storage effects, the latter can be stud-ied with time-variable flow rates of fluid production.
Numerical Simulation of the Process ofMud-Filtrate InvasionAn adaptation of INVADE, developed by Wu et al. (2002), is usedto simulate the process of mud-filtrate invasion. Simulations in-clude the dynamically coupled effects of mudcake growth andmultiphase, immiscible mud-filtrate invasion. In simple terms, theflow rate of mud-filtrate invasion depends on both mud and rockformation properties. This approach differs from the proceduredescribed by Gk et al. (2006), who considered stationary com-posite zones and assumed single-phase flow. By coupling the in-vasion algorithm with the flow simulator, our inversion method isnot restricted to discontinuous fluid saturation gradients and, moreimportantly, it does not assume that the invaded zone across thestraddle-packer interval is immobile nor stays under constant fluidsaturation during the test (i.e., our inversion approach remainsaccurate in cases of significant fluid cleanup). The INVADE al-gorithm assumes that the rock formation is drilled with a water-based mud (WBM).
Our base-case model replicates the conditions of an invadedzone through the injection of brine into the formation during 1.5days. This value, as well as other assumptions on rock formationand mud properties, was arbitrarily chosen to illustrate the methodproposed in this paper rather than to describe a specific situation.Additional assumptions include the values of brine salinity, equalto 5,000 ppm (1.75 lbm/STB) and formation water salinity, equalto 120,000 ppm (42.06 lbm/STB). Table 1 summarizes the prop-erties of the assumed mud (and mudcake). Fig. 2 describes the as-sumed water/oil relative permeability and capillary pressure curves.
Fig. 1Configuration of the base-case straddle-packer, awireline formation tester consisting of two vertical observationprobes and a dual-packer module.
Fig. 2Water/oil relative permeability and capillary pressurecurves assumed in the numerical simulations of both mud-filtrate invasion and formation-tester measurements.
340 September 2007 SPE Journal
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