Models for deep bed filtration in the injection of seawater with solid inclusions depend on an empirical filtration function that represents the rate of particle retention. This function must be calculated indirectly from experimental measurements of other quantities. The practical petroleum engineering purpose is to predict injectivity loss in the porous rock around wells. In this work, we determine the filtration function from the effluent particle concentration history measured in laboratory tests knowing the inlet particle concentration. The recovery procedure is based on solving a functional equation derived from the model equations. Well-posedness of the numerical procedure is discussed. Numerical results are shown.
Abstract. Deep bed filtration of particle suspensions in porous media occurs during water injection into oil reservoirs, drilling fluid invasion of reservoir production zones, fines migration in oil fields, industrial filtering, bacteria, viruses or contaminants transport in groundwater, etc. The basic features of the process are particle capture by the porous medium and consequent permeability reduction.Models for deep bed filtration contain two quantities that represent rock and fluid properties: the filtration function, which is the fraction of particles captured per unit particle path length, and formation damage function, which is the ratio between reduced and initial permeabilities. These quantities cannot be measured directly in the laboratory or in the field; therefore, they must be calculated indirectly by solving inverse problems. The practical petroleum and environmental engineering purpose is to predict injectivity loss and particle penetration depth around wells. Reliable prediction requires precise knowledge of these two coefficients.In this work we determine these quantities from pressure drop and effluent concentration histories measured in one-dimensional laboratory experiments. The recovery method consists of optimizing deviation functionals in appropriate subdomains; if necessary, a Tikhonov regularization term is added to the functional. The filtration function is recovered by optimizing a non-linear functional with box constraints; this functional involves the effluent concentration history. The permeability reduction is recovered likewise, taking into account the filtration function already found, and the functional involves the pressure drop history. In both cases, the functionals are derived from least square formulations of the deviation between experimental data and quantities predicted by the model.
Severe fall of injectivity happens with the reinjection of produced water which contains oil droplets and solid particles, with the injection of sea water in offshore fields which contains organic and mineral inclusions, and in a general case of injection of a poor quality water. The mathematical model contains two empirical functions - filtration coefficient versus concentration of deposited particles and velocity and formation damage function versus concentration of deposited particles. Two inverse problems for determination of these two functions from the laboratory coreflood test are formulated. The first problem is determination of a filtration coefficient from the concentration history on a core outlet. The algorithm of solution is given, and the laboratory data treatment is presented. The second problem is determination of a formation damage function from the pressure drop history on a core. These two methods allows to determine from laboratory test the information necessary for prediction of well impairment. Introduction Injectivity reduction with the injection of water which contains solid and liquid inclusions takes place to some degree in most waterflooding projects. It becomes an important issue with respect to waterflooding of offshore fields using the sea water which usually contains different organic matters and solids. Facilities for raw sea water treatment are very limited by limitations for operations in sea platforms, so often a poor quality water is used in offshore waterflood projects. Reinjection of produced water which contains oil droplets and solid particles happens in several onshore fields and is often accompanied by injectivity decline. Amongst the advantages of produced water reinjection are that the water quality treatment for produced water is less than that for raw water from other sources; also produced water is usually compatible with the reservoir fluids and do not cause scale problems7,8. Nevertheless, oil droplets and solid particles may be present in the produced water, which often causes severe formation damage. Presently produced water reinjection is under consideration in offshore fields due to the possible environmental impact of the alternative of sea disposal. In designing a waterflood project, the level of water treatment necessary to minimise formation damage must be assessed, particularly whether solids or oil droplets should be removed and by how much. The different water treatment options are to be considered, so it is important to know the performance of an injector as a function of quality of injection water. Therefore, substantial efforts have been done in modelling of injectivity decline with injection of water with solid particles and oil droplets. The flow of solid-containing suspensions have been studied in many engineering branches, including the petroleum one, and was widely exposed in the literature, while there has been little published on the aspect of formation damage resulting from the flow of oily water. The basic mathematical model for deep filtration with particle retention consists on the mass balance equation and the kinetic equation for clogging1–5. The analytical model for diffusion-free flow was developed in the work1 under the assumptions that accumulation of suspended particles can be ignored and that the suspended concentration distribution is steady state. The simplified assumption that filtration coefficient is constant is proposed in other works3,4. In this case the direct problem of prediction of concentation of particles and pressure drop on the core allows for simple analytical solution.
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