A NEW APPROACH TO PREDICTING MISCIBLE FLOODPERFORMANCE

N. UDEY T. J.T. SPANOS

this article begins on the next page FFPETROLEUM SOCIETY OF CIM and AOSTRA PAPER NO. CIM/ AOSTRA 91 -5 THIS IS A PREPRINT - SUBJECT TO CORRECTION A NEW APPROACH TO PREDICTING MISCIBLE FLOOD PERFORMANCE BY N. Udey University of Alberta T.J.T. Spanos University of Alberta PUBLICATION RIGHTS RESERVED THIS PAPER IS TO BE PRESENTED AT THE CIWAOSTRA 1991 TECHNICAL CONFERENCE IN BANFF, APRIL 21-24,1991. DISCUSSION OF THIS PAPER IS INVITED. SUCH DISCUSSION MAY BE PRESENTED AT THE TECHNICAL MEETING AND WILL BE CONSIDERED FOR PUBLICATION IN CIM JOURNALS IF FILED IN WRITING WITH THE TECHNICAL PROGRAM CHAIRMANPRIOR TO THE CONCLUSION OF THE MEETING. Abstract Laboratory breakthrough curves obtained from field cores are used to characterize the dispersion of miscible displacement. The analysis presented is based upon a set of equations for miscible flow which have been recently proposed by the authors. A solution to these equations is applied to breakthrough curves. This allows one to determine the underlying functions of the proposed equations. The kinematic behaviour of miscible floods and their rate of dispersion are examined; in particular R is shown that the theory proposed here implies thatthe standard dispersion coefficient must be scale dependent. Comparison with laboratory results indicates that this prediction is consistent with the observed discrepancies between the "old, convection-diffusion theory and laboratory experiments. Introduction Predictions of the evolution and performance of miscible floods are performed by using computer simulations. The two basic elements of any successful simulation are a dynamical theory of miscible displacement and a corresponding knowledge of field characteristics. To obtain an accurate prediction we need a good dynamicaltheory as well as accurate field data. Such data typically consists of values for porosity, grain size, dispersion coefficients, permeability, etc. A typical procedure to obtain this data is to extract a core sample from the field and use R to measure the required data; in particular, the core can be used in a miscible displacement experiment to obtain a breakthrough curve. The interpretation of a breakthrough curve depends upon the theory which will be used to simulate the miscible flood; there are, broadly speaking, two types of theory used to model miscible displacement. By comparingand contrasting these two theories, we will show how to improve our analysis of breakthrough curves. Hopefully, this improvement will lead to more accurate predictions of miscible flood performance.

PETROLEUM SOCIETY OF CIM and AOSTRA PAPER NO. CIM I AOSTRA 91-5

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THIS IS A PREPRINT· SUBJECT TO CORRECTION

A NEW APPROACH TO PREDICTINGMISCIBLE FLOOD PERFORMANCE~

BYN.Udey

Univenlty af A1bor1.1

T.J.T. SpanosUnlvaralty of Alberta

PUBUCATION RIGHTS RESERVEDTHIS PAPER IS TO BE PRESENTED AT THE CIMIAOSTRA 1991 TECHNICAL CONFERENCE IN BANFF, APRIL21-24,1991. DISCUSSION OF THIS PAPER IS INVITED. SUCH DISCUSSION MAY BE PRESENTED AT THETECHNICAL MEETING AND WILL BE CONSIDERED FOR PUBLICATION IN CIM JOURNALS IF FILED INWRITING WITH THE TECHNICAL PROGRAM CHAIRMAN PRIOR TO THE CONCLUSION OF THE MEETING.

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Abstract

Laboratory breakthrough curvesobtained from field cores are used tocharacterize the dispersion of miscibledisplacement. The analysis presented isbased upon a set of equations for miscibleflow which have been recenlly proposed bythe authors. A solution to these equations isapplied to breakthrough curves. This allowsone to determine the underlying functions ofthe proposed equations. The kinematicbehaviour of miscible floods and their rate ofdispersion are examined; in particular it isshown that the theory proposed here impliesthat the standard dispersion coefficient mustbe scale dependent. Comparison withlaboratory results indicates that this predictionis consistent with the observed discrepanciesbetween the "old" convection-dillusiontheory and laboratory experiments.

Introductfo n

Predictions of the evolution andperformance of miscible floods are performed

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by using computer simulations. The twobasic elements of any successful simulationare a dynamical theory of miscibledisplacement and a correspondingknowledge of field characteristics. To obtainan accurate prediction we need a gooddynamical theory as well as accurate Iielddata. Such data typically consists of valuesfor porosity, grain size, dispersioncoefficients, permeability, etc. A typicalprocedure to obtain this data is to extract acore sample from the Iield and use it tomeasure the required data; in particular, thecore can be used in a miscible displacementexperiment to obtain a breakthrough curve.The interpretation of a breakthrough curvedepends upon the theory which will be usedto simulate the miscible flood; there are,broadly speaking, two types of theory usedto model miscible displacement. Bycomparing and contrasting these twotheories, we will show how to improve ouranalysis of breakthrough curves. Hopefully,this improvement will lead to more accuratepredictions of miscible flood performance.

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Dynamics and Flow Regimes

we will show how the breakthrough curve canbe "inverted" to obtain the fractional 1I0w andrelated functions. These results may then beused in a corresponding simulator to modelthree dimensional miscible floods.

Consider a miscible disptacement where anin-situ fluid (fluid 1) is being displaced byanother (fluid 2). The density of fluid i =1 ..2 isp, and it's viscosity is 11'. The porosity 01 theporous medium is TJ and it's permeability is K.Both 01 these quanmies are assumedconstant, i.e. a homogeneous medium. Theconfiguration of this syslem is encapsulatedin the fractional concentration, CI2(;(,t). whichis a function 01 space and time; il is defined asthe ratio of the concentration of lIuid 2 to thetotal concentration. The flow rate;; is related

to the Darcy velocity qby;; = q",. Forincompressible fluids

The first type of theory for miscibledisplacement is the standard theory found intextbooks on fluid llow (e.g. Bear,1972). Thistheory describes the dynamics of the fluid byintroducing a dispersion tensor in analogy toFick's law for diffusion. When thisdescription is combined with the equation ofcontinuity (conservation of mass) thedynamics is embodied in the well knownconvection-diffusion equation. In this case,the analysis of breakthrough curves isdirected towards determining the dispersiontensor. There is however, a long standingpuzzle using this approach. The value of thedispersion needed by a simulation to matchactual field scale miscible displacements isusually much larger than the dispersion valueobtained from the breakthrough curves(Pickens and Grisak,1981); therefore thedispersion is apparently scale dependent.The usual explanation for this discrepancy isthat the dispersion is being affected in someunknown way by field scale heterogeneities.We have found a simple explanation for thisscale dependence. We will present thisexplanalion below and show it is operativeeven in homogeneous media.

V· q=O ,

and qsatisfies Darcy's law:

(1 )

where D is the dispersion tensor.

de - - -~ + ;;,VC'2 = v·( D,V(cf2)) ,ill (3)

When immiscible equations are usedto model the flow, the relevant evolutionequation for saturation is

The effective viscosity 11 is assumed to be afunction of Ihe composition of the lIuid, i.e.the fractional concentration. To complete thedescription, we need an evolution equationfor the fractional concentration. Thisequation will depend on what theory we use.

In the case of standard miscibletheory, the equation of evolution is theconvection-diffusion equation.Breakthrough curves are often conductedwith fluids closely matched in density so thatthe total density is essentially constant; thedisplacing fluid is marked by a tracer. In thissituation the convection-diffusion equation is

(2)

.. K - -q = - __ (V P - P g) .IJ.{C12)

The other type of theory describesthe flow using a set of equations forimmiscible two phase flow Traditionally, theequations used are those of the Buckley­Leveren solution to Muskat's equations(Koval,1963; Todd and Langstaff, 1972). Inthis case, the analysis of breakthrough curvesis directed towards finding relativepermeability curves. This is normally done byassuming some analytic lorm lor the relativepermeability curves (e.g. Jankovic,1986) andthen finding parameter values which historymatch the production curve (the integral 01the breakthrough curve). In recent years ithas been recognized that Muskat's equationsare not eXlensible 10 three dimensions and toovercome this limitation, a new theory of two­phase immiscible flow has been proposed(de la Cruz and Spanos, 1983;Whitaker, 1986; Kalaydjian, 1987; Spanosel.al.,198B). We have recently applied thisnew theory to miscible displacement (Udeyand Spanos,1990) and the solution we haveobtained is summarized in the appendix. Itwill be extensively used in this paper; inparticular, it is used to analyze a typicalbreaklhrough curve. This analysis willincorporate new technique to obtain thetheory's underlying functions as opposed tothe history matching approach. In essence,

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